Beam Stress & Deflection Calculator
CE Style
Inputs
Steel ≈ 200 GPa, Aluminum ≈ 70 GPa
Section Properties
Beam Results
Assumes prismatic beam, small deflection linear theory, and ideal supports. For safety-critical work, include load factors, stress concentrations, shear deformation if deep, and check code requirements.
How to Use the Beam Stress & Deflection Calculator
Quickly size simple beams. Pick a loading case, enter section geometry and material stiffness, then read bending stress and tip/midspan deflection.
1) Choose Units & Loading Case
- Units: SI (mm, N, GPa) or Imperial (in, lbf, Mpsi).
- Case (supported in the tool):Simply Supported + Mid-Point Load,Simply Supported + UDL,Cantilever + End Load,Cantilever + UDL.
2) Select Section & Enter Geometry
- Rectangle: width b, height h.
- Round: diameter d.
- Hollow Round: outer Do and thickness t (the tool computes Di).
The tool computes area
A, second moment I, and distance to the extreme fiber c for the chosen section.3) Enter Material & Loads
- E (Elastic Modulus): steel ≈ 200 GPa, aluminum ≈ 70 GPa (or Imperial equivalents).
- Point Load P or UDL w depending on the case.
- Length L is the clear span (support-to-support or fixed-to-free).
4) Review Section Properties
- A (area) for weight and bearing checks.
- I (second moment of area) and c (neutral axis → extreme fiber) used for stress and deflection.
Formulas: rectangle
I = b·h³/12, round I = π·d⁴/64, tube I = π(Dₒ⁴ − Dᵢ⁴)/64.5) Read the Results
- Mmax, Vmax and location (midspan or fixed end).
- σmax (bending) using
σ = M·c/I. - δmax (deflection) from classic closed-form solutions:SS + mid-load:
δ = P·L³ / (48·E·I)
SS + UDL:δ = 5·w·L⁴ / (384·E·I)
Cantilever + end-load:δ = P·L³ / (3·E·I)
Cantilever + UDL:δ = w·L⁴ / (8·E·I)
Outputs show stress in MPa and deflection in mm for SI (or psi and in for Imperial).
Quick Checklist
- Correct loading case selected
- Span L and loads P/w are realistic
- Material modulus E matches your alloy/temper
- Section dimensions oriented correctly (height drives stiffness)
- Check σmax ≤ allowable (yield/FS)
- Check δmax ≤ service limit (e.g., L/250)
- Add safety factors; consider shear deformation for deep beams
- Real supports/fixity may differ from ideal cases
FAQ & Tips
My measured deflection is higher.
Actual boundary conditions are rarely perfectly fixed or pinned; connections add flexibility. Verify E and I, and include self-weight if significant.
Which way should a rectangle face?
Put the larger dimension as the height (in bending direction) to maximize I and reduce deflection.
Is shear included?
These formulas use Euler–Bernoulli (small deflection, negligible shear). For short/deep beams, Timoshenko shear effects may be non-negligible.
Allowable deflection?
Common service limits are L/360 to L/240 for floors; choose per your application or code.
Copy-Paste Mini Workflow
1) Select units and loading case
2) Pick section and enter dimensions
3) Enter E, span L, and P or w
4) Read I, c, then σ_max and δ_max
5) Compare against allowable stress and deflection limits; iterate