How It Works

Combined Bending and Torsion Stress

Rotating shafts experience complex loading from multiple sources simultaneously:

Bending Stress - From belt/gear forces, mounted weights, or cantilever loads. Creates tension on one side and compression on the other.
Torsional Shear Stress - From transmitted torque (power = torque x speed). Creates shear stress that varies from zero at center to maximum at surface.
Axial Stress - From thrust loads in helical gears, propellers, or axial forces.

The critical location is always at the shaft surface, where both bending and torsional stresses are maximum.

Von Mises vs Tresca Yield Criteria

Von Mises (Distortion Energy) - Most accurate for ductile metals. Predicts yielding when equivalent stress reaches yield strength. Formula: sqrt(sigma^2 + 3*tau^2)

Tresca (Maximum Shear) - More conservative (15% lower allowable). Predicts yielding when max shear stress reaches Sy/2. Used in some pressure vessel codes.

ASME Stress Concentration Factors

Real shafts have keyways, shoulders, and grooves that concentrate stress. ASME B106.1M recommends:

Kb = 1.5 for keyways (bending)
Kb = 2.0 for sharp shoulders
Kt = 1.0-1.25 for torsion (less sensitive)

Design Guidelines for Shaft Sizing

Starting Point

Use d = (32*M / (pi*Sy/SF))^(1/3) for bending-only, then add 10-20% for torsion.

Deflection Check

For gears: slope < 0.001 rad at mesh. For bearings: slope < 0.0005 rad at bearing.

Critical Speed

Operating speed should be < 70% or > 130% of first critical speed.

Quick Select - Common Applications

Shaft Stress Calculator

Calculate combined bending and torsion stresses in rotating shafts. Includes von Mises equivalent stress and ASME shaft design criteria.

Unit System

Material

Shaft Type

Outer Diameter (D)

Bending Moment (M)

N-m

Torque (T)

N-m

Axial Force (F) - Optional

Material Yield Strength (Sy)

MPa

ASME Stress Concentration Factors

Kb (bending)

Kt (torsion)

✓

Design Acceptable

Safety Factor > 2.5

Stress Results

-- MPa

Von Mises Equivalent Stress

Stress vs Allowable

Actual:

Allowable:

Safety Factor (yield) --

Bending Stress (sigma_b) --

Torsional Shear Stress (tau) --

Axial Stress (sigma_a) --

Max Shear Stress (Tresca) --

ASME Equivalent Stress --

ASME Safety Factor --

Section Modulus (Z) --

Polar Moment (J) --

Cross-sectional Area --

Reference Formulas

Bending Stress:

sigma_b = 32M / (pi * D^3) [solid]

Torsional Shear:

tau = 16T / (pi * D^3) [solid]

Von Mises:

sigma_vm = sqrt(sigma^2 + 3*tau^2)

ASME (B106.1M):

sigma_eq = sqrt((Kb*sigma_b)^2 + 3*(Kt*tau)^2)

Design Guidelines

Application	Min Safety Factor
General machinery	2.0 - 3.0
Known loads, mild shock	1.5 - 2.0
Heavy shock/impact	3.0 - 4.0
Fatigue loading	2.5 - 4.0
Human safety critical	4.0+