Lug Strength: Axial, Transverse and Oblique Load (AFFDL-TR-69-42 Section 9)

Lug Strength by AFFDL-TR-69-42 Section 9

Method, Assumptions and Checks

Problem Statement

A lug is a plate with a hole through which a pin carries load into it. It can fail in several ways at once: the pin can tear out the material ahead of the hole (shear-out), crush or tear the lug in bearing, break the net section beside the hole in tension, or, when the load comes from the side, push out through the thin edge of the lug. The pin itself can shear or bend, a bushing can crush, and the shank behind the lug (the tang) can break. None of these is predicted well by the textbook formulas for a hole in a plate: the bearing stress is far from uniform, the net-section stress concentration depends on how much the material can yield before it breaks, and a pin that bends shifts the bearing load toward the faces of the lugs. The method used here replaces those formulas by efficiency factors measured on test lugs.

The tool follows Section 9 of the Air Force Stress Analysis Manual, AFFDL-TR-69-42 (Maddux, Vorst, Giessler and Moritz, Air Force Flight Dynamics Laboratory, 1969), which codifies the method published by Cozzone, Melcon and Hoblit in Product Engineering (May 1950) and by Melcon and Hoblit (June 1953). The manual's nomenclature, figure numbers and equation numbers are used throughout, and its worked examples (Section 9.6 for the static joint, 9.14.1.3 for the design procedure and 9.18 for fatigue) are reproduced by the tool's defaults or its test script. The allowables are those of MIL-HDBK-5J and may be filled from your saved materials.

Nomenclature

D is the hole diameter (the bushing outside diameter when a bushing is fitted), D_p the pin diameter, w the width of the lug at the hole, t its thickness, e the edge distance from the hole centre to the end of the lug along its axis and a = e − D/2 the distance from the edge of the hole, w_T the width of the tang. F_tu and F_ty are the ultimate and yield strengths along the lug axis, F_tux and F_tyx the cross-grain values (the material ahead of the hole is loaded across the grain), E the modulus and ε_u the ultimate strain (the handbook elongation divided by 100 is a fair estimate). Subscript 1 denotes the female (outer) lugs of a double-shear joint, 2 the male (inner) lug, B a bushing, p the pin and T the tang.

Simple and Advanced Modes

Simple mode checks one lug. The pin is assumed rigid, so the bearing load is spread uniformly across the lug thickness, which is the condition on which the strengths of Section 9.3 are based; the pin itself, the mating lugs and any bushing are not analysed, and the transverse strength is that of a concentric lug. The lug may be in double shear, where it is loaded symmetrically and carries no moment, or in single shear, where the offset of the two pin loads bends it by a moment M that the analyst enters (Figure 9-6). The load P is the load on this lug. The lug's inputs are the inner-lug fields of the Advanced mode and carry the subscript 2.

Advanced mode analyses the whole joint lug by lug, as in the manual's Table 9-1: a double-shear joint (Figure 9-5, two female lugs each carrying P/2 and the male lug carrying P), a single-shear joint (Figure 9-6, each lug carrying P) or a multiple-shear connection (Figure 9-13), each lug with its own material, dimensions and optional bushing, and the pin in shear and bending. The 9.14 design procedure and the 9.16 press-fit stresses are offered only in this mode.

Applied Loads and Margins of Safety

The manual gives allowables and leaves the applied loads to the analyst. The entered load is a limit or an ultimate load at an angle α from the lug axis (or as axial and transverse components). With the factor of safety FS, the yield factor FS_yld (1.0 for the Air Force and the FAA, 1.15 where an agency requires it) and the fitting factor FF (1.15 under FAR 25.625), a limit load P gives the applied ultimate load P_U = FF FS P and the applied yield load P_Y = FF FS_yld P; an ultimate load gives P_U = FF P and P_Y = FF FS_yld P / FS. Every margin of safety is the allowable divided by the applied load (or stress), minus one. The factors enter only the applied loads: no allowable is factored. Margins checked against ultimate allowables are labelled ULT, those against yield allowables LIM.

The factor 1.304 that appears in many of the manual's equations is 1.5 / 1.15. Where the ultimate strength of a material is more than 1.304 times its yield strength, an ultimate allowable could be reached with the material yielded at limit load; the manual then uses 1.304 times the yield allowable as the ultimate allowable, so that a part sized at ultimate does not yield at limit.

Axial Load: Bearing and Shear-Out (9.3.1)

Shear-out, bearing and the hoop tension ahead of the pin are treated as one failure mode with a single allowable load coefficient K. For D/t of 5 or less K is read from Figure 9-2 at e/D; for larger holes from the Figure 9-3 family, which also depends on D/t because a thin lug wrinkles ahead of the pin. Below e/D = 1.5 the failure is shear-out and the coefficient multiplies a/D:

F_bruL = K (a/D) F_tux   (9-1a),   F_bryL = K (a/D) F_tyx   (9-2a)

and above it the failure is bearing, F_bruL = K F_tux (9-1b), F_bryL = K F_tyx (9-2b). The allowable ultimate bearing load is P_bruL = F_bruL D t (9-3a) when F_tux ≤ 1.304 F_tyx, otherwise 1.304 F_bryL D t (9-3b). The manual also limits the bearing stress to the MIL-HDBK-5J bearing allowables at e/D = 2.0; when they are entered (or filled from a saved material) P_bruL/(D t) is capped at F_bru and at 1.304 F_bry.

Axial Load: Net Section (9.3.2) and Design Strength (9.3.3)

The net section through the hole carries the load at an average stress that depends on how much the material can yield at the hole edge before it breaks. The net-tension stress coefficient K_n is read from Figure 9-4 as a function of D/w, of F_ty/F_tu (three panels) and of F_tu/(E ε_u) (seven curves); the tool interpolates between the bracketing curves and panels. Then F_nuL = K_n F_tu (9-4), F_nyL = K_n F_ty (9-5) and P_nuL = F_nuL (w − D) t (9-6a), or 1.304 F_nyL (w − D) t (9-6b) when F_tu > 1.304 F_ty. The lug's allowable ultimate load is the smaller of the bearing and net-section values, P_uL = min(P_bruL, P_nuL) (9-7). The manual gives the yield allowables as stresses only; the tool forms the yield loads F_bryL D t and F_nyL (w − D) t in the same way and takes the smaller as P_yL.

Bushings (9.3.4, 9.3.5)

A bushing's allowable bearing yield stress is the compressive yield strength of its material, F_cyB, and its ultimate is 1.304 F_cyB (9-8); a bushing through the full thickness carries P_uB = 1.304 F_cyB D_p t (9-9), and the lug-bushing strength is the smaller of the lug and bushing values (9-10). Under transverse and oblique load the bushing is checked by its axial calculation (9.7.2, 9.10.2).

Transverse Load (9.7)

A lug loaded across its axis is more redundant than one loaded along it and has more than one failure path. The manual folds the shape of the lug into one effective edge distance

h_av = 6 / (3/h₁ + 1/h₂ + 1/h₃ + 1/h₄)

where h₁ to h₄ are the distances of Figure 9-7 (h₁ the most critical region, toward the shank on the loaded side, which is why it is weighted three times; h₃ the least distance from the hole to the edge). For a concentric lug with parallel sides h_av/D follows from e/D alone (Figure 9-10), which is what Simple mode uses. The transverse coefficients K_tru and K_try are read from Figure 9-8 at h_av/D, and F_bruLtr = K_tru F_tux (9-28), F_bryLtr = K_try F_tyx (9-29), P_truL = F_bruLtr D t or 1.304 F_bryLtr D t (9-30a, 9-30b).

Oblique Load (9.10)

A load at an angle α is resolved into its axial component P cos α and transverse component P sin α (Figure 9-11), and the lug is checked by the interaction of Melcon and Hoblit,

R_a^1.6 + R_tr^1.6 = 1   (9-31)

with R_a the axial component over P_uL and R_tr the transverse component over P_truL. The margin of safety is 1/(R_a^1.6 + R_tr^1.6)^0.625 − 1, which equals the allowable oblique load P_αL over the applied load minus one; at α = 0 it is P_uL/P − 1. The same interaction is formed with the yield loads for the LIM margin. Figure 9-12 is drawn with each lug's load line and applied point.

Double-Shear Joint (9.4, Advanced mode)

The joint strength from the lugs and bushings is the smaller of twice the outer lug-bushing strength and the inner one (9-11). The pin carries P_usp = 1.571 D_p² F_sup in double shear (9-12). With the bearing uniform across each lug its maximum bending moment is (P/2)(t₁/2 + t₂/4 + g) (9-13), its failing moment 0.0982 k_bp D_p³ F_tup (9-14) with the plastic bending coefficient k_bp (1.56 for a ductile pin), and the pin bending load P_ubp follows (9-15). If P_ubp is at least the lug-bushing strength or the pin shear strength the pin is strong and the joint allowable is the smaller of those two (9-19a). Otherwise the pin is weak: it deflects, the bearing load crowds toward the shear faces over effective widths b₁ and 2b₂, and its bending strength rises until the joint is equally critical in pin bending and in the lugs, the balanced design of equations 9-16 to 9-18; the allowable is then the smaller of the pin shear strength and the balanced-design bending load (9-19b). The tangs are checked in uniform tension when 9-19a governs (9-20) and for the combined tension and bending of the off-centre bearing load when 9-19b governs (9-21, 9-22, with the tang coefficient k_bT).

Single-Shear Joint (9.5)

In single shear each lug carries the whole load and the offset of the two loads bends the lugs; the end moments satisfy M + M₁ = P(t + t₁)/2 when the lugs touch, the split following from the stiffness of the supports. The bearing stress is distributed as in a rectangular section of width D and depth t under the load and the moment, so the maximum lug bearing stress is P/(D t) + 6M/(k_bL D t²) (9-23) and the outer-fibre net-section stress P/((w − D)t) + 6M/(k_bL (w − D)t²) (9-24), compared with the axial allowables of 9.3. In Advanced mode the moment is split between the lugs by their thicknesses unless a share is entered, and the bushing (9-25), pin shear (9-26) and pin bending (9-27) follow; in Simple mode the moment on the one lug is entered and is factored with the load (M_U = M P_U/P). Under oblique load the bearing allowable is P_αL/(D t) and the net-section term uses the axial component (9.12).

Multiple-Shear Connection (9.13, Advanced mode)

For 5 to 11 alternating lugs (Figure 9-13) Table 9-2 distributes the load: each lug of thickness t′ carries P₁ except the two outer ones, which carry C P₁ and must be at least C t′ thick, and each lug of thickness t″ carries P₂; the table also gives the largest pin shear load and the effective bearing offset b of the largest pin moment P₁b/2. The distribution assumes plastic lugs and an elastic pin. Each strength is expressed as the joint load at which that element reaches its allowable.

Side Load for Misalignment

Lugs are normally checked for a minimum side load that accounts for misalignment of the parts: the NASA Astronautic Structures Manual (TM X-73305, Section B 2.1.0 and its Figure B 2.1.0-1) applies 0.10 P along the pin axis and checks the lug by beam formulas. This load is not the in-plane transverse load of 9.7: it bends the lug about its thin direction. The tool treats the lug as a cantilever from the pin to its root section, taken as the tang section w_T × t at the entered distance L from the hole centre, and adds at the corner of that section the lug's axial stress, the out-of-plane bending stress of the side load and the in-plane bending stress of the lug's transverse load:

f = P_a/(w_T t) + 6 S_U L/(k_bS w_T t²) + 6 P_tr L/(k_bS w_T² t)

compared with F_tu. The whole side load is put on each lug checked: in a clevis a good design clamps the pin in only one of the female lugs, so the two are not preloaded against each other, and either may be the clamped one. Near the hole the side-load moment is small, so the root rather than the net section through the hole is the critical place.

Lugs of Low Elongation (9.15)

The efficiency curves were obtained on ductile lugs. When the ultimate strain is below 5 %, the bearing allowables (9-1 to 9-3 and 9-28 to 9-30) are multiplied by B/B_.05, where the ductility factor B is read from Figure 9-17 at ε_y/ε_u (the yield strain taken as F_ty/E + 0.002) and B_.05 is the value for the same material with ε_u = 0.05; the plastic bending coefficients of the tang and of a single-shear lug become 1.5 B, at most 1.4. The net section is unchanged, because Figure 9-4 already depends on the elongation. The tool applies this automatically.

Press-Fit Bushings (9.16, Advanced mode)

An interference fit makes the lug a thick ring under internal pressure and the bushing a thick ring under external pressure. The pressure follows from the condition that the increase of the ring bore plus the decrease of the bushing outer radius equals the interference (9-36 to 9-38), and gives the hoop, radial and shear stresses at the bore of the lug and in the bushing. The tangential stress in the lug is limited by the manual to 0.50 F_ty in aluminum alloys and 8,000 psi in magnesium alloys against stress corrosion, and to 0.25 F_tu in steel heat treated above 200 ksi against hydrogen embrittlement. The interference stresses do not depend on the applied load, so this margin is labelled static.

Lug Design Procedure (9.14, Advanced mode)

For a symmetrical double-shear joint under a static axial load the manual gives an optimised design: the likely pin failure mode from equation 9-32 and Figure 9-14 (the density ratio of pin and lug materials matters because a heavier pin should be smaller), first approximations to the pin diameter (9-33 or 9-35), the optimum e/D from Figure 9-15, the lug thicknesses, the pin diameter that prevents bending failure (9-34), the final pin diameter rounded up to a standard size (9-36), and a reduced edge distance (Figure 9-16) or width where the bushing or the bearing governs. The resulting dimensions can be entered in the form to analyse the designed joint in full.

Fatigue (9.17, 9.18)

The manual's fatigue method is limited to 2024-T3 and 7075-T6 aluminum lugs under axial load, with dimensional ratios inside the region of Figure 9-22 (outside it the manual asks for tests). Three parameters are read from Figure 9-23: k₁ from a/c, k₂ from D/c and k₃ from the hole size D in inches, where c = (w − D)/2 (Figure 9-21). Their product selects an R = 0 life curve in Figure 9-24 (7075-T6) or 9-25 (2024-T3). The curve is extended to static failure by a straight line from the point where it reaches 0.4 F_nuL to 0.5 F_nuL at one cycle, F_nuL being the net-section allowable of 9.3.2. The amplitudes read from it are placed on the R = 0 line of a Goodman diagram and joined to F_nuL on the mean-stress axis by straight constant-life lines. The life of an applied cycle (net-section stresses f_max and f_min) is read where its constant-life line meets the R = 0 line, and the allowable stresses at a required life are read back from the same diagram. The manual's example gives 8 × 10⁴ cycles against tests of 8.6 × 10⁴.

Accuracy and Limitations

The manual's curves were digitized from the scanned report to about ±0.02 in the coefficients, so a value may differ from the manual's printed example in the third digit (the 9.6 example: joint allowable 37,790 lb against the manual's 37,900 lb). Readings beyond the ends of a figure use the end value, and the page says so. The method covers lugs with a single pin; it does not cover lugs with flanges, tapered thickness or multiple holes, composite lugs, or fatigue of other materials. The cross-check lines quote the original curves of Cozzone, Melcon and Hoblit and of the NASA manual for comparison only.