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Machine Element Simulator

Ball Bearing Hertz Contact Stress Simulator — Inner & Outer Race Stress

Real-time Hertz contact stress for ball-and-race contact. Compute the per-ball maximum load from a Stribeck distribution and visualize the peak contact pressure on the concave inner race and the convex outer race.

Parameters
Total bearing load F_total
kN
Ball diameter D_b
mm
Number of balls Z
balls
Pitch diameter D_p
mm

Material fixed to bearing steel (E*=115.4 GPa, nu=0.3). Contact angle a=0 (deep-groove ball bearing); race conformity (groove curvature) is not modelled.

Results
—
Max load per ball F_max
—
Inner race stress p_max,i (concave)
—
Outer race stress p_max,o (convex)
—
Contact radius a (inner, sphere-on-plane)
Bearing cross section and contact points

Outer race, inner race and ball cross section / red dot = outer-race contact (higher stress), orange = inner-race contact / D_p = pitch diameter, D_b = ball diameter

Theory & Key Formulas

In a radially loaded ball bearing the total load is not shared equally among the balls. Stribeck's analysis shows that the stress is concentrated on the single most heavily loaded ball.

Maximum load per ball (alpha is the contact angle, alpha = 0 for a deep-groove bearing):

$$F_\text{max} = \frac{5\,F_\text{radial}}{Z\,\cos\alpha}$$

Equivalent radius of curvature for the concave inner-race contact and the convex outer-race contact:

$$\frac{1}{R_{eq,i}} = \frac{1}{r_b} - \frac{1}{R_i},\qquad \frac{1}{R_{eq,o}} = \frac{1}{r_b} + \frac{1}{R_o}$$

Peak Hertz pressure for a sphere-on-plane contact (E* is the equivalent modulus):

$$p_\text{max} = \frac{1}{\pi}\left(\frac{6\,F_\text{max}\,E^{*2}}{R_{eq}^{\,2}}\right)^{1/3}$$

Contact radius:

$$a = \left(\frac{3\,F_\text{max}\,R_{eq}}{4\,E^*}\right)^{1/3}$$

The outer race has the smaller equivalent radius of curvature, so its contact patch is smaller and its peak contact pressure is higher than the inner race.

What is the ball bearing Hertz contact stress simulator

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If a ball bearing just has balls rolling between two races, why does the contact stress get so absurdly high?
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Roughly speaking, because the ball and the race only touch over a tiny patch a few hundred microns across. Two curved surfaces meeting at a point or along a thin ellipse have to carry several kN per ball, and the resulting pressure shoots up to the GPa range almost immediately. Look at the inner-race stress card with the default settings: about 5.9 GPa, well above the bulk yield strength of steel.
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Wait, above the yield strength? Then how does the bearing not just smash itself?
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In a Hertz contact the stress state is almost pure triaxial compression, so yielding is governed by the maximum shear stress, not by simple uniaxial tension. With a peak pressure of 5 to 6 GPa the maximum shear is only about 30 percent of that, around 1.5 to 2 GPa. As long as that stays within the allowable shear stress of through-hardened bearing steel (around HV 700), no plastic flow occurs. That is exactly why bearing rings and balls are heat-treated so hard.
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OK. But then why does the inner-race stress differ from the outer-race stress? It's the same ball!
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That is the most interesting part of this tool. On the inner race the ball (convex) sits inside a concave groove, so the two curvatures partially cancel and the contact patch is fairly wide. On the outer race a convex ball pushes against a convex track from the inside, so the equivalent radius of curvature is small and the patch is narrow. The simulator shows about 8.6 GPa on the outer race and 5.9 GPa on the inner race, a 1.45 factor. The outer race is the critical surface for fatigue life.
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Can I just add more balls to lower the stress?
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You can, but not as much as you might hope. The per-ball load drops with 1/Z, but Hertz stress scales with load to the 1/3 power, so doubling Z only cuts the stress by a factor of $2^{-1/3}$ which is about 0.79. Try Z = 16 in the simulator and you go from 5.9 to about 4.7 GPa, only a 20 percent reduction. To really drive the stress down, increase the ball diameter D_b: that grows the contact patch directly while leaving the per-ball load unchanged.

Frequently asked questions

This tool uses a sphere-on-cylindrical-surface contact and only the curvature along the major axis of the contact ellipse, i.e. a sphere-on-plane Hertz approximation. Real bearings have a "race conformity" (groove radius / ball radius, typically 0.52 to 0.54) that produces a truly elliptical contact. Real stresses are usually 60 to 80 percent of the values this tool returns, so the tool is conservative. It is well suited to early sizing and trend studies; use a vendor catalog C and an L10 calculation for the final design.
In a deep-groove ball bearing a is near zero and radial load goes almost directly into the balls. Angular contact ball bearings and tapered roller bearings have contact angles of 15 to 40 degrees, which lets them carry axial (thrust) load as well. In that case the per-ball maximum load grows as F_max = 5*F_radial / (Z*cos a) and so does the stress. As an approximation in this deep-groove tool, you can enter F_total / cos a as the equivalent radial load to study an angular contact case.
For high-hardness bearing steel (SUJ2 / 52100 at HRC 58 to 62) the rules of thumb are roughly 4.2 GPa for a static design with no permanent deformation, about 4.0 GPa at the L10 rolling-contact-fatigue criterion, and 2.5 to 3.0 GPa for very long life (L1) requirements. The default 5.9 / 8.6 GPa results of this tool therefore correspond to overloaded conditions: either double the ball diameter, increase the ball count, or halve the load. Some wind-turbine main bearings do run above 4 GPa for short overloads at the cost of life.
Three effects lower the stress. (1) The equivalent radius of curvature R_eq grows directly with D_b, enlarging the contact patch. (2) The inner race radius R_i = (D_p - D_b)/2 shrinks, so the concave conformity is slightly tighter. (3) The outer race radius R_o = (D_p + D_b)/2 grows, making the convex contact gentler. Going from D_b = 12 mm to 18 mm in the simulator drops the outer-race stress from about 8.6 GPa to about 6.5 GPa. The trade-off is that the overall bearing envelope and the maximum number of balls also change.

Real-world applications

Automotive wheel hub bearings: Wheel hubs use heavily loaded angular contact ball bearings or tapered roller bearings. During cornering, the radial load is combined with a lateral (thrust) component and per-ball peak contact stresses commonly reach 3 to 4 GPa. To stay inside the allowable range, ball diameter, ball count and groove geometry are tuned carefully. Pushing the load slider above 20 kN in this tool puts you into roughly the same stress regime as a real passenger-car wheel bearing.

Wind turbine main shaft bearings: Multi-megawatt wind turbines use main bearings several meters in diameter. The fluctuating wind load and the huge rotor mass push inner-race contact stresses momentarily above 3 GPa, and that repeated loading drives the rolling contact fatigue (RCF) life of the bearing. Simple calculators like this one are used to get a feel for the trends; final design relies on finite element analysis and detailed contact mechanics.

Machine tool spindles: In high-speed spindles, centrifugal force adds a contact pressure between the balls and the outer race on top of the externally applied load. Above 10,000 rpm the centrifugal component can match or exceed the external load, and the operating stresses must be assessed for the whole regime. The "outer-race stress > inner-race stress" rule shown by this tool is exactly why outer-race fatigue dominates in high-speed spindle bearings.

Robot joint bearings: Small cross-roller bearings and thin-section angular contact ball bearings in cobot joints are weight-constrained, so both ball diameter and ball count are limited. Yet they must carry significant moment loads, so the per-ball peak load can exceed 50 percent of the catalog rating. Try the simulator with Z = 5 to 8 balls and D_b = 3 to 6 mm to see how brutally stress-limited those small bearings really are.

Common misconceptions and cautions

The most common mistake is to assume that the total bearing load is shared equally by all Z balls. Because of internal clearance and elastic deformation, only the balls in the lower load zone actually carry the load, and even there the shares are very unequal. Stribeck's analysis gives F_max = 5*F_radial/(Z*cos a), about five times the naive average F_radial/Z. The simulator deliberately shows that increasing the ball count does not collapse the stress immediately. The design starting point is always "the most heavily loaded single ball".

The next common pitfall is to treat inner and outer race contacts as if they had the same stress. They do not. The outer-race contact has the smaller equivalent radius of curvature, so its peak pressure is 20 to 50 percent higher. In this tool p_max,o is always greater than p_max,i. As a result, fatigue cracks on the outer race typically appear first, which is why outer-race material quality and surface finish are critical. Conversely, you can use the pitch diameter D_p to redistribute stress between the two races and balance their fatigue lives.

Finally, do not read this tool's peak stress directly as a "bearing life". The Hertz pressure p_max is an instantaneous value, while the bearing life (such as L10) scales with the inverse 9th power of p_max for ball bearings. Doubling the stress cuts the life by a factor of about 512; conversely, reducing the stress from 4 GPa to 3 GPa increases the life by roughly a factor of 10. After computing a stress here, always pair the result with an L10 life calculation (see the rolling-bearing simulator) to check that the stress level meets the life requirement. Stress alone is not enough to judge a bearing design.

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