Shrink Fit Analysis

Shrink Fit Analysis (structures/fem/shrink-fit-analysis)

Compute the closed-form stress state of a bushing, sleeve or bearing race pressed or shrunk into a lug, hub or race: the two-cylinder Lame radial compliance of the interference fit converts the total radial interference delta into the interface contact pressure p, the bore radial and hoop stresses of both members follow from the exact Lame extreme-fiber values, and the von-Mises yield margin of each bore closes with the governing member and the maximum allowable radial interference before that bore yields. Deterministic closed-form core only in pure Python, stdlib only: no FEA, no iteration, no plasticity. This leaf owns the fitted-joint pre-stress territory that sibling leaves fence off: it does NOT do deformation-driven fastener hole-fill and head-geometry checks (solid-rivet-installation-quality, whose hole-fill quick reference declares interference fits out of scope), finite element contact enforcement machinery (contact-analysis, which computes contact mechanics quantities, not the closed-form stress solution, and lists bushing-sleeve interfaces only as finite element application examples), pin-loaded lug proportioning under an axial load with no pre-stress from a pressed-in bushing (lug-joint-analysis), thin-shell external-pressure stability (cylindrical-shell-buckling) and membrane-theory pressure domes (pressure-bulkhead). It pairs with structures/fem/lug-joint-analysis for the load path that hosts the fitted bushing and with structures/fem/contact-analysis for the finite element side of the same interfaces.

Domain quick reference

• Interface contact pressure from radial interference (the two-cylinder Lame form, a paraphrase of the standard Shigley/Juvinall class press fit relation): p = delta / [ (r_c / E_i) * ((r_c2 + r_i2) / (r_c2 - r_i2) - nu_i) + (r_c / E_o) * ((r_o2 + r_c2) / (r_o2 - r_c2) + nu_o) ], where r_i is the inner member bore, r_c the interface radius and r_o the outer member outer radius, delta the total RADIAL interference, E and nu the member moduli and Poisson ratios.
• Displacement forms behind the formula: the inner member (bore r_i, interface radius r_c) carries only the external pressure p, so its inward interface displacement is u_inner = -(p r_c / E_i) * ((r_c2 + r_i2) / (r_c2 - r_i2) - nu_i); the outer member (bore r_c, outer radius r_o) carries only the internal pressure p, so its outward bore displacement is u_outer = +(p r_c / E_o) * ((r_o2 + r_c2) / (r_o2 - r_c2) + nu_o). Compatibility delta = u_outer - u_inner gives the formula above.
• Sign convention: the COMPRESSED inner member carries the -nu_i term and the EXPANDED outer member the +nu_o term (the two nu terms cancel only for identical materials). Swapping the two signs is the common transcription error and shifts p by several percent (17 percent over-read on the worked example).
• Critical bore planes (exact Lame extreme-fiber values): the inner member bore (r = r_i) is a free surface, sigma_r = 0.0 there, and its hoop is the most compressive value in the assembly, sigma_theta = -2.0 * p * r_c2 / (r_c2 - r_i2). The outer member bore (r = r_c) carries sigma_r = -p and the maximum tensile hoop sigma_theta = p * (r_o2 + r_c2) / (r_o2 - r_c2). The inner member interface hoop at r_c, -p (r_c2 + r_i2) / (r_c2 - r_i**2), is smaller in magnitude than its bore hoop and is not the critical location.
• Von-Mises yield margin (plane stress, sigma_z = 0, distortion energy): sigma_vm = sqrt(sigma_theta2 - sigma_theta * sigma_r + sigma_r2) and margin = sy - sigma_vm, positive below yield.
• Governing member and allowable interference: every stress is linear in p (elastic, small strain), so the assembly scales linearly to the yield point of the governing (least-margin) bore: governing_contact_pressure = p * sy_gov / sigma_vm_gov and allowable_interference = delta * sy_gov / sigma_vm_gov.
• Units: all radii share one length unit, all stresses one stress unit and all moduli and pressures the same stress unit. In the worked example: mm for radii and delta, MPa for stress, modulus and pressure.
• FAR-25 and CS-25 frame the airframe bushing-in-lug context; the relations above are standard engineering methodology, summary-only per standards-map.yaml.

Workflow

1. Fix the assembly geometry, materials and interference: inner bore r_i, interface radius r_c and outer radius r_o in one length unit, the total radial interference delta > 0, the member moduli E_i and E_o, Poisson ratios nu_i and nu_o in (0, 0.5) and yield strengths sy_i and sy_o in one stress unit.
2. Convert the radial interference into the interface contact pressure: run contact_pressure (the Lame radial-compliance pass) on delta, r_i, r_c, r_o, E_i, nu_i, E_o, nu_o to get p. Cross-check the sign convention by re-deriving both compliance terms by hand.
3. Recover the critical bore stress state: run stress_distributions on p, r_i, r_c, r_o (the bore-stress pass) for the inner member bore radial and hoop stresses (free surface at sigma_r = 0.0, most compressive hoop) and the outer member bore stresses (sigma_r = -p, maximum tensile hoop).
4. Form the von-Mises yield margins of both bores: run von_mises_margin (the yield-margin pass) on each bore plane stress state, with sy_i for the inner member and sy_o for the outer member, and read sigma_vm and margin = sy - sigma_vm.
5. Close with the governing member and the allowable radial interference: run allowable_interference (the governing-member pass) on the full input set including sy_i and sy_o, and read governing_member ("inner" or "outer"), governing_contact_pressure and allowable_interference, the radial interference that takes the governing bore exactly to yield.
6. Verify the closed-form identities: confirm the compatibility identity delta = u_outer - u_inner at the returned contact pressure from the displacement forms above, confirm p scales linearly with delta at fixed geometry and materials, and confirm that at delta equal to the returned allowable interference the governing bore margin returns to zero.
7. Confirm the deterministic checks: rerun the offline contract test scripts/test_shrink_fit_analysis.py and confirm all 33 methods pass (deterministic, stdlib math only, no RNG).

Worked example

Steel bushing pressed into an aluminum lug (the corpus geometry): inner member steel bushing E_i = 207000 MPa, nu_i = 0.30, Sy_i = 620 MPa with bore r_i = 6 mm and interface (outer) radius r_c = 8 mm; outer member aluminum lug E_o = 71000 MPa, nu_o = 0.33, Sy_o = 276 MPa with bore r_c = 8 mm and outer radius r_o = 16 mm; total radial interference delta = 0.02 mm (real module outputs):

• contact_pressure(0.02, 6.0, 8.0, 16.0, 207000.0, 0.30, 71000.0, 0.33) = 56.91381 MPa, inside the 50 to 150 MPa sanity band for a 0.02 mm interference over 6 to 16 mm radii in steel on aluminum. The swapped-sign variant of the same formula returns about 66.6 MPa, a 17 percent over-read.
• stress_distributions(56.91381, 6.0, 8.0, 16.0): inner member bore sigma_r = 0.00000 MPa and sigma_theta = -260.17743 MPa (compression, the largest hoop magnitude in the assembly); outer member bore sigma_r = -56.91381 MPa and sigma_theta = +94.85635 MPa (tension). The interface hoop of the bushing at r_c is -203.26361 MPa, smaller in magnitude than its bore hoop.
• von_mises_margin(620.0, 0.0, -260.17743): sigma_vm = 260.17743 MPa, margin = 359.82257 MPa (steel bushing bore, comfortable).
• von_mises_margin(276.0, -56.91381, 94.85635): sigma_vm = 132.79889 MPa, margin = 143.20111 MPa (aluminum lug bore). The lug bore is the governing location: the soft aluminum sees only 2.3333 p von-Mises, but its yield strength is less than half the steel's.
• allowable_interference(0.02, 6.0, 8.0, 16.0, 207000.0, 0.30, 71000.0, 0.33, 620.0, 276.0): governing_member "outer", governing_contact_pressure = 118.28571 MPa, allowable_interference = 0.04157 mm. The 0.02 mm design interference holds a 1.07 margin ratio on the lug (276 / 132.79889) and the fit can take 0.04157 mm before the aluminum lug bore yields.
• Compatibility identity at the returned p: u_outer(r_c) = +0.01280 mm and u_inner(r_c) = -0.00720 mm, so u_outer - u_inner = 0.02000 mm, exactly the input delta (roundoff 1e-15 class).
• Identical-members check (E_i = E_o, nu_i = nu_o, sy_i = sy_o): the governing member flips to "inner", because the inner bore hoop factor 2 r_c2 / (r_c2 - r_i2) = 4.5714 exceeds the outer bore von-Mises factor sqrt(1.66672 + 1.6667 + 1) = 2.3333.

Verification

• Confirm contact_pressure(0.02, 6.0, 8.0, 16.0, 207000.0, 0.30, 71000.0, 0.33) returns 56.91381 MPa within 1e-4, and doubling delta doubles p exactly (linear scaling).
• Confirm stress_distributions returns the four documented bore-stress values within 1e-4 with sigma_r continuous across the interface at -p, inner bore sigma_r = 0.0 exactly (free surface) and the inner bore hoop as the largest hoop magnitude in the assembly.
• Confirm the von-Mises margins 359.82257 MPa (inner) and 143.20111 MPa (outer) within 1e-3 and the uniaxial identity sigma_vm = |sigma| at sigma_r = 0.
• Confirm allowable_interference returns governing_member "outer", governing_contact_pressure = 118.28571 MPa and allowable_interference = 0.04157 mm, and flips to "inner" for identical members.
• Confirm the compatibility identity delta = u_outer - u_inner reproduces the input delta within 1e-12 at the returned p, and that at delta equal to the returned allowable interference the governing bore margin returns to 0 within 1e-9.
• Confirm ValueError rejection of every non-physical input class: delta zero or negative, r_i zero, r_c <= r_i, r_o <= r_c, zero or negative moduli, Poisson ratios at 0, 0.5 and above, and non-positive yield strengths, across every function.
• Run the deterministic contract test offline: python3 scripts/test_shrink_fit_analysis.py (33 tests, sub-second).

Related leaves

• structures/fem/lug-joint-analysis: the pin-loaded fitting that hosts the pressed-in bushing, proportioned under an axial load with no pre-stress from the fit.
• structures/fem/contact-analysis: finite element contact enforcement for the same bushing-sleeve interfaces, where this leaf's closed-form stress solution is the analytical cross-check.
• structures/fem/curved-beam-analysis: curved member stress analysis for the lug bodies that carry fitted bushings.
• structures/fem/pressure-bulkhead: membrane-theory pressure domes, outside this leaf's thick-wall radial-interference model.
• structures/fem/cylindrical-shell-buckling: thin-shell external pressure stability of the same cylinders, outside this leaf's elastic stress state.

Pitfalls

• Swapping the nu signs: the compressed inner member takes -nu_i and the expanded outer member +nu_o; transposing them over-reads the contact pressure by 17 percent on the worked example (66.6 MPa against 56.9 MPa).
• Reading the interface hoop as the inner member critical stress: the bushing interface hoop at r_c (-203.26 MPa) is smaller in magnitude than its bore hoop (-260.18 MPa); the bore is the critical plane.
• Expecting the soft member to govern by strength alone: the aluminum lug governs here (margin 143.2 MPa against 359.8 MPa) even though its von-Mises factor 2.3333 p is half the steel bore's 4.5714 p, because its yield strength is less than half the steel's. With identical members the inner bore governs instead.
• Comparing stresses across members without a shared unit scheme: all radii share one length unit and all stresses, moduli and pressures one stress unit; mixing mm with m or MPa with Pa silently shifts the contact pressure.
• Treating the fit as a thin-shell problem: the Lame thick-cylinder radial compliance needs r_c**2 terms in both members, not the thin shell hoop-only membrane result.
• Using exact float equality on computed sums: assert module outputs with a tolerance (the contract test uses assertAlmostEqual with an explicit delta or math.isclose throughout).

Behavior contract (gate 3)

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

python3 scripts/test_shrink_fit_analysis.py

The test covers the worked-example anchors (contact pressure 56.91381 MPa within 1e-4 and the 50 to 150 MPa sanity band, bore hoop -260.17743 MPa and +94.85635 MPa within 1e-4), the sign-convention guard against the swapped-nu variant, linear p-delta scaling, the exact bore-stress dict keys and the free-surface sigma_r = 0.0, the von-Mises yield margins of both bores within 1e-3 with the uniaxial and equibiaxial identities, the governing-member pass results (outer on the worked geometry within 1e-3 / 1e-5, inner for identical members), the yield-crossing identity at the allowable interference, the compatibility identity delta = u_outer - u_inner within 1e-12, determinism, and ValueError rejection of every non-physical input class across all four functions.

Compliance

• Standards referenced, not reproduced: FAR-25 and CS-25 frame the airframe bushing, bearing and lug fit context (standards-map.yaml ids, both reference-only, the fem-pack convention used by lug-joint-analysis); the Lame relations above are standard engineering methodology (Shigley/Juvinall class press-fit relations), summary-only, never verbatim standard text.
• compliance: STANDARDS-REF, gated: false.