This page summarizes a Spring 2011 project in which a partner and I were to investigate some type of recurring mechanical phenomena and draw conclusions from a statistical analysis. Of course this goal can be satisfied by studying almost any mechanical system or object. Because of the prevalence of threaded fasteners, we were quite interested in forming a general consensus on the overall quality of production, or conformance to existing standards, by a random set of known and unknown fastener manufacturers. Can we assume that any mass-produced fastener is perfect? This takes into consideration too many variables: material, thread form, head size, surface finish, etc. We selected a particularly important feature, the 'pitch diameter,' to serve as our feature of interest.
In this experiment, pitch diameter measurements for different sizes of American National
and Unified threaded fasteners were compiled for statistical analysis. The hypothesis was that
the pitch diameters of many of these fasteners would not conform to the dimensions given by
current ANSI/ASME screw thread standards. Using a micrometer, and a method of
measurement known as ‘the three wire method’, nearly 200 bolts were carefully measured to a
resolution of .0001”. Of these fasteners, 25, or 12.4%, were outside their allowable tolerance
range and thus the hypothesis was successfully verified. The measurements conformed to a
normal distribution, and as a result, it was determined that any randomly selected bolt could be
assumed, with 90.53% average confidence level, to have a pitch diameter within its tolerance
range. Additional analysis revealed correlations between manufacturing accuracy and fastener
material or coating, namely that zinc-plated or low-grade fasteners are most likely to be out of
Our hypothesis is concerned with a dimension called the 'pitch diameter'. It is the
imaginary circle appearing approximately halfway between the major and minor diameters of
any thread form and describes the 'intersection of engagement’ for mechanical components. It
is not only a dimension of great importance for mating (internal and external) threaded
fasteners, but also for gears. Such dimensions are standardized by the ANSI/ASME societies,
and are listed in handbook tables. The fasteners used in this experiment are of the American
National and Unified (UN) form, otherwise known as ‘English’ (non-metric) type. Of the four
sample groups, the conventional callout suffix of UNC 2A or UNF 2A, which defines the class of
fit and whether internal/external, has been dropped for ease of reading.
Because the pitch diameter is not a directly measurable feature, the ‘three wire method’
is used. It entails a micrometer reading over a fastener, which has been sandwiched between
three thin and solid wires of a known diameter as shown in Figure 1. The thread wires are small
enough to fall between the crests of the thread, making contact halfway down the angled
surfaces. A measurement thus taken ‘over the wires’ will include the wires themselves, and so a
supplementary calculation must be performed. This is discussed later in the Theory section.
The practical importance of this experiment is great because most people generally
assume all fasteners to be dimensionally accurate. Good fit and strength of engagement is
ensured when male and female threads interface properly. This can include a lack of ‘jiggle’
when hand-tightening, as well as not binding up. Proper fastener performance is generally
governed by the pitch diameter, and is regulated by a relatively tight tolerance as shown in
Table 1 for the fasteners tested in this experiment.
The experiment had a fairly narrowed focus of only four ‘everyday’ fastener sizes: 10-32,
1/4-20, 5/16-18, and 3/8-16. For each size, 45 to 60 samples were measured and recorded for
later statistical analysis. Besides proving the hypothesis, this data allowed for various analyses
such as normal distribution plots, % rejection rates, and confidence intervals.
Following the experiment, the superficially large micrometer readings (since the wires are included in the measurement) must be corrected by an equation, which is derived of the geometric relations between the round wires and the triangular thread form. The shape of the threads is that of a complex helicoid, and thus the resting location of the wires is affected not only by the relative diameter, but also by the lead angle of the screw. It is because of this complication that there are at least three available equations of varying precision, which relate the measurement (M) to the actual pitch diameter (PD). These equations will either assume no lead angle (simplified), an approximation, or an exact compensation for lead angle. The formula for screw-thread lead angle is:
(1) Lead Angle = tan-1 [ (1/Threads Per Inch) ÷ (Mean PD) ]
Consequently, the lead angles for our fasteners fall in the range of 3.2-4.3 degrees, and
the need for lead angle compensation becomes negligible. The simplified equation was selected
for our calculations and is given here in (2). Because many parameters associated with Unified
National thread forms are constant, a simplified version is given in (3).
(2) PD = M + TcosA - W(1+cscA)
(3) PD = M + 0.866025 – 3W
Also, because the wire diameter W is known for each of the four different fastener sizes
being measured, equation (3) may be reduced to a single constant relating the pitch diameter
PD to measurement M. This is convenient because the pitch diameter can extracted from any
measurement by simple subtraction. This relationship between measurements over wires
and the actual pitch diameter measurement is demonstrated in the table below.
DESIGN OF THE EXPERIMENT
In order to accurately measure the pitch diameter of a thread form, there are many methods available. An incomplete list is presented below.
• Three wires: This is the method used in this experiment, and is very common.
• Pitch micrometer: A preferred, although expensive alternative.
• Optical comparator: More involved, and requiring expensive and large equipment.
• Plug gages: A quick, reliable method used on internal and external threads.
We chose the "three wire method" because prior experience had proven its reliability, and additionally, the equipment was available and highly portable. Another advantage was that the inclusion of many different sizes of wires allowed two inspectors to work simultaneously. Pictures of the method in action can be found in Figure 7 and 8 in the Appendix. The primary measuring instruments used were two Mitutoyo micrometers (a very reputable brand), selected for their clutch thimbles. The clutch allowed each micrometer’s thimble to ‘slip’ when a certain torque had been applied, thus reducing the subjective effects of human operation. Each of these micrometers was capable of resolving a measurement to 0.0001” across a span of 0-1”.
All other details of the experimental design were associated with the minimization of
systematic and random error. The steps that were taken are outlined as follows:
• Micrometers were periodically re-calibrated against high-grade gage blocks.
• The thread wire diameters were carefully measured to ensure the most accurate calculations on pitch diameter.
• Measuring equipment was kept at thermal equilibrium prior to the experiment.
• Micrometer stands were used to minimize human contact with the measuring instrument. Heat transfer caused concern for material expansion after calibrating with gage blocks at ambient (approx. 60 degree F) temperature.
• Fasteners, which may have been used in the past, were cleaned or blown off with compressed air or a brush.
Also, we recognize that although our instruments are capable of resolving to 0.0001”, we are subject to external conditions that may negate the need for such precision. For instance, multiple readings on the same fastener may differ slightly due to manufacturing inconsistencies. We did in fact plan to supplement our experiment with such a trial, and the results are presented in the Results section.
APPARATUS AND TEST PROCEDURE
The following equipment and supplies were required:
• Two ‘Mitutoyo’ mechanical micrometers of .0001" resolution with torque-limiting clutch
• One set of ‘Pee-Dee’ precision ground thread wires
• One set of ‘Flexbar’ rubber thread wire holders
• Gage blocks for instrument calibration
• Micrometer stand
• Compressed air
• Nylon or wire brush
• Computers with Microsoft Excel
• Recent copy of Machinery’s Handbook, with standard thread dimension tables
• Bolts of sizes 10-32, 1/4-20, 5/16-18, 3/8-16
Example procedure for the case of a 3/8-16 fastener:
1. Select the three .040” diameter thread wires from the Pee-Dee wire kit, which were predetermined for use with this fastener’s particular thread pitch.
2. Assemble two of the three wires in the convenient Flexbar ‘wire-holder’ (blue rubber thing shown in picture below) which is affixed to the non-moving anvil of the micrometer.
3. Twist the micrometer’s thimble until the gap is slightly larger than required to fit the 3/8-16 bolts plus one additional wire’s width.
4. Using your right hand, begin to close the thimble down. Your left hand must carefully hold the fastener and the last wire in position until contact is made.
5. Once the micrometer has the bolt sandwiched between the wires, take a few more seconds to verify that the wires are centered on the faces of the micrometer’s anvils, and that the free wire is diametrically opposed and centered in relation to the other two wires. Make any necessary adjustments.
6. Slightly open the micrometer, and then make the final measurement by using light controlled force to close down until the torque-limiting clutch is activated. Take your hands off the micrometer.
7. Carefully take the reading off the micrometer’s primary scale and vernier scale.
8. The actual pitch diameter can be checked, if necessary, by subtracting the corresponding
The experiment yielded a total of 199 pitch diameter measurements, divided almost evenly between four common sizes of threaded fasteners (primarily bolts). For the sizes of 10- 32, 1/4-20, 5/16-18, and 3/8-16, separate statistical analyses were performed. Of immediate interest are the normal curves generated when using the Gaussian probability density function to describe the dispersion of pitch diameter measurements. The use of this function makes the reasonable assumption that the distribution is symmetric about the mean. Plots of the relative frequency of pitch diameters for each of the four sample groups are given below, with vertical red lines to indicate tolerance limits.
Approximately 12.6% of all measured bolts were found to be out their allowable tolerance range as specified in current ASME/ANSI tables for screw thread standards. This result proves the hypothesis to be correct beyond a reasonable doubt.
A confidence level was calculated for each sample group in order to find the likelihood that a randomly selected bolt would be in tolerance. The 10-32 sample group revealed the lowest confidence level of 74%, since it had the most rejects. The other three samples of bolts produced more desirable percentages. The ¼-20 confidence level is the best at 98%, followed by 5/16-18 and then 3/8-16. These confidence levels are 96.8% and 93.33% respectively.
Furthermore, the data showed that the mean measurement for each size was nearly the same as the ideal mean pitch diameter with one exception. For the 1/4-20 bolts, the group appeared centered around 0.2155” in pitch diameter, which is nearly 0.001” larger than expected. Despite this abnormality, this size of fastener had only two rejects -- the fewest of all four sample groups. More specific results pertaining to individual fastener sizes are presented in the table below.
In order to understand the results, additional information concerning the brand, coating, and grade of the fastener can be inspected. This data was not needed to prove the hypothesis but was recorded during execution of the experiment. For instance, organizing fastener size alongside the coating or finish reveals the trend presented below.
There is apparently a correlation between rejected fasteners that received a protective zinc coating and pitch diameters that are out of tolerance. Samples with the zinc coating came from a minimum of 5 different manufacturers and therefore this result cannot be attributed to error in a single production batch. Conversely, uncoated stainless steel fasteners represent the sample group with the lowest percentage of rejects. Overall, plain stainless steel bolts and those with a black oxide coating are most likely to be dimensionally accurate.
Finally, a correlation between bolt grade and number of rejects was explored in the 5/16- 18 sample set. It appears that higher grade bolts are more accurately manufactured. However, because the sample size for each grade is so small, the results may not be representative of the population. The results are presented below.
Finally, to demonstrate repeatability of the results, a single bolt was selected for multiple
measurements. A 1/4-20 ‘Unbrako’ brand bolt was selected for this experiment and the results
are presented below in Table 6. For ten readings along the length of the bolt, the standard
deviation for pitch diameter measurement was only 0.0001”. In conclusion, if the bolt is clean
and all other errors notwithstanding, any single measurement would be reasonably
representative of the entire bolt.
Four of the most common and popular bolt thread sizes were measured for pitch diameter accuracy. The allowable dimensional tolerances are set forth and standardized by the ANSI/ASME. For each of the four sizes, 45-60 bolts were measured and compared to the maximum and minimum allowable values. As suspected, not every bolt fell within the tolerance range. The number of rejects varied from 4-17% with no direct correlation between the size of the bolt and the percent rejected. The percent rejection from smallest to largest bolt size was 13.3%, 4.2%, 16.7%, and 15.2% respectively. Initially, it was suspected that the smaller bolts would yield higher rejection rates simply because the pitch diameter tolerance range is much narrower. The results indicate that small fasteners are no more difficult to manufacture than larger ones!
The bolt coatings and grade of each size were also compared to the rejection percent. Uncoated stainless, galvanized, and black oxide coated bolts ranged from 8-10% in rejection. This is not a great reject number, because filtering out 10% of all inspected bolts carries significant time and money loss. The highest reject rate based on coating was 27% and was found for zinc-plated bolts – a surprisingly high percentage!
Furthermore, there appears to be a correlation between the bolt grade and number of rejects. The higher the grade, the more likely the bolt is acceptable. Within the 5/16-18 sample group, Grade 2 bolts had 22.7% rejection, while Grade 5 and 8 had 9-10% rejected. Some of this has to do with the cost and care taken by the manufacturer when making higher grade bolts. It is intuitive that cheaper, weaker bolts would not be made to the same level of quality control as those made for more demanding applications. This result, therefore, was anticipated.
Measurements on a single bolt (in ten different places) were recorded to check the variance of the threads across each bolt and to get an idea for presence of random error. We found that, for a random 1/4-20 black oxide coated bolt, the pitch diameter didn’t deviate much. For instance, the standard deviation was only .0001” for all ten measurements. This low deviation, coupled with good instrument calibration, indicated that our experimental setup was solid and that our rejection rates were accurate.
For the 199 sample fasteners that were measured, an average of 12.4% were outside the pitch diameter tolerances set forth by the ANSI/ASME standards for American National and Unified thread forms. We found that this number can be correlated to coating or grade, but not the nominal fastener size. Zinc plated bolts, and Grade 2 bolts had the highest rejection percentages of the all sample groups. These percentages were 27% and 22.7% respectively. The 12.4% average rejection percentage confirms our hypothesis that fasteners are not entirely manufactured to conform to standard specifications. For any randomly selected bolt, confidence levels for choosing a good bolt are in the mid to high 90% range. Thus, we feel confident that a randomly selected bolt will work for any ordinary task being performed. Furthermore, in consideration of the 12.4% rejection average, we conclude that we should always use higher grade or non-zinc plated bolts produced by a trusted manufacturer when pitch diameter accuracy is required.
Oberg, Eric, Franklin D. Jones. Machinery’s Handbook 28th Edition. New York, NY: Industrial Press, 2008.