Stiffness is not strength, and usually it is the one that governs

A floor beam can be twice as strong as it needs to be and still fail its design check, because it bounces when somebody walks across it. A glazed façade can be supported by a beam that will never break and still shatter, because the beam sags onto the glass.

Strength and stiffness are separate properties, they are limited by separate criteria, and for a great many members it is the second that decides the size. That is not a subtlety at the edge of the subject — it is the ordinary case for anything spanning more than a few metres.

Two limits, two rates #

Both quantities grow with span, and they grow at different powers, which is the whole reason they swap places.

For a uniformly loaded simply supported beam, the bending moment goes as the square of the span:

$$M = \frac{wL^2}{8}.$$

The deflection goes as the fourth power:

$$\delta = \frac{5wL^4}{384EI}.$$

So doubling the span quadruples the moment and multiplies the deflection by sixteen. Utilisation of the strength limit rises as $L^2$ and utilisation of the deflection limit as $L^4$, and a curve rising as the fourth power will overtake one rising as the square.

Where it overtakes depends on the section and the limit, and for ordinary steel floor beams it happens at spans of roughly six to eight metres. Below that, strength governs. Above it, deflection does, and it does so increasingly emphatically.

Stronger steel does not help #

The deflection formula contains $E$ and $I$. It does not contain the yield stress.

That is the fact with the most practical consequence in this essay. Structural steels of grade S275 and S355 have the same modulus of elasticity — 210 GPa, both of them — and differ only in the stress at which they yield. Substituting the stronger grade into a deflection-governed beam changes nothing whatsoever.

The same holds within a material family generally. All steels have essentially the same stiffness; all aluminium alloys have about a third of it, regardless of alloy; concrete’s stiffness depends on its strength but far more weakly than its strength does. Improving material strength is a strength strategy and never a stiffness one.

What does help is $I$, and the ways to increase the second moment of area are all geometric: deeper section, more material at the extremes, or a different arrangement of the same steel.

The crossover is not a span, it is a span-to-depth ratio #

“Six to eight metres” is the usual way the crossover is quoted and it is the wrong unit for it. The condition has a closed form, and what comes out is not a length.

Take a beam worked to its full bending stress $\sigma$ and ask what its deflection then is. With $\delta = 5wL^4/384EI$, $M = wL^2/8$, $\sigma = M/Z$ and $Z = I/(d/2)$, the load and the second moment both cancel:

$$\frac{\delta}{L} = \frac{5}{24}\cdot\frac{\sigma}{E}\cdot\frac{L}{d}$$

Set that equal to a limit of $1/n$ and the crossover is

$$\frac{L}{d} = \frac{24E}{5n\sigma}$$

A span-to-depth ratio, containing no load and no span. For steel at a service stress around 245 N/mm² against a limit of $L/360$, it is 11.4. Build a beam shallower than a eleventh of its span and deflection governs; deeper, and strength does. Against the slacker $L/250$ it is 16.5.

Two readings follow. Ordinary steel floor beams run at span-to-depth ratios around 20, which is comfortably past both — the crossover is not a boundary most beams sit near, it is one nearly all of them are on the far side of. The “six to eight metres” figure is really a statement about the depths people are allowed, not about spans.

And the ratio scales as $E/\sigma$, which is why the material question has the shape it does. Aluminium has a third of steel’s modulus at a comparable working stress, so its crossover ratio is a third — an aluminium beam is deflection-governed at a depth where a steel one is not, and the two cannot be compared span for span at all.

The other term in that expression is $n$, the limit’s own denominator, and it moves the answer as much as the material does.

What the deflection limit protects #

The limits — span over 250, over 360, over 500 — look arbitrary and each protects something specific.

Finishes. Plaster cracks, brittle partitions split, and glazing is stressed if the structure supporting it moves. The relevant number is the deflection occurring after the finish is installed, which is why limits are often written against the imposed load only, with the dead-load deflection having already happened.

Drainage. A flat roof that sags collects water, and the water is a load, which increases the sag, which collects more water. Ponding is a genuine instability with the same runaway structure as second-order effects, and roofs have collapsed from it — an instability of a beam that was never in danger of buckling.

Appearance. A visibly sagging beam is alarming whether or not it is safe. Around span over 250 is where a horizontal line starts to look wrong to a person standing under it.

Fit. Doors bind, lift guides go out of tolerance, and machinery mounted on a moving floor misaligns.

Vibration. The stiffest limit of all, and the one increasingly likely to govern. A floor’s natural frequency falls as its deflection rises — roughly $f \approx 18/\sqrt{\delta}$ with $\delta$ in millimetres — so a floor deflecting 20 mm has a frequency around 4 Hz, which is exactly the range human walking excites. Long-span lightweight floors are routinely designed by vibration rather than by anything else.

Precamber, and the limit that is not really about deflection #

If a beam is going to sag, it can be built curved upward by the amount it will sag, so that it ends up straight.

Precamber is standard for long-span steel and for bridges, and it is a complete answer to the appearance and fit criteria: a beam cambered for its dead load is level once the dead load is on. It is not an answer to the others. A cambered beam still deflects under imposed load, still ponds if the ponding is driven by live water, and still has whatever natural frequency its stiffness gives it.

That distinction is the useful test of what a deflection limit is really protecting. If precambering satisfies it, the limit was about position. If it does not, the limit was about stiffness, and only more $I$ will do.

Precamber is the deflected shape itself, built in upside down before the load arrives — the second integral of the moment diagram, drawn the other way up and cast or welded into the member.

Making it stiffer, in order of cost #

For a member that fails deflection, the options are not equal.

Deepen it. Stiffness goes as the cube of the depth for a solid section, so a small increase in depth is worth a great deal. Almost always the cheapest fix and almost always constrained by something architectural.

Reduce the span. Deflection goes as the fourth power, so an intermediate support is enormously effective — halving the span cuts the deflection by sixteen. Usually the cheapest fix of all when it is possible at all.

Make it continuous. A beam continuous over its supports deflects far less than a series of simple spans, because the hogging at the supports curves the ends back. Continuity gives roughly a fifth of the simply supported deflection for the same span and load.

Add material at the extremes. Cover plates on the flanges of a beam are a targeted way of raising $I$ without changing depth.

Change material. Rarely useful, for the reasons above, unless the change is large — timber to steel, or steel to a composite section.

Prop it during construction. For a composite beam, propping means the concrete’s stiffness is available for the dead load as well as the imposed, which can halve the total deflection.

What is actually being integrated #

The deflection of a beam is not an independent calculation. It is the moment diagram, integrated twice.

The chain from load to deflection is one function and four integrations, and that is why anything reducing the area under the moment diagram reduces the movement as well as the stress. Continuity, an overhang and an extra support all work on both at once.

Strength is a statement about the peak of the stress block in a section; stiffness is a statement about the second moment of area behind it, and the two are different properties of the same geometry. The distinction is visible in the two ratios: strength uses the section modulus and stiffness uses the second moment, and they differ by one power of depth. That single power is what makes the two limits diverge with span.

The vibration check is the deflection check #

The rule of thumb quoted above — that a floor’s natural frequency is about $18/\sqrt{\delta}$ — looks like an empirical correlation. It is not. It is an identity, and deriving it takes three lines and explains why the two criteria can never be traded against each other.

The fundamental frequency of a simply supported beam carrying a mass $m$ per unit length is

$$f = \frac{\pi}{2}\sqrt{\frac{EI}{mL^4}}.$$

The deflection of the same beam under its own weight is $\delta = 5wL^4/384EI$ with $w = mg$. Rearranging the second for $EI/mL^4$ and substituting into the first,

$$f = \frac{\pi}{2}\sqrt{\frac{5g}{384\,\delta}} = \frac{17.8}{\sqrt{\delta}},$$

with $\delta$ in millimetres. The span has gone. The section has gone. The material has gone. Every property of the beam has cancelled except the deflection it produces under its own weight, and the constant is $\pi$, $g$ and the coefficient $5/384$ arriving together.

Two things follow that are worth more than the formula. The first is that frequency and deflection are the same measurement, so a floor cannot be made to vibrate less without being made to deflect less — there is no separate lever. A floor at $10$ mm of self-weight deflection has a frequency near $5.6$ Hz; at $20$ mm, near $4.0$; at $40$ mm, near $2.8$. Since walking excites the range around $1.5$ to $2.5$ Hz and its second and third harmonics reach $5$ to $7$, a floor deflecting more than about $15$ mm under permanent load is being tuned into the excitation rather than away from it.

The second is that this is why floors have got worse. Longer spans, lighter construction and the removal of partitions all raise the self-weight deflection while doing nothing to the excitation, and the frequency follows down the curve. A traditional short-span floor with heavy finishes and blockwork partitions was accidentally excellent by this measure. Nothing about the strength calculation registers any of that.

Two questions, two sets of loads #

The deepest reason strength and stiffness cannot be collapsed into one check is that they are asked about different loads, and the difference is deliberate.

An ultimate check asks whether the structure collapses. The consequence of getting it wrong is people dying, so the loads are multiplied by partial factors — typically 1.35 on permanent load and 1.5 on imposed — and the material strength is divided by another. The result is a calculation about a load case that will almost certainly never occur, which is the point: the margin is what makes collapse improbable rather than merely unlikely.

A serviceability check asks whether the structure is unpleasant, unusable or damaging to what it carries. The consequence of getting it wrong is expense and complaint. So the loads are unfactored, the material properties are the real ones, and the combinations are often less onerous — an imposed load reduced to the fraction likely to be present for long periods, rather than the peak used for strength.

That is why the two checks can give opposite answers about which beam is bigger, and why a member can be at 60 per cent of its strength utilisation and fail its deflection check outright. They are not two views of one calculation; they are two calculations about two different futures, with different probabilities attached and different amounts of caution built in.

The framework also explains something that puzzles people arriving at the subject: why a deflection limit is a nice round fraction with no derivation behind it, while a strength calculation is carried to three significant figures. The strength calculation is defending against something quantifiable and catastrophic. The deflection limit is a proxy for a set of consequences — cracked plaster, a binding door, an anxious occupant — that are not reducible to a number at all, and $L/360$ is the profession’s accumulated judgement rather than a result.

What the proxy costs #

A limit that stands in for something else can be satisfied while the thing it stands for is not, and the deflection limit has two well-known versions of this.

The first is that it is a limit on a total where several of the real criteria concern increments. Plaster does not care what the beam did before the plaster was applied; it cares what happens afterwards. A door does not bind because the floor sagged during construction; it binds because the floor moved after the frame was fitted. Codes handle this by writing limits against the imposed load alone, which is the right instinct, and the sequencing still has to be got right on site — a finish applied before the dead load is complete inherits deflection that the calculation assumed it would miss.

The second is precamber, which satisfies the limit exactly where the limit is a proxy for position and not at all where it is a proxy for stiffness. A cambered beam has the same $I$ it always had, so it has the same frequency, the same response to a person walking and the same additional sag under imposed load. It looks level, which was one of the four things being protected and is not the others.

Precamber has a failure mode of its own that follows from the same observation. The camber is built in for a predicted dead load, and if the prediction is high — a screed thinner than specified, a finish lighter than assumed — the beam never comes down to level and is left permanently hogged. That is a defect nobody can fix, produced by a calculation that was conservative in the direction conservatism usually helps.

Both are instances of the same warning. Every serviceability limit is a number standing in for a judgement, and the number can be met by means that do not deliver the judgement. Which is why the useful question when a deflection check fails is not “how can this be made to pass” but “which of the four things was this limit protecting” — because the answer decides whether more depth, a camber, a different sequence or a different floor is the honest response.

Where the model stops #

Elastic behaviour. The deflection formulae assume the material stays elastic, which for a serviceability check is nearly always true — that is what makes serviceability separable from strength.

Short-term stiffness. Concrete creeps, and its long-term deflection can be two or three times its short-term value. Timber creeps too, and both are handled by an effective modulus rather than by the real one. A steel beam does not creep, which is one of the reasons steel deflection checks are simpler.

Uncracked sections. A reinforced concrete beam cracks in service, and its stiffness after cracking is a fraction of the gross value. Which stiffness to use is one of the harder questions in concrete design.

Isolated members. A floor’s actual movement depends on the whole system, which is usually redundant — the slab, the beams, the connections, and any partitions that turn out to be structurally active. A beam-by-beam calculation is an approximation to something two-dimensional.

Static behaviour. Deflection under a static load is not the same question as how a floor responds to a person walking, and the vibration check is a separate calculation with a separate criterion.

There is a distortion this whole field cannot escape, and it is worth stating in numbers: a beam at a limit of span over 360 has moved by 0.28 percent of its span. Drawn honestly at a span of 700 pixels, that is two pixels — thinner than the line used for the beam. Every deflected-shape figure ever published exaggerates by a factor of a hundred or more, and the exaggeration makes deflection look like a large visible event when the entire subject concerns movements that are almost invisible.

The ladder from here #

Later rungs: the standard deflection formulae and where they come from. Limit states, and why serviceability and ultimate are checked separately. Deflection limits by finish type. Precamber. Continuity and its stiffness benefit. Creep in concrete and timber. Cracked-section stiffness. Ponding instability. Floor vibration and the response-factor method. And the general question of what a structure is actually for, which is where the whole distinction between strength and serviceability comes from.

The first deflection limits were written for plaster ceilings, which crack at about a three-hundredth of the span. Plaster ceilings are now uncommon and the number has outlived them.