BIZOMAT 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). |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Shaft diameter
- What it is
- The diameter of the shaft between the bearings, treated as uniform along the span.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 2000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 70 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Deflection and slope fall with the fourth power of diameter; stress with the cube.
- Get it right
- If the shaft is stepped, using the smallest diameter gives a conservative answer.
What this means: Bearing span
- What it is
- The distance between the centres of the two bearings that support the shaft, treated as simple supports.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 20000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 600 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Measure between bearing centres, not between shaft shoulders or ends.
What this means: Elastic modulus
- What it is
- The elastic (Young’s) modulus of the shaft material, from the material data sheet or a materials reference.
- Units and format
- Enter a number in GPa: gigapascals (1 GPa = 1000 MPa). The form accepts 0.000001 GPa to 400 GPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 206 GPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Deflection and slope are inversely proportional to it.
- Get it right
- Enter GPa, not MPa: the tool multiplies by 1000.
Loads (0 = none; positions from bearing A)
What this means: Load 1
- What it is
- The first transverse point load on the shaft, such as a gear separating force or belt pull, acting at right angles to the shaft in the plane checked.
- Units and format
- Enter a number in N: newtons. The form accepts -10000000 N to 10000000 N. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 5000 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Its share of deflection and slope is proportional to it; the shares of all loads add.
- Get it right
- Loads in different planes cannot be added here; check each plane separately. A negative value acts in the opposite direction.
What this means: Load 1 position
- What it is
- Where load 1 acts, measured along the shaft from the centre of bearing A.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 20000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 200 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- It must lie strictly between the bearings, greater than 0 and less than the span.
What this means: Load 2
- What it is
- The second transverse point load on the shaft, in the same plane as load 1.
- Units and format
- Enter a number in N: newtons. The form accepts -10000000 N to 10000000 N. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 3000 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Its share of deflection and slope is proportional to it; the shares of all loads add.
- Get it right
- Enter 0 if there is no second load; a negative value acts in the opposite direction to a positive one.
What this means: Load 2 position
- What it is
- Where load 2 acts, measured along the shaft from the centre of bearing A.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 20000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 400 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- It must lie between the bearings; it is ignored when load 2 is 0.
What this means: Load 3
- What it is
- An optional third transverse point load on the shaft, in the same plane as the other loads.
- Units and format
- Enter a number in N: newtons. The form accepts -10000000 N to 10000000 N. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 N as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter 0 if there is no third load; a negative value acts in the opposite direction.
What this means: Load 3 position
- What it is
- Where load 3 acts, measured along the shaft from the centre of bearing A.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 20000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 300 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- It must lie between the bearings; it is ignored when load 3 is 0.
Your limits
What this means: Largest deflection at a load (gear mesh)
- What it is
- The largest deflection you accept at a load point, often set by gear mesh quality. Deflection at each load is checked against it.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.13 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Take it from your gear design basis or the gear supplier; it applies at the loads, not mid-span.
What this means: Largest slope at a bearing
- What it is
- The largest shaft slope, the angle of tilt, you accept at a bearing, from the bearing maker’s misalignment limit for the bearing type.
- Units and format
- Enter a number in rad: rad. The form accepts 0.000001 rad to 0.1 rad. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0.001 rad as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Enter radians, not degrees or milliradians: 1 mrad = 0.001 rad.
What this means: Allowable bending stress
- What it is
- The bending stress you will accept in the shaft, from your design basis with fatigue and stress concentrations allowed for.
- Units and format
- Enter a number in MPa: megapascals (1 MPa = 1 N/mm²). The form accepts 0.000001 MPa to 2000 MPa. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 80 MPa as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- The tool checks nominal stress at the loads only; keyways and shoulders need their own fatigue check.
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