Authored by: Edited by Jessica Shapiro Key Points: Resources: "Joint Decisions," MACHINE DESIGN, April 1, 2005 "How Tight is Right?" MACHINE DESIGN, August 23, 2001 
Measuring torque when installing threaded fasteners is the best indicator of future joint performance, right? Actually, bolt tension is a better performance indicator, but measuring torque is far easier to do.
Bolt tension is created when a bolt elongates during tightening, producing the clamp load that prevents movement between joint members. Such movement is arguably the most common cause of structural joint failures. The relationship between applied torque and the tension created is described by the relationship:
T = K × D × F
where T = torque, K = nut factor, sometimes called the friction factor, D = bolt diameter, and F = bolt tension generated during tightening. This expression is often called the shortform equation.
The nut factor
The nut factor, K, consolidates all factors that affect clamp load, many of which are difficult to quantify without mechanical testing. The nut factor is, in reality, a fudge factor, not derived from engineering principles, but arrived at experimentally to make the shortform equation valid.
Various torquetension tests call for controlled tensioning of a threaded fastener while monitoring both torque and tension. At a specified torque or tension, the known values for T, D, and F are inserted into the shortform equation to permit solving for K.
Many engineers use a single value of K across a variety of threadedfastener diameters and geometries. This approach is valid to some extent, because an experimentally determined nut factor is by definition independent of fastener diameter. But to truly understand the factors involved, it is helpful to compare the shortform equation with a torquetension relationship derived from engineering principles.
Several of these equations are common, especially in designs that are primarily used in the E.U. Each produces similar results and takes the general form:
T = F × X
The long way T_{in} = F_{P} × [(P/2 π) + ( μ_{t} × r_{t}/cosβ ) + (μ _{n} × r_{n})] The following table defines each variable and shows that, although the longform equations appear unrelated, they are actually different forms of the same equation and yield similar results. Each term calculates a length which, when multiplied by the force generated by bolt tension, generates a moment or torque. They represent reaction torques resisting the input torque and must sum to equal that input torque. The term containing the variable P is the clamp load on thread pitch’s inclined plane. The second term — involving μ_{t}, μ_{G}, or μ_{th} — is resisting torque caused by thread friction, and the last — using μ_{n}, μ_{K}, or μ_{b} — is a similar resisting torque generated by friction between the nut or head face and mating surface. The value of each term indicates the relative influence of each of these friction factors. For example, applying the Motosh equation to an M121.75 angehead screw with friction coefficients μ_{t} and μ_{n} equal to 0.15, shows that each additional 1,000 N of tension produces 0.28 Nm of reaction torque from clamp load on the thread pitch, 0.93 Nm of reaction torque from thread friction, and 1.35 Nm of reaction torque from friction under the nut. Total torque is then 2.56 Nm. These values break out to 10.9%, 36.3%, and 52.8%, respectively, of the total torque, confirming the commonly heard assertion that only 10 to 15% of the input torque goes toward stretching the bolt. 
where X represents a series of terms detailing fastener geometry and friction coefficients. These relationships are often referred to as longform equations. (Three of the most widely used longform equations are discussed in the sidebar, The long way.)
To understand how the nut factor compares to the terms in the longform equations, let’s consider the socalled Motosh equation:
T_{in} = F_{P} × [(P/2 ) + (μ_{t} × r_{t}/cos ) + (μ_{n} × r_{n})] where T_{in} = input torque, F_{P} = fastener preload, P = thread pitch, μ_{t} = friction coefficient in the threads, r_{t} = effective radius of thread contact, μ = half angle of the thread form (30° for UN and ISO threads), μ_{n} = friction coefficient under the nut or head, and r_{n} = effective radius of head contact.
The equation is essentially three terms, each of which represents a reaction torque. The three reaction torques must sum to equal the input torque. These elements, both dimensional and frictional, contribute in varying degrees to determining the torquetension relationship, the purpose of calculating nut factor.
The impact of variables
So how do design decisions influence the nut factor that defines the torquetension relationship? Specifically, engineers may wonder how valid a nut factor determined from a torquetension test on one type of fastener is for other fastener geometries.
The shortform equation is structured so that the fastener diameter, D, is separate from the nut factor, K. This implies that a nut factor derived from torquetension tests on one fastener diameter can be used to calculate the torquetension relationship for fasteners with a different diameter.
Like the nut factor itself, however, this approach is only an approximation. The accuracy of using a nominal fastener diameter, D, to apply a constant nut factor across a range of fastener sizes depends on the extent to which fastener diameter affects reaction torque.
Because some variables that affect boltedjoint design have more of an impact than others, it makes sense to examine them individually.
Clearancehole diameter, for instance, is directly tied to nominal diameter. The longform equations calculate that swapping a closediametricalclearance hole with one 10% larger leads to a 2% reduction in bolt tension for a given torque. Enlarging the fastenerhead bearing diameter by 35%, say by replacing a standard hexhead fastener with a hexflangehead, cuts bolt tension by 8% for a given torque.
Both bearing diameter and hole clearance generally scale linearly with bolt diameter, so their relative importance remains the same over a range of fastener diameters. However, different head styles or clearances change the reaction torque from underhead friction because the contact radius changes.
Doubling the thread friction coefficient on its own, say by changing the finish or removing lubricant, reduces bolt tension by 28% for a given torque. Engineers should note that the thread and underhead friction coefficients are often assumed to be equal for convenience in test setups and calculations. ISO 16047 estimates that this assumption can lead to errors of 1 to 2%.
Thread pitch tends to be more independent of nominal diameter than the other variables. Increasing just thread pitch by 40% cuts tension 5% for a given torque. However, the reactiontorque term containing the thread pitch, P — P/2 π — does not contain a friction coefficient. Therefore, as fastener diameter and, consequently, friction increase, the relative importance of thread pitch falls.
Motosh

D_{IN}/V_{DI}

ISO 16047

Description

T_{in}

M_{A}

T

Input torque

F_{P}

F_{V}

F

Fastener preload or tension

P

P

P

Thread pitch

r_{n}

D_{Km}/2

(D_{o} + d_{h})/4

Effective radius of head contact

μ_{n}

μ_{K}

μ_{b}

Coefficient of friction under head

μ_{t}.

μ_{G}

μ_{th}

Coefficient of friction in threads

r_{t}

d_{2}/2

d_{2}/2

Effective radius of thread contact (half the threadpitch diameter)

β

Halfangle of thread form (30° for UN and ISO threads)

There is a maximum deviation of 4.2% between the results of the short and longform equations for standard pitch, metric, hexheadcap screws with constant and equal coefficients of friction. As friction coefficients increase, there’s less error in assuming the nut factor varies directly with nominal diameter because the impact of thread pitch, the most independent variable, falls.
So, if all else is held constant, it’s reasonable to apply a nut factor calculated at one fastener diameter across a range of fastener sizes. For best results, engineers should base testing on the weighted mean diameter of the fasteners for which the nut factor will be used.
Many engineers find it expedient to apply a single nut factor across even greater fastener ranges, such as those with different head styles or clearance diameters. Applying the nut factor to joints where geometric variables other than nominal diameters are changing causes the shortformequation results to differ from the longform results by up to 15%.
Long or short?
So given all the discussion about which variables impact the nut factor and when it is reasonable to use a constant nut factor across a range of fasteners, it may seem as if the longform equations get engineers closer to the “right” answer thanks to their fundamental correctness. However, longform equations, even when derived from engineering principles, use assumptions and approximations that may make them no more correct than an equation using an experimentally derived nut factor.
The potential errors in both types of equations pale compared to the variations in realworld joints. Benchtop torquetension testing shows approximately 10% variations within samples even when all fasteners and bearing materials are the same. According to both short and longform equations, there should be no variation at all. Friction coefficients seem to vary from sample to sample.
The situation is even worse in productionrepresentative joints. One injoint torquetension test with bolt tension measured in real time using ultrasonic pulseecho techniques revealed variations inherent in how components fit together. A sixbolt pattern magnified the effects of imperfect contact between bolts and the bearing surface and produced a 60% sampletosample variation because both geometric and friction variables were changing over the sample set.
The bottom line is that neither the nut factor nor the longform equation’s friction coefficients can be reliably established using reference tables. Only testing accurately determines friction conditions. As we have seen, these are too sensitive to component and assembly variation to be determined by analysis alone.
Testing reliably converts input torque to induced tension, letting engineers determine mean friction coefficients or nut factors and the distributions about those means. Tests may also help engineers identify inherent shortcomings in joints that need to be revised to make them more reliable.