How to use this calculator
- Pick the load case. Choose simply supported with a central load, or a cantilever with an end load.
- Enter the diameter. Enter the round shaft diameter d. The area moment of inertia is I = π·d⁴/64.
- Enter the span and load. Enter the span (or free length) L and the point load F.
- Enter Young's modulus. Enter E — about 200 GPa for steel, 69 GPa for aluminium.
- Read the results. Read the maximum deflection, the maximum bending stress and the area moment of inertia.
How it works
The shaft is treated as a slender beam in bending. The stiffness of the round solid
section is its area moment of inertia:
I = π · d⁴ / 64
Because I scales with the fourth power of diameter, a small change in
diameter has an outsized effect on both deflection and stress.
For a simply supported shaft with a central point load the maximum
deflection (at mid-span) is δ = F·L³ / (48·E·I) and the maximum bending
moment is M = F·L/4. For a cantilever with the load at
the free end the deflection is δ = F·L³ / (3·E·I) and
M = F·L — far more flexible for the same length.
The maximum bending stress is at the outer fibre:
σ = M·c / I, with c = d/2. Compare it with the material's
allowable stress, and separately check the deflection against a stiffness limit.
Worked example
Verified against the live calculator
Take a 25 mm steel shaft (E = 200 GPa) on a 500 mm span, simply supported,
with a 1,000 N central load. The section stiffness is
I = π × 25⁴ / 64 ≈ 19,174.76 mm⁴. The deflection is
δ = 1000 × 500³ / (48 × 200000 × 19174.76) ≈ 0.6791 mm. The bending moment
is M = F·L/4 = 125,000 N·mm, so the bending stress is
σ = 125000 × 12.5 / 19174.76 ≈ 81.49 MPa. Those are the numbers the
calculator shows for these inputs.
Switch the same shaft to a cantilever with the 1,000 N at the free end
and it becomes much more flexible: δ = F·L³ / (3·E·I) ≈ 10.87 mm with
M = F·L = 500,000 N·mm and σ ≈ 325.96 MPa.
Frequently asked questions
How do I calculate the deflection of a shaft?
Model the shaft as a beam. For a simply supported shaft with a central point load the maximum deflection is δ = F·L³ / (48·E·I); for a cantilever with an end load it is δ = F·L³ / (3·E·I). Here F is the load, L the span (or free length), E Young’s modulus and I = π·d⁴/64 the area moment of inertia of the round solid section. For a 25 mm steel shaft on a 500 mm span carrying 1,000 N, δ ≈ 0.68 mm.
What is the area moment of inertia of a round shaft?
For a solid round shaft of diameter d the area (second) moment of inertia about a diameter is I = π·d⁴/64. Because it scales with the fourth power of diameter, a small increase in diameter sharply reduces deflection and bending stress — doubling the diameter cuts deflection to one-sixteenth.
How do I find the bending stress in a shaft?
The maximum bending stress is σ = M·c/I, where M is the maximum bending moment, c = d/2 is the distance to the outer fibre and I = π·d⁴/64. The moment is M = F·L/4 for a simply supported central load and M = F·L for a cantilever end load. Compare σ with the material’s allowable stress (yield divided by your safety factor).
Should I size a shaft for stress or for deflection?
Both. Check that the bending stress σ stays below the allowable stress, and separately that the deflection δ stays under a serviceability limit (for shafts a common rule of thumb is about 0.0008–0.0012 mm per mm of span, and far tighter near gears or bearings). Stiffness, not strength, often governs shaft sizing.
What are the limits of this calculation?
It uses elastic, small-deflection beam theory (Euler–Bernoulli) for a single point load on a prismatic, constant-section round shaft, and ignores self-weight, shear deflection and the effect of bearings. Real shafts with steps, keyways, fillets and press-fits behave differently — treat the result as a first-order estimate, not a final design.
Does this work in metric and imperial?
Yes — enter the diameter, span and load in mm/N or in/lbf and Young’s modulus in GPa or Mpsi. Deflection is shown in mm or in, bending stress in MPa or ksi, and the moment of inertia in cm⁴ or in⁴. Toggle SI/Imperial in the header.
Method & assumptions
- Elastic, small-deflection beam theory (Euler–Bernoulli) — valid while the shaft stays well below yield and deflections are small relative to the span.
- A single point load on a prismatic, constant-section round solid shaft; self-weight, shear deflection and distributed loads are ignored.
- Idealised supports — a perfect pin/roller pair (simply supported) or a perfectly rigid fixed end (cantilever); real bearings add their own flexibility.
- Stress concentrations at steps, keyways, fillets and press-fits are not included — apply a stress-concentration factor for fatigue.
- Check the result against both an allowable stress and a deflection limit; for shafts, stiffness often governs over strength.