Gear Contact Stress (Hertz)
Calculate the Hertzian contact stress on gear tooth surfaces, σ_H = C_p·√(F_t ÷ (b·d·I)), from the elastic coefficient C_p (√MPa, a function of the pair's elastic moduli), the tangential force F_t (N), the face width b (mm), the pinion pitch diameter d (mm) and the geometry factor I (dimensionless). This is the basis of SURFACE FATIGUE (pitting) design — the second fundamental gear failure mode, distinct from bending breakage. When two teeth touch, the contact is practically a LINE, and even moderate loads create very high contact stresses (hundreds of MPa) in the tiny contact area, per Hertz theory. Under repeated cycles, these stresses cause sub-surface fatigue that flakes off small bits of material, forming craters (pitting) that progress, destroy the tooth profile, generate noise and vibration and lead to failure. The elastic coefficient C_p gathers the materials' elastic properties (steel-steel, steel-bronze, etc.), and the geometry factor I, the curvature and contact ratio. Contact stress is compared with the material's pitting resistance (which depends strongly on surface HARDNESS — so gears are often case-hardened). It is one of the two central AGMA criteria. Enter the elastic coefficient, tangential force, face width, diameter and geometry factor.
Result
—
Gear contact stress (Hertz)
The contact stress (Hertzian) on the tooth surface is σ_H = C_p·√(F_t ÷ (b·d·I)), computed from the elastic coefficient C_p (a function of the elastic moduli of the mating materials), the tangential force F_t, the face width b, the pinion pitch diameter d and the contact geometry factor I. This is the basis of design against surface fatigue (pitting) — the second fundamental gear failure mode, distinct from breakage by bending. When two teeth touch, the contact is practically a line, and even moderate loads generate very high contact stresses (hundreds of MPa) over that tiny contact area, according to Hertz theory. Under repeated cycles, those stresses drive sub-surface fatigue that tears small flakes of material away, forming pits (pitting) that spread, destroy the tooth profile, generate noise and vibration and lead to failure. The elastic coefficient C_p gathers the elastic properties of the materials (steel-steel, steel-bronze and so on), and the geometry factor I captures curvature and contact ratio. Contact stress is compared with the pitting resistance of the material, which depends heavily on surface hardness — which is why gears are so often case-carburized and surface-hardened, hard on the outside and tough on the inside. It is one of the two central AGMA criteria. Enter the elastic coefficient, the tangential force, the face width, the diameter and the geometry factor.
Related Tools
Gear Tooth Bending Stress (Lewis)
Calculate the bending stress at a gear tooth root by the Lewis equation, σ = F_t ÷ (b·m·Y), from the tangential force F_t (N), the tooth face width b (mm), the module m (mm) and the Lewis form factor Y (dimensionless, a function of tooth count). Lewis's 1892 equation was the first rational treatment of gear-tooth strength and is still the basis of BENDING design. It models the tooth as a cantilever beam fixed at the root: the tangential force transmitted between teeth (from the torque) creates a bending moment that tends to break the tooth at the root — the catastrophic failure where a tooth cracks and snaps. The form factor Y accounts for tooth geometry (gears with more teeth are 'fatter' at the root and stronger, higher Y). The computed stress is compared with the material's bending fatigue strength (with safety factors), since gears endure millions of cycles. The basic Lewis formula is then refined by the AGMA standard with stress-concentration, dynamic, load-distribution and surface-condition factors. It is one of the two fundamental gear design criteria (the other is contact stress). Enter the tangential force, face width, module and Lewis form factor.
Young Equation
Compute γSL = γSV − γLV·cos(θ) from Young's equation.
Gear Dynamic Load
Calculate the effective dynamic load on gear teeth, F_d = F_t·K_v, from the nominal tangential force F_t (N) and the dynamic factor K_v. The dynamic load is the REAL tangential force the teeth bear in operation, larger than the nominal force (simply torque over radius) because of the dynamic effects of meshing at speed. These effects — vibrations, contact impacts, tooth deflections under load and manufacturing errors — make the instantaneous tooth load fluctuate and peak above the average, especially at high pitch-line velocities. The dynamic factor K_v (from Barth or other formulas) quantifies this amplification. The dynamic load is then used in strength checks: in Lewis bending stress (risk of tooth breakage at the root) and Hertzian contact stress (risk of surface fatigue and pitting). Using the nominal load without amplifying by the dynamic factor would underestimate the demands and lead to undersized gears that fail prematurely by fatigue. It is an essential, classic step in gear design. Enter the nominal tangential force and the dynamic factor.
Gear Base Pitch
Calculate the base pitch of an involute gear, p_b = π·m·cos(φ), from the module m (mm) and the pressure angle φ (degrees). The base pitch is the distance between two homologous flanks of consecutive teeth, measured along the base circle (or, equivalently, along the line of action) — different from the circular pitch (π·m), measured on the pitch circle. The base pitch is a FUNDAMENTAL property of involute meshing for an elegant reason: for two meshes to transmit motion correctly, they must have the SAME base pitch — it is the conjugacy condition of involute profiles. Moreover, the base pitch appears directly in the CONTACT RATIO (the average number of teeth in simultaneous contact, found by dividing the line-of-action length by the base pitch): a contact ratio above 1 (ideally above 1.4) ensures there is always at least one tooth pair meshed, transmitting motion continuously and smoothly, without impacts. The base pitch is also the basis of checking gears 'over two pins' or by span measurement (W over teeth), classic dimensional-control methods. It is an essential parameter in gear geometry and metrology. Enter the module and the pressure angle.
Gear Tooth Bending Safety Factor
Calculate the bending safety factor of a gear tooth, FS = σ_perm ÷ σ, from the material's allowable (permissible) bending stress σ_perm (MPa, the bending fatigue strength with its factors) and the acting bending stress σ (MPa, from Lewis/AGMA from the load). It is the final bending-design check: the material strength must exceed the demand with adequate margin. Gears run for millions to billions of cycles, so the allowable stress is the material's bending FATIGUE strength (at the design-life cycle count), adjusted by reliability, temperature and life factors. Required safety factors depend on criticality and uncertainty (typically 1.5-3 for bending). If FS is insufficient, the module is increased (bigger, stronger teeth), the face width, or a better material/heat treatment used. A tooth breaking by bending fatigue is a CATASTROPHIC, sudden failure (unlike pitting, which gives progressive signs), since the broken tooth comes loose and can damage the whole drive — so bending is designed with generous margins. With the pitting (contact) safety factor, it defines the gear's robustness. Enter the allowable stress and the acting stress.
Weber Number
Compute the Weber number We = (ρ·v²·L)/σ.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.