This page consists of descriptions of mistakes involving formulas and calculations that I have found in engineering textbooks about threaded fasteners. |
Even the ASME specification (ASME B1.1-1989, Unified Inch Screw Threads) has some questionable digits. The trouble exists for sizes #6 and larger in the very lengthy Table 3A - Limits of Size for Standard Series Threads (UN/UNR) (pages 23-47). Most of this table is copied in Machinery's Handbook in the table Standard Series and Selected Combinations -- Unified Screw Threads (Table 4, pages 1654-1676, 25th edition). The trouble is in the figures for minimum minor nut (internal) thread diameter. Notice that for sizes #6 and larger, this dimension is given to thousandths of an inch for class 2B, and to ten-thousandths of an inch for class 3B. But notice that in every case, the class 3B figure is just the class 2B figure with a zero tacked on as the ten-thousandths digit. Our calculator follows the formulas in the ASME specification and gives the result rounded to ten-thousandths of an inch. In about 90% of cases, this will not match the figure in the ASME table because the last digit in the ASME table is a bogus zero. Legitimate calculations to ten-thousandths of an inch appear in The New American Machinist's Handbook (Table 20, pages 10-40 and 10-41). (The title of that table says "Class 2B", but the Class 2B minimum minor nut (internal) thread diameter applies to Class 3B as well.) Legitimate calculations to the ten-thousandth of an inch also appear in Handbook of Fastening and Joining of Metal Parts. For example, use our calculator to calculate dimensions for 8UN 3.000-8 (3") Class 3A-3B. The minimum minor nut diameter provided by our calculator is 2.8647 inches. The number in the ASME table is 2.8650. But notice that the Class 2B figure immediately above this in the ASME table is 2.865 and the Class 3B figure is just the Class 2B figure with an extra zero added, and notice that this is true for all numbers in that column of the table for all sizes #6 and larger. Looking in the Handbook of Fastening and Joining of Metal Parts, Table 1.7, page 20, we find the correct figure 2.8647.
Occasionally, our calculator produces dimension values that differ from those in the ASME table in that the last digit is off by one. For instance, the minimum major diameter of a UNC 0.25-20 bolt is 0.2407 inches according to our calculator, but 0.2408 inches according to the ASME table and books such as Machinery's Handbook that copy the ASME table. Assuming one considers the formulas described in the ASME standard to be correct, our calculator produces the correct result rounded to the nearest ten-thousandth of an inch. At this point the reader may expect that the explanation is that the formulas produce a number like 0.24074999 and the discrepancy is due to a combination of insignificant errors in calculation and rounding. Actually, without rounding, the calculated number is not something like 0.24074999. We believe we know what shortcut was used in the calculations to produce these discrepancies in the ASME table. The practice may seem sloppy to someone with a Pentium-equipped Windows-equipped computer, but the latest edition of the standard was published in 1989 when computers were still relatively user-hostile, and we suspect that in fact ASME has not bothered to re-check the thousands of numbers in their tables since they were originally calculated for the introduction of the UN standard in 1949. The people who did the calculations for the tables back then may not have had any kind of calculating devices to work with, in which case it would not be at all surprising if they used all possible shortcuts in their calculations. Usually no discrepancy results, and we have never found the discrepancy to be more than a ten-thousandth of an inch. We recognise that some users may prefer to be consistent with the ASME tables, even if there are occasional inaccuracies. If enough customers express an interest, we could add an option to our software to reproduce the calculations from 1949. Users may want to be aware that although our software reports dimensions rounded to a ten-thousandth of an inch, we actually use our more precise unrounded calculated values to calculate stress areas.
Chapter 38 of Blake consists of examples of threadform calculations. The solution to Design Problem 1 ends with a formula for Knmax, but the formula actually represents Knmin. Also, the question mentions the "maximum material condition for the unified screw threads" which is irrelevant to the question, and even if it were relevant, it cannot be calculated because the question does not state the class of fit so there may or may not be an allowance.
Design Problem 2 concerns "minimum pitch diameter of external thread" "at maximum material condition". For an external thread, when we are talking about a maximum material condition, we get maximum diameters, not minimum. The solution ends with a formula for Esmin, but actually this formula represents Esmax, and even then it is only correct for Class 3A which has no allowance.
Design Problem 3 concerns an external thread stripping area. To solve this, Blake relies on his solutions to the two previous problems and gets an incorrect result of 1.134 sq.in. This mistake is especially odd since the same calculation is done correctly earlier in the book. Table 1 on page 36 says that the external thread stripping area for UNC 3/4-10 2A is 1.21 sq.in. per inch of engagement. The problem assumes that the length of engagement is 0.75 inches, so according to this table the answer to this problem should be 0.75*1.21=0.908 sq.in. This is much closer to the answer provided by our calculator (0.913 sq.in.). Comments on the help page which are duplicated further down this page explain why rounding errors are a significant problem in calculations of thread stripping areas. Blake finally gets back on track in Design Problem 4 where his results agree with Table 8 on page 84 of the ASME specification and also with our calculator.
On page 2.17 of the U.S. Department of Standards Screw-Thread Standards
for Federal Services, the formula for pitch diameter tolerance has
the cube root of length of engagement where it should have the
square root of length of engagement. On page 2.20, the
description of minor diameter tolerances for internal threads
for class 3B has an incorrect formula. Instead of
Machinery's Handbook (25th edition) is inconsistent regarding notation. On page 1407, tensile stress area is denoted by the symbol As, but on page 1416 As means bolt thread shear stress area, and tensile stress area is denoted by At. The notation on page 1407 is consistent with the current edition of Appendix B of the ASME specification, but page 1416 is not. Bickford (page 67, 3rd edition) gives some translations between old and new notation.
Bickford (Table 3.4, page 74) and Blake (Table 1, pages 36-37) have different results for the tensile stress area of UN 4-8. Using our calculator or the ASME tables, we see that Blake is correct, Bickford has a typo.
Table 1-2 on page 1-25 of Standard Handbook of Fastening and Joining
says that assuming a nut factor (torque coefficient) of 0.15,
inducing a preload (clamp load) of 1,500 pounds in a UNF 0.25-28 bolt
requires 55 inch-pounds of torque. The correct torque is
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. 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.
The only point about our calculator that we consider potentially troublesome is the one-inch diameter bolt from the obsolete American National standard. This bolt has 14 threads per inch. None of the UN one-inch bolts have 14 threads per inch, but this bolt was so widely used before the UN standard was introduced that one-inch-diameter 14-thread-per-inch bolts remain in use today. In the UN standard, a bolt with a one-inch diameter and 14 threads per inch is a "special" (UNS) size. Our calculator provides UNS calculations for any bolt diameter and number of threads per inch that the user cares to specify. The rules for calculating allowance and tolerances for UNS sizes differ from those for UNC and UNF sizes. Our calculator follows the UNS rules when calculating UNS parameters, but for the size UNS 1-14, many manufacturers and even ASME Table 3A actually follow the UNC/UNF rules because they provide parameters closer to the obsolete American National parameters.