Beam Deflection Formulas and a Worked Cantilever Example
Beam deflection is one of the most-run checks in mechanical and structural work, and one of the easiest to get wrong on units. Here are the standard formulas and a worked cantilever example you can reproduce.
Standard maximum-deflection formulas
- Cantilever, end point load P:
δ = P·L³ / (3·E·I) - Cantilever, UDL w:
δ = w·L⁴ / (8·E·I) - Simply supported, central point load P:
δ = P·L³ / (48·E·I) - Simply supported, UDL w:
δ = 5·w·L⁴ / (384·E·I)
where E is Young's modulus, I the second moment
of area, and L the span.
Worked cantilever example
Steel cantilever, rectangular section 50 mm wide × 100 mm deep, length 2 m, end load 5 kN.
I := b·h³ / 12 = 0.05·0.1³/12 ≈ 4.17×10⁻₆ m⁴E := 200 GPaδ := P·L³ / (3·E·I) = (5\,kN · (2\,m)³) / (3 · 200\,GPa · 4.17×10⁻₆\,m⁴) ≈ 16 mm
Then check against the allowable, e.g. L/250 = 8 mm—here
the section is too flexible and must be revised.
The unit traps
- I in mm⁴ with E in GPa and L in m—the classic mismatch. Unit-aware math removes it (details).
- UDL as total load vs. load per length—different formulas, same symbol.
- Deflection limit relative to span—document the criterion, do not hard-code a number.
Keep it reviewable
Because the section property feeds the deflection feeds the acceptance check, this is a chained derivation best kept on one sheet with the assumptions visible (best practices), not scattered across spreadsheet cells (why).
Open Calculeaf and build this beam deflection check with units carried through every step.