Thread Engagement and Strength: Method

Thread Strength: Method and Conventions

What the tool does

A threaded joint can fail in three ways: the bolt breaks across its threaded section, the bolt thread strips out of the nut, or the nut (or tapped-hole) thread strips. The tool evaluates all three for a Unified or ISO metric thread from the public-domain Screw-Thread Standards for Federal Services (NBS Handbook H28, 1969 Part I for the Unified series and the strength appendix, 1957 Part III for the metric series), prints the thread dimensions and limits of size with the handbook formulas substituted, and reports the length of engagement that makes the bolt break before either thread strips. Its default is a 1/2-13 UNC-2A steel bolt of 120 ksi tensile strength in a 2B nut of the same material engaged over 0.5 in, with an 8 kip tensile load and an 85 ksi proof strength for the proof and torque lines; the same analysis of an M12 × 1.75 bolt at the basic profile is the metric twin in the test script.

Basic profile and stress area

The 60° profile of H28 Table 2.1 (the ISO 68 profile) has the sharp-V height H = 0.866025 p. The basic pitch diameter is E = D − 0.649519 p (3H/4 below the major diameter), the minor diameter of the internal thread K_n = D − 1.082532 p (5H/4 below D, the flat H/4 at the root of the external thread), and the minor diameter of the external thread in its rounded-root design form K_s = D − 1.226869 p (17H/12). These are the columns of H28 Tables 2.8 (UNC), 2.9 (UNF) and 2.10 (UNEF), which the tool reproduces to the printed digits for every row. The lead angle at the pitch diameter is tan λ = p/(πE). The tensile stress area is the H28 Appendix A5 formula A_t = 0.7854 (D − 0.9743/n)², the area of a circle midway between the pitch and minor diameters; for metric threads the tool uses the ISO 898-1 form A_s = 0.7854 (D − 0.9382 p)², which is exactly (π/4)[(d₂ + d₃)/2]² of the basic profile (0.9382 = (0.649519 + 1.226869)/2). The 1/2-13 UNC value 0.1419 in² and the 1/4-20 value 0.0318 in² are printed in Table 2.8; M12 × 1.75 gives 84.27 mm², the 84.3 mm² of ISO 898-1.

Classes, allowance and tolerances (Unified threads)

H28 section 2, paragraphs 5 to 7: the pitch-diameter tolerance is T = C (0.0015 ∛D + 0.0015 √L_e + 0.015 ∛p²) with C = 1.5, 1.0, 0.75 for classes 1A, 2A, 3A and 1.95, 1.30, 0.975 for 1B, 2B, 3B (the internal factor is 30 percent larger because internal threads are harder to make). The L_e in the formula is the length of engagement the standard tables are based on: the diameter D for the UNC, UNF, 4UN, 6UN and 8UN series and nine pitches for UNEF and the 12UN to 32UN series; the tabulated values hold for engagements up to 1.5 D (15 p), and the handbook raises the tolerance to 125 percent for engagements up to 3 D (30 p) and 150 percent beyond, which the tool applies to the entered engagement and says so. The allowance of classes 1A and 2A is 0.3 times the class 2A tolerance (C = 0.3 in the same formula); class 3A has none. The major-diameter tolerance of the external thread is 0.060 ∛p² for classes 2A and 3A and 0.090 ∛p² for class 1A. The minor-diameter tolerance of the internal thread follows Table 2.20: for classes 1B and 2B, 0.25 p − 0.4 p² at 0.25 in and larger (0.15 p coarser than 4 tpi), and below 0.25 in 0.05 ∛p² + 0.03 p/D − 0.002, not more than 0.394 p nor less than 0.25 p − 0.4 p²; for class 3B at all sizes 0.05 ∛p² + 0.03 p/D − 0.002, not more than 0.394 p nor less than 0.23 p − 1.5 p² (80 to 13 tpi) or 0.120 p (12 tpi and coarser). The 1969 printing of the 3B lower bound reads "0.23 − 1.5 p²"; the p is a misprint, since only 0.23 p − 1.5 p² is dimensionally consistent and reproduces Table 2.21.

Limits of size: the external thread's maximum major and pitch diameters are the basic values less the allowance and its minima are those less the tolerances; the internal thread's minimum minor and pitch diameters are basic and its maxima are those plus the tolerances; the internal major diameter has only a minimum (D). The handbook tabulates the allowance and tolerances to four decimals and forms the limits from the rounded values (1/2-13 UNC-2A: 0.4985 − 0.0109 = 0.4876 for the minimum major diameter, where the unrounded subtraction gives 0.48765); the tool does the same, so its Table 1 is the printed Table 2.21 line for line. The external minor diameter is given for reference as E_s,max − 0.57735 p (the rounded-root design form of Table 2.21) and E_s,min − 0.649519 p (a flat root, par. 6.3). Metric threads are analysed at the basic profile: the ISO 965 tolerance classes (6g, 6H and the rest) are cited only, and the page says that the shear areas at basic dimensions are slightly unconservative against a toleranced pair.

Thread shear areas

H28 Appendix A5 (p. A5.04): the external thread shears along the cylinder of the maximum minor diameter of the internal thread and the internal thread along the cylinder of the minimum major diameter of the external thread, each over the length of engagement: A_ss = π n L_e K_n,max [1/(2n) + 0.57735 (E_s,min − K_n,max)] and A_sn = π n L_e D_s,min [1/(2n) + 0.57735 (D_s,min − E_n,max)]. The bracket times n is the fraction of the cylinder occupied by thread material: one half at the pitch line, plus 0.57735 n times the radial distance from it because each 60° flank widens the tooth by tan 30° per unit of radius. At the basic profile the fractions are 0.75 and 0.875, giving the handbook's 0.75 π K_n L_e and 0.875 π D L_e; the tolerances reduce them (0.574 and 0.733 for the default 2A / 2B pair). The test script integrates the 60° thread form numerically on a fine grid and recovers both closed forms to six figures.

Strengths, the break-before-strip rule and the length of engagement

The bolt's tensile strength is P_t = A_t F_tu; the stripping strengths are P_ss = A_ss F_su,ext and P_sn = A_sn F_su,int. Where no shear strength is entered the tool takes F_su = 0.6 F_tu, the approximation NASA RP-1228 (p. 21) gives for alloy and carbon steels (0.55 F_tu for stainless steels); for aluminium or plastic tapped holes enter the handbook value (MIL-HDBK-5J Table 3.6.2.0(b) gives F_su 27 ksi for 6061-T651 plate at F_tu 42 ksi, A basis, the tool's parent-material default). The design rule of H28 is that the bolt should break before either thread strips, so the margins M.S. = P_ss/P_t − 1 and P_sn/P_t − 1 should be positive. The required engagement follows the handbook: for materials of equal strength, failing together near the pitch diameter, L_e = 4 A_t/(πE); in general the shear area must be twice the tensile stress area, a factor from the NBS tests of 1929 (1.7 to 2.0 for steel and brass screws and nuts, taken as 2 for a small margin), so L_e = 2 A_t/(A_ss per unit length) governed by the bolt thread or 2 A_t/(A_sn per unit length) by the nut thread. For a nut or tapped material weaker than the bolt, Table A5.2 items 12 to 14 compare the ratio of shear areas R₁ with the ratio of strengths R₂ and multiply the bolt-thread length by R₁/R₂ when the material is the weaker; written with the shear strengths this is J = A_ss F_su,ext/(A_sn F_su,int) and L_e,req = L_e,ext × max(1, J). The tool also prints the lengths obtained with the entered shear strengths in place of the factor 2 (A_t F_tu over the shear area per unit length times F_su), which is the pure break-before-strip condition with no added margin; the H28 value is the recommended one.

Applied load, proof load and torque

With a tensile load P the margins are strength/(n_d P) − 1 for the bolt tension, both stripping modes and the proof load P_p = A_t F_p, where n_d is the entered design factor (1.0 compares the strengths with the load as entered; the H28 engagement rule carries its own factor independently). The torque line T = K F_i D for a preload of 0.75 P_p is the nut-factor relation only; the Bolt Preload and Torque tool holds the friction equation, the grade tables, the joint stiffness and the fatigue checks.

Threaded inserts

A wire-thread (helical coil) or solid bushing insert gives a soft parent material a strong internal thread. Its outside thread is a larger thread of the same pitch, nominally 1/8 in bigger in diameter than the bolt thread, so that, in NASA RP-1228's words, a 10-32 bolt in an insert is the equivalent of a 5/16-18 nut. The tool treats the insert as a second engagement: the bolt thread in the insert's internal thread (insert material) over L_e, and the insert's outside thread in the parent material over the insert length, each with the same shear-area formulas; the required insert length uses the bolt's stress area because the whole joint should still break the bolt first, and the governing strength is the least of the five. The outside thread is chosen from the standard series as the nearest size; the exact STI tapped-hole dimensions of MS 33537 are cited and may be entered as a custom thread when known.

Sources and copyright

Every formula and table value comes from the public-domain NBS Handbook H28 (1969 Part I and 1957 Part III) and NASA RP-1228, filed with the page renders in References/Thread-Strength/. ASME B1.1 and B1.13M, ISO 68, 261, 965 and 898-1, SAE J429, FED-STD-H28/2B and MS 33537 are cited only: no table of theirs is reproduced; the SAE grade 5 strengths (120 ksi tensile, 85 ksi proof) and the ISO 898-1 stress areas appear as single cited values. The standard-series sizes and pitches are dimensions of fact.

Limitations

Static strength only: no fatigue, no thread bending or dilation of thin nuts, no load-distribution effects along the engagement (the handbook's shear areas assume the whole engagement carries the load), no effect of the nut's material yield on the shear cylinder (the handbook notes the shearing diameter moves toward the weaker material), no combined torsion and tension on the bolt section (H28 A5.03 gives the combined-stress formulas for the installation torque), no pipe, Acme or buttress threads, and no tolerance classes for metric threads.