Deflection of a Simply Supported Beam

Finds how far a beam supported at both ends sags under load. Choose between a single load at the centre and a load spread evenly along it. Enter a concentrated load in kN, or a distributed load in kN per metre.

A beam resting on a support at each end is a simply supported beam. Loading it makes the middle sag, by an amount that depends on how the load is applied.

Here are the cases of a single load at the centre and a load spread evenly along the span.

\delta = \dfrac{P L^3}{48 E I} \qquad \delta = \dfrac{5 w L^4}{384 E I}

$\delta$ is the deflection
$P$ is a concentrated load and $w$ a distributed load per metre
$L$ is the span between supports
$E$ is Young's modulus and $I$ the second moment of area

$E$ measures the stiffness of the material and $I$ the contribution of the cross-section's shape. Their product is the flexural rigidity, the beam's resistance to bending.

Span dominates everything

The crucial feature is that the span is cubed, and to the fourth power under a distributed load.

Double the span under a central load and the deflection rises eightfold. Doubling the load merely doubles it. Distance costs far more than weight, which is why long spans are difficult to design.

The cross-section behaves similarly. For a rectangle the second moment of area is width times height cubed divided by twelve, so height enters as a cube. That is why the same piece of timber is vastly stiffer on edge than laid flat.

Example

The default input applies 5 kN at the centre of a 3.64 m span. The Young's modulus of 8 GPa is typical of timber, and a second moment of 10000 cm⁴ corresponds to a beam 105 mm wide.

The deflection is about 6.28 mm and the maximum bending moment is $PL \div 4$ , exactly 4.55 kN·m. Each support carries 2.5 kN.

Dividing the span by the deflection gives about 580. Building practice limits deflection to a three-hundredth of the span, so this beam has ample margin.

Points to watch

The Young's modulus of timber ranges from about 6 to 12 GPa depending on species and grade. Use the figure for the material actually specified.

Under sustained load, timber goes on deflecting slowly for years. This creep is generally taken to roughly double the final deflection, and it is not included here.

A real design must also check bending and shear stresses, not deflection alone.