What is the Lewis bending stress formula for gears?

The Lewis bending stress formula is sigma = Wt / (b x m x Y), where Wt is the tangential force on the gear tooth, b is the face width, m is the module, and Y is the Lewis form factor that depends on the number of teeth and pressure angle.

How does AGMA gear stress analysis differ from Lewis equation?

AGMA analysis extends the Lewis equation by including correction factors: Ko (overload), Kv (dynamic), Ks (size), Km (load distribution), and KB (rim thickness). This gives more accurate real-world stress predictions: sigma = Wt x Ko x Kv x Ks x Km x KB / (b x m x J), where J is the AGMA geometry factor.

What is a safe factor of safety for gear design?

For gear design, a safety factor (FOS) of 1.5 to 2.0 is typical for bending stress and 1.0 to 1.5 for contact stress. Critical applications like aerospace may require FOS above 2.5. The FOS is calculated as allowable stress divided by actual calculated stress.

Gear Strength Simulator

Lewis & AGMA Bending • Contact Stress • Safety Factor — Spur • Helical — Simulate • Explore • Practice • Quiz

Mode

Wt0 N

Bending0 MPa

Contact0 MPa

FOS Bend0

FOS Contact0

Gear Type

Module (mm) 2

Pinion Teeth 20

Gear Teeth 40

Face Width (mm) 30

RPM (pinion) 1500

Power (kW) 5

Material

Tangential Force

0 N

Bending Stress

0 MPa

Contact Stress

0 MPa

FOS Bending

FOS Contact

Pitch Line Vel

0 m/s

Gear Ratio

0 :1

Pitch Dia (P)

0 mm

Gear Strength Analysis — Lewis Bending and AGMA Contact Stress

Gear strength analysis is a critical step in mechanical design that ensures gear teeth can withstand the forces transmitted during power transmission. The two primary failure modes are tooth bending fatigue and surface contact fatigue (pitting). Engineers evaluate both using standardised methods to determine whether a gear pair will operate safely under its intended load and speed conditions. This simulator lets you compute bending stress, contact stress, and safety factors for spur and helical gear pairs in real time.

The Lewis bending stress equation, developed by Wilfred Lewis in 1892, treats the gear tooth as a cantilever beam: σ = W_t / (b × m × Y), where W_t is the tangential load, b is the face width, m is the module, and Y is the Lewis form factor. The form factor depends on the number of teeth and the pressure angle, and it accounts for the tooth geometry at the weakest cross-section. While the Lewis equation provides a useful baseline, it does not account for dynamic loads, load distribution across the face width, or manufacturing tolerances.

AGMA Stress Analysis Method

The American Gear Manufacturers Association (AGMA) extended the Lewis approach by introducing correction factors for real-world operating conditions. The AGMA bending stress equation is σ = W_t × K_o × K_v × K_s × K_m × K_B / (b × m × J), where K_o is the overload factor (accounts for shock loads), K_v is the dynamic factor (accounts for tooth-to-tooth speed variations), K_s is the size factor, K_m is the load distribution factor, K_B is the rim thickness factor, and J is the AGMA geometry factor. For contact stress, the AGMA equation is σ_c = C_p × √(W_t × K_o × K_v × K_s × K_m × C_f / (b × d × I)), where C_p is the elastic coefficient and I is the geometry factor for pitting resistance.

Safety Factors in Gear Design

The factor of safety (FOS) is the ratio of allowable stress to actual calculated stress. For bending: FOS = σ_allowable / σ_actual. Material selection is crucial — hardened steel gears can withstand bending stresses of 250–400 MPa and contact stresses of 1000–1500 MPa, while cast iron and bronze have significantly lower limits. A minimum FOS of 1.5 for bending and 1.0 for contact is typical for industrial applications. The gear module, face width, and number of teeth are the primary design variables that engineers adjust to achieve adequate safety margins.

Who Uses This Simulator?

This gear strength simulator is designed for mechanical engineering students studying machine design, power transmission trainees, workshop instructors teaching gear analysis, and practising engineers performing preliminary gear sizing. It provides instant visual feedback on stress distribution and safety factors without requiring expensive FEA software or manual calculations.

Explore Related Simulators

If you found this gear strength simulator helpful, explore our Gear Train Calculator, Power Screw Calculator, Shaft Torsion Simulator, and Bearing Selection Tool for more hands-on practice.