Compound Cylinder (Shrink Fit)

stressmass-cost

Verified build 10 relations · 7 identities proven · 2 modeling steps · 3 parity samples

The thick-walled cylinder ends on a refusal: the bore shear always exceeds the pressure, so past $p = \sigma_y/2$ no amount of wall contains the pressure elastically — the widget there diverges and says so. This page is the classical cheat. Machine the liner’s outside a few microns larger than the jacket’s inside, heat the jacket, drop it on, let it cool. The interference $\delta$ generates a contact pressure at the interface:

$p_c = \frac{E\,\delta}{r_c}\,\frac{(r_c^2 - r_i^2)\,(r_o^2 - r_c^2)}{2\,r_c^2\,(r_o^2 - r_i^2)}$

and that contact pressure squeezes the liner bore into hoop compression — $-2 p_c r_c^2/(r_c^2 - r_i^2)$ , about $-69$ MPa at this page’s defaults — parked exactly where service pressure will later hurt most. Pressurize the assembly and (same metal throughout) the wall responds as one monobloc cylinder with the frozen assembly field riding on top: the bore’s tension must first spend the compression before it can start doing damage.

Nothing is free — the relief the liner enjoys is tension the jacket prepaid at the interface. The page therefore carries two margins, and the knobs slide the danger between them:

Under-shrunk: the bore governs. Small $\delta$ leaves the wall nearly monobloc; $\mathrm{SF}_i$ is the binding margin, $\mathrm{SF}_c$ loafs.
Over-shrunk: the interface governs. Every micron of interference lowers the bore shear and raises the interface shear. Push far enough and the bore stays in net compression at full pressure — at which point a tensile bore margin is meaningless, and the widget refuses that one readout (the jacket margin and the assembly-compression warning carry the story instead). Keep pushing and assembly alone yields something — before any pressure exists.
Balanced: equal margins. The two shears are linear in $p_c$ , one falling, one rising, so they cross exactly once; the $\delta_{bal}$ readout names that crossing for the current pressure. The defaults sit exactly on it: $\mathrm{SF}_i = \mathrm{SF}_c = 1.39$ at 100 MPa — a pressure that the same envelope as a one-piece wall would barely survive at $\mathrm{SF} \approx 1.0$ .

The headline is the ceiling. Balance the fit, put the interface at the geometric mean $r_c = \sqrt{r_i\,r_o}$ (the defaults do: $\sqrt{40 \times 90} = 60$ mm), and the elastic capacity becomes $\sigma_y\,(1 - r_i/r_o)$ — against the monobloc’s $(\sigma_y/2)(1 - r_i^2/r_o^2)$ , a gain of $2/(1 + r_i/r_o)$ : two shells approach twice the ceiling as walls grow thick. Both the optimum location and the doubling are machine-proven in the test pipeline, not quoted. This is 19th-century gun-barrel arithmetic, and it is why built-up construction exists.

Two quiet notes worth knowing. First, there is no Poisson knob on this page: with both members the same metal, $\nu$ cancels exactly from the closure — the derivation below proves the cancellation with $\nu$ left symbolic (different metals would keep all four of $E_i, \nu_i, E_o, \nu_o$ ). Second, the shrink fit is secretly a chaining story: the fit turns microns into a pressure, $\delta \mapsto p_c$ , and that pressure is exactly the kind of typed port — pascals, pressure_stress — that another THING’s bore could someday consume. The rotating disk left its bore free for precisely this reason.

Try it

Material

T3, bare flat sheet 0.010-0.128 in. thick, AMS 4037 / AMS-QQ-A-250/4 (MIL-HDBK-5J Table 3.2.3.0(b1), p. 3-71)

Bound properties of 2024-T3 aluminum sheet (bare)
E	10.5 Msi	typical	mil-hdbk-5j
sigma_y	47 ksi	design min.	mil-hdbk-5j
rho	0.1 lb/inch**3	typical	mil-hdbk-5j

Inputs

Bore radius (liner inner) (r_i)

Interface radius (r_c)

Outer radius (jacket outer) (r_o)

Radial interference (delta)

Working internal pressure (p)

Shrink-fit contact pressure: —
Assembly hoop at the bore (residual): —
Assembly hoop at the jacket bore: —
Bore hoop at full pressure: —
Bore shear at full pressure (signed): —
Interface shear at full pressure: —
Safety factor — liner bore (first yield): —
Safety factor — jacket bore (first yield): —
Balanced-fit interference: —
Wall mass per unit length: — kg/m

3 materials in the database are not listed here: no published value in our cited sources for every property this THING needs.

Materials modeled here: 2024-T3 aluminum sheet (bare) 304 stainless steel 6061-T6 aluminum 7075-T6 aluminum AISI 1045 medium-carbon steel AISI 4340 low-alloy steel (Ni-Cr-Mo) ASTM A36 structural steel (hot-rolled) C26000 Cartridge Brass (70/30) Nylon 6/6 (PA66), unfilled Ti-6Al-4V

Governing relations

p_c = \frac{E\,\delta}{r_c}\,\frac{(r_c^2 - r_i^2)\,(r_o^2 - r_c^2)}{2\,r_c^2\,(r_o^2 - r_i^2)}

Assumes: both members the same elastic material — Poisson's ratio cancels exactly from the closure (the derivation proves this with ν left symbolic); different metals would keep it; equal member lengths, ends free (plane stress); interference small next to the radii; the radial interference δ, not the diametral value — half the difference of the as-machined diameters · Valid while: The interface has met (or passed) the bore — no liner remains, and every number on this page is geometry-free fiction until r_i < r_c again. The jacket has no wall — the outer radius must clear the interface. Every readout is refused until r_c < r_o again.

Source: Timoshenko, S., Strength of Materials, Part II: Advanced Theory and Problems, 3rd ed., Van Nostrand, 1956 (first published 1930) — ch. VI "Deformations Symmetrical About an Axis", Art. 40 (the thick-walled cylinder: Lamé field, internal/external pressure, displacements) and Art. 41 "Stresses Produced by Shrink Fit" (eq. 181: the same-material contact pressure; built-up cylinders under internal pressure behave as one solid wall with shrink stresses superposed; gun barrels; pre-yielding a solid tube as the analogous trick).

\sigma_{\theta,i}^{res} = -\,\frac{2\,p_c\,r_c^2}{r_c^2 - r_i^2}

Assumes: the liner alone is a Lamé cylinder under external pressure p_c — hoop compression peaks at ITS bore, exactly where the service pressure will later hurt most · Valid while: Assembly alone has driven the liner bore to compressive yield (Tresca) — before any service pressure exists. The elastic shrink-fit prediction, and the residual compression it promises, stops being the truth here: back the interference off.

\sigma_{\theta,c}^{res} = p_c\,\frac{r_o^2 + r_c^2}{r_o^2 - r_c^2}

Assumes: the jacket alone is a Lamé cylinder under internal pressure p_c — hoop tension peaks at its bore, the interface; the jacket pays for the liner's relief in advance · Valid while: Assembly alone yields the jacket bore (Tresca) — the as-built contact pressure falls below this elastic prediction as the metal redistributes. Done by accident this is an oversized fit; done on purpose — pre-yielding the wall to leave the bore in residual compression — it is autofrettage, with pressure standing in for the liner.

\sigma_{\theta,i} = p\,\frac{r_o^2 + r_i^2}{r_o^2 - r_i^2} + \sigma_{\theta,i}^{res}

Assumes: same-material superposition — under working pressure the assembled wall carries the one-piece (monobloc) Lamé field of a cylinder r_i → r_o, and the frozen-in assembly stresses ride on top (Timoshenko, Art. 41); the interface stays closed — internal pressure presses the liner INTO the jacket, so the fit cannot separate in service

\tau_i = \frac{p\,r_o^2}{r_o^2 - r_i^2} - \frac{p_c\,r_c^2}{r_c^2 - r_i^2}

Assumes: the monobloc bore shear p·r_o²/(r_o²−r_i²) — the thick-walled cylinder's governing number — minus the shear relief bought by the fit; signed by convention, and honest only on the tension side — the in-plane spread it measures governs while the total bore hoop stays tensile, i.e. while τ_i > p/2; the scoped envelope below refuses the bore margin at exactly that boundary (σ_θ,i = 0) rather than let this number mislead · Valid while: The liner bore has reached first yield in service (Tresca, 2τ = σ_y). A ductile wall redistributes beyond this line — this elastic page ends here.

\tau_c = \frac{p_c\,r_o^2}{r_o^2 - r_c^2} + \frac{p\,r_i^2\,r_o^2}{r_c^2\,(r_o^2 - r_i^2)}

Assumes: the jacket bore (the interface) is the jacket's worst point — assembly tension plus the working field's spread there; both terms pull the same way, so over-shrinking moves the failure point outward to the interface; hoop is tensile and radial compressive at every knob state here, so this spread is the honest Tresca shear with no sign caveat · Valid while: The jacket bore has reached first yield in service (Tresca, 2τ = σ_y) — the fit is carrying more than the jacket can elastically hold. Less interference, a larger jacket, or a stronger material.

\mathrm{SF}_i = \frac{\sigma_y}{2\,\tau_i}

Assumes: maximum-shear-stress (Tresca) criterion against first yield at the liner bore — same convention as the thick-walled cylinder this page extends; meaningful only while the bore carries net hoop tension; the scoped envelope below refuses the readout (and only this readout) when the fit holds the bore in compression · Valid while: Over-shrunk for this pressure — even at full p the liner bore sits in net hoop compression, so a tensile first-yield margin there is meaningless and this readout is refused. The binding checks are compressive yield at assembly (see that warning) and the jacket margin, which now governs.

Source: Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §3-16 "Press and Shrink Fits" (radial interference, the general two-material contact-pressure relation and its same-material collapse), §3-14 (stresses in pressurized cylinders), and §5-4 (maximum-shear-stress theory, S_sy = 0.5 S_y).

\mathrm{SF}_c = \frac{\sigma_y}{2\,\tau_c}

Assumes: Tresca against first yield at the jacket bore — under-shrunk walls fail at the liner bore, over-shrunk walls fail here; the balanced fit makes the two margins equal

\delta_{bal} = \frac{2\,p\,r_c\,r_o^2\,(r_c^2 - r_i^2)}{E\,\left(2\,r_o^2\,r_c^2 - r_c^4 - r_o^2\,r_i^2\right)}

Assumes: the interference at which the two Tresca shears are equal at the current pressure — τ_i falls and τ_c rises linearly with contact pressure, so the crossing is unique and closed-form; no new physics enters, only algebra over the two cited shears — Timoshenko motivates the favorable distribution qualitatively (Art. 41) but does not state this form; the equal-shear crossing is re-derived in pipeline/tests/ and the closed form is machine-checked against the authored solutions; the denominator 2r_o²r_c² − r_c⁴ − r_o²r_i² = r_c²(r_o²−r_c²) + r_o²(r_c²−r_i²) is positive for every legal geometry, so a balanced fit always exists; with the interface at r_c = √(r_i·r_o), this fit maximizes the elastic pressure capacity — the classical two-shell gun-barrel optimum (proven in pipeline/tests/)

\mu = \rho\,\pi\,(r_o^2 - r_i^2)

Assumes: prismatic two-piece annulus, uniform density — the interface adds no material; the compound wall weighs exactly what the monobloc wall of the same envelope weighs

Source: Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — prismatic members and section properties (annular area, mass per length).

Derivation

Steps marked modeling step are where physics enters by citation — every other line is machine-proven to follow from them, and the cited models are independently re-derived in the test pipeline where possible. See what is and isn't machine-verified.

u_{j} = \frac{r_{c} \left(\nu p_{c} + \frac{p_{c} \left(r_{c}^{2} + r_{o}^{2}\right)}{- r_{c}^{2} + r_{o}^{2}}\right)}{E}

1. The modeling step: each member alone is the thick-walled cylinder's Lamé field. The jacket carries the contact pressure p_c on its bore, so its bore hoop stress is p_c(r_o²+r_c²)/(r_o²−r_c²) with radial −p_c — and plane-stress Hooke turns that face's hoop strain into a radial displacement, u = (r_c/E)(σ_θ − ν σ_r). The jacket bore grows by u_j. (The test suite re-derives the field and this displacement from equilibrium, compatibility, and Hooke independently.) — elasticity: Lamé field + plane-stress Hooke at the jacket bore modeling step

u_{l} = - \frac{r_{c} \left(- \nu p_{c} + \frac{p_{c} \left(r_{c}^{2} + r_{i}^{2}\right)}{r_{c}^{2} - r_{i}^{2}}\right)}{E}

2. The liner sees the same p_c from outside: a Lamé cylinder under external pressure. Its outer face carries hoop −p_c(r_c²+r_i²)/(r_c²−r_i²) and radial −p_c, so the same Hooke line moves that face inward — u_l is negative. The two displaced faces must land on a common circle. — elasticity: the same field with the pressure outside modeling step

\delta = u_{j} - u_{l}

3. Closure: the jacket bore grows, the liner face shrinks, and together they absorb exactly the machined mismatch δ. Watch ν: the jacket contributes +ν·p_c·r_c/E, the liner −ν·p_c·r_c/E — same metal, equal and opposite, gone. This line is verified with ν left fully symbolic, so the cancellation is machine-proven for every ν in the registry, not sampled at one value. (Different metals keep their ν's: Shigley's §3-16 general form has all four of E_i, ν_i, E_o, ν_o.) — closure: the faces share one circle — and ν cancels

p_{c} = \frac{E \delta \left(- r_{c}^{2} + r_{o}^{2}\right) \left(r_{c}^{2} - r_{i}^{2}\right)}{2 r_{c}^{3} \left(- r_{i}^{2} + r_{o}^{2}\right)}

4. Solve the closure for the contact pressure: Timoshenko's eq. 181, Shigley's §3-16 same-material form. Microns of interference make tens of MPa — at this page's defaults, 30 µm of squeeze buys 19.2 MPa of contact pressure. This is why shrink fits are specified in tenths of a thousandth and why the parts are assembled with heat or liquid nitrogen, not a bigger hammer. — solve the closure for p_c (Timoshenko eq. 181)

\sigma_{ti res} = - \frac{2 p_{c} r_{c}^{2}}{r_{c}^{2} - r_{i}^{2}}

5. The prize. Evaluate the liner's external-pressure field at its own bore: hoop compression of −2p_c·r_c²/(r_c²−r_i²) — at the defaults, −69 MPa parked exactly where the thick-walled cylinder hurts most. The factor 2r_c²/(r_c²−r_i²) > 2 means every pascal of contact pressure buys MORE than two pascals of bore relief. — evaluate the liner field at the bore

\sigma_{ti tot} = \frac{p \left(r_{i}^{2} + r_{o}^{2}\right)}{- r_{i}^{2} + r_{o}^{2}} + \sigma_{ti res}

6. Now pressurize. Same metal on both sides of the interface means the assembled wall responds to p exactly as one monobloc cylinder r_i → r_o — "the stresses produced by this pressure are the same as in a cylinder with a solid wall" (Timoshenko, Art. 41) — and the frozen assembly field rides on top. The bore's service tension starts from −69 MPa instead of zero: at the defaults it climbs only to +80 MPa where the monobloc bore would already read +149 MPa. — superposition: monobloc working field + frozen residual

2 \tau_{bore} = p + \sigma_{ti tot}

7. The bore's Tresca pinch: radial stress there is the full −p, so the governing spread is σ_θ,i − (−p). The fit cannot touch the radial term — relief arrives only through the hoop. (While σ_θ,i stays positive this is the true maximum shear at the bore; when over-shrinking drives σ_θ,i negative the page refuses the bore margin rather than quietly reporting the wrong criterion — see the scoped envelope.) — maximum shear at the liner bore

2 \tau_{iface} = \frac{2 p r_{i}^{2} r_{o}^{2}}{r_{c}^{2} \left(- r_{i}^{2} + r_{o}^{2}\right)} + p_{c} + \sigma_{tc res}

8. The bill arrives at the interface. The jacket bore stacks its assembly tension σ_θ,c^res on the working field's spread there, plus its own radial −p_c: every micron of interference RAISES this shear while it LOWERS the bore's. The wall has two candidate failure points and the knobs slide the danger between them. — maximum shear at the jacket bore

E \delta_{bal} \left(- r_{c}^{4} + 2 r_{c}^{2} r_{o}^{2} - r_{i}^{2} r_{o}^{2}\right) = 2 p r_{c} r_{o}^{2} \left(r_{c}^{2} - r_{i}^{2}\right)

9. Both shears are linear in p_c, one falling and one rising, so they cross exactly once: set τ_i = τ_c, solve for p_c, and map back through the closure. The result is the balanced interference — the fit that makes liner and jacket reach first yield together, which is the most pressure this geometry can elastically hold. Put the interface at the geometric mean r_c = √(r_i·r_o) and the balanced wall's capacity is σ_y(1 − r_i/r_o): against the monobloc's (σ_y/2)(1 − r_i²/r_o²) that is a gain of 2/(1 + r_i/r_o) — TWO shells approach TWICE the ceiling as walls grow thick (machine-proven in the test suite, along with the optimum location itself). The default knobs sit exactly at this balanced optimum; nudge δ either way and watch one margin sag. — balance: equal margins, the two-shell optimum

How it fails

The widget’s two margins are first yield at the liner bore and at the jacket bore, for a two-piece elastic wall under steady pressure. Real compound cylinders also fail on the way to existing, and in service by ways the elastic page cannot see:

Assembly is the first proof test. The jacket goes on hot (or the liner goes in cold) and the interference is committed the moment the temperatures equalize. Get $\delta$ wrong by a few microns — a tolerance class, a thermal miscalculation — and assembly alone yields the liner bore in compression or splits a brittle jacket in hoop tension. The widget’s two assembly warnings mark where elasticity ends on each side; a brittle jacket’s actual fracture line is a different criterion (peak tension against ultimate strength) that this yield-based page does not carry. Pressed (rather than thermal) assembly adds galling: the faces weld and tear locally, and the “fit” becomes a gouge.
Microns are the currency, scatter is the tax. Standard force-fit classes specify interference as a range; the contact pressure inherits that range linearly. A fit specified at the balanced optimum can leave the factory under-shrunk at one tolerance limit and bore-compressed at the other — high-pressure practice treats $p_c$ as a distribution and proof-tests every assembly.
Heat undoes what heat did. The jacket was expanded onto the liner thermally; any service condition that heats the jacket faster than the liner (external fire, friction at the OD) re-expands it and bleeds off $p_c$ — the compound wall silently degrades toward the monobloc one whose refusal this page exists to escape. Sustained temperature adds creep relaxation: the squeeze leaks away over years even at uniform temperature.
Spin loosens the grip. On rotors, centrifugal growth opens the bore of the outer member faster than the shaft grows — the bored disk’s own field working against the fit. Shrink-fitted wheels and couplings have a release speed at which $p_c$ reaches zero and the joint lets go; it is a design number, not an accident.
The interface frets. Under cyclic load the faces micro-slip, and fretting initiates fatigue cracks exactly at the press-fit edge — the classic seat-of-failure for railway axles since the 19th century. Residual bore compression, by contrast, is fatigue’s enemy: like autofrettage, the fit holds the crack-driving cycle partly shut at the bore.
Past first yield, the wall does not burst — it autofrettages itself. Both margins here end at first yield; a ductile wall redistributes and holds well beyond. The deliberate version pressurizes a monobloc tube until the bore yields, then releases, leaving residual compression with no second part at all — Timoshenko notes the trick in the same article as the shrink fit. Stacking shells keeps raising the elastic ceiling toward the classical fully plastic limit $\sigma_y \ln(r_o/r_i)$ — a closed-end (axial stress intermediate) result; with this page’s open ends the elastic stack saturates at $\sigma_y$ instead, which is exactly the bound the two-shell optimum approaches. Either way that territory belongs to the autofrettaged wall, and burst itself is governed by ultimate strength and crack behaviour, not by these margins.

Rotating Disk with a Central Bore
Drill the smallest possible shaft hole through a spinning disk and the peak stress exactly doubles — not "roughly increases": doubles, in the limit of a vanishing bore. The solid flywheel's optimistic numbers meet the hole every real rotor needs.
- stress
- energy-storage
- mass-cost
Thick-Walled Cylinder (Lamé)
Where the thin-wall pressure vessel hands off: the exact elastic field for any wall thickness. The stress piles up at the bore and decays as 1/r² — and the bore shear always exceeds the pressure, so past p = σ_y/2 no amount of thickness can contain it elastically.
- stress
- mass-cost
Thin-Walled Pressure Vessel (Cylinder)
A pressurized tube with closed ends — scuba tank, boiler, rocket stage. The hoop stress is exactly twice the longitudinal stress, which is why sausages split lengthwise; and because the relations are undirected, the same widget runs backwards as a design tool: pick a safety factor and the wall thickness falls out.
- stress
- mass-cost
Cantilever Beam (End Load)
A beam fixed at one end, loaded at the other — the fruit-fly of structures. One widget shows why stiffness (E) and strength (σ_y) are independent axes: swap steel for titanium and deflection goes UP while the safety factor also goes up.
- stress
- mass-cost
Composite Bar (Core + Sleeve)
A solid core inside a concentric sleeve, bonded between rigid end plates and pushed by a centric axial load. The two materials must stretch together, so the load splits in proportion to each member's axial stiffness A·E — and the build solves that coupled 2×2 share exactly. Swap the sleeve's metal and watch the load migrate to the stiffer member.
- stress
- mass-cost
Eccentric Column (Secant Formula)
Load a column even slightly off-axis and the clean buckling story dissolves: it bows from the first newton, stress grows faster than load, and the Euler limit survives only as the asymptote the deflection chases. Because nothing here is linear, the safety factor must be taken on the LOAD — the page solves that transcendental equation live, by bracketed root-finding.
- stability
- stress
- mass-cost

Chains with

Outputs whose SI dimension and quantity kind match another THING's input — the only wires the planner's connectionLegal accepts (invariant 2, computed at build time, not hand-listed). Wire these on the chaining demo.

Composite Bar (Core + Sleeve)
- delta_bal L
Impact Loading (Falling Mass, Energy Method)
- delta_bal b
- delta_bal d
- delta_bal h
- delta_bal L
Symmetric Two-Bar Truss
- delta_bal d
- delta_bal L
Thermal Assembly (Two-Segment Bar Between Rigid Walls)
- delta_bal L_1
- delta_bal L_2
Cantilever Beam (End Load)
- delta_bal b
- delta_bal h
- delta_bal L
Circular Plate under Uniform Pressure (Clamped vs Simply Supported)
- delta_bal a
- delta_bal t
- p_c q
- p_c sigma_allow
- sigma_tc_res q
- sigma_tc_res sigma_allow
- sigma_ti_res q
- sigma_ti_res sigma_allow
- sigma_ti_tot q
- sigma_ti_tot sigma_allow
- tau_bore q
- tau_bore sigma_allow
- tau_iface q
- tau_iface sigma_allow
Curved Beam in Bending (Winkler — Crane Hook, C-Clamp, Press Frame)
- delta_bal r_i
- delta_bal r_o
- delta_bal w
Fixed-Fixed Beam (UDL)
- delta_bal b
- delta_bal h
- delta_bal L

+ 25 more THINGs its outputs can legally feed (showing the first 8 in course order).

Sources

Timoshenko, S., Strength of Materials, Part II: Advanced Theory and Problems, 3rd ed., Van Nostrand, 1956 (first published 1930) — ch. VI "Deformations Symmetrical About an Axis", Art. 40 (the thick-walled cylinder: Lamé field, internal/external pressure, displacements) and Art. 41 "Stresses Produced by Shrink Fit" (eq. 181: the same-material contact pressure; built-up cylinders under internal pressure behave as one solid wall with shrink stresses superposed; gun barrels; pre-yielding a solid tube as the analogous trick).
Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §3-16 "Press and Shrink Fits" (radial interference, the general two-material contact-pressure relation and its same-material collapse), §3-14 (stresses in pressurized cylinders), and §5-4 (maximum-shear-stress theory, S_sy = 0.5 S_y).
Gere, J. M., & Goodno, B. J., Mechanics of Materials, 9th ed., Cengage, 2018 — prismatic members and section properties (annular area, mass per length).