KEVOS® MAKEOS · Strength
Shaft Deflection and Bearing Slope (Up to Three Loads)
How far does this shaft bend under its gear and pulley loads, how much do the ends tilt in the bearings, and is the bending stress acceptable?
Stiffness check of a simply supported round shaft carrying up to three transverse point loads in one plane: bearing reactions, bending moment and stress at the loads, deflection at each load and the largest deflection along the span by superposition, and the slope at each bearing, against your limits for gear mesh deflection, bearing misalignment and stress (for one load, the reactions, moment and largest deflection match the beam cases tool).
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Shaft diameter | mm | Diameter of the shaft (uniform). |
| Bearing span | mm | Distance between the bearing centres. |
| Elastic modulus | GPa | Elastic modulus of the shaft. |
| Loads (0 = none; positions from bearing A) | ||
| Load 1 | N | First load (gear force, belt pull) at right angles to the shaft. |
| Load 1 position | mm | Its distance from bearing A. |
| Load 2 | N | Second load (0 if none); use a negative value for a load in the opposite direction. |
| Load 2 position | mm | Its distance from bearing A. |
| Load 3 | N | Third load (0 if none). |
| Load 3 position | mm | Its distance from bearing A. |
| Your limits | ||
| Largest deflection at a load (gear mesh) | mm | Largest deflection you accept at a gear (a common guide for spur gears is about 0.13 mm; your value). |
| Largest slope at a bearing | rad | Largest slope you accept at the bearings (from the bearing maker for the bearing type). |
| Allowable bending stress | MPa | Allowable bending stress (your design value, allowing for fatigue). |
Method and formulas
- Reactions R_A = Σ P (L − a) ÷ L, R_B = Σ P a ÷ L; bending stress σ = 32 M ÷ (π d³).
- Deflection at x from each load: for x ≤ a, y = P b x (L² − b² − x²) ÷ (6EIL), b = L − a; for x > a, y = P a (L − x)(2Lx − a² − x²) ÷ (6EIL); summed.
- Slopes θ_A = Σ P b (L² − b²) ÷ (6EIL), θ_B = Σ P a (L² − a²) ÷ (6EIL).
- Example: a 70 mm shaft deflects 0.11167 mm at load 1 and tilts 0.67731 mrad at bearing A; the stress is 25.737 MPa.
Limitations
- Uniform shaft, simple supports, loads in one plane; combine two planes as vectors.
- Bearing and housing flexibility add to the slope and deflection.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check a gear shaft for mesh deflection
- Check bearing tilt against the bearing’s misalignment limit
- Choose a shaft diameter for stiffness
Who it is for
- Design engineers
- Maintenance engineers
- Power transmission engineers
What you get
- Reactions and bending stress
- Deflection at the loads and the largest deflection
- Slope at each bearing
Interpreting the result
- Stiffness rises with the fourth power of diameter: a small diameter increase cuts deflection a lot.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 257–277, 295–303. Topic reference; no source tables or recommended settings are reproduced.
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Open the Shaft Deflection and Bearing Slope (Up to Three Loads) calculator
