Rolling-Contact Bearing Life (Load–Life + Weibull Reliability)

dynamicsreliability

Verified build 5 relations · 4 identities proven · 2 modeling steps · 6 parity samples

A rolling-contact bearing does not have a strength you compare a load against. It has a rating, and the rating is a statement about a population: the basic dynamic load rating $C_{10}$ is the radial load at which 90% of a large batch of identical bearings survive one million revolutions before the races and balls spall from rolling-contact fatigue. Load a bearing below $C_{10}$ and it lasts longer; load it above and it lasts less — and the trade is steep. This page turns the two numbers stamped in the catalog, $C_{10}$ and the static rating $C_0$ , into a life you can read in hours, and turns the reliability you demand into the life you actually get.

The load–life law is a power law, and the exponent is the whole story

Rolling-contact fatigue life scales as the load ratio raised to a fixed exponent, calibrated so that the rating load gives exactly the rating life:

$L_{10} = 10^{6}\left(\frac{C_{10}}{P}\right)^{a}$

The $10^{6}$ is not decoration — it is the definition of $C_{10}$ (the load for $10^6$ revolutions at 90% reliability, ISO 281 / ABMA), so it belongs in the relation, not in the readout. The exponent $a$ is where ball and roller bearings part company, and it is a cited constant, not a knob:

Ball bearings make point contact, and $a = 3$ . Halve the load and life multiplies by $2^3 = 8$ .
Roller bearings make line contact, spreading the stress, and $a = 10/3$ . The same halving multiplies life by $2^{10/3} \approx 10$ .

Switch the configuration and watch the rated-life bar jump for the identical loads — that gap is the difference between point and line contact. The lesson bearing designers live by falls straight out of the exponent: a modest reduction in load buys a large increase in life, so oversizing a bearing pays back far faster than intuition suggests.

From revolutions to hours

A life in revolutions is turned into a life in time by the speed. Each revolution is $2\pi$ radians, so $L_{10}$ revolutions at angular speed $\omega$ (rad/s) — equivalently $n$ (rev/min) — take

$L_{10h} = \frac{2\pi\,L_{10}}{3600\,\omega} = \frac{L_{10}}{60\,n}$

hours, with $L_{10}$ in plain revolutions in both forms (the $2\pi$ trades revolutions for radians; the $3600$ and $60$ trade seconds and minutes for hours). Shigley prints the same fact as $L_{10h} = 10^{6}L_{10}'/(60\,n)$ with the life $L_{10}'$ expressed in millions of revolutions — the catalog convention — so his $10^{6}$ merely unpacks the Mrev; feeding this page’s plain-revolution $L_{10}$ into that form would overshoot by a factor of a million. This page keeps the engine in SI and lets the readout show hours, so the arithmetic you can check by hand and the number an engineer quotes are visibly one thing. Bearing life is quoted in hours because that is what a maintenance schedule is written in; the page adds the hour and million-revolution (Mrev) display units for exactly this readout.

Reliability is bought with life

The rated life $L_{10}$ is the life 90% of bearings beat. Ask for a higher survival fraction and the life you can count on drops, because you are now designing for the unluckier tail of the population. The scatter in rolling-contact fatigue life is described by a three-parameter Weibull distribution of the dimensionless life $x = L/L_{10}$ :

$R = \exp\!\left[-\left(\frac{x - x_0}{\theta - x_0}\right)^{b}\right]$

Inverting it for the life at a chosen reliability gives the reliability life factor

$x(R) = x_0 + (\theta - x_0)\left(\ln\tfrac{1}{R}\right)^{1/b}$

and the usable life is $L_R = x(R)\,L_{10}$ . At $R = 0.90$ the factor returns $\approx 1$ — a self-consistency check that $L_{10}$ really is the 90% life. Push the reliability knob to 0.99 and the factor falls to about $0.22$ : demanding that 99% survive instead of 90% cuts the usable life to roughly a fifth of the catalog number. That is not pessimism, it is the geometry of the tail. The Weibull fit is cited only on its calibrated domain, so below $R = 0.90$ the reliability-adjusted readouts are refused (the adjusted-life bar is withheld and a banner explains why) while the rated life $L_{10}$ keeps standing — a scoped refusal, the same machinery the eccentric column uses to hand off between two models on one page.

The Weibull parameters are cited data, carried as constants

The numbers $x_0 = 0.02$ , $\theta = 4.459$ , $b = 1.483$ are not universal — they are a manufacturer’s fit to a specific bearing family (Shigley’s Table 11-6, the 02-series set on the $10^6$ -revolution rating basis). They appear on the page the way the standard gravity does on the critical-speed page: labeled, sourced values that are never sliders. Reading them off the constants panel is the point — a reliability number that traces to a cited fit is honest in a way a magic coefficient buried in a formula is not.

Every other stress page here has a material axis. This one deliberately does not, and the reason is worth stating: the bearing steel is already inside the catalog ratings. $C_{10}$ and $C_0$ are measured on the finished bearing — its steel, its heat treatment, its geometry, its surface finish all baked in — so exposing a separate “material” would double-count the very thing the rating encapsulates. Like the planetary gearset, this is a geometry-and-catalog THING: the honest user axes are the loads, the speed, and the reliability goal, and the page teaches what the catalog number means rather than pretending to recompute it. The bearing’s material story lives one page over, in the shafts it carries.

Set the catalog ratings and the equivalent load, pick a speed and a reliability, and toggle ball versus roller. The two bars are on a logarithmic scale so the power law is legible at a glance: one decade of life is a fixed width, and halving the load slides the ball bar a full three doublings (8×) to the right. Drive the load past the static rating $C_0$ and it warns — that is a different failure (brinelling, covered in the failure note). The radial load $P$ here is exactly the gear separating force a spur gear pair throws onto its shaft, and the shaft it rides is the combined bending-and-torsion shaft — an explicit hand-off the chaining work will wire together.

Try it

Configuration

Constants

Cited physical constants used on this page
x0	Weibull guaranteed-life x₀	0.02	shigley
theta	Weibull characteristic life θ	4.459	shigley
b	Weibull shape parameter b	1.483	shigley

Governing relations

L_{10} = 10^{6}\left(\dfrac{C_{10}}{P}\right)^{a}

Assumes: the basic dynamic load rating C₁₀ is defined as the load that gives a rated life of one million revolutions at 90% reliability (ABMA/ISO 281 rating basis; physics enters here by citation); fatigue-limited rolling contact; a = 3 for ball bearings (point contact), a = 10/3 for roller bearings (line contact), a cited per-configuration constant, not a knob · Valid while: The applied load exceeds the static load rating C₀. Above C₀ the rolling elements brinell (permanently dent the races) and the rotating-contact fatigue model no longer describes the failure — the life shown assumes no static overload.

Source: Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — Ch. 11 (Rolling-Contact Bearings): §11-3 (the load–life relation L₁₀ = (C/P)^a, with a = 3 for ball and 10/3 for roller bearings, C₁₀ rated at 10⁶ revolutions), §11-4 (reliability and life via the three-parameter Weibull distribution), and Table 11-6 (Weibull parameters for two manufacturers' rating lives).

L_{10h}\,\omega = 2\pi\,L_{10}

Assumes: each revolution is 2π radians, so a life of L₁₀ revolutions at speed ω takes L₁₀·2π/ω seconds — the SI form of Shigley's L_h = 10⁶·L₁₀/(60 n); the 60 is a display artifact, not a residual

x_R = x_0 + (\theta - x_0)\left(\ln\tfrac{1}{R}\right)^{1/b}

Assumes: the dimensionless life x = L/L₁₀ follows a three-parameter Weibull distribution, R = exp[−((x − x₀)/(θ − x₀))^b]; inverting it gives the life factor x(R) for a chosen reliability (Shigley §11-4; parameters from Table 11-6, Manufacturer 2); x(0.90) ≈ 1 by construction — the rating life L₁₀ is itself the 90%-reliability life, so at R = 0.90 the factor returns essentially the rated life · Valid while: The Weibull reliability fit is cited only for R ≥ 0.90 (the catalog rating basis). Below it the reliability-adjusted life would be an extrapolation, so it is refused — but the rated-reliability life L₁₀ still stands. Reliability must be below 1. As R → 1 the model demands the survival of the very weakest bearing and the adjusted life collapses toward the guaranteed-life floor x₀·L₁₀; R = 1 is not a designable target, so the adjusted-life readouts are refused.

L_R = x_R\,L_{10}

Assumes: the life a fraction R of the population survives is the reliability factor times the rated life

L_{Rh}\,\omega = 2\pi\,L_R

Assumes: the reliability-adjusted life expressed in hours, the same rev-to-time conversion as L₁₀

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.

L_{10} = 1000000 \left(\frac{C_{10}}{P}\right)^{a}

1. Start from the catalog. The basic dynamic load rating C₁₀ is defined so that a group of bearings run at exactly that load reaches a rated life of one million revolutions with 90% of them surviving. Rolling-contact fatigue then follows a load–life power law: life scales as (C₁₀/P) raised to a. For ball bearings a = 3 (point contact), for roller bearings a = 10/3 (line contact). Halving the load multiplies a ball bearing's life by 2³ = 8. This is the one modeling step — the rating and the exponent are cited from ISO 281 / Shigley §11-3. — load–life rating law (cited): L₁₀ = 10⁶(C₁₀/P)^a modeling step

\omega t_{10} = 2 \pi L_{10}

2. Convert revolutions to time. One revolution is 2π radians, so a life of L₁₀ revolutions at angular speed ω lasts L₁₀·2π/ω seconds. Bearing tables usually print this as L_h = 10⁶·L₁₀/(60 n) with n in rev/min — the same fact, with the 60 and the 10⁶ folded in as display conversions. Here the engine works in SI and those constants live in the readout, not the relation. — revolutions to time: each rev is 2π rad

R = e^{- \left(\frac{- x_{0} + x_{R}}{\theta - x_{0}}\right)^{b}}

3. Real bearings scatter around L₁₀. Model the dimensionless life x = L/L₁₀ as a three-parameter Weibull variable: the surviving fraction is R = exp[−((x − x₀)/(θ − x₀))^b], with a guaranteed life x₀ (below which essentially none fail), a characteristic life θ, and a shape b. The values x₀ = 0.02, θ = 4.459, b = 1.483 are the 02-series manufacturer set from Shigley Table 11-6 — cited data, carried as labeled constants. — three-parameter Weibull life distribution (cited) modeling step

\frac{- x_{0} + x_{R}}{\theta - x_{0}} = \log{\left(\frac{1}{R} \right)}^{\frac{1}{b}}

4. Invert the distribution for the life at a chosen reliability. Take logs twice: ln(1/R) = ((x − x₀)/(θ − x₀))^b, so (x − x₀)/(θ − x₀) = (ln 1/R)^{1/b}. At R = 0.90 the right side is ≈ 0.219 and x ≈ 0.994 — the factor returns essentially the rated life, which is the consistency check that L₁₀ really is the 90% life. (The pipeline re-derives this inversion symbolically from the Weibull CDF.) — solve the Weibull CDF for the life factor

x_{R} = x_{0} + \left(\theta - x_{0}\right) \log{\left(\frac{1}{R} \right)}^{\frac{1}{b}}

5. Rearranged, the reliability life factor is x(R) = x₀ + (θ − x₀)(ln 1/R)^{1/b}. It falls as R rises: demanding that 99% survive instead of 90% cuts the usable life to about a fifth of the rated value. Reliability is bought with life. — the reliability life factor x(R)

L_{R} = L_{10} x_{R}

6. The life a fraction R of the population reaches is that factor times the rated life, L_R = x(R)·L₁₀, and in hours L_R·2π/ω. These are the two numbers a bearing selection actually turns on: the catalog gives L₁₀, the application sets the reliability, and the product is what you can count on. — reliability-adjusted life, in revolutions and hours

How it fails

A bearing rated for a life does not fail the way a beam fails. There is no single load it cannot carry and no moment of sudden collapse — the number this page computes is a statistical prediction about fatigue, and understanding what it does and does not promise is the whole point.

The modeled failure: rolling-contact fatigue

Every time a ball rolls over a point on the race, that point feels a pulse of Hertzian contact stress, with the maximum shear a fraction of a millimetre below the surface. Millions of those pulses grow a subsurface crack until a flake of steel breaks away — a spall. Once one spall forms the bearing gets noisy, runs rough, and quickly tears itself up. $L_{10}$ is the number of revolutions at which 10% of a population has spalled; equivalently, the reliability the catalog quotes is $R = 0.90$ . This is the failure the load–life law describes, and nothing else.

$L_{10}$ is a distribution, not a guarantee

The most common misreading of a bearing rating is to treat $L_{10}$ as “the life.” It is not. Ten percent of bearings fail before $L_{10}$ — some far before — and the median life $L_{50}$ is several times longer, about $3.5\,L_{10}$ for the Weibull fit this page uses (the steep shape $b$ pulls the median in from the shallower $\approx 5\times$ rule of thumb). A single bearing has no deterministic life at all; only a population does. That is exactly why the reliability adjustment exists: if losing 10% is unacceptable, you design to $L_R$ at a higher $R$ and accept the shorter usable life. Reading a 90%-reliability rating as a promise that this bearing will reach it is the error the whole Weibull apparatus is built to correct.

Brinelling: a different failure the fatigue math cannot see

Push the load past the static rating $C_0$ and a wholly different failure appears — one the rotating-fatigue model on this page knows nothing about. Under a static or shock load the balls press so hard they permanently dent the races, leaving a pattern of shallow craters called brinelling. It is instantaneous, not cumulative; it can happen to a bearing that is not even turning — a bad press-fit during assembly, a machine dropped in shipping, or the ground-in false brinelling and fretting corrosion that vibration produces in a bearing parked under load. The widget warns when $P > C_0$ because past that point the life it reports is meaningless: the bearing is damaged before fatigue ever gets a vote.

What this page deliberately leaves out

The catalog life is an ideal — it assumes clean, adequate lubrication, correct mounting, and tolerable temperature. In service the biggest life-killers are usually none of the things modeled here:

Lubrication and contamination. A single hard particle rolled into a raceway starts a dent that becomes a spall; a starved or overheated film lets metal touch metal. ISO 281’s modified rating life folds these into a factor $a_{\text{ISO}}$ that can swing the real life by an order of magnitude in either direction. That factor is out of scope here — named, not modeled — and it is why field lives so often fall short of the catalog number.
The equivalent load $P$ itself. A real bearing sees combined radial and thrust load, and converting that to the single equivalent $P$ needs the bearing’s $X$ and $Y$ factors, which switch on the load ratio. This page takes $P$ as given; that conversion is a natural next capability.
Misalignment, speed limits, and cage wear, each its own failure mode outside the fatigue law.

So read the life here as the clean-running, correctly-mounted ceiling — the best case the geometry allows. The shaft the bearing supports carries its own story: keep the running speed clear of the shaft’s critical speed, and remember that the radial load driving the fatigue is the same separating force a spur gear pair delivers.

Shaft Critical Speed (Rayleigh + Dunkerley)
Spin a shaft fast enough and it whips: at its critical speed the rotor whirls in resonance with its own static sag, ω_c = √(g/δ_st). Gravity sets the sag but cancels out of the answer — the critical speed is pure stiffness over inertia. Dunkerley's estimate folds in the shaft's own mass and is provably never higher than Rayleigh's.
- dynamics
- stress
Stepped Shaft — Shoulder-Fillet Stress Concentration
A shoulder where a shaft steps from a large diameter D to a small diameter d concentrates stress in the fillet of radius r. The peak stress is the nominal stress times a geometric factor K_t read from a cited chart — pure geometry: the material changes the safety factor but never K_t.
- stress
Axial Disk Clutch / Brake (Uniform Wear vs Uniform Pressure)
The torque an axial plate clutch can pass depends on an assumption you cannot see: how the contact pressure is distributed across the friction annulus. A new, rigid clutch presses uniformly; a worn-in one wears until pressure ∝ 1/r, concentrating load at the inner edge. This page shows both torque predictions side by side — never picking a winner — with the worn-in model always giving the smaller (safe) number, and the r_i = r_o/√3 that squeezes the most torque from a given lining.
- torque-power
Belt Drive (Flat Belt / Capstan)
Friction compounding like interest: every degree of wrap multiplies the tension a belt can hold, e^μθ in total — the same exponential that lets a sailor check a ship with two turns of rope. At speed, centrifugal relief steals tension back, so every belt has a power ceiling.
- torque-power
Bolted Joint with Gasket (External Tensile Load)
A preloaded bolt clamping a gasketed joint, then pulled by an external tensile load. The bolt and the members act as two springs in parallel, so the external load does NOT all go to the bolt — it splits by stiffness. The build solves the coupled bolt/member force system exactly and refuses the moment the members go slack.
- stress
Helical Compression Spring
A torsion bar wound into a package: push on the coil and the wire twists. G sets the rate, σ_y sets the margin, and the geometry trades them against three envelopes — coil bind, buckling, and a spring index you can actually wind.
- stiffness
- 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.

Euler Column (Buckling)
- x_R K
Planetary (Epicyclic) Gearset
- L_R N_p
- L_R N_s
- L10 N_p
- L10 N_s
Spur Gear Pair (Lewis Bending)
- L_R N_g
- L_R N_p
- L10 N_g
- L10 N_p
Stepped Shaft — Shoulder-Fillet Stress Concentration
- x_R Dd
- x_R rd
Axial Disk Clutch / Brake (Uniform Wear vs Uniform Pressure)
- L_R N
- L10 N
Helical Compression Spring
- L_R N_a
- L10 N_a

Sources

Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — Ch. 11 (Rolling-Contact Bearings): §11-3 (the load–life relation L₁₀ = (C/P)^a, with a = 3 for ball and 10/3 for roller bearings, C₁₀ rated at 10⁶ revolutions), §11-4 (reliability and life via the three-parameter Weibull distribution), and Table 11-6 (Weibull parameters for two manufacturers' rating lives).
Juvinall, R. C., & Marshek, K. M., Fundamentals of Machine Component Design, 5th ed., Wiley, 2011 — ch. 14 (rolling-element bearings): the same catalog load–life relation and the reliability adjustment for lives other than the 90% rating.
ISO 281:2007, Rolling bearings — Dynamic load ratings and rating life; and ABMA Standards 9 and 11 — the standards that define the basic dynamic load rating C and the L₁₀ = (C/P)^p rating-life framework used throughout the bearing industry.