Spur Gear Calculator: Load Capacity per DIN 3990 / ISO 6336

Formulas and fundamentals

Gear geometry

The geometry follows the involute chain: from normal module mn, tooth numbers z1/z2 and helix angle β one obtains the transverse module, pitch circles and base circles:

The profile shift sum defines the working pressure angle α_wt via the involute function, which yields the working centre distance:

In the inverse mode the calculator determines the required profile shift sum x1 + x2 from a given centre distance.

Contact ratio and limits

The transverse contact ratio ε_α is the ratio of the length of the path of contact to the base pitch and describes how many tooth pairs are in mesh on average. It must exceed 1; typical spur gear values are 1.1 to 1.9. For helical gears the overlap ratio is added:

The calculator also checks undercut (limit tooth number, minimum profile shift), the pointed tooth limit of the tip thickness and the tip clearance. Tip circles are output without tip shortening; the tip shortening factor k is reported separately.

Root interference (tip corner meshing)

The flank is a usable involute only between the root form circle d_Ff and the tip circle; below that lies the root fillet cut by the tool. The mesh, however, already starts at the root usable circle d_Nf, which is set by the tip circle of the mating gear. If the tip corner of the mating gear reaches below d_Ff, it runs into the root fillet instead of the involute. The condition applies to both gears:

The root form circle follows from the usable tool addendum hN. It is the same quantity that appears in the limit tooth number equation and in the x_min equation (for DIN 867 hN = 0.99997·mn, which is why x_min = 1 − z·sin²α/2 works there):

With the distance e of the form point from the base circle tangency point T, the root form circle and the root usable circle follow, the latter from the path of contact T1T2 and the tip circle of the mating gear:

What is assessed is not the radial difference but the reserve Δ_ξ along the line of action. Near the base circle the radial sensitivity approaches zero: a textbook zero profile shift pair with mn = 2, z = 18/71 has only 17 µm radial reserve but 0.5 mm along the line of action. The limit is Δ_ξ ≥ max(0.02·mn; 0.05 mm), derived from a tolerance stack of centre distance deviation, runout, pitch deviation and scatter of the form circle. If interference occurs, the usable transverse contact ratio ε_α,nutz drops below the geometric value; the geometric value is then too favourable and the load capacity calculation is void, because it assumes undisturbed involute meshing.

Limits of this check: nominal geometry only without tolerances, root side only (tip chamfers and tip relief are not modelled), external gearing only, hobbing with a rack type tool without protuberance and without grinding stock only (a grinding step raises d_Ff and can disturb a design that computes green), tool addendum taken equal to the dedendum of the basic rack. With pronounced undercut the closed form d_Ff equation is not exact; the calculator then falls back to "not verified" instead of claiming a verdict. This has two consequences. First, ε_α,nutz is then only a lower bound, so the design is judged against the geometric contact ratio and no blocking error is derived from the uncertain figure. Second, one bound still holds, because the usable involute can start no lower than the base circle: if even that upper bound of the reserve is negative, the interference is certain and is reported as such, although its magnitude remains open. For helical gears the usual transverse section approximation per ISO 21771 applies.

Tooth root capacity

The tooth root quick check follows DIN 3990-3 / ISO 6336-3. The nominal tooth root stress uses the form factor YFa and the stress correction factor YSa from the 30° tangent procedure of Method B, applied at the tooth tip (Method C variant with contact ratio factor Y_ε):

With the force factors KA, KV, KFβ and KFα the safety factor is:

Flank capacity

The flank quick check per DIN 3990-2 / ISO 6336-2 models the Hertzian contact pressure at the pitch point, with zone factor ZH, elasticity factor ZE (computed from Young's modulus and Poisson's ratio), contact ratio factor Z_ε and helix factor Z_β:

The safety factor is SH = σHlim/σ_H; the softer material governs. The standard switch accounts for the definition difference Z_β = √cos β (DIN 3990) versus 1/√cos β (ISO 6336).

Deliberate simplifications of the quick check: dynamic factor fixed at KV = 1.2 (valid up to about v = 10 m/s), face load factors KHβ/KFβ from a quality grade table, KHα = KFα = 1, single contact factor ZB/D = 1 and endurance limit without life, lubricant and size factors. The material values σHlim/σ_Flim are estimates from published scatter bands, not curve values per ISO 6336-5. For a documented proof per full Method B the standard itself is required.

Worked example

Reference example: a spur gear pair with mn = 3 mm, z1 = 20, z2 = 60 (u = 3), b = 40 mm transmits P = 7.5 kW at n1 = 1450 rpm. This gives T1 = 49.4 Nm, Ft = 1646 N and v = 4.56 m/s. The geometry yields ε_α = 1.67; for gear 1 the factors are YFa = 2.80 and YSa = 1.55.

With KA = 1.25 (moderate shocks), KV = 1.2 and quality 7-8 (KHβ = 1.5 / KFβ = 1.25) and case hardened 16MnCr5 on both gears (σHlim = 1470, σ_Flim = 430 N/mm²) the results are: σ_F = 78 N/mm² and SF = 11.0 at the pinion, σ_H = 598 N/mm² and SH = 2.46 at the flank. As is typical for case hardened gears the flank is the governing criterion while the tooth root has large reserves - both safety factors are well above the required minimum values.

Frequently asked questions

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