Beam Deflection — Simply Supported, Centre Load

Also known as simply supported deflection · PL³/48EI

\delta = \frac{P L^{3}}{48 E I}

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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Beam Deflection — Simply Supported, Centre Load explained

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

\delta = \frac{P L^{3}}{48 E I}

Where

$\delta$ = Maximum deflection (m)
$P$ = Point load (N)
$L$ = Span (m)
$E$ = Young's modulus (kPa)
$I$ = Area moment of inertia (mm⁴)

Missing one of these? Work it out first, then come back

Maximum deflection — Beam Deflection — Simply Supported, Uniform Load
Point load — Max Bending Moment — Centre Point Load, Max Moment — Simple Beam, Off-Centre Point Load
Span — Max Bending Moment — Centre Point Load, Max Bending Moment — Uniform Load
Young's modulus — Young's Modulus (E = σ/ε), Axial Deformation (δ = PL/AE)
Area moment of inertia — Bending Stress (σ = Mc/I), Beam Deflection — Simply Supported, Uniform Load

Part of this formula set

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.