Lug Joint Analysis

Lug Joint Analysis (structures/fem/lug-joint-analysis)

Use when you must analyze a metallic pin-loaded lug fitting under an axial pin load: the bearing stress on the hole, the net section tension stress across the lug width, the tearout shear stress on the two planes tangent to the hole running to the outer contour, the margin of each mode against the material allowables, the governing failure mode (lowest margin), and the governing-mode capacity map over the edge distance ratio e/D (short-lug net tension, intermediate tearout, long-lug bearing). This leaf implements the round-end lug convention w = 2e in pure stdlib Python, SI units, offline and deterministic. It pairs with the single-row bolted joint analysis for fiber-reinforced panels (no lug proportioning there) and with the finite element contact boundary analysis for pin-to-hole contact detail. It does not size pin-jointed frames: beam-frame-analysis and truss-analysis own those structures.

Domain quick reference

• Round-end lug: hole of diameter D centered in a round head of radius e, so the lug width is w = 2e; thickness t carries axial pin load P. Edge distance e must exceed D/2 and width w must exceed D.
• Bearing stress on the hole: sigma_b = P / (D t).
• Net section tension across the remaining width: sigma_nt = P / ((w - D) t), net section width (w - D).
• Tearout shear on the two planes from the hole tangent to the outer contour: sigma_te = P / (2 t L_te), with each shear plane length L_te = sqrt(e^2 - (D/2)^2).
• Margin per mode: m = allowable / applied - 1, using the bearing ultimate F_bru for bearing, the tension ultimate F_tu for net section tension and the shear ultimate F_su for tearout.
• Governing mode: smallest margin; the lug passes when min margin >= 0.
• Per-mode allowable load (load that makes that mode margin zero): bearing F_bru D t, net section tension F_tu (w - D) t, tearout F_su 2 t L_te; the limiting mode has the smallest capacity.
• Allowables are material inputs, for example MMPDS chapter 9 lug ultimate data for the lug alloy (referenced by name, never reproduced). FAR 25.307 frames proof-of-structure fitting substantiation context.
• Governing behavior over e/D for typical aluminum allowables: net section tension below about 1.03, tearout between about 1.03 and 1.74, bearing above about 1.74.
• Units SI throughout: N, m, Pa.

Workflow

1. Fix the lug geometry: hole_diameter_m, thickness_m, lug_width_m, edge_distance_m, and the axial pin load load_n, then get the applied mode stresses with lug_stresses (returns bearing_pa, net_tension_pa, tearout_pa, tearout_plane_length_m, net_section_width_m).
2. Choose the material allowables f_tu_pa, f_su_pa, f_bru_pa from the lug alloy data, and form the per-mode margins with lug_margins.
3. Run the full margin check with lug_analysis: applied stresses, margins, governing_mode, min_margin, passes, plus e_over_d, d_over_t and the tearout plane geometry.
4. For a margin-of-safety style capacity statement, run lug_allowable_capacity at the round-end geometry (w = 2e is assumed inside) to get the per-mode allowable loads, the limiting mode and the limiting capacity.
5. For the short-lug / intermediate / long-lug map, run lug_governing_map with the material allowables at fixed D and t; it sweeps e/D from 0.6 to 2.5 with e = ratio * D and w = 2e and returns the governing mode and limiting capacity per sample.
6. Confirm the deterministic checks with the contract test scripts/test_lug_joint_analysis.py.

Worked example

7075-T6 lug: D = 20 mm, t = 12 mm, e = 24 mm (e/D = 1.2), w = 48 mm, F_tu = 572 MPa, F_su = 331 MPa, F_bru = 1050 MPa, P = 90 kN. Real module outputs:

• Stresses: bearing 375.0 MPa, net section tension 267.857 MPa, tearout 171.881 MPa with L_te = 21.817 mm and net section width 28 mm.
• Margins: bearing +1.800, net section tension +1.135, tearout +0.926.
• Governing mode tearout, min margin +0.926, passes True.
• Same lug at P = 200 kN: tearout stress 381.958 MPa, margin -0.133, governing stays tearout, passes False.
• Allowable capacities: bearing 252000 N, net section tension 192192 N, tearout 173318 N; limiting mode tearout at 173318 N.
• Governing map (D = 20 mm, t = 12 mm, 7075 allowables): net section tension governs below e/D about 1.03, tearout governs the middle band (for example 168060 N at e/D = 1.17), bearing governs above e/D about 1.74 at the constant 252000 N. All three modes govern across the sweep.

Pitfalls

• Analyzing a non-round-end lug with the round-end convention: the governing map and lug_allowable_capacity assume w = 2e; a lug with width other than twice the edge distance needs the explicit geometry inputs, not the round-end shortcuts.
• Missing a degenerate edge relation: the hole edge distance must exceed D/2 and the width must exceed D (tearout planes need room to run); degenerate e/D or w/D relations raise ValueError.
• Reading the margin of the wrong mode: the governing mode is the LOWEST margin, not the largest stress - in the worked 90 kN case tearout governs (+0.926) even though bearing stress (375 MPa) exceeds tearout stress (171.9 MPa), because each margin uses its own allowable.
• Quoting capacity from a single mode: the lug's limiting capacity is the smallest per-mode allowable load (tearout 173318 N in the worked example); the governing mode changes with e/D, so a fixed-mode capacity statement only holds at one geometry.
• Sizing lug fittings with a frame solver: pin-loaded lug proportioning with bearing, net tension and tearout is this leaf; beam-frame-analysis and truss-analysis size pin-jointed frames and do not cover lug fittings.
• Forgetting the applied-load direction: the stresses and margins here assume an AXIAL pin load on the lug; oblique pin loading changes the net-section and tearout paths.

Verification

• Confirm lug_analysis(90000, 0.020, 0.012, 0.048, 0.024, 572e6, 331e6, 1050e6) returns bearing margin 1.8, net section tension margin 1.135, tearout margin 0.926, governing tearout and passes True.
• Confirm the same lug at 200000 N gives tearout margin -0.133 and passes False with tearout still governing.
• Geometry identity: at e/D = 1 the tearout plane length is sqrt(3)/2 * D = 0.8660254 D and the net section width is 2e - D.
• Capacity identity: lug_allowable_capacity per-mode loads equal the applied load that drives that margin to zero (checked for all three modes in the contract test), and the limiting mode matches the governing mode of lug_analysis at the same geometry.
• Governing map: all three modes appear as governing over [0.6, 2.5], capacity is monotonically non-decreasing in e/D and the bearing capacity is constant in e/D.
• Rejection: negative load, non-positive dimensions, edge distance not above D/2, width not above D and non-positive allowables all raise ValueError in every entry point that uses them.
• Determinism: pure stdlib, no random numbers, identical floats run-to-run.

Related leaves

• structures/composites/composite-bolted-joints: the single-row bolted joint sibling for fiber-reinforced panels; this leaf is the metallic lug with e/D proportioning, head geometry and tearout planes from the hole contour, with no load-sharing split concept.
• structures/fem/contact-analysis: the finite element contact boundary sibling for pin-to-hole bearing detail.
• structures/fem/beam-frame-analysis and structures/fem/truss-analysis: pin-jointed frame analysis siblings; they do not size lug fittings.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_lug_joint_analysis.py

The test covers the 7075-T6 worked lug anchors (stresses, margins, governing tearout, pass/fail at 90 kN and 200 kN), the e/D = 1 geometry identity, per-mode allowable capacity and margin-zero identities, the three-mode governing sweep with monotone capacity, the convenience dict keys, and ValueError rejection of negative loads, non-positive dimensions, degenerate e/D and w/D relations and non-positive allowables.

Compliance

• Standards referenced by name only, not reproduced: MMPDS chapter 9 lug ultimate data heritage (mmpsd) and FAR 25.307 proof-of-structure context for fitting substantiation; the stress and margin relations above are standard engineering methodology, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.