Lug Joint Strength — Maddux Section 9

Double-shear clevis & tang · axial / transverse / oblique · v1.0
Unsaved
Loading Python environment (numpy + matplotlib)…
Initializing Pyodide…
Joint Configuration
Picking a size auto-fills Dp, race OD/W, ball OD/W, gaps, and lug width below. Tick Override spherical-bearing dims above to keep your manual values when changing sizes.
Pin
Lugs — Geometry
e/D apply to all:
Typical: 1.5 (min), 2.0 (preferred), 2.4 (Maddux K-peak). Below 1.5, bearing capacity drops sharply.
Outer 1CenterOuter 2
e (in)
t (in)
w (0=auto)
L (in)
wt (in)
g gap (in) —
Lugs — Material
Outer 1CenterOuter 2
Material
Bushings / Spherical
Outer 1CenterOuter 2
Bushing (set above)
OD (in)
Race W (in)
Ball Db (in) — —
Ball W (in) — —
Governing
—
Min MS
—
Loads & Factors
Margins of Safety
Force Capability vs e/D
e/D capability chart will render here.
Geometry Verification Outer plan:
Diagram will render here.
Net-section Kn vs w/D (Bruhn-style)
Kn chart will render here.
Curves use the same Kn_lookup the per-lug net-section margin uses (Maddux Fig 9-4 polynomial family), plotted with the standard Bruhn x-axis convention (w/D, not D/w). Vertical markers show the current w/D on each lug; the Kn shown next to each marker is the value fed into that lug's net-section calc. Materials with missing εu are skipped. Lugs with w/D > 5 are clamped to the right edge of the chart (Kn ≈ 1 in that regime — ample ligament).
Materials to overlay
How the Kn curve is generated for a material

Goal. Kn is the empirical net-section knockdown applied to the ligament strength when the lug fails by tearing across the net section. It is not a constant: brittle materials, narrow ligaments, and high-yield-relative-to-ultimate materials all knock the section harder. The curve plotted above is Kn(w/D) for one specific material, evaluated at every w/D from 1.0 to 5.0.

Three things index a material into the family:

  • Fty / Ftu — how close yield is to ultimate. Higher = less plastic reserve = harder knockdown.
  • Ftu / (E · εu) — a brittleness index. Numerator is ultimate stress. Denominator is roughly the stress the material would reach if it stayed elastic up to its strain at failure. The ratio rises as materials become more brittle (low εu) or relatively low-stiffness for their strength. High = harder knockdown.
  • D / w — the geometry. Wider hole relative to lug width = less ligament = harder knockdown. The chart's x-axis is the inverse (w/D) per Bruhn convention.

The data structure: 3 master charts × 7 sub-curves.

Maddux Fig 9-4 is published as three master charts, one for each Fty/Ftu ∈ {0.6, 0.8, 1.0}. Inside each master chart, seven sub-curves are plotted, one for each representative value of Ftu/(E·εu) ∈ {0.0, 0.1, 0.2, 0.4, 0.6, 0.8, 1.0}. Each sub-curve is a function of D/w over (0, 1).

The 21 sub-curves were digitised from Maddux's published figure and fit with polynomials in D/w (degree 4–6 depending on curve shape). The coefficient tables live in the Python source as KN_M06, KN_M08, KN_M10 — one row per Ftu/(E·εu) bin within each master.

Bilinear lookup, evaluated at the lug's actual D/w.

  1. Compute the material's Fty/Ftu and Ftu/(E·εu). Clamp them to the supported domain [0.6, 1.0] and [0.0, 1.0] respectively (any material outside these ranges sits at the nearest extrapolation).
  2. For each of the 3 masters: evaluate all 7 sub-curve polynomials at the lug's D/w → that gives 7 Kn values for that master, one per Ftu/(E·εu) bin.
  3. Linear interp across those 7 points at the material's actual Ftu/(E·εu) → one Kn per master.
  4. Linear interp across the 3 masters at the material's actual Fty/Ftu → the final Kn.

To draw the curve for a material on this chart: the lookup above is repeated at 81 D/w points spanning the visible w/D range (D/w = 1/w/D), and the resulting Kn(D/w) is plotted with the chart's x-axis re-indexed as w/D. Per-lug vertical markers show the lug's actual w/D, with the Kn the engine actually used in that lug's net-section margin (so the marker reads the same number you'll find in the calc sheet).

Worked example — 7050-T7451 (default outer-lug material):

F_ty = 64,000 psi   F_tu = 74,000 psi   E = 10.45e6 psi   eu = 0.05

  F_ty / F_tu          = 64000 / 74000           = 0.865
  F_tu / (E · eu)       = 74000 / (10.45e6 · 0.05) = 0.142
  D / w (default lug)  = 0.625 / 1.20            = 0.521

  Step 1: bilinear in (F_ty/F_tu = 0.865, F_tu/(E·eu) = 0.142)
          interpolates between the F_ty/F_tu = 0.8 and 1.0 master charts,
          and within each between the F_tu/(E·eu) = 0.1 and 0.2 sub-curves
  Step 2: each of those 4 polynomials is evaluated at D/w = 0.521
  Step 3: bilinear-blended -> K_n ≈ 0.95

Reference: Maddux et al. (1969) AFFDL-TR-69-42, Section 9, Figure 9-4. The polynomial coefficients are documented in Lugs/lug_calculator_documentation.md §8.3 and the engine implementation is the Kn_lookup() function in this file's PYTHON_CODE block.

Sensitivity sweep chart
Variable
Min
Max
Sweep chart will render here.
Calculation sheet (auditable)
(computing…)
Theory and limitations
Methodology: Maddux et al., Stress Analysis Manual, Section 9 (AFFDL-TR-69-42, 1969). Public-domain USAF reference. Three-element clevis-and-tang topology (Outer 1 / Center / Outer 2); each outer lug carries half the joint load (modified by the optional center moment).

Failure modes computed:
  • Lug bearing (axial) — Eq 9-1, 9-2, 9-3 with the four-way MIN: K-curve ult, K-curve yield×1.304, MMPDS-table ult, MMPDS-table yield×1.304. K-curve allowables are scaled by the §9.15 ductility factor B(εu) for low-elongation lug materials (B = 1 for εu ≥ 5%).
  • Lug net-section (axial) — Eq 9-4, 9-5, 9-6 with Kn from Maddux Fig 9-4 (three Fty/Ftu master charts × seven Ftu/(E·εu) sub-curves; bilinear interpolation across the 7×3 grid evaluated at the lug's D/w). Net-section allowable is also scaled by B(εu) per §9.15.
  • Lug transverse — Eq 9-28, 9-29, 9-30 with Ktru/Ktry from Fig 9-8.
  • Lug oblique — Eq 9-31 with exponent 1.6: (P/Pu,L)1.6 + (Ptr/Ptru,L)1.6 = 1.
  • Pin shear — Eq 9-12 generalised for hollow pin: Pus,p = 2·(π/4)·(Dp2−Dp,ID2)·Fsu,p.
  • Pin bending — Eq 9-15 generalised for hollow pin. Ductile pins (εu,p ≥ 5%) use the geometric solid-circular plastic shape factor kbp = 16/(3π) ≈ 1.70 (Bruhn Sec C3.7, Niu Ch. 9). Brittle pins (εu,p < 5%) get the §9.15 cap kbp = min(1.5·B, 1.4) instead of the previous binary fall-through to 1.0.
  • Balanced design — Eq 9-16/9-17/9-18: pin allowable used for the margin is max(Pub,p, Pub,p,max). Both forms are valid lower bounds on pin-bending capacity per Maddux: 9-15 assumes uniform bearing pressure (centroid at full touter/2), 9-16 accounts for the load redistribution that physically occurs as the pin bends and the bearing pressure peaks toward the gap edge (centroid moves inboard, shorter moment arm, higher pin allowable). The weak-pin flag (Pub,p < min(PuLB, Pus,p)) is shown for reference, but no longer gates which equation is used — the higher-allowable form is always taken, and lug bearing remains capped separately by Pbru,L. b1,min, 2b2,min exposed alongside.
  • Bushing bearing — Eq 9-8 (Fbru,B = 1.304·Fcy,B) with engagement length MIN(Lbush, tlug).
  • Lug root / tang combined — combined σaxial + σbend,trans + σbend,side + σbend,eccen at the lug-to-parent transition. The eccentricity bending term is Maddux Eq 9-24: when balanced-design pin redistribution narrows the effective bearing engagement (b1,min or 2b2,min < tlug), the bearing centroid shifts toward the gap by Δe = (t−beff)/2 and applies an eccentricity moment to the tang root. Brittle lugs get the §9.15 cap kbT = min(1.5·B, 1.4) in place of the previous binary 1.4 / 1.0.
Margin form: MS = Pallow / Papplied / (FS · FF · ECF) − 1.

Improvements over the baseline workbook (documented in detail in Lugs/excel_pin_bending_change_justification.md):
  • Eq 9-16 wired into the pin-bending margin as max(Pub,p, Pub,p,max). The baseline workbook computes Pub,p,max at C138 but never uses it (cell C139 is a literal copy of C136 = Pub,p). b1,min, 2b2,min exposed. Closes a typical 5−500% conservatism gap vs FEM, depending on pin/lug stiffness ratio.
  • kbp = 1.70 for ductile pins (was 1.56). Bruhn / Niu / Stenman all use 1.70 (= 16/(3π), the geometric plastic shape factor for a solid circle). The 1.56 number was Cozzone-1950's strain-hardening derate, more conservative than necessary once kbp is already gated on εu ≥ 5%. ~9% pin-bending allowable bump.
  • Tang Eq 9-24 eccentricity term added to the lug-root combined-stress check. Catches the case where balanced-design pin redistribution forces the bearing load centroid inboard and applies an extra bending moment to the tang. Was identified as Priority 1 in lug_references_comparison.md §1.4.
  • Maddux §9.15 ductility B-factor applied to lug bearing & net-section allowables, plus the kbp/kbT 1.5·B caps for low-εu materials. Baseline applies no §9.15 derate at all.
  • kbp/kbT gated on εu (V.2.2 fix; baseline hardcodes 1.56/1.4 with no gate).
  • Default FSult = 1.50 (FAA standard) rather than 1.25 (project-specific).
Limitations: D/t ≤ 5 only (K-curve quadratic; Fig 9-3 family for D/t > 5 not yet digitised — tool warns). Symmetric-hole only (no eccentric correction per Ekvall 1986). No explicit lubrication-hole derate. The B(εu) curve is a conservative piecewise-linear approximation of Maddux Fig 9-17 (linear from 0.5 at εu=0 to 1.0 at εu=5%) rather than the exact published curve. For single-shear configurations use the dedicated single-shear tool.

References: Maddux et al. (1969) AFFDL-TR-69-42 §9; Stenman (2008) RPI MEng project; Cozzone, Melcon, Hoblit (1950) Product Engineering 21; Melcon, Hoblit (1953) Product Engineering 24; Ekvall (1986) J. Aircraft 23(5).