Deflection, Creep and Vibration: What Governs a Span

The fourth power, and what it means for long spans

Deflection under a uniformly distributed load grows with the FOURTH power of the span. Doubling a span multiplies its deflection by sixteen while multiplying its bending moment by only four — so the two checks diverge rapidly as members get longer.

The consequence is a crossover. Short members are governed by STRENGTH: they will break before they sag noticeably. Long members are governed by STIFFNESS: they will sag unacceptably long before they are anywhere near breaking. Somewhere in between the governing check changes hands, and which side of it a member sits on determines what to do about it.

This is why a joist table has two columns for the same timber, why a long-span truss is deeper than its moment demands, and why the answer to "the beam does not work" depends entirely on WHICH check failed. Strengthening a stiffness-governed member is money spent on the wrong property.

Grade does nothing; depth does everything

The modulus of elasticity is a property of the MATERIAL, not of its strength grade. Structural steel has essentially one modulus whatever its yield strength, so specifying a higher grade to cure a deflection problem achieves precisely nothing — the member is stronger and exactly as flexible.

The same is true within a species of timber to a lesser degree, and within a concrete strength class to a lesser degree still; concrete's modulus does rise with strength, but only with its square root, so doubling the strength buys about forty per cent more stiffness for a great deal more cement.

What works is geometry. The second moment of area grows with depth cubed, so fifty per cent more depth is more than three times the stiffness. That is the lever, and it is why deflection problems are solved by deeper sections, closer spacing, or a shorter span — and by nothing else.

Which deflection, and what the limit protects

A deflection limit is meaningless without saying which deflection it applies to, and codes give several. TOTAL deflection includes the dead load, and matters for clearances, drainage and appearance. LIVE-LOAD deflection excludes it, and matters for the finishes — because a plaster ceiling is applied after the dead load is already on and only ever sees what comes afterwards. INCREMENTAL deflection is what happens after a specified stage, which is the one a brittle partition actually experiences.

So a member is routinely checked two or three times against different limits, and passing one says nothing about the others. The limits themselves are ratios of span, and they differ by what is being protected: a plastered ceiling gets a tighter limit than an open soffit, a roof draining to a low point gets a tighter one than a pitched roof.

On a low-slope roof the limit exists to prevent PONDING, and it is qualitatively different from the others: water collecting in a sagging bay adds load, which deepens the sag, which collects more water. That feedback does not settle at a new equilibrium if the roof is flat enough, which is why deck limits can look tight for a structure carrying nothing brittle at all.

Creep: the deflection that arrives later

Timber and concrete continue to deflect under sustained load long after it is applied. The mechanism is different in each — moisture movement and fibre relaxation in timber, gradual deformation of the cement paste in concrete — but the consequence is the same: the deflection measured on the day the props come out is not the one the finishes will live with.

The codes handle it with a long-term multiplier applied to the dead-load portion, and it is not a small correction. For timber the long-term deflection under sustained load can approach twice the immediate one, and for concrete more. Only the SUSTAINED part of the load creeps; a live load that comes and goes does not.

Steel does not creep at ambient temperature, which is one of the reasons steel sections can be sized on immediate deflection alone and timber cannot. It is also why a timber floor that felt fine on handover can feel different two heating seasons later without anything having changed.

Vibration is not deflection, and a passing floor can still be wrong

A floor that satisfies every deflection limit can still be unpleasant to walk on, and the reason is that vibration is a different question. What matters is the floor's natural FREQUENCY and its response to footfall — not its static sag under a load nobody is applying.

Frequency depends on stiffness and mass together: stiffer raises it, heavier lowers it. Human walking produces harmonics in a band that a lightweight long-span floor can fall squarely into, at which point ordinary footsteps drive the floor at something near its own frequency and the response builds. This is the mechanism behind almost every complaint about a bouncy floor, and it has nothing to do with strength.

Adding MASS is the counter-intuitive fix that often works: it lowers the frequency, which sounds wrong, but it also reduces the amplitude the same footfall produces. Adding stiffness works by moving the frequency above the driving harmonics. Which of the two is right depends on where the floor currently sits, and a frequency estimate on its own does not say — modern assessment methods compute the RESPONSE, in acceleration, against what people actually notice.

The calculators here estimate a fundamental frequency, which is the screening step. A floor comfortably above the walking harmonics needs no more; one near them needs a response calculation rather than a reassurance.

Camber hides deflection; it does not remove it

CAMBER is a deliberate upward curvature built into a member so that, once the dead load is on, it settles to level. It is a cosmetic and functional device — it protects drainage falls and stops a soffit looking sagged — and it adds no stiffness whatsoever. The member deflects exactly as much as it would have; it simply starts higher.

Because it is set against a PREDICTED dead load, camber is wrong whenever that prediction is. A beam cambered for a slab that turns out thinner finishes humped, and an excessively cambered member is a genuine problem for floor flatness that cannot be fixed by adding load. Common practice is therefore to camber for a proportion of the dead load rather than all of it, and to leave short members uncambered.

Trusses carry the same logic with an extra term: their deflection includes the take-up of the connections as well as the elastic deformation of the members, so a fabricated camber has to allow for both. This is one of the reasons a truss's deflection comes from the truss designer's own output rather than from a beam formula — the joints are part of the answer.