Structural Engineering & Statics
where the load goes
Deflection

The angle nobody limits

Every serviceability rule in this collection limits a displacement. What a bearing, a joint and a cladding gap actually have to accommodate is an angle — and the angle is locked to the displacement by a coefficient that contains no material, no section and no span.

Assumes Stiffness is not strength, and usually it is the one that governs, The area of a diagram is a rotation and Span to the fourth, which is why spans are short.

18 min read 9 figures Which failure arrives firstThe load must go somewhereGeometry beats material

Every serviceability check in this collection is a limit on a distance. Span over three hundred and sixty, span over five hundred, twenty millimetres, ten. The quantity that appears in the calculation is a displacement, the quantity in the rule is a displacement, and the two are compared.

The things at the ends of the beam do not care about the displacement. A bearing has to rotate. An expansion joint has to open by whatever the rotation opens it. A cladding panel fixed to the beam’s top flange near a support gets dragged sideways. A nominally pinned connection is asked to turn, and turns while resisting.

All of those are angles, and no rule anywhere states a limit on one.

One coefficient, and nothing else in itThe deflected shapes of one beam under four load cases, each scaled so that its mid-span deflection is the same, with the tangent at the left-hand support drawn on each. The end rotation is that deflection times a coefficient that depends only on the shape of the load: 3.20 for a uniform load, 3.00 for a load at mid-span, 2.99 for a triangular load, 3.60 for a load on half the span. Every material property, every second moment and the span itself cancel out of the ratio θL/δ, so a beam at any deflection limit has an end rotation that is known before anything about it is: at L/360 it is 8.89 milliradians, or 0.51 of a degree.024681012-0.15-0.1-0.0500.050.10.15along the span (m)deflection, each scaled to the same mid-span drop3.20 · a uniform load3.00 · a load at mid-span2.99 · a triangular load3.60 · a load on half the spanθ = C · δ/L, and the same δ gives 1.21× the rotation across these
Fig. 1 Four load cases on one beam, each scaled so that its mid-span deflection is the same, with the tangent at the support drawn on each. The four end rotations are different, and the ratio that turns one into the other contains nothing about the beam.

Which free body produced the number

Take the deflected shape as the second integral of the curvature and evaluate two things from it.

The slope at the left support is the area of the M/EIM/EI diagram between the support and the point of zero slope, and the mid-span deflection is the first moment of that same diagram about the support. Both are integrals of the same function over the same interval, and their ratio therefore contains that function’s shape and nothing else.

For a uniform load on a simply supported span,

θend=wL324EI,δmid=5wL4384EI,θLδ=38424×5=165\theta_{end} = \frac{wL^3}{24EI}, \qquad \delta_{mid} = \frac{5wL^4}{384EI}, \qquad \frac{\theta L}{\delta} = \frac{384}{24\times 5} = \frac{16}{5}

EE, II, ww and LL all cancel. For a point load at mid-span, θ=PL2/16EI\theta = PL^2/16EI and δ=PL3/48EI\delta = PL^3/48EI, so the ratio is exactly 3. A triangular load gives 224/75 at its light end and 256/75 at its heavy one — the two ends differ by exactly 8/7, which is a fact about the shape of the load rather than about the beam.

The coefficient is between 3 and 3.6 for every ordinary load case, which makes a useful rule of thumb possible: θ≈3.2 δ/L\theta \approx 3.2\,\delta/L, near enough always.

The area is the rotation, and its first moment is the movementA 6 m cantilever under a tip load of 10, with the M/EI diagram beneath it. The shaded area is 180.00, which by the first theorem is the change of slope along the whole member. Its centroid is at 2.000 m, and the first moment about the tip is 720.00 — which by the second theorem is the deviation from the tangent, and for a cantilever that tangent is horizontal, so it is the deflection itself. Integrating the curvature twice instead gives 720.00.10centroid at 2.00 mM/EIarea = 180.00 · first moment = 720.00by double integration: 720.00
Fig. 2 The two integrals the ratio is made of. The area of the curvature diagram is a rotation and its first moment is a deflection, so a single diagram supplies both — and the coefficient linking them is a property of that diagram’s shape.

What the number turns out to be

Take a beam at a deflection limit of span over 360 — the ordinary limit for a member carrying a brittle finish.

θ=3.2×1360=0.0089 radians=0.51∘\theta = 3.2 \times \frac{1}{360} = 0.0089\ \text{radians} = 0.51^\circ

Half a degree, for a beam of any span, any depth, any material, at that limit. A beam at span over 250 turns 0.73° and one at span over 500 turns 0.37°.

Those are not small numbers in the places they act. On a beam 700 mm deep the top of the section moves 6.2 mm horizontally relative to the bottom — which is where a bolted end plate is, where a cladding bracket is, and where the beam meets a column that is not moving. Half a degree at a bearing 400 mm wide lifts one edge 3.5 mm off the other.

The whole quantity is fixed before anything about the design is chosen, and it is invisible in the calculation that fixed it.

The angle is what the bearing has to allowThe end of a 12 m beam 700 mm deep, rotated by 42.9 milliradians — the rotation a uniform load actually produces here, drawn at 3 times its size. A rotation about the bearing moves the top of the section 30.0 mm horizontally, which is what an expansion joint, a cladding gap or a nib has to accommodate and what a deflection limit never mentions. A beam held down at the top of its section instead of at its bearing is a beam with an axial force in it that the analysis does not contain: at 0.0429 radians the restraint is stiff and the movement is small, which is exactly the combination that produces a large force.the bearing30.0 mm at the top faceθ = 42.9 mrad = 2.46° — drawn 3× overat the L/360 deflection limit the rotation would be 8.9 mrad, or 0.51°
Fig. 3 The rotation drawn at forty times its size, and the movement it produces at the top of a 700 mm section. Nothing in the beam’s design contains this number; it follows from the deflection limit, which was chosen for a completely different reason.

The bearing, which is the honest case

A bridge bearing is the one place where the rotation is designed for explicitly, and the reason is that it cannot be avoided: something has to be provided that permits the rotation, and its permitted rotation is a stated property.

An elastomeric bearing accommodates rotation by compressing more on one side than the other, and it has a limit — beyond it, one edge lifts off and the bearing is being loaded on part of its area. A pot bearing has a defined rotation capacity. A plain sliding bearing needs a curved surface if the rotation is more than trivial.

The design rotation is not only the beam’s own. It is the sum of the rotation under load, the rotation from any camber that has been built in, a construction tolerance for the bearing being set out of level, and — the largest term on a long span — the rotation from the temperature gradient through the deck. Codes require the sum, plus an allowance, and the allowance is often the same size as the calculated value.

Cambered against the wet loadA 12 m composite beam whose flexural rigidity rises from 94 to 260 kN·m² when the slab sets, so the first two loads are carried by the bare steel and the rest by the composite section. Fabricated with 25.3 mm of camber, it moves through -20.7, 0.0, 7.5, 12.7 mm as the four stages arrive — 0.0 mm on the day the slab is poured, and 12.7 mm at the end, which is one part in 947 of the span. The largest curvature it ever has is 25.3 mm of hog, and it has that with nothing on it. Every shape is drawn at the same exaggeration and the drawing is a diagram of a proportion: the vertical scale is 119484 times the horizontal.levelas fabricated: 25.3 mm of camber-20.70.07.512.7self-weight of the steel: 4.6 mm on EI = 94wet concrete: 20.7 mm on EI = 94finishes and services: 7.5 mm on EI = 260imposed load: 5.2 mm on EI = 260
Fig. 4 The shape the beam was built to, which is one of the terms in the bearing’s rotation. A cambered beam arrives on site already rotated at its ends, in the opposite sense to the one the load will produce, and the bearing has to accept both extremes.

The connection that was called a pin

In a building the rotation is usually not designed for at all, because the connection is drawn as a pin and pins do not resist rotation.

They do. A fin plate, a web cleat, a flexible end plate — every one of them has a real rotational stiffness, and the beam is going to turn through its 8.9 milliradians regardless. The moment that appears is the rotation times the stiffness, and it is present in a joint that was designed for shear alone.

The classification exists precisely because of this. A connection is “nominally pinned” if its stiffness is low enough that the moment it attracts is small compared with the beam’s — which is a statement about a ratio, so the same connection can be a pin on a flexible beam and a semi-rigid joint on a stiff one.

Two consequences follow that are easy to miss. The moment is largest where the beam is stiffest, because a stiff beam rotates less but attracts moment in proportion to its own stiffness relative to the joint’s. And the moment goes into the column as an eccentricity moment whether or not the beam design accounted for it.

A joint is springs in seriesThe five components of an end-plate joint, with each bar the flexibility it contributes. The column flange in bending is 39.62% of the total on its own, and doubling its stiffness raises the joint's by a factor of 1.25 — while doubling the stiffest component buys 1.05. The joint's rotational stiffness is 30818.34 kN·m per radian.flexibility contributed by each componentthey add, so the softest dominates — Sj = 30818.34 kN·m/radwhat doubling it buyscolumn web in shear21.89%×1.12column web in compression11.55%×1.06column flange in bending39.62%×1.25end plate in bending18.09%×1.1bolts in tension8.85%×1.05
Fig. 5 Where a connection’s stiffness comes from — several components in series, so the softest one governs. A “pin” is a joint whose series stiffness happens to be small, and the number is computable rather than nominal.
Moment against rotation, for three real jointsThree connections on one plot, with the classification boundaries for a beam of EI/L = 3500 drawn as rays through the origin. web cleats is semi-rigid, flush end plate is semi-rigid, extended end plate is rigid. The boundaries are multiples of EI/L, so the same joint is rigid on a short stiff beam and semi-rigid on a long slender one.00.0050.010.0150.020.0250.03050100150200rotation, radiansmoment, kN·mrigid abovepinned belowweb cleats — semi-rigidflush end plate — semi-rigidextended end plate — rigid
Fig. 6 Three real connections, their stiffnesses spanning a factor of twenty-five, drawn against the rotation a beam demands. Where a connection’s curve crosses the beam’s demand line is the moment that connection actually carries — and the leftmost one crosses at a moment small enough to ignore, which is the whole content of the word “pinned”.

The rotation at a support that is continuous

At an internal support of a continuous beam the two spans rotate together, and the relative rotation is zero by definition — that is what continuity means.

What is not zero is the rotation of each span relative to the support. A column at that support is being asked to accommodate the rotation of the joint, and the moment it takes is the joint rotation times the column’s stiffness. That is the ordinary frame problem, and it is worth naming here because it is the same angle appearing in a third role: as a demand at a bearing, as an unwanted moment at a pin, and as the shared rotation of a frame joint.

The distinction between the three is not the angle. It is what is on the other side of it.

The moment in 2 continuous spansThe bending moment in a continuous beam, solved by the stiffness method. The peak sagging moment is 253.1, and over the interior support a hogging moment of 450.0 appears — a moment no simply supported span carries at all, because continuity is what puts it there.moment253.1 sagging450.0 hoggingreactions 112.5 375.0 112.5 — the inner supports carry far more than a sharethe continuous case needed stiffness; the comparison did not
Fig. 7 Continuity as a rotation constraint. The support moment is exactly the moment required to make two spans that would rotate by 8.9 milliradians each, rotate by the same amount instead. Every indeterminate structure on this site is a rotation being enforced somewhere.

Two more places the angle arrives

Cladding and glazing. A panel spanning between two floors is fixed to the beam at its head, and the beam’s end region rotates. If the fixing is at the top of the section and the panel is stiff, the rotation drags the panel sideways by θ\theta times the depth — 6 mm on the beam here. Glazing gaskets accommodate a few millimetres; a rigid stone panel accommodates none. The movement joint that exists at every floor level in a facade is sized by a sum of movements in which this one is rarely the largest and is never absent.

Drainage. A flat roof falls toward its outlets by a designed gradient, typically 1 in 80 — 12.5 milliradians. A beam whose end rotation is 8.9 milliradians is tilting the slab it carries by nearly three quarters of the design fall, in whichever direction the beam happens to be deflecting. Falls are laid to the deflected shape or the water goes somewhere else, and what happens when it does not run off is a separate and worse problem.

Deflection goes as the fourth power of the spanDeflection against span for a constant load intensity and section, with two slower relationships drawn faintly behind it for comparison: the load itself, which grows in proportion to the span, and the bending moment, which grows as its square. Doubling the span multiplies the deflection by sixteen, while the moment only quadruples.11.522.533.54050100150200250300span, relative to the first1×5×16×81×256×moment: the squareload: the first powerdeflection: the fourth
Fig. 8 The relationship the deflection limit is fighting. Deflection grows as the fourth power of span and rotation as the third, so the rotation-to-deflection ratio falls as 1/L1/L — which is exactly the cancellation that makes the coefficient a pure number and makes θ=3.2δ/L\theta = 3.2\delta/L true at every span.

Why no rule exists

It is worth asking why a quantity this consequential has no limit written against it anywhere, because the answer is instructive rather than an oversight.

A limit needs a consequence to be limited. A deflection limit exists because a deflection cracks a partition, ponds water, alarms an occupant or spoils a line. Each of those is a phenomenon at the mid-span of a member, and each has a rough threshold that experience has settled.

A rotation’s consequences are all at the ends, and every one of them belongs to somebody else’s component. The bearing has its own rotation capacity, stated by its manufacturer and checked by whoever specifies it. The connection has its own classification. The cladding has a movement allowance in its own specification. Each of those components carries a limit, and the beam designer’s job is to supply the number rather than to satisfy a rule about it.

That division works when the number is passed on and fails silently when it is not — which is the ordinary failure mode of any quantity that belongs to an interface. The beam is designed by one person against a displacement limit; the connection is designed by another against a shear; the facade is designed by a third against a movement allowance derived from a frame model that had pinned joints in it. Nobody has got anything wrong, and the rotation has not been transmitted anywhere.

The practical remedy is small and is rarely done: state the end rotation on the drawing, next to the reaction. It is one number, it follows from a calculation that has already been made, and it is the only thing the three interfaces above actually need.

Where the model stops

The beam is simply supported and prismatic. A continuous beam’s end rotation at an outer support is smaller than a simple span’s, by a factor that depends on the far end’s restraint; a cantilever’s tip rotation obeys a different coefficient entirely.

The rotation is elastic and instantaneous. A concrete beam’s long-term deflection is two or three times its immediate one, and so is its rotation — so the bearing detail has to accept a movement that arrives over years and does not come back.

And the coefficient assumes the load is symmetric. For an asymmetric load the two ends differ, as the triangular case shows: 224/75 at one end and 256/75 at the other, which is a 14 per cent difference in a quantity often taken as one number.

The same angle, arriving as a force

There is one more place the rotation turns up, and it is the one where it stops being a serviceability matter.

If the end of a beam is restrained against rotating — a stiff connection, a deep end plate, a beam cast into a wall — the rotation does not happen, and what appears instead is a moment. Its size is the rotation the beam wanted multiplied by the stiffness of whatever stopped it, which for a stiff restraint is a large number multiplied by a small one and is not small.

This is an imposed deformation rather than a load, and it has the property all imposed deformations have: the force it produces is proportional to the stiffness of the restraint, so making the connection stronger makes the force larger. A beam whose ends are held rigid develops wL2/12wL^2/12 at each end, which is 133 kNm on the beam here — from a beam that was designed as simply supported and detailed with a connection somebody made generous.

The general rule this belongs to is worth stating once more because it is counter-intuitive every time: for a load, strength helps; for an imposed deformation, flexibility helps. The end rotation is a deformation the beam is going to undergo, and a detail that resists it is a detail that will be asked for a force.

What the pictures cannot show

The rotation is drawn forty times over. At true scale the tangent and the beam would be indistinguishable, which is the honest reason nobody sets a limit on it: it does not look like anything.

Nor can any figure here show the sum that a real bearing is designed for. The load rotation is one term among five, and the others — camber, temperature gradient, construction tolerance, long-term creep — arrive from four different calculations and are added at the end by somebody who has to remember that they belong together.

A third omission is the sign. Every rotation drawn here is one direction, and the beams that matter most for this are the ones that reverse: a bridge under a passing vehicle rotates one way and then the other, a floor under a moving imposed load does the same on a smaller scale, and a bearing or a gasket asked for a rotation in both senses is being asked for twice the range the single figure shows.

The assumption the figure rests on

The deflection limit is treated as though it fixed the rotation, and it does — for the load case the limit applies to. But limits apply to different load cases: total load for some purposes, imposed load only for others, imposed load after the finishes were installed for the case that actually cracks a partition. Each of those is a different deflection and therefore a different rotation, and the rotation that matters at a cladding fixing is the one measured from the moment the cladding was fixed, not from the beam’s unloaded shape.

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. 9 Which limit arrives first, which is the question the whole serviceability apparatus exists to answer. For most long-span members it is deflection, and this essay is about the consequence nobody checks once that answer is reached — the angle that came with it, at whichever end of the beam somebody has detailed something.

The ladder from here

Later rungs on this anchor: the rotation demand at a bridge bearing, as the sum of five terms with different signs and different time scales. The nominally pinned connection’s moment, computed rather than assumed away, and what it does to the column. Rotation compatibility in a semi-continuous frame, where the joint’s stiffness is a design variable rather than a nuisance. The rotation of a cantilever tip, whose coefficient is different and whose consequence is a facade. And the same angle at the other end of the subject — the rotation capacity a plastic hinge has to deliver, which is the same quantity in radians being asked for by a completely different question.

Where this goes next

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 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. real Mpeak 32.0a unit load, here and nowhere elseunit mM × marea ÷ EI = 213.33the unit load is the only place the question 'deflection where?' is asked One deflection, without solving everything To find how far one point of a structure moves, put an imaginary force of one unit there, multiply two moment diagrams together, and integrate. The answer arrives without ever solving for the deflected shape. 00.0050.010.0150.020.0250.030.0350.040.0450.05050100150200rotation, radiansmoment, kN·mrigid abovepinned belowweb cleats — pinnedflush end plate — semi-rigidextended end plate — semi-rigid Neither pinned nor rigid, which is every real connection Frame analysis offers two options for a joint and reality supplies a continuum between them. Worse, the boundaries are not properties of the connection at all — the same end plate is rigid on a short stiff beam and semi-rigid on a long slender one. 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.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

The objects this essay names

Each one links to every other essay that touches it.

BearingCladdingCompatibilityCurvatureDeflectionEnd rotationExpansion jointImposed deformationJoint stiffnessMoment area methodRestraintServiceabilitySpan to depth ratioStiffnessTolerance