Shaft Bending Fundamentals
Shaft deflection is critical in rotating machinery design. Excessive deflection causes bearing misalignment, seal wear, vibration, and premature failure. The deflection is governed by the Euler-Bernoulli beam equation:
EI * d^2y/dx^2 = M(x)
- E = Elastic modulus (steel: 200 GPa, 29 Mpsi)
- I = Second moment of area = pi*d^4/64 for solid shaft
- y = Deflection at position x
- M(x) = Bending moment at position x
Shaft stiffness is proportional to diameter to the fourth power - doubling diameter increases stiffness by 16x.
Support Conditions
The boundary conditions at bearings determine the shaft's deflection behavior:
- Simply Supported: Both bearings act as pinned supports allowing free rotation but no vertical displacement. Most common configuration for between-bearing loads.
- Cantilever (Fixed-Free): One end rigidly fixed (typically at gearbox or motor), other end free. Used for short stub shafts.
- Overhung Load: Load applied beyond the outboard bearing. Common for pump impellers, fans, and pulleys. Critical case for deflection.
Deflection Limits
Industry standards specify maximum allowable deflections:
- General machinery: L/3000 to L/5000
- Gear shafts: 0.005" per inch of gear face width
- Pump shafts: 0.002" to 0.004" at mechanical seal
- At bearings: Slope < 0.001 rad (ball), < 0.0005 rad (roller)
Superposition Principle
For shafts with multiple loads, the total deflection is the sum of individual load effects:
delta_total = delta_1 + delta_2 + delta_3 + ...
This is valid for linear elastic behavior with small deflections. Each load's contribution is calculated independently using beam formulas, then summed algebraically.
- Downward forces create positive deflection
- Upward forces create negative deflection
- Critical deflection may occur at load locations or between them
Critical Speed
Shafts have natural frequencies that cause resonance when operating speed matches. The first critical speed (Rayleigh approximation):
Nc = (60/2*pi) * sqrt(g / delta_static)
Operating speed should be:
- Stiff shaft: < 0.5 Nc (subcritical)
- Standard: < 0.7 Nc or > 1.4 Nc
- Avoid: 0.7 Nc to 1.4 Nc (resonance zone)
Common Formulas
- SS, center load: delta = PL^3/(48EI)
- SS, load at 'a': delta = Pa^2*b^2/(3EI*L)
- Cantilever, end load: delta = PL^3/(3EI)
- Overhung, load at 'a': delta = Pa^2(L+a)/(3EI)
Quick Select - Common Shaft Configurations
Shaft Deflection Calculator
Calculate deflection for shafts with multiple loads using superposition. Includes critical speed analysis and deflection limit checking.
Shaft Diagram
Deflection Results
Deflection Limit Checker
Design Guidelines
| Application | Deflection Limit |
|---|---|
| General machinery | L/3000 to L/5000 |
| Gear shafts | 0.005" per inch face width |
| Pump shafts (at seal) | 0.002" to 0.004" |
| Ball bearing slope | < 0.001 rad (0.057 deg) |
| Roller bearing slope | < 0.0005 rad (0.029 deg) |
| Critical speed margin | < 0.7Nc or > 1.4Nc |
Reference Formulas
Simply Supported - Load at distance 'a':
delta = P*a*b*(L^2 - a^2 - b^2) / (6*E*I*L)
Cantilever - End load:
delta = P*L^3 / (3*E*I)
Overhung Load:
delta = P*a^2*(L+a) / (3*E*I)
Critical Speed (Rayleigh):
Nc = 946 / sqrt(delta_mm) RPM