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Assembly Method: model & verification

Version 1.0.3 · 6 November 2026. Preliminary elastic and thermal assessment for smooth, concentric, uniform isotropic cylinders with unconstrained ends. Bearings receive separate guidance only. No DIN 7190 compliance claim.

Mechanical calculation

All lengths are millimetres; elastic moduli and stresses MPa. Define shaft OD dₛ, shaft ID dᵢ, housing ID dᵦ and OD dₒ. qₛ = (dᵢ/dₛ)²; qᵦ = (dᵦ/dₒ)². Shaft compliance Cₛ = [(1+qₛ)/(1−qₛ)−νₛ]/Eₛ; housing compliance Cᵦ = [(1+qᵦ)/(1−qᵦ)+νᵦ]/Eᵦ. Pressure p = δ/(dₛCₛ+dᵦCᵦ), where δ is diametral interference. At equal interface diameters this is the conventional Lamé fit equation. Small differences use each component’s basic diameter in its own compliance, a first-order free-surface compatibility approximation.

The main ISO engine supplies each component’s exact permissible sizes at 20 °C. Mechanical interference at initial temperatures is computed from thermally adjusted size limits, clipped at zero. Geometry scales with its own material expansion. Force F = μπ(dₛ+dᵦ)Lp/2, divided by 1000 for kN. This estimates full-engagement sliding force for the entered friction coefficient, not the full insertion-force curve or breakaway force.

At the interface, radial stress is −p in both parts. Housing bore hoop stress is p(1+qᵦ)/(1−qᵦ); shaft surface hoop stress is −p(1+qₛ)/(1−qₛ). At a hollow shaft’s free bore, radial stress is zero and hoop stress is −2p/(1−qₛ). A solid shaft has uniform radial and hoop stress −p. Plane-stress von Mises stress is √(σθ²−σθσr+σr²). The shaft maximum includes its inner wall. Material screening takes the worse of initial-temperature and final 20 °C maximum-interference results. It is not an operating-temperature or service-load assessment.

Yielding invalidates elastic predictions; flagged values remain visible for screening only. Limits must be supplied by the user. Representative presets (steel: E 210 GPa, ν 0.30, α 12; austenitic stainless: 193 GPa, 0.30, 17; aluminium: 70 GPa, 0.33, 23 µm/m·°C) are illustrative engineering assumptions, not verified properties of a particular grade or heat treatment. No yield strength is guessed. Properties must be justified for the temperature range.

Rough surfaces, unknown surface conditions and unconfirmed geometry block mechanical results. No roughness correction is silently applied. The app excludes diameter mismatch or interference above 1% of diameter and wall diameter ratios above 0.98; these are conservative software screening bounds, not DIN applicability criteria. Slots, keyways, stepped sections, constraints, end effects, axial friction stress, galling, buckling, plasticity and residual stress are not modelled. Below-capacity force does not establish safe assembly.

Thermal calculation

A free dimension at temperature T is D₂₀[1+α(T−20)]. Maximum-interference assembly uses maximum shaft and minimum bore. Required clearance c means Bmin[1+αᵦ(Tᵦ−20)] − Smax[1+αₛ(Tₛ−20)] ≥ c. Heating-only holds the shaft at its stated initial temperature; cooling-only holds the housing at its stated initial temperature. Combined results sample five shaft temperatures between its initial and minimum allowed temperatures and solve the required housing temperature, without cooling the housing below its initial value. The final sample tests the extreme feasible combination, so a no-combination result is not caused merely by sparse sampling. Optional user-selected temperatures are checked directly.

All geometric thermal comparisons use each actual size limit, including independent basic diameters. Uniform temperatures, constant expansion coefficients and free expansion are assumed. Results are geometric possibilities within user limits, not installation approval. Heat transfer during handling, gradients, temperature-dependent properties and condensation are not calculated. The −200 to 500 °C software envelope is not permission to use a material at those temperatures. Heating cannot remove the interference stresses after equalisation. Displayed temperatures are rounded; retain practical handling clearance.

Verification

Independent hand-derived fixtures: for a solid 50 mm shaft, 100 mm housing OD, E=200 GPa in both, ν=0.3, δ=0.05 mm, L=30 mm and μ=0.1: pressure 75 MPa, force 35.342917 kN, housing hoop 125 MPa and equivalent stress 175 MPa. A 25 mm shaft bore changes pressure to 60 MPa and peak shaft equivalent stress to 160 MPa. With Eₛ=210 GPa, Eᵦ=70 GPa, νᵦ=0.33 and a solid shaft, pressure is 7000/223 MPa.

The Jönköping University worked 60/118 mm example at 0.072 mm interference was independently recalculated: p=93.423154 MPa, housing hoop=158.576846 MPa and equivalent=220.656409 MPa. Its displayed 93.4, 159 and 221 MPa are used only at their approximate reporting precision; the exact equations govern this implementation. Roughness-corrected minimum-pressure and torque results in that example are not used as fixtures for this smooth-surface model.

Thermal check: a 50.000 mm bore and 50.050 mm shaft at 20 °C, α=12 µm/m·°C and 0.020 mm assembly clearance require a bore temperature of 136.666667 °C if the shaft stays at 20 °C. Automated tests also cover cooling, mixed coefficients, initial temperatures, both temperature limits, zero and transition cases, invalid geometry, yield screening, backups and bearing-mode isolation. This verifies equations and software behaviour; physical assembly validation has not been performed.

Technical sources

Bearing scope and future module

Version 1 does not apply ISO shaft/bore tolerances to manufactured bearings, or calculate their mounting force, heating temperature or internal clearance. The selected ring and both-ring warning guide the user to the relevant manufacturer procedure. A future module can add bearing references, tolerance classes, internal clearance and manufacturer-specific restrictions without changing the ISO engine.

Automatic geometry and friction

New assessments start with an assumed solid shaft, engagement length equal to shaft diameter and housing OD twice bore diameter. These are editable starting values, not an inferred design. Press feasibility is withheld until actual geometry is selected. Press capacity remains a user equipment constraint.

Reference friction uses preliminary longitudinal press ranges for an uncoated steel shaft from MITcalc section 2.6: steel housing dry 0.07–0.16, greased 0.05–0.12; aluminium housing dry 0.05–0.09, greased 0.04–0.06. Force bounds combine friction and interference bounds. Unsupported pairs require manual input; stainless is not treated as carbon steel. Values are estimates, not measured coefficients or guaranteed bounds. Old saved values stay manual.