Beam Deflection Formula, Limits, and Calculators
Beam deflection is the distance a beam moves under load. The formulas below are the usual maximum-deflection cases for a prismatic beam in bending. Each one is also a worksheet you can open and change.
Beam deflection formulas
- Cantilever, end point load P:
δ = P L³ / (3 E I). Open the calculator. - Cantilever, uniform load w:
δ = w L⁴ / (8 E I). Open the calculator. - Simply supported, central point load P:
δ = P L³ / (48 E I). Open the calculator. - Simply supported, uniform load w:
δ = 5 w L⁴ / (384 E I). Open the calculator. - Fixed-fixed, uniform load w:
δ = w L⁴ / (384 E I). Open the calculator.
E is Young's modulus, L is the span, and
w is load per unit length. A total load written as
W is not the same symbol: divide by the span before using a
w formula.
What is I in beam deflection?
I is the second moment of area of the cross-section, about
the axis of bending. For a rectangle of width b and depth
h, I = b h³ / 12. Depth is cubed, so a small
change in section depth moves the deflection a lot. Keep I in
m⁴ when E is in pascals and L is in metres.
Mixing mm⁴ with GPa and metres is the usual unit error.
L/360, L/240, and L/180
A deflection limit is a fraction of the span. L/360 means the
allowed movement is the span divided by 360. L/240 allows more
movement than L/360; L/180 allows more again;
L/500 allows less. Floors are often checked near
L/360 for live load and L/240 for total load.
Roofs are often looser. The number is a serviceability criterion from the
code or the project brief, not a strength check.
Compare the calculated δ with
δallow = L / n. A
beam deflection check
keeps that comparison on the same sheet as the formula.
Worked cantilever
Steel cantilever, 50 mm wide by 100 mm deep, 2 m long, 5 kN at the tip.
I = 0.05 × 0.1³ / 12 ≈ 4.17×10⁻⁶ m⁴E = 200 GPaδ = 5 kN × (2 m)³ / (3 × 200 GPa × I) ≈ 16 mm
L/250 = 8 mm, so this section is too flexible for that limit
even though the stress may still be acceptable. Increase the depth before
chasing the load.
Timber beams
The same deflection formulas apply. What changes is E for the
timber grade and the span limit the code asks for. Bending capacity is a
separate check:
timber beam allowable bending moment.
Open any of the calculators above in Calculeaf to carry the units through the derivation instead of reconciling them by hand. More sheets are in the example library.