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Machine Elements · Chapter 9 of 10 · Intermediate

Rolling-Contact Bearings

Bearings are not designed, they are selected. A bearing has a finite fatigue life that falls steeply with load, and the catalog rating plus one power law turns a load into a life.

Readiness check

This chapter selects a bearing from a load and a life. Tick only what you can do closed-notes.

Evaluate a power law such as (C/P)^a.
Convert rev/min and hours to total revolutions.
Recall that fatigue life falls with load.
Read a catalog value with units.
Rearrange an equation to solve for a rating.

0 or 1 weak itemsContinue with this chapter.

2 weak itemsRevisit fatigue life in Chapter 5.

3 or more weak itemsPractise unit conversions before the worked examples.

The core idea

A rolling bearing fails by surface fatigue, so it has a statistical life that scales as a steep power of the load. One catalog number, the dynamic load rating, anchors that curve.

L = (C/P)^aa = 3 (ball), 10/3 (roller)C = P (L_D/L_rating)^1/a

Balls or rollers running on hardened races eventually spall from contact fatigue, so a bearing has a rated life L₁₀, the life 90 percent of a batch will exceed. The basic dynamic load rating C is the load that gives the rated life of one million revolutions; from it, any other load gives a life L = (C/P)^a, where the exponent a is 3 for ball bearings and 10/3 for roller bearings. Because a is large, halving the load multiplies the life roughly eightfold. Design runs the relation backwards: from a target life, find the rating C the catalog must meet.

The skill works when: you use a consistent rating basis and the right exponent for the bearing type.

The skill breaks down when: revolutions and hours are mixed, or a ball exponent is used for rollers.

The concept. Balls roll between hardened inner and outer races under a radial load. Surface fatigue sets a statistical life that falls as a steep power of the load, anchored by the catalog rating C.

The skills, taught in order

Five skills cover bearing types, the load-life law, the rating and equivalent load, the reliability adjustments, and the selection procedure.

9.1 Bearing types

Deep-groove ball bearings take radial and some thrust load and run at high speed; cylindrical roller bearings carry heavier radial loads; tapered roller and thrust bearings handle large axial loads. The type sets the load-life exponent and the load each direction can take.

9.2 The load-life equation

Life and load trade off as L = (C/P)^a, with L in millions of revolutions, C the basic dynamic load rating, and P the equivalent load. The exponent a is 3 for ball bearings and 10/3 for roller bearings, so life is extremely sensitive to load.

Bearing	Exponent a	Best at
Ball	3	moderate load, high speed
Cylindrical roller	10/3	heavy radial load
Tapered roller	10/3	combined radial and thrust

9.3 Rating and equivalent load

Catalogs list the basic dynamic load rating C for a rated life of one million revolutions. When a bearing carries both radial and thrust load, an equivalent radial load P = XF_r + YF_a stands in for the combination, with X and Y from the catalog.

9.4 Reliability and application factors

The rated life L₁₀ is for 90 percent survival; a higher reliability needs a life adjustment factor that lowers the allowable life. A service or application factor inflates the design load to cover shock and vibration the steady analysis misses.

9.5 Selecting a bearing

Selection runs the law backwards: from the design life L_D and load, compute the required rating C = P (L_D/L_rating)^1/a, then pick the smallest catalog bearing whose C meets or exceeds it. Bore size and speed limits then finalise the choice.

Engineering connection: the bearing loads come from the shaft of Chapter 6 and the gear forces of Chapter 10, closing the power-transmission loop.

Worked example 1: life of a ball bearing

A deep-groove ball bearing has a basic dynamic load rating C = 26.9 kN. It carries a radial load of 3.0 kN at 1800 rev/min. Find its rated life in revolutions and in hours.

Figure 1. The load-life curve falls steeply: at a load far below the rating C, the life is many times the rated million revolutions.

ProblemFind the rated life of the bearing in Figure 1, in revolutions and hours.
Given / findC = 26.9 kN, P = 3.0 kN, a = 3, n = 1800 rev/min. Find L₁₀ and the life in hours.
AssumptionsPure radial load; ball bearing (a = 3); rating C is for one million revolutions; steady operation.
ModelApply L = (C/P)^a in millions of revolutions, then convert with the speed.
EquationsL = (C/P)³L_hours = L × 10⁶/(60n)
SolveL = (26.9/3.0)³ = (8.97)³ = 721 million revolutions. In hours: 721×10⁶/(60×1800) = 721×10⁶/108 000 = 6680 hours.
CheckThe load is about one ninth of the rating, and 9³ ≈ 730, so the life is roughly 730 times the rated million revolutions, consistent with the cube law.
ConclusionA modest load relative to the rating buys an enormous life, because the exponent is three. This steepness is the defining feature of bearing selection.

Result. L₁₀ = 721 million revolutions, about 6680 hours at 1800 rev/min.

Worked example 2: selecting a bearing for a design life

A ball bearing must carry 4.0 kN at 1200 rev/min for a design life of 10 000 hours. Find the required basic dynamic load rating C.

Figure 2. Selection runs the load-life law in reverse: the design life in revolutions and the load set the minimum catalog rating the bearing must have.

ProblemFind the required rating C for the bearing in Figure 2.
Given / findP = 4.0 kN, n = 1200 rev/min, design life 10 000 h, a = 3, rating life 10⁶ rev. Find C.
AssumptionsPure radial load; ball bearing; 90 percent reliability (no extra life factor).
ModelConvert the design life to revolutions, then C = P (L_D/L_rating)^1/a.
EquationsL_D = 60 n × hoursC = P (L_D/10⁶)^1/3
SolveL_D = 60 × 1200 × 10 000 = 7.2×10⁸ rev = 720 million. C = 4.0 × (720)^1/3 = 4.0 × 8.96 = 35.9 kN.
CheckThe required rating is about nine times the load, matching the cube-root of 720 ≈ 9. Choose the smallest catalog bearing with C ≥ 35.9 kN.
ConclusionDesign fixes the life and load and solves for the rating, the catalog quantity. A service factor on P would raise C further.

Result. The bearing needs a rating C ≥ 35.9 kN.

Misconceptions and diagnostics

Mistake	Symptom	Diagnostic question	Correction
Wrong exponent	Roller life computed with a = 3	"Is this a ball or roller bearing?"	Use a = 3 for balls, 10/3 for rollers.
Mixing revolutions and hours	Life off by factors of thousands	"Is L in millions of revolutions?"	Convert hours to revolutions with 60n.
Ignoring thrust load	Life over-predicted on a combined load	"Is there an axial load too?"	Use the equivalent load P = XF_r + YF_a.
No service factor	Early failure under shock loads	"Is the load steady or shocky?"	Inflate P with an application factor for shock and vibration.

Practice ladder

Level 1 · Direct skill

A ball bearing with C = 20 kN carries 4 kN. Find its life in millions of revolutions.

Show answer

L = (20/4)³ = 5³ = 125 million revolutions.

Level 2 · Mixed concept

The bearing of Worked Example 1 has its load doubled to 6 kN. By what factor does the life change?

Show answer

Life scales with P⁻³, so doubling the load cuts life by 2³ = 8 times: from 721 to about 90 million revolutions.

Level 3 · Independent problem

A cylindrical roller bearing (a = 10/3) with C = 40 kN carries 8 kN. Find its life in millions of revolutions.

Show answer

L = (40/8)^10/3 = 5^3.333 = 213 million revolutions. The roller exponent makes life even more load-sensitive than a ball bearing.

Level 4 · Transfer to real engineering

Find a product with a known bearing (a skateboard, a fan, a bike hub). Estimate its load and speed and reason about why it lasts as long as it does.

What good work looks like

A load well below the bearing's rating, a recognition that the cube law makes the life very long at light load, and an awareness that contamination or misalignment, not fatigue, usually ends such bearings.

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Check that I used the right exponent and a consistent rating basis."

"Give me three duty cycles; I will find the required rating for each."

"What is the bearing life?" Applying the load-life law yourself is the skill.

"Pick a bearing for me." Computing the required C and choosing is the point.

Portfolio task

Select a bearing for a real duty: state the load, speed, and target life, compute the required rating, and identify a catalog bearing that meets it.

Must include: a load and speed, a design life in revolutions, a required rating C, and a candidate bearing.

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. Write the load-life equation.

L = (C/P)^a, with L in millions of revolutions.

2. Give the exponents for ball and roller bearings.

a = 3 for ball, 10/3 for roller.

3. What is the basic dynamic load rating C?

The load that gives the rated life of one million revolutions.

4. How do you select a bearing?

From the design life and load, compute the required C, then pick a catalog bearing that meets it.

5. What does an equivalent load handle?

Combined radial and thrust loading, via P = XF_r + YF_a.

TodayFinish this quiz and Levels 1 and 2 of the ladder.

+1 dayRe-derive the life and the required rating from a blank page.

+3 daysSelect a bearing for a new duty cycle.

+7 daysCarry the bearing loads into gears, Chapter 10.

+30 daysCombine shaft, bearing, and gear into one drive.

Textbook mapping

Item	Mapping
Primary source	Budynas and Nisbett, Shigley's Mechanical Engineering Design, Chapter 11 (Rolling-Contact Bearings)
Cross-reference	Norton, Ch. 11 · Machine Elements, Ch. 6 (shafts)
Core topics	9.1 Bearing types · 9.2 Load-life equation · 9.3 Rating and equivalent load · 9.4 Reliability factors · 9.5 Selecting a bearing
Engineering connection	Bearing loads come from the shaft and gear forces of the surrounding chapters.
Read next	Chapter 10: Spur and Helical Gears.