by Dr. Roy Maltby

The calculations on the UN page are meant to address three potential types of joint failure - stripping of the bolt thread, stripping of the nut thread, and breakage of the bolt. It is generally considered good design practice to use a sufficiently long length of engagement that if a joint is subjected to excessive stress, the first failure to occur will be a broken bolt. A broken bolt is considered to be a more desirable failure than a stripped thread since a broken bolt is relatively easy to notice and easy to replace. If a bolt is subjected to too much tensile stress, it will probably break obviously as soon as the stress is applied. But if a thread is overstressed, failure is likely to occur only at the start of the thread (where the stress is highest) at first, but then gradually propagate along the thread so that the joint fails at some random later time.

In some joints there are shear stresses that must be analysed. For instance, if a nut and bolt are used to hold together two metal plates, and forces act to slide these plates in opposite directions, this creates shear stress in the bolt. This calculator makes no provision for such analysis. Another possible fastener failure is twisting off the head of a machine screw by applying too much torque. This calculator makes no provision for that analysis either. This calculator also makes no provisions for analysing the effects of vibrations, fatigue, or temperature variations.

The terms "nut" and "bolt" are somewhat abused throughout these pages. "Nut thread" is used to mean "internal thread", and "bolt thread" is used to mean "external thread", even though these threads may be used on threaded fasteners other than nuts and bolts.

# Input Parameters

The threads that you will find on any nut and bolt purchased at any hardware store in the US, UK, or Canada are Unified threads (in fact, usually Unified Coarse (UNC) threads) with classes of fit 2A and 2B. The threadform diameter refers to the diameter of the threaded part of the bolt measured at the peak of the thread (to within some amount of tolerance and allowance).

Parts with classes of fit 3A and 3B fit more snugly than parts with classes of fit 2A and 2B. It is more difficult (i.e. expensive) to make 3A/3B threads but some engineers think the snug fit results in better performance. However, other engineers doubt that it really helps (Blake, page 24). One may wonder how meaningful it is to say that 3A/3B fit more snugly than 2A/2B since the tolerances actually allow a 3A/3B connection to be looser fitting than a 2A/2B connection. It is normal to use 3A/3B threads in aerospace applications, and 2A/2B everywhere else. Actually, Blake (page 15, 3rd edition) says more specifically that UNJ with class of fit 3A/3B is normal for aerospace applications. The UN calculator is not suitable for UNJ calculations.

For a plated or coated 2A bolt, the maximum bolt dimensions apply to the state of the bolt before plating, and the maximum bolt dimensions with plating are those for Class 3A. Class 2AG means class 2A, except for the preceding rule. That is, if a 2AG bolt is plated, the maximum dimensions after plating are the usual maximum dimensions for Class 2A (before plating).

1A/1B can be looser fitting than 2A/2B. (The allowance is the same but the tolerance is more.) They are for military use in conditions where parts may need to be assembled quickly even with dirty or slightly damaged threads. We have included 1A/1B threads in the calculator because it was easy to do so, but they are such a tiny minority of the UN threads in use that they are almost trivia. Actually, the ASME specification for UN threads is somewhat sketchy regarding 1A/1B threads. We expect that there is a better military specification for 1A/1B threads, but we have not investigated since these threads are used on such a tiny minority of fasteners.

Tensile Strength

The tensile strength (aka "ultimate strength") is the maximum tensile stress that can be induced in a material. A material also has proof strength, yield strength, and shear strength, but for the sake of simplicity, our calculator only uses tensile strength. The default tensile strength we use is 60,000 p.s.i. since that is the lowest tensile strength given in Machinery's Handbook for any steel (page 1414, 25th edition). It is generally accepted design practice that any nut should have at least as much tensile strength as the bolt it is used with. The calculator will determine the appropriate torque required to induce the desired tensile stress that you enter. If you leave desired tensile stress at zero, the calculator will replace it with one half of the bolt's tensile strength. Machinery's Handbook (page 1399, 25th edition) recommends inducing tensile stress equal to 90% of yield strength if the joint will be permanent, 75% of yield strength if the joint will be reused. (Bickford suggests 60% as a conservative amount (page 426, 3rd edition).) For all the grades of steel in Machinery's Handbook (page 1414, 25th edition), the yield strength is at least 55% of the tensile strength.

If a static load is present, the calculation to determine the appropriate torque to induced the desired tensile stress assumes that the tensile stress in the bolt equals the sum of the tensile stress due to preload (caused by tightening the nut) and the tensile stress due to the static load. The true tensile stress is almost certainly less than this sum, so this assumption can be considered conservative in that it protects against the danger of excessive tensile stress which can break the bolt. On the other hand, this assumption defeats possible protection against the danger of an under-tightened nut. There is no easy answer here. Tricky tradeoffs like this are among the reasons people get licensed engineers to design machinery.

Nut Factor

The nut factor is most often assumed to be 0.2 for steel nuts and bolts tightened without lubrication. Nut factors for many other materials and lubricants are given by Bickford (page 231, 3rd edition), and a few are given by Blake (page 155) and in Machinery's Handbook (page 1399, 25th edition).

The nut factor is not simply a coefficient of friction. The nut factor combines a number of parameters, some of which are determined by the geometry of the threadform, and some of which can only be determined by experimenting with the materials involved. These pages follow the school of thought that it is simpler and probably more reliable to use nut factors determined by experiment rather than separately trying to determine by experiment the factors that contribute to the nut factor and assuming that the nut factor formula which combines them truly represents all the factors that determine the nut factor. For what it is worth, if one assumes that the relevant coefficients of friction are equal and makes certain other simplifying assumptions, the nut factor formula says that the nut factor is equal to the coefficient of friction multiplied by 1.275. Bickford (page 233, 3rd edition) suggests estimating the nut factor as the coefficient of friction plus 0.04, but he also warns: In critical applications you should always develop your own nut factors by an appropriate experiment. Tables in Machinery's Handbook (pages 1410 and 1411, 25th edition) show that these rules for estimating the nut factor based only on the coefficient of friction are not very accurate.

Length of Engagement

Generally, you can leave this parameter alone and let the calculator fill in the standard length of engagement for the threadform size you have specified. If the true length of engagement is different from the standard length of engagement, the output parameters are not affected unless the difference is quite large. If the length of engagement is about 50% more than the standard length of engagement for the given thread size, then the tolerances of the thread dimensions are increased, and if the length of engagement is more than about triple the standard length of engagement, the tolerances are increased even more. This calculator does not reduce tolerances for very short lengths of engagement since the ASME specification is unclear about this.

The usefulness of this parameter is minimal, but unfortunately it is not quite possible to do away with it altogether. The UN threadform specification uses "length of engagement" as a parameter in determining the tolerance of pitch diameters. But, in fact, the "length of engagement" in the tolerance formula does not represent the actual length of engagement. It represents the "standard" length of engagement for the given thread size. The formula only changes if the actual length of engagement differs drastically from the "standard" length of engagement. Even then, the formula is adjusted by multiplying the entire tolerance by a constant, and the "length of engagement" term remains equal to the "standard" length of engagement. For most sizes, the "standard" length of engagement used in the tolerance formula is equal to the size (nominal bolt thread diameter) of the threadform, even though the thickness of a "regular" hex nut is 0.875 times this amount. In other cases the "standard" length of engagement is 9 times the pitch.

The ASME specification suggests that the usual pitch diameter tolerance does not apply if the length of engagement is less than five times pitch, but does not say what adjustment to the tolerance is necessary. We do not mind ignoring this issue since applications where one might consider using such a short length of engagement must be rare. At the end of Section 5.6.1 in the ASME specification, reductions in the tolerance of the minor diameter of the nut thread for unusually short lengths of engagement (less than 2/3 of the bolt diameter) are clearly stated, but then apparently contradicted at the end of Section 5.8.2 where it says "The formulas are suitable for general applications having lengths of engagement up to 1.5 diameters."

If you are using a nut and bolt simply to hold things together with no outside forces acting on the joint, then you have a static load of zero. The static load is the load from outside sources that the joint will be subjected to along the bolt axis. For instance, if a bolt sticks vertically into a ceiling and supports the weight of a chandelier, the weight of the chandelier is a static load.

# Output Parameters

Diameters, Allowance, Pitch

The pitch is the distance along the bolt from any point to the corresponding point on the next thread. It is equal to one inch divided by the number of threads per inch. We do not use the word "lead" on these pages since all the calculations on these pages assume single-start threads so the lead is equal to the pitch.

Caution: The UNS calculator allows any combination of thread size and number of threads per inch. It is possible to enter a combination that corresponds to any standard size in series UNC, UNF, 4UN, 6UN, or 8UN, but the resulting calculations will not correctly describe a UNC, UNF, 4UN, 6UN, or 8UN thread. This is because the standard length of engagement for those series is equal to the threadform diameter, but for UNS it is nine times the pitch, and this affects the allowance and all dimension tolerances.

For class 2A, if there is a coating or plating, the description of thread dimensions above applies to the state of the bolt without this finish and the coating or plating is allowed to increase diameter by as much as the allowance. For classes 1A, 3A, and 2AG, the maximum diameters for points on the bolt thread apply to the finished bolt.

Tensile Stress

The tensile stress area is the area of a cross-section of the bolt according to the familiar formula of pi multiplied by the square of half the diameter. Of course, the diameter of a bolt depends on where you measure it, and this is how we get the three different tensile stress areas on the output page: usual, severe, and root. In the usual calculation of the tensile stress area, the number used for the diameter is the mean of the maximum pitch and root diameters of the bolt. In the "severe" variation, the number used for the diameter is the mean of the minimum pitch and root diameters. In the "root" variation, the number used for the diameter is the minor diameter of the basic threadform. This area is called Ar in tables of threadform data, instead of As which is used for the usual calculation. The severe calculation is also denoted by As. The differences between the usual, severe, and root figures for all the other calculations are strictly a result of using different diameters in the tensile stress area calculations. The severe and root variations are supposed to be conservative, but the next paragraph describes reasons to doubt this. Bickford (page 31, 3rd edition) mentions another variation - calculating the tensile stress area using the maximum bolt pitch diameter. He says this is used by the military for certain bolts designed to withstand very high stress. Usually the usual calculation is appropriate, and Bickford (page 31, 3rd edition) goes so far as to say Most people will never use the alternatives.

Machinery's Handbook (page 1416, 25th edition) recommends using the severe formula with materials rated to withstand 100,000 psi or more. Bickford (page 31, 3rd edition) says that other sources recommend using the severe formulas only with materials rated to withstand 180,000 psi or more and observes that the severe formula for tensile stress area has been dropped entirely from the relevant appendix to the ASME specification of UN threads. We provide the "conservative"/severe calculations because many people want to follow Machinery's Handbook and so will want these calculations. One questionable aspect to this approach is that this "conservative" tensile stress area leads to shorter recommended lengths of engagement to protect the threads from stripping which obviously is less conservative than the usual calculation. Changing the formula for tensile stress area also affects the calculation of preload which produces a dubious effect on the calculation of the torque required to induce the preload and reduces the estimate of thread shear stress for both nut and bolt threads. The same issues are present when using the thread root area as the tensile stress area.

Caution: Tightening the nut also induces a couple of shear forces which are often ignored in calculating suitable torque. For the sake of simplicity, the calculator does not deal with these forces, but if your application is critical or perhaps if you want to keep stress below yield as well as below breaking point, obviously you should take these into account. To be conservative, it is generally assumed that bolt materials can withstand only about half as much shear stress as tensile stress. The von Mises formula for combined tensile stress (page 1402 of Machinery's Handbook 25th edition) is a formula that is something like a weighted sum of tensile stress and shear stress to provide a number representing total stress which can be compared with the rated strength of the material. Tightening the nut induces shear stress by twisting the bolt. Depending on the circumstances, the bolt may not untwist itself immediately when the wrench is removed so that some of this stress persists, at least briefly. (Page 198 of Bickford, 3rd edition has an interesting discussion of how this can result in the joint tightening itself and increasing preload by maybe 2% after torque is removed. Usually one expects preload to diminish slightly after torque is removed as the materials relax.) Shear stress also exists as a result of the force (not torque) exerted on the wrench handle by the mechanic. The mechanic pushes on the wrench handle in a particular direction with a particular force and this creates a shear force acting on the bolt in the same direction. There may be other shear stress when the bolt is in use and the calculator makes no provision for that either. For instance, if a nut and bolt hold together two steel plates, and forces act to slide the plates in opposite directions, this induces shear stress in the bolt.

Recommended Torque to Induce Desired Stress

The calculator subtracts the stress due to static load from the desired tensile stress to determine the appropriate stress due to preload (in psi). Multiplying this by the tensile stress area provides the preload (in pounds). The preload is multiplied by the nut factor and thread size (nominal major diameter of the bolt) to determine the required torque. This is the usual procedure for calculating appropriate torque when there is no static load. When there is a static load, the next section explains how this procedure may underestimate required torque.

Some users may prefer to specify a torque and ask the calculator for the maximum static load that can be used with that torque. We have not set up the calculator to work that way because we do not consider that to be the most logical method. Of course, if you insist on doing your calculations that way, you can adjust the static load input parameter until the calculator recommends the torque that you want to use. Machinery's Handbook (page 1415, 25th edition) has an interesting discussion related to this. The Handbook mentions a study regarding the amount of torque used by "experienced mechanics" in tightening certain joints. The study found that experienced mechanics apply torque that is roughly in proportion to the diameter of the bolt. In reality, to induce consistent stress due to preload, the applied torque must be roughly proportional to the cube of the diameter of the bolt. In other words, if you double the diameter of the bolt, you need eight times the torque to induce the same stress. (You can verify this by trying different bolt sizes in this calculator, or by observing that in the Wrench Torque table on page 1400 of Machinery's Handbook, 25th edition, the numbers in the m column are all close to 3.) This suggests that "experienced mechanics" do not tighten large bolts nearly as much as they could. As for small bolts, the study found that experienced mechanics tend to break bolts up to a half inch in diameter. If you have to rely on mechanics who work like this, this calculator is not really designed to help you. The Handbook recommends formulas for calculating permissible static load (working strength of the bolt) when torque is applied as this study describes. (Actually, the Handbook says the mechanics used "pull", not torque, that was in proportion to the diameter of the bolt. If "pull" means force applied at the end the wrench handle, and if they used bigger wrenches for bigger bolts, say a handle length in proportion to bolt diameter, then the torque for large bolts would be more than the preceding comments indicate, but still well short of that required for the tightest connection.)

Caution: The calculations described in the foregoing paragraphs are based on the assumption that shear load is spread uniformly over the shear stress area. In reality, the start of the engagement takes more than its share of shear load. This is due partly or mainly to the fact that the bolt is loaded in tension and the nut is loaded in compression. In other words, forces acting on the bolt tend to stretch it, and forces acting on the nut tend to squash it. You should be especially suspicious of any calculation that recommends a particularly long length of engagement, since after a certain point, adding turns of thread in the engagement may not significantly increase the true load capacity of the engagement.

The apparent error in Machinery's Handbook may be considered immaterial in the following sense. Machinery's Handbook (page 1416, 25th edition) provides a formula for calculating a ratio J to be used to determine whether the nut thread will strip before the bolt thread in cases where the nut material is not as strong as the bolt material. This formula involves the nut and bolt thread shear stress areas and so involves a length of engagement. But the areas are used in ratio in this formula so that the length of engagement cancels itself. The formula provides the same result whatever length of engagement is used. In fact, we could use just the nut and bolt thread shear stress areas in a single turn of thread and get the same result. Simpler yet, we could leave the length of engagement out of the calculations altogether which would amount to calculating shear stress area per inch of length of engagement as in Blake's table.

Another problem which our calculator does not take into account is nut dilation. Since the sides (flanks) of the threads are sloping, a nut may react to the load placed on it by dilating - that is, stretching outward from its center in every direction. If this happens, the nut and bolt shear stress areas will be reduced - possibly to something worse than the worst case calculation that our calculator and Blake's table both provide. The "worst case" calculations are based on the worst case according to the dimensions at the time of manufacturing.

Belaboring the point about accuracy further... Bickford's table seems to amplify inaccuracies in Blake's table since it appears each value in Bickford's table is derived by multiplying the corresponding entry in Blake's table by a reasonable length of engagement (the nominal threadform diameter). Where Blake's values are rounded to a hundredth of a square inch, rounding can change his calculated result (which already is affected by the rounding errors described above) by about 0.005 square inches. Then Bickford multiplies these results by a threadform diameter up to 4 inches, so that this error increases to as much as about 0.020 square inches. So where Bickford gives a number with two decimal places, it should not be a surprise if the last digit differs from the result produced by our calculator, or even if it differs from the result produced by using the rounded diameter values in the ASME tables.

# Bibliography

Bickford, John H.,
An Introduction to the Design and Behavior of Bolted Joints, 3rd Edition
Marcel Dekker, Inc, 1995.

Blake, Alexander,
Marcel Dekker, Inc., 1986.

Oberg, Erik; Jones, Franklin D.; Horton, Holbrook L.; Ryffel, Henry H.;
Machinery's Handbook, 25th Edition,
Industrial Press, Inc., 1996.

Parmley, Robert O.,
Standard Handbook of Fastening and Joining, Second Edition,
McGraw-Hill, 1989.

American Society of Mechanical Engineers,