Static load rating and static safety

What the static rating protects against

A bearing that stands still, oscillates slowly or takes a heavy shock does not fail by fatigue first. It fails by permanent dents in the raceways, which later make it run rough and noisy. ISO 76 builds on the experience that a total permanent deformation of about 0.0001 times the element diameter, at the centre of the most heavily loaded contact, does no harm in most applications (Introduction). Tests showed that this deformation appears at a calculated Hertz contact stress of

• 4 600 MPa for self-aligning ball bearings,
• 4 200 MPa for all other ball bearings,
• 4 000 MPa for all roller bearings.

The basic static load rating C0 is the load that produces exactly this stress in the most heavily loaded contact (definitions 3.2 for radial and 3.3 for thrust bearings). For a single-row angular contact bearing the radial rating refers to the radial part of the load that displaces the rings purely radially. The German edition DIN ISO 76:2009 is identical to ISO 76:2006 with its Amendment 1 included.

The standard's equations

ISO 76 turns the stress limits into closed formulas:

• radial ball bearings: $C_{0r}=f_0\cdot i\cdot Z\cdot D_w^2\cdot \cos \alpha$ (Eq. (1)), with f0 from Table 1;

• thrust ball bearings: $C_{0a}=f_0\cdot Z\cdot D_w^2\cdot \sin \alpha$ (Eq. (4));

• radial roller bearings (Eq. (7)):

  $$C_{0r}=44\cdot (1-\frac{D_{we}\cdot \cos \alpha }{D_{pw}})\cdot i\cdot Z\cdot L_{we}\cdot D_{we}\cdot \cos \alpha$$

• thrust roller bearings: the same form with 220 and sin α (Eq. (11)).

The note to Table 1 states the basis: Hertz contact with E = $2.07\cdot {10}^5$ MPa and ν = 0.3, and a load distribution in which the most heavily loaded ball of a radial bearing carries $5\cdot F_r/(Z\cdot \cos \alpha )$ (thrust: $F_a/(Z\cdot \sin \alpha )$). Table 1 applies to grooves no wider than 0.52·Dw on the inner and 0.53·Dw on the outer ring (5.1.1). Wider grooves need a smaller f0; tighter grooves do not necessarily raise the capacity.

Static equivalent load and static safety

For combined loads ISO 76 defines the static equivalent load

$P_{0r}=X_0\cdot F_r+Y_0\cdot F_a$, but not less than Fr (Eq. (2), (3)),

with X0 and Y0 from Table 2 (ball) and Table 3 (roller); thrust bearings use Eq. (5), (6), (12) and (13). The static safety factor is

$s_0=C_0/P_0$ (Eq. (14) radial, (15) thrust).

Clause 9 gives experience-based minimum values (Tables 4 and 5). For ball bearings they run from about 1 for normal, smooth running to 2 where quiet running and high running accuracy matter, with at least 1.5 for pronounced shock loads of unknown size. Roller bearings need higher values, from about 1.5 to 3, and at least 3 when the shocks are not known; thrust spherical roller bearings at least 4. The guide in the app summarises this as s0 ≥ 1–2 for normal service. ISO 76 also notes that the permissible P0 can be below, equal to or above C0, depending on the demands on smooth running and friction.

How the app computes C0

The app does not use the tables. It applies the definition directly:

1. Hertz point or line contact at the bearing's actual groove radii (or roller geometry) gives the element load Q0 at which the most heavily loaded contact, inner or outer, reaches the stress limit.
2. $C_{0r}=0.2\cdot i\cdot Z\cdot Q_0\cdot \cos \alpha$ for radial bearings, which is the standard's $5\cdot F_r/(Z\cdot \cos \alpha )$ basis turned round; $C_{0a}=Z\cdot Q_0\cdot \sin \alpha$ for thrust bearings.

Steel bearings are rated with the standard's E = 207 000 MPa and ν = 0.3, whatever moduli you enter, as long as every modulus lies within 15 % of 207 000 MPa. Hybrid bearings (ceramic elements) keep their own moduli; their own standard, ISO 20056, was not available. With these constants the method reproduces f0 of ISO 76, Table 1 to half a printed digit and the constants 44 and 220 of Eq. (7) and (11) within 0.5 %, as axm verify shows ("Bearings: ISO 76 static load ratings").

For the bearing-6207 template (grooves 0.52/0.53·Dw) the inner contact governs: Q0 = 8 499 N per ball at 4 200 MPa, so C0 = 0.2 · 9 · 8 499 N = 15 298 N.

How the app computes s0

The app does not form P0 from X0 and Y0. It solves the internal load distribution under the actual loads (with clearance, contact angle change and tilt; see Internal load distribution), takes the largest contact stress pmax, and compares it with the ISO 76 limit σlim:

• point contact (balls): $s_0={({\sigma }_{\lim }/p_{\max })}^3$
• line contact (rollers): $s_0={({\sigma }_{\lim }/p_{\max })}^2$

The exponents follow from Hertz theory: the element load grows with p³ in a point contact and with p² in a line contact. s0 is therefore the factor by which the most heavily loaded element's load could grow before its contact reaches the limit. Under the load distribution that ISO 76 assumes, this equals C0/P0. ISO 76, Clause 1 itself says its equations do not give satisfactory results when misalignment, preload or a very large clearance change the usual load distribution; the stress-based s0 takes these effects into account.

For the template, pmax = 2 130 MPa on the inner ring, so $s_0={(4200/2130)}^3=7.669$. The classical route gives less: P0 = Fr = 2 500 N (Eq. (3) governs here) and C0/P0 = 6.12, because the standard assumes a most loaded ball of 5·Fr/Z = 1 389 N, while the solved distribution, with the axial load turning the balls to about 13°, gives 1 118 N.

Other results from the same bearing:

With a duty cycle the static safety is that of the worst load case. A template duty cycle with a 1.5 × peak case reports s0 = 5.02 for that case, and that value is checked. A bearing that the shaft leaves without load has an unlimited static safety (inf).

The check

Minimum static safety s₀ under Requirements (limits.minStaticSafety) sets the limit; the template uses 2. The check bearings.<id>.minStaticSafety fails when s0 falls below it. To raise s0, lower the peak load, choose a bearing with more or larger elements, use a roller bearing for heavy static loads, or reduce clearance and tilt so that more elements share the load.

Limits of the method

• s0 judges the contacts only. Ring, cage and housing strength and the fit are not part of it.
• ISO 76 excludes designs or conditions in which the contact area is considerably truncated, for example a contact ellipse running over the groove edge under high axial load (Clause 1). The app does not check for this.
• The stress limits assume hardened bearing steel of current quality (Clause 1). The app does not lower σlim or C0 for a ring hardness below 58 HRC; only C is reduced.
• Elements running directly on a shaft or in a housing are covered only when that raceway is equivalent to a bearing ring (Clause 1).

From the command line

axm solve p.axm.json --detail --json

summary.staticSafety, summary.maxContactStress and summary.staticRating (C0) hold the values; detail.maxStress names the element and ring.

Related

• Rating life (ISO 281)
• Hertz contact and subsurface stress
• Internal load distribution (ISO/TS 16281)
• Bearings
• Checks and safeties