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Beam Deflection: Simply Supported, Centre Load

Also known as simply supported deflection · PL³/48EI

δ=PL348EI\delta = \frac{P L^{3}}{48 E I}
sig figs

Worked example: 20 kN at midspan of 4 m → 32 mm sag. Press Try an example to run it live, then adjust anything.

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See also

Beam Deflection: Simply Supported, Uniform Load: Both use maximum deflection and beam span.Cantilever Deflection: End Load: Both use maximum deflection and point load.Max Bending Moment: Centre Point Load: Both use point load and beam span.Max Moment: Simple Beam, Off-Centre Point Load: Both use point load and beam span.Support Reaction: Simple Beam, Off-Centre Point Load: Both use point load and beam span.Beam Reaction by Moment Equilibrium: Two Point Loads: Both use point load.
3 more relatedFixed-End Moment: Fixed-Fixed Beam, Uniform Load: Both use beam span.Max Bending Moment: Uniform Load: Both use beam span.Support Reaction: Simple Beam, Uniform Load (R = wL/2): Both use beam span.
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Beam Deflection: Simply Supported, Centre Load explained

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Strength decides whether a beam breaks; stiffness decides whether anyone will walk on it happily. A 20 kN load at the centre of a 4 m steel span with I = 4.167 × 10⁻⁶ m⁴ deflects δ = 20 000 × 4³ ÷ (48 × 200 × 10⁹ × 4.167 × 10⁻⁶) = 0.032 m: 32 mm, or L/125, far past any serviceability limit even if the stresses are fine. Codes usually cap live-load deflection at L/360 for plastered ceilings (the number dates from the era when plaster cracked visibly at about that curvature) and L/240 overall.

The L³ is the headline: doubling the span multiplies the sag by eight, and the only real cures are more depth (I goes as h³) or a shorter span. Note that E and I always appear together as the product EI, the flexural rigidity, which is why "use stronger steel" never fixes a bouncy floor, since all structural steels share E = 200 GPa. Remember to enter I as a plain number in m⁴, and to superimpose cases: a beam carrying both its own uniform weight and a point load deflects by the sum of 5wL⁴/384EI and PL³/48EI.

Beam Deflection: Simply Supported, Centre Load formula

δ=PL348EI\delta = \frac{P L^{3}}{48 E I}
Where
  • δ\delta= Maximum deflection (m)
  • PP= Point load (N)
  • LL= Span (m)
  • EE= Young's modulus (kPa)
  • II= Area moment of inertia (mm⁴)
In this toolkit

Beam bending and deflection

9 formulasMaximum moment, bending stress by Mc/I or M/S, and the deflection of simply supported and cantilever beams under point and uniform loads.

Revised 16 October 2026
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formula.expert. (2026, October 16). Beam Deflection: Simply Supported, Centre Load Calculator. https://formula.expert/solvers/beam-deflection-center-load

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