Structural Engineering & Statics
where the load goes
Deflection

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

A beam can be nowhere near failure and still be unusable, because it has moved too far. For most long-span members that limit arrives first, and stronger steel does not help at all.
11 min read 7 figures Which failure arrives firstScale changes everything

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.

Which limit arrives firstUtilisation of the strength limit and of the deflection limit, against span. Strength grows as the square of the span and deflection as the fourth power, so the two cross — and past the crossing a beam is sized by how far it moves rather than by what it can carry.0.60.811.21.41.61.8200.511.5span, relative to the firstthey cross heredeflection runs out at 1.40strength runs out at 1.54the limitstrengthdeflection
Fig. 1 Utilisation of the strength limit and of the deflection limit, against span. The two grow at different rates, so they cross — and past the crossing the beam is sized by how far it moves.

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=wL28.M = \frac{wL^2}{8}.

The deflection goes as the fourth power:

δ=5wL4384EI.\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 L2L^2 and utilisation of the deflection limit as L4L^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.

Deflection goes as the fourth power of the spanDeflection against span for a constant load intensity and section, with a straight line for comparison. Doubling the span multiplies the deflection by sixteen, while the bending moment only quadruples.11.522.533.54050100150200250300span, relative to the first1×5×16×81×256×moment: the squareload: the first powerdeflection: the fourth
Fig. 2 Deflection against span at constant load intensity and section, with the square-law and linear comparisons drawn faintly. The gap between the curves is what makes the crossover inevitable.

Stronger steel does not help

The deflection formula contains EE and II. 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 II, 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 same material, four waysFour cross-sections of identical area, so identical weight and cost, with the second moment of area computed from each profile's own geometry. Only the arrangement differs, and the stiffest is many times the flattest.the same, laid flatI = 0.06 × 10⁶1.0× the firstsquareI = 0.75 × 10⁶13.3× the firsttall rectangleI = 10.00 × 10⁶177.8× the firstI-sectionI = 24.29 × 10⁶431.8× the firstevery section here has an area of 3000 — only the shape differsthe bar is the second moment of area, to scale
Fig. 3 Four sections of identical area. For a deflection-governed member the ranking here is the whole design decision, because the material property that would otherwise distinguish them is identical in all four.

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≈18/δf \approx 18/\sqrt{\delta} with δ\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 II will do.

The deflected shape is the moment, integrated twiceA loaded beam with its deflected shape above and its bending-moment diagram below. The shape was obtained by integrating the moment twice and fitting the constants to the supports; the vertical scale is exaggerated enormously.the largest movement, at x = 4.00momentdrawn at roughly three hundred times the real deflection —a beam at its serviceability limit moves about a three-hundredth of its span
Fig. 4 The deflected shape of a beam, obtained by integrating the moment diagram twice. Precamber is this curve, built in upside down before the load arrives.

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 II 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.

Load, shear and moment — a simple spanThe applied load, the shear force it produces and the bending moment that follows, drawn one above another to the same horizontal scale. Shear is the integral of the load and moment is the integral of shear.4 per unit lengthshear16.0moment32.0 at x = 4.00the moment peaks exactly where the shear passes through zero
Fig. 5 The moment diagram, which is the input to the deflection calculation. The deflected shape is its second integral, so anything that reduces the area under this curve reduces the movement as well as the stress.

What is actually being integrated

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

The deflected shape is the moment, integrated twiceA loaded beam with its deflected shape above and its bending-moment diagram below. The shape was obtained by integrating the moment twice and fitting the constants to the supports; the vertical scale is exaggerated enormously.the largest movement, at x = 4.00momentdrawn at roughly three hundred times the real deflection —a beam at its serviceability limit moves about a three-hundredth of its span
Fig. 6 A loaded beam with its deflected shape above and its bending-moment diagram below. The shape is the second integral of the moment, with the constants fixed by the supports.

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.

Bending is a push and a pullA section carrying a bending moment, with the stress at every height computed as the moment times the distance from the neutral axis divided by the second moment of area. It is compression above and tension below, and zero exactly at the neutral axis.neutral axiscompressiontensionI = 29.97 × 10⁶Z = 299.7 × 10³peak stress 200.2σ = M y ÷ I, at every height
Fig. 7 The stress block in a section. Strength is a statement about the peak of this distribution; 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.

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.

The figures have 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.

Where this goes next

11.522.533.54050100150200250300span, relative to the first1×5×16×81×256×moment: the squareload: the first powerdeflection: the fourth Span to the fourth, which is why spans are short Doubling a span multiplies its deflection by sixteen. No other relationship in ordinary structural work is that steep, and it is the reason long spans are always a different kind of structure. neutral axiscontribution of each striptotal I = 79.86 × 10⁶the outer strips do almost all of the work The material far from the middle does nearly all the work A strip of steel contributes to bending stiffness in proportion to the square of its distance from the centre. Move the same steel outward and the section gets stiffer for nothing. simply supportedstatics alonesag 32.0propped at one endneeds stiffnesssag 18.0hog 32.0built in at both endsneeds stiffnesssag 10.7hog 21.3the load never changes; only what is holding the endsthe built-in case peaks at two-thirds of the simple span's moment One support too many, and what it costs to know Add a redundant restraint and the load has two routes to the ground. Equilibrium cannot say how it splits, and the answer turns out to depend on stiffness — which is a different kind of question.