Errors in Textbooks |

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.

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.

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.

Comments by Blake (page 160) suggest that every pound of static load reduces preload by a pound, or at least that one should make that assumption to be conservative. Bickford (pages 434 and 435, 3rd edition) says that this assumption is not strictly correct, but often is close to the truth. Machinery's Handbook (page 1415, 25th edition) says only that the tensile load will be more than either of preload or static load, but less than their sum. Blake says that a static load (working load) is typically about 0.3 to 0.8 times the preload and says that the safest approach is use large enough bolts with enough preload so that the static load is small compared with the preload.

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.

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.

The basic profile of a paired nut and bolt is a theoretical profile along which the nut thread and bolt thread meet. Of course, tolerances are involved in any kind of manufacturing, so we do not expect a nut and bolt to meet exactly along the basic profile. For the nut thread, the basic profile represents the smallest-diameter thread permitted. Tolerances for various points along the nut thread profile indicate how much bigger in diameter than the basic profile the true profile is allowed to be. The allowance is a minimum required gap between the nut and bolt thread profiles, or perhaps we should say that the allowance is double the minimum gap since the allowance is applied to diameters and a diameter of the joint passes through two allowances - one on each side of the bolt. For a bolt thread of class 3A (which has zero allowance), the basic profile represents the largest-diameter thread permitted. For classes 1A and 2A, subtract the allowance from the diameter at every point on the basic profile to get the profile representing the largest-diameter thread permitted. Tolerances for various points along the bolt thread profile indicate how much smaller in diameter than this profile (or the basic profile for class 3A) the true profile is allowed to be. This description is adequate for most purposes, but a complete description would also mention that both bolt and nut threads are allowed to have deeper roots than the basic profile indicates. The ASME specification says that tolerances for the diameters at the thread roots for both nut and bolt threads are "for reference only". Presumably, this means there is no need to enforce such tolerances because they do not affect the fit of threads. Whatever the reason, the given diameters are based on the root of the bolt thread being above the root of the theoretical sharp-V thread root by only one eighth of the sharp-V thread height (instead of one quarter for the basic profile) and the root of the nut thread being below the root of the theoretical sharp-V thread root/crest by only one twenty-fourth of the sharp-V thread height (instead of one eighth for the basic profile).

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.

We consider two sources of tensile stress in a cross-section of the bolt. One is the preload which is caused by tightening the nut which causes the bolt to stretch. The preload is the force (measured in pounds) with which the bolt tries to pull itself back to its original length. This force remains present in the bolt after we have finished tightening the nut and the wrench has been removed. This is the source of clamping force. The other source of tensile stress is the static load. If the bolt takes some weight or other constant force pulling along its axis, this is a static load. If the nut and bolt are used only to provide clamping force, the static load is zero. The tensile stress area is the area of a cross-section of the bolt. The tensile stress due to preload is the preload divided by the tensile stress area. The tensile stress due to static load is the static load divided by the stress area. The tensile stress sum as provided by the calculator is the sum of these. This sum can be considered conservative in that it probably overestimates the actual stress and so increases protection against the danger of breaking a bolt. In another sense, the sum is not conservative because it results in recommended torque that is probably less than the true maximum possible and so fails to protect against the danger of an undertightened nut. This may be all right if you consider preventing the bolt from breaking to be a higher priority than providing maximum possible clamping force. There is no easy answer here. Tricky trade-offs like this are one reason people hire licensed engineers to design machinery. More detailed comments are given above in the paragraphs about the Tensile Strength input parameters.

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.

One may wonder whether it is overly simplistic to calculate stress
simply by dividing load by stress area since stress throughout
the bolt will vary. For instance, stress is higher than average
at thread roots. Bickford (page 29, 3rd edition) provides some
useful comments about this. We do not need to worry about this
issue because bolt manufacturers base their ratings on the simple
form of stress calculation described above. If a manufacturer
shows you a bolt that has a tensile stress area of one square inch and
says the bolt can withstand tensile stress of 60,000 pounds per square
inch, the manufacturer is saying that in laboratory tests, this type of
bolt was able to take a load of 60,000 pounds without breaking. This
test probably induced stresses higher than 60,000 psi in parts of the
bolt, but the strengh rating is based only on the load and tensile stress
area. That is, manufacturers state their ratings with the
assumption that you are going to use exactly the calculations described
above to calculate the load a bolt can take.

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.)

If the load on the joint is too great, and the bolt thread is the weakest part of the joint, the nut will be pulled off the bolt, stripping off the bolt threads as it goes leaving only a stub of thread on the bolt. Since the bolt threads stick out from the bolt, the preload and static load on the bolt are shear load for the threads. If the bolt thread is sheared off, the remaining stub of thread will have a major diameter equal to the minor diameter of the nut thread. Consider an imaginary cylinder having this diameter and whose height is equal to the joint's length of engagment. If this cylinder is positioned around the bolt in the engagement, it will intersect the bolt thread in the material that would be exposed if the thread strips. The area where this cylinder intersects the bolt thread is the bolt thread shear stress area. That is, the material in this intersection is the material that must withstand the shear stress. One of the main guidelines in designing joints is that any length of engagement should be long enough that thread failure will not occur before the bolt breaks. So we want the length of engagement to be long enough and the shear stress area big enough that the load required to shear the thread is more than the load required to break the bolt. Following the standard formulas, we assume that the bolt material can withstand only half as much stress in shear as it can in tension. This means that we want the thread shear stress area to be at least double the bolt's tensile stress area. (Bickford mentions studies that have found that bolt materials generally have a true yield strength about 55% to 70% of their tensile strength (page 95, 3rd edition) and provides a table of tensile and yield strengths of some materials (page 103, 3rd edition).) The "Minimum Engagement to Protect Bolt Thread" on the output page is the length of engagement required to make the bolt thread shear stress area double the tensile stress area. The "Minimum Engagement to Protect Nut Thread" is the length of engagement required so that the nut thread is not the weakest part of the connection. If the nut material and bolt material have the same tensile strength, this is the length of engagement that makes the nut thread shear stress area double the tensile stress area. If there is any logic to the ASME specifications, it seems that any standard-sized nut should provide a length of engagement that satisfies these requirements, but we have not verified that the specifications do this.

The parameters calculated for shear in the nut thread are derived similarly to those for the bolt thread, but in the case of shear of the nut thread, the remaining thread stub will have minor diameter equal to the major diameter of the bolt thread, so we use that diameter in the calculations. The geometry involved guarantees that whatever length of engagement the calculations are based on, the resulting nut thread shear area will be more than the bolt thread shear area, hence shear stress in the nut will be less than shear stress in the bolt, and the nut thread will not strip before the bolt thread assuming the nut material has as much shear strength as the bolt material, which it should if proper design guidelines are followed. If the internal thread is actually on a tapped hole and not a nut, then this calculation takes on more significance since there are not such clear rules about materials.

**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.

**Who is right? **
On the output page, we provide the nut and bolt thread shear stress
area associated
with the length of engagement that you specify on the input page.
This is recommended by Blake (pages 40 and 175) and maybe Bickford
(Table 3.4, page 74), but apparently not by Machinery's Handbook
(page 1416, 25th edition).
Bickford (Table 3.4, page 74) gives bolt thread shear stress areas
assuming the length of engagement is equal to the thread size, and
Blake (Table 1, pages 36-37) gives thread stripping areas
per inch of engagement for both nut and bolt threads.
(Bickford and Blake have different results for the tensile stress
area of UN 4-8. Using our calculator, we see that Blake is correct,
Bickford has a typo.)
Machinery's Handbook makes a different recommendation which we find
incomprehensible, and Bickford seems to follow Machinery's
Handbook when he describes these calculations
(pages 66-69, 3rd edition), apparently contradicting his choice of length
of engagement in his table of shear stress areas.
Machinery's Handbook (page 1416, 25th edition) clearly states
that the nut and bolt thread shear stress areas should be calculated using
the minimal length of engagement to prevent the bolt thread stripping
described in the previous paragraph.
In addition to the obviously
questionable relevance of such a calculation, it is redundant
in that we already know that it in every case it will provide a
bolt thread shear stress area exactly double the tensile stress area
and a bolt thread shear stress exactly half the tensile stress.
Appendix B to the ASME specification (which is not officially part
of the specification) says that the shear stress area is calculated
using the "length of engagement". It seems obvious that this should
mean the actual length of engagement in the joint since no other
length of engagement is referred to in the Appendix, but comments
in other references make us wonder whether an earlier edition of the
specification created some confusion since it also provided a calculation
for the minimum length of engagement to prevent stripping of the bolt thread.
We only have the current edition of the specification so we can only
guess about this.

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.

The ASME specification does not provide tables of thread shear stress areas, and it seems the only tables of calculated values in most other references are ones that are copied from the ASME specification. The only tables of thread shear stress areas that we have found are in Blake and in Bickford. Bickford's table seems to be derived from Blake's table, judging partly by the fact that Bickford credits Blake as a source for his table. Meticulous users will test our calculator's results against Blake's and wonder why there is often some discrepancy. The source of the discrepancy seems to be rounding. We are able to reconstruct Blake's calculations using the major, minor, and pitch diameter limits in the ASME tables which are rounded to the nearest thousandth or ten-thousandth of an inch. The tables may seem to provide plenty of precision considering the magnitude of the diameters, but the effect of this rounding is greatly increased by the fact that the formulas for shear stress areas use a difference between two diameters. The difference between the diameters is about a quarter of the sharp-V thread height, so less than one quarter of the pitch, (sharp-V thread height is half of the square root of 3 multiplied by the pitch, roughly 0.866 times pitch) and using the tabulated values this calculation is affected by two rounding errors. The resulting percentage error can be much worse than the percentage error due to rounding in a single tabulated dimension. For instance, for a 4-8 UN thread, a quarter of a pitch is 1/32 inch - about 1/128 of the diameter (depending on which diameter one uses). So rounding of the diameters in the tables can produce more than a hundred times as much percentage error in this difference as there is in the tabulated diameters individually. Our calculator calculates thread dimensions to the maximum accuracy possible in an Active Server Page and does not round any values used in subsequent calculations. If you are concerned about the differences between Blake's calculations and ours, then probably you should be quite concerned about shear stress areas in general. We say the accuracy of dimensions in the ASME tables is not enough for highly accurate calculations of shear area, but, in fact, the accuracy of dimensions in the tables is probably about the best you could expect in any actual manufacturing of nuts and bolts. So if discrepancies between Blake's calculations and ours are worrisome, this can be a reminder that the thread shear area is based on a very narrow band of material whose dimensions cannot be controlled to a high degree of accuracy. We realise that some users will prefer Blake's calculations to ours since sticking with the rounded thread dimensions in the ASME tables provides maximum consistency if not maximum accuracy.

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.

Marcel Dekker, Inc, 1995.

Blake, Alexander,

Marcel Dekker, Inc., 1986.

Oberg, Erik; Jones, Franklin D.; Horton, Holbrook L.; Ryffel, Henry H.;

Industrial Press, Inc., 1996.

Parmley, Robert O.,

McGraw-Hill, 1989.

American Society of Mechanical Engineers,

ASME, 1990.

Department of Commerce,

McGraw-Hill, 1956.

McGraw-Hill, 1955.

McGraw-Hill, 1959.