Lug and Pin Analysis — Air Force Method (AFFDL-TR-69-42, Bruhn D1): Lug Bearing, Net Section, Transverse and Oblique Strength, Pin Shear and Bending, Press-Fit Bushings

This page follows the Air Force Method, set out in Section 9 of the Stress Analysis Manual of the Air Force Flight Dynamics Laboratory (AFFDL-TR-69-42). Bruhn Chapter D1 reproduces the same curves, so the two names describe one method. It is empirical: the interaction between bearing, shear-out and hoop tension is messy enough that the manual rolls it into coefficients read off test-derived charts rather than deriving each mode separately.

A pin-loaded lug can fail by tension across the net section, shear-out along two planes ahead of the hole, bearing against the pin, hoop tension across a single plane, or by buckling out of plane (“dishing”) if it is thin. The method folds the middle three — bearing, shear-out and hoop tension — into a single allowable, because in a real lug they are not independent.

The bearing allowable carries the combined bearing, shear-out and hoop tension behaviour in a single coefficient, read from Figure 9-2 as a function of e/D and D/t.

Net-section tension is checked separately, with the net-tension coefficient K_n from Figure 9-4 — a knockdown for the stress concentration at the hole that depends on the material's ultimate and yield strengths, modulus and ultimate strain.

In both, \( F_{u} \) is not simply F_tu: the manual caps the effective ultimate at \( \min\!\left(F_{tu},\; 1.304\,F_{ty}\right) \), which binds on materials with a high yield-to-ultimate ratio. That cap is applied here and shown in the calculation steps.

In a single-shear joint the two members are offset, so the load path is eccentric: the pin carries the whole load on one plane and the offset drives a bending moment through the joint. A double-shear (clevis) joint splits the load either side of the inner lug, halving the load on each pin plane and keeping the load path balanced. This is why the pin shear condition below changes the answer, and why the clevis gap sets the pin bending arm.

A hinge or a stacked fitting puts n lugs on one pin, interleaved from the two sides (Figure 9-13). The side with more lugs has (n + 1)/2 of them, the other (n − 1)/2. The manual (§9.13) does not ask for a pin-deflection analysis. It gives a load split (Table 9-2) that, at ultimate, leaves the pin almost straight. It assumes plastic lugs and an elastic pin.

• The inner lugs on the side with more lugs, all of thickness t′, carry P₁ each.
• The two outermost lugs on that side carry only C P₁. They may be as thin as C t′.
• Every lug on the other side, of thickness t″, carries P₂ = P/[(n − 1)/2].

So P = P₁[(n + 1)/2 − 2 + 2C]. The pin sees a peak shear k_sP₁ on one plane and a moment M = P₁b/2:

Choose Multiple under Pin shear condition, set the lug count, and enter t″ as the thickness and t′ as the other side’s inner lugs. The outermost lugs default to the minimum C t′. Each lug is then checked at its own load, with its own bearing coefficient: a thin outermost lug has a large D/t and can govern even though it carries least. The pin is checked against one shear plane, 0.785 D²F_su, and against M_u = 0.0982 k_bD³F_tu. The gap does not enter. Tangs are not checked in this mode.

Loading a lug across its axis rather than along it engages a different, more redundant load path, so it gets its own coefficient family from Figure 9-8. That curve is driven by an effective edge distance h_av, averaged from four radial distances between the hole and the lug boundary — which means a proper transverse check needs more of the lug outline than a single edge distance describes.

A load at an angle is resolved into its axial and transverse components and recombined against the two pure allowables by a power-law interaction, with an exponent of 1.6:

\( \left(\dfrac{P_{ax}}{P_{u}}\right)^{1.6} + \left(\dfrac{P_{tr}}{P_{tru}}\right)^{1.6} = 1 \)

The exponent matters. A plain ellipse (exponent 2) reads about 9% high at 45°, in the unconservative direction. The “allowable vs load angle” chart above is this envelope, sliced along the applied angle.

Each lug — the inner lug and the outer lugs of a clevis, or the two lugs of a single-shear joint — is worked through the full method:

• Axial bearing — \( P_{bru} = K_{br}\,F_{u}\,D\,t \), with K_br from Figure 9-2 at D/t ≤ 5 and Figure 9-3 above. This one coefficient covers bearing, shear-out and hoop tension together, so there is no separate tear-out check.
• Axial net section — \( P_{nu} = K_{n}\,F_{u}\,(W - D)\,t \), with K_n from Figure 9-4 at the material’s F_ty/F_tu and F_tu/(Eε_u).
• Transverse — \( P_{tru} = K_{tru}\,F_{tu}\,D\,t \) from Figure 9-8, at the effective edge distance h_av of Figure 9-10.
• Oblique — the two combined by the 1.6-power interaction at the applied angle.
• Fatigue (optional) — life of a 2024-T3 or 7075-T6 lug under constant-amplitude axial load, §9.17, from Figures 9-23 to 9-26.

F_u = min(F_tu, 1.304 F_ty) throughout. A bushing is checked at 1.304 F_cy on its bore. In double shear the joint allowable is equation 9-19: the lower of the inner lug-bushing strength and twice the outer, pin shear, and pin bending — with the balanced-design value of equation 9-16 when the pin is weak in bending. In single shear the lug moment amplifies the bearing and net-section stress (equations 9-23 to 9-27). Lugs under 5% elongation are reduced by the §9.15 ductility factor.

MS = P_allow/P − 1. A margin at or above +0.15 shows as PASS, between 0 and +0.15 as MARGINAL, and below zero as FAIL. Those bands are a display convention on this page, not a certification requirement — the margin your programme demands is the one that governs.

The coefficients are read off digitized charts and interpolated linearly between curves, so expect agreement with a careful hand reading to about the width of a pencil line — a percent or two. Outside a chart’s range the nearest edge is held and the calculation steps say so.

• The outer lugs (double shear) or mating lug (single shear) share the width, edge distance and material of the lug entered. A clevis with different outer-lug geometry needs a second run with those values.
• Figure 9-10 assumes a concentric lug, whose head radius is both e and W/2; the smaller of the two is used. For any other outline, measure h₁ to h₄ as in Figure 9-7 and enter them under Advanced method options; the page then computes h_av directly instead of reading the chart.
• In single shear the lug moment is split equally, M = P(t + t₁)/4, which assumes neither member restrains the joint. Enter a moment from your own load path if the structure carries part of it.
• K_n uses the published elongation as the ultimate strain ε_u. Where none is published, the least ductile curve (F_tu/Eε_u = 1.0) is used, which is conservative.
• Not checked: out-of-plane buckling of thin lugs and yield-load limits. Lug tangs (§9.4.4) are checked in double shear only, as the manual gives them. Fatigue (§9.17) covers constant-amplitude axial load on 2024-T3 and 7075-T6 lugs only; see Lug fatigue.

Properties come from the same MIL-HDBK-5 dataset behind the Material Property Lookup and Multi-Material Comparison, so F_tu, F_ty, F_su and F_bru are published allowables rather than typical values. The short list covers the common aerospace picks; ticking Full MIL-HDBK-5 database opens every published condition — alloy, product form, temper, thickness band and A/B/S basis — along with grain direction and a service temperature that derates the strengths off the effect-of-temperature curves.

Beyond the strengths, the method reads the modulus E and elongation (for K_n, the pin plastic bending coefficient and the §9.15 low-ductility check), F_cy for a bushing, and the e/D = 2.0 bearing allowables, which cap the lug bearing (P_bru/Dt may not exceed F_bru or 1.304 F_bry). A service temperature derates the yield-type values along with F_ty. Every one of these is named in the calculation steps.

A bushing pressed into a lug with negative clearance squeezes the hole outward. Treating the lug head as a thick ring and the bushing as a thick tube, the contact pressure is set by the two radial displacements adding up to the interference (§9.16, equations 9-36 to 9-38):

with A the bushing bore radius, B its outside radius, C the lug outer radius and D the hole radius; δ is the radial interference, half the diametral interference entered above. The hole sees a hoop tension \( F_t = p\,(C^2+D^2)/(C^2-D^2) \) and the bushing bore a hoop compression \( 2pB^2/(B^2-A^2) \).

The manual limits the sustained hoop stress rather than letting it run to yield: 0.50 F_ty for aluminum alloys and 8,000 psi for magnesium (stress corrosion), and 25% of F_tu for steel heat treated above 200 ksi (hydrogen embrittlement). The bushing bore is checked against its compressive yield. A moderate press fit helps fatigue, because the hoop preload cuts the stress range under oscillating loads.

Section 9.17 of the manual predicts the fatigue life of 2024-T3 and 7075-T6 aluminum lugs under constant-amplitude axial load. It is an empirical fit to lug tests, not a stress-concentration calculation. Three factors from Figure 9-23 describe how severe a lug is, and their product k₁k₂k₃ picks one curve from a family of R = 0 life curves (Figure 9-24 for 7075-T6, 9-25 for 2024-T3). A higher product is a better lug and sits on a higher curve.

• k₁ is read against a/C, where a is the end distance from the edge of the hole to the end of the lug and C the side distance from the edge of the hole to the side (Figure 9-21). For the lug drawn here, a = e − D/2 and C = (W − D)/2.
• k₂ is read against D/C.
• k₃ is read against the hole diameter D in inches: a size effect.

For 2024-T3 a single line serves for both k₁ and k₂. The method is only claimed inside the shaded region of Figure 9-22: C/D from 0.3 to 2.0 and a/C from 0.8 to 2.0, with the corner cut at a/C + C/D = 3.5. Outside it the manual asks for tests, and the page says so.

The printed curves stop at a few hundred to a thousand cycles. To reach static failure, the manual replaces the top of the chosen curve with a straight line on the semi-log axes. The line runs from 0.5 F_nu at one cycle down to where the curve crosses 0.4 F_nu. Here F_nu is the net-tension ultimate stress of §9.3.2, K_nF_u. At R = 0, 0.4 F_nu is the alternating stress whose peak is 0.8 F_nu, the manual’s average yield for these alloys.

The mean stress is then handled with a Goodman diagram (Figure 9-26). Each constant-life line runs from the R = 0 point (f_a0, f_a0) to (F_nu, 0). Instead of drawing the diagram, the page solves that line for the R = 0 amplitude equivalent to the applied cycle, then reads its life off the curve:

When a required life is entered, the curve gives f_a0 at that life. The same Goodman line then gives the largest peak stress allowed at the applied stress ratio R = f_min/f_max:

The tile’s margin is f_max,allow/f_max − 1, a margin on stress, not on life. All stresses are net-section stresses on the inner lug, P/((W − D)t), with the hole diameter taken as the bushing OD when a bushing is fitted.

Check against the manual (§9.18). The example is a concentric 7075-T6 lug with a = C = 0.344 in and D = 0.437 in.

• Figure 9-23 gives k₁ = 1.0, k₂ = 1.33 and k₃ = 0.99. The digitized charts give 1.008, 1.322 and 0.996.
• With F_nu = 70 ksi and a cycle of 27,700/18,470 psi, the manual reads N = 8×10⁴ from its Goodman diagram. The test result was 8.6×10⁴. The same inputs here give 7.7×10⁴.
• The manual’s table of f_a against N for this lug is matched to within 0.3 ksi at every life except 10⁴ cycles (12.7 against 13.5 ksi). That is the tolerance of reading the scan.

Limits. Constant-amplitude, axial loading only: no spectrum, no transverse or oblique fatigue load, no fretting allowance. A compressive minimum load extrapolates the Goodman line below R = 0, where the tests did not go. The single-shear bending moment is not included. Any other alloy is read on the curve family chosen, and the page flags that as an estimate. A press fit generally improves fatigue life, but this method does not credit it.

The calculation engine (js/lug-math.js) is run in automated tests against the worked example in §9.6 of the manual, the NASA reproduction of Bruhn’s peaking-factor chart, and hand calculations of the press-fit equations.

The peaking factor γ of the Bruhn pin-bending option is Bruhn’s Fig. D1.15 as reproduced in the NASA Astronautic Structures Manual, traced from the NASA scan. Its dashed starting portions are, in the manual’s words, not substantiated by test data.

Section 9.6 of AFFDL-TR-69-42 works a complete double-shear joint by hand. It is worth following once, because every step on this page is one of these steps. The two outer (female) lugs are 2024-T351 plate, the inner (male) lug 7075-T651, each with an aluminum-bronze bushing on a 3/4″ 4130 steel pin; the gap between lugs is g = 0.10″. The page reproduces the manual’s numbers in its automated tests; they are shown below side by side.

The pin governs, and it does so by bending, not shear: at 37,900 lb the joint is worth only 86% of the lug-bushing strength. The balanced-design widths say the same thing from the other side: lugs thicker than b₁ = 0.33″ and 2b₂ = 0.65″ carry no extra load with this pin, so a lighter design would thin the lugs or a stronger one would enlarge the pin. The manual prints the inner lug’s net section as 112,313 lb using 1.304 F_ty, but F_tu < 1.304 F_ty for this 7075, so equation 9-6a (K_nF_tu) applies; bearing governs that lug either way.

This joint cannot be loaded straight into the calculator: the page uses one material and one lug outline for the inner and outer lugs, and its materials come from MIL-HDBK-5 design allowables rather than the manual’s Table 9-1. Running the inner lug as entered and the outer lug as a second run with its own geometry reproduces the same steps.

1. Air Force Flight Dynamics Laboratory (1986). Stress Analysis Manual, AFFDL-TR-69-42, Section 9, Lug Analysis. Open Section 9.
2. Melcon, M. A. and Hoblit, F. M. (1953). “Developments in the analysis of lugs and shear pins.” Product Engineering, 24(6), 160–170. The source of the Air Force method’s coefficient charts.
3. Cozzone, F. P., Melcon, M. A. and Hoblit, F. M. (1950). “Analysis of lugs and shear pins made of aluminum or steel alloys.” Product Engineering, 21(5), 113–117.
4. Bruhn, E. F. (1973). Analysis and Design of Flight Vehicle Structures. Jacobs Publishing. Chapter D1, lugs and shear pins, including the pin-bending peaking factor.
5. NASA (1975). Astronautic Structures Manual, NASA TM X-73305, Volume I, Section B2, Lugs and Shear Pins. Open Section B2 (PDF, 27.8 MB). Volumes II and III cover stability, thermal stress and composites.
6. Department of Defense (2003). MIL-HDBK-5J, Metallic Materials and Elements for Aerospace Vehicle Structures. The design allowables used here.
7. ESDU 91008 (1991). Design against fatigue: lugs, and related ESDU static-strength data items. A widely used alternative to the Air Force method.