The formula
q = min(8·σ·W / L², 384·E·I / (5·L³·limit))
Where it comes from
A beam is checked twice and the two checks give different numbers. Strength says the stress must stay below the allowable value: the mid-span moment is q·L²/8 and the stress is that moment over the section modulus. Deflection says it must not bend more than a given fraction of the span, and there the span is cubed, so it grows much faster. Past a certain span deflection takes over, and a beam that "carries the load" gets rejected anyway.
How to work it out by hand
- Look up the section modulus W and second moment of area I
- Allowable moment: multiply W by the allowable steel stress
- Load by strength: eight times that moment over the span squared
- Load by deflection: 384·E·I over 5·L³ times your limit
- The allowable load is the smaller of the two
What is worth knowing
The deflection limit is not arbitrary: L/300 on a six-metre beam is two centimetres, and that alone cracks a partition sitting on top. With brittle partitions the limit drops to L/400 or L/500, and deflection then governs from much shorter spans. It is also worth looking at how far apart the two figures come out: if they are close, the section is well chosen; if one is double the other, there is material going to waste on one side. And the allowable stress is a field because each code sets it together with its safety factors — it is not a property of the steel.