Beam Deflection: How to Calculate How Much a Beam Bends Under Load
Every loaded beam sags. For a simply supported beam the maximum deflection is PL³ ÷ 48EI under a central point load and 5wL⁴ ÷ 384EI under a uniformly distributed load, where E is the material's elastic modulus and I the section's moment of inertia. A 4 m steel beam with I = 8,000 cm⁴ deflects under 1 mm under a 10 kN center load, far inside the L/360 limit of 11.1 mm. This guide covers the formulas, two worked examples and how to judge whether the result is acceptable.
Why Deflection Matters
Deflection is the distance a beam moves from its unloaded line, measured where it moves most: at midspan for a symmetric load on a simply supported beam. A beam can have a large margin against breaking and still deflect enough to crack plaster, jam doors or make a floor feel springy. That is the difference between the ultimate limit state (will it break?) and the serviceability limit state (will it work and feel solid?). Codes cap deflection for exactly this reason, so many beams are sized by stiffness rather than strength.
The Variables
Load. A point load P in newtons, or a uniformly distributed load w in newtons per meter of span (not the total load).
Span L, in meters, between supports. It is the dominant factor: deflection grows with the cube of span for a point load and the fourth power for a distributed load, so doubling the span multiplies deflection by 8 or 16.
Elastic modulus E, the material's stiffness. Structural steel is about 200 GPa and aluminum about 69 GPa, so steel is nearly three times as stiff; timber is lower still.
Moment of inertia I (second moment of area) describes how the cross-section's shape resists bending. Material far from the neutral axis counts most, which is why beams are deep rather than square and why an I-beam outperforms a solid bar of the same weight.
How to Calculate Beam Deflection
For a simply supported beam (resting on a support at each end):
- Point load at midspan: δ = P × L³ ÷ (48 × E × I)
- Uniformly distributed load: δ = 5 × w × L⁴ ÷ (384 × E × I)
Worked example 1: point load. A simply supported steel beam:
- P = 10,000 N
- L = 4 m
- E = 200 GPa = 200 × 10⁹ Pa
- I = 8,000 cm⁴ = 0.00008 m⁴
2. Denominator: 48 × 200 × 10⁹ × 0.00008 = 768,000,000
3. δ = 640,000 ÷ 768,000,000 = 0.000833 m = 0.83 mm
Worked example 2: distributed load. The same beam carrying 5,000 N per meter (20 kN in total):
1. Numerator: 5 × 5,000 × 4⁴ = 25,000 × 256 = 6,400,000
2. Denominator: 384 × 200 × 10⁹ × 0.00008 = 6,144,000,000
3. δ = 6,400,000 ÷ 6,144,000,000 = 0.00104 m = 1.04 mm
The Beam Deflection Calculator handles both cases, plus a moment applied at one end, in SI base units: load in N (N/m for a distributed load), span in m, modulus in Pa and moment of inertia in m⁴, returning meters. Choose the distributed load and enter 5,000, 4, 200,000,000,000 and 0.00008 to reproduce the 0.00104 m (1.04 mm). For the reverse question, how much uniform load a section can carry before reaching L/360, the Structural Beam Load Calculator works in feet, in⁴ and psi and also covers fixed-fixed and fixed-pinned supports.
Checking Serviceability Limits
A deflection number means nothing until it is compared with an allowable limit, expressed as a fraction of span: commonly L/250 for general beams and L/360 for floors carrying brittle finishes such as plaster. For L = 4 m = 4,000 mm:
- L/250 = 16 mm
- L/360 = 11.1 mm
Common Mistakes
Mixing units. The SI form wants newtons, meters, pascals and m⁴; the calculator wants N/m, m, MPa and cm⁴. A millimeter-versus-meter slip changes the answer by orders of magnitude. Convert I carefully: 1 cm⁴ = 10⁻⁸ m⁴.
Entering total load as w. The distributed-load formula uses load per meter. A 20 kN total on a 4 m span is w = 5,000 N/m.
Using the wrong load case. A distributed load and a point load of the same total magnitude deflect differently (1.04 mm versus 1.67 mm for 20 kN on this beam).
Confusing strength with stiffness. A beam that passes a bending-stress check can still fail a deflection check. Verify both.
Assuming simple supports. The formulas here assume a pin at each end. Fixed ends, overhangs and continuous spans need their own equations, and these small-deflection formulas apply to slender beams only.
Related guide: Soil Bearing Capacity: How to Calculate the Load Your Soil Can Carry covers what happens to the load once it reaches the supports.