Beam Deflection

How far does a beam bend under a point load or an evenly spread load — a floor beam on two supports, or a balcony, diving board or shelf bracket fixed at one end only? Choose the support, solve for the deflection, the load, the span, E or I, and read the shear force, bending moment, deflection and stress at any point along the beam.

StructuresBendingStiffness

Inputs

Input examples

Support

Load case

Click Solve next to the quantity you want to find: it moves into the result below. Then fill in the others. Use the chips to fill in a material (E) or a standard section (I).

Material

Section

Point load F

Span L

Young’s modulus E

Second moment of area I

Maximum deflection δ

Section depth h — for the bending stress

Position x along the beam — from the left support; or click the diagrams

What Determines Beam Deflection?

A beam resists bending through its bending stiffness E·I: E is the stiffness of the material, I describes how the material is distributed in the cross-section. Material far from the centre counts much more — that is why steel beams have an I-shape.

The span has a dramatic effect: deflection grows with L³ for a point load and L⁴ for a uniform load. Doubling the span makes the beam sag 8 to 16 times as much. Floors are usually limited to L/360 so they don’t feel bouncy or crack finishes.

A cantilever is fixed at one end only. The wall has to resist both a vertical force and a bending moment, so the largest moment and stress are at the wall, and the beam is bent the other way (hogging: the top in tension). With nothing holding up the free end it bends far more: a point load at the tip gives 16 times the deflection of the same load at the centre of an equally long beam on two supports. Cantilevers are usually limited to L/180.

I × 2deflection halves

L × 2δ × 8 (point)

cantilever× 16 vs. two supports

L/360, L/180typical limits

Live Diagram

exaggerated

unloaded beam

deflected shape (exaggerated)

load

position x

Shear, Moment & Deflection

click to move x

Result

Enter the values above

Standard Cases

Support & load	δ_max	M_max

Second Moment of Area

Rectangle

Height counts cubed: a plank on edge is far stiffer than the same plank lying flat.

Example:

Solid and Hollow Circle

A tube puts its material far from the centre, so it is nearly as stiff as a solid bar at a fraction of the weight.

Example:

Serviceability Check

A common deflection limit for floors carrying finishes. Roofs often allow L/250, cantilevers about L/180.

Example: a 4 m floor beam may sag at most 11.1 mm.

Sources & Verification

References

R. C. Hibbeler, Mechanics of Materials in SI Units, 10th ed., Pearson, 2018 — Ch. 6 (Bending), Ch. 12 (Deflection of Beams and Shafts) and App. C (Slopes and Deflections of Beams).
J. M. Gere & B. J. Goodno, Mechanics of Materials, 9th ed., Cengage Learning, 2018 — Ch. 4 (Shear Forces and Bending Moments) and Ch. 9 (Deflections of Beams).