Rotating Disc & Ring: Stress, Growth and Burst Speed

The model solves axisymmetric radial equilibrium for the actual thickness $h(r)$:

Plane-stress elasticity gives $\sigma_r=\frac{E}{1-\nu^2}[u'+\nu u/r-(1+\nu)\alpha\Delta T]$ and $\sigma_\theta=Eu/r+\nu\sigma_r-E\alpha\Delta T$.

With $x=r/b$, $d=u/b$, $H=h/h(a)$ and $f=xH\sigma_r/E$, the shooting equations are

Fourth-order Runge–Kutta integrates separate pressure, rotation and thermal load bases on a logarithmic radial grid. Linear shooting enforces zero rim traction. The annular bore has $\sigma_r=-p$; a solid disc uses the regular centre solution $u(0)=0$ and $\sigma_r(0)=\sigma_\theta(0)$. These equations are derived here from equilibrium and Hooke’s law; the uniform-disc limits are checked against the classical rotating-disc solutions of Timoshenko and Goodier and the university reference.

Shrink fit uses $p=\max[0,(\delta+u_{h,\mathrm{free}}-u_{d,\mathrm{free}}(a))/(C_d+C_h)]$. For a solid plane-stress hub, $C_h=a(1-\nu_h)/E_h$ and $u_{h,\mathrm{spin}}=\rho_h\omega^2a^3(1-\nu_h)/(4E_h)$. Separation is the first zero contact pressure on increasing speed. Contact can re-engage if the hub grows faster; pressure and limit searches include that transition.

The thickness-weighted average hoop stress is $\bar\sigma_\theta=\int_a^b h\sigma_\theta\,dr/\int_a^b h\,dr$. Half-disc force balance also gives $\bar\sigma_\theta=[\rho\omega^2\int r^2h\,dr+a h(a)p]/\int h\,dr$. Burst is the first speed satisfying $\bar\sigma_\theta=k_bS_u$ (NASA SP-8052, printed p.81, PDF p.93, equations 82–84). $k_b=1$ is the ideal ductile average-stress criterion; a lower factor must come from relevant evidence. First local yield instead solves $\max\sqrt{\sigma_r^2-\sigma_r\sigma_\theta+\sigma_\theta^2}=S_y$. Bracketed scalar searches operate separately on either side of a contact transition.

Autodesk AVE-21/22 (PDF pp.113–116) provides the independent conical-disc FEM benchmark and identifies Roark’s sixth-edition rotating-member case. NASA CR-135033, printed p.28 / PDF p.39, describes a modified Hallinan method with additional notch and ductility effects; those proprietary modifications are not part of this calculator.

Homogeneous, isotropic, linearly elastic material with constant properties across each part; small strains; axisymmetric geometry, thickness and prescribed temperature; steady rotation and free axial faces. A free rim and either a free bore, a unilateral frictionless radial fit, or a regular solid centre. The solid hub shares rotational speed but may have different elastic and thermal properties. Interference is small enough to use the nominal bore radius in hub geometry. Disc temperature is prescribed linearly in radius; no heat-transfer solution is implied. The thermal changes refer to the same stress-free state as the interference. Tensile stress and outward displacement are positive. Steel preset strengths follow the table’s thickness ranges (up to 80 mm); other products and temperatures need custom data.

Plane stress suits thin discs with smooth thickness variations. A local shoulder, hub end effect or steep surface slope needs 3D analysis even if the radial ODE converges. A custom thickness profile is represented by nine stations with PCHIP interpolation; increasing the integration grid cannot recover features missing between those stations. The default 512 intervals can be doubled under Advanced. A solid centre is initialized at r/b = 10⁻⁶ with the regular limiting solution; the omitted core load is negligible for smooth finite thickness and temperature. Field values are solved at grid nodes; plotted curves use those nodes, and reference read-offs use PCHIP.

The average-stress burst criterion assumes enough ductility to redistribute stress; it is not appropriate for brittle materials, defects or notch-controlled failure. A burst factor of 1 is an ideal comparison, not a validated overspeed rating. Do not interpret the yield or burst estimate as permission to operate or spin-test a rotor. Local stresses and growth above first yield are elastic extrapolations. Thermal strain redistributes local stress without changing the average resultant of a free disc; thermal weakening is covered only by the strengths you enter. Fit release is an elastic estimate; hub yielding, axial restraint, slip torque and friction are not checked. The contact status describes radial contact only.

Source correction: NASA TN D-8304 (1976) is Kilgore’s cryogenic wind-tunnel report, not a Hallinan burst reference. The burst criterion here is supported by NASA SP-8052; Hallinan context is from NASA CR-135033. No fracture, plastic-flow, fatigue, rotor-dynamics or containment assessment is performed.