Deflection — where it appears
Named by 34 essays across 8 fields — each of them below, with the objects they name alongside it.
The deflection that arrives three years late
A concrete beam that passes every check on the day it is built goes on deflecting for a decade, and ends up three times where it started. Nothing about the load changed, and nothing about the strength was ever in question.
The redistribution nobody chose
A beam designed as simply supported, on connections that are not pins, has end moments the analysis never predicted and a mid-span moment smaller than it was sized for. Usually that is safe. It is never intentional, and there is one direction in which it is not safe at all.
The period nobody chose
Every structure has a natural period, it decides the answer to every question in this field, and no drawing anywhere records it. It is a consequence of a mass picked for one reason and a stiffness picked for another — and it has already been computed, by the serviceability check.
The deflection that is not bending
Engineer's beam theory computes a deflection as the second integral of a moment, and that calculation silently drops a term. The beam also shears, and the shear deflection is not a correction to the curve — it is a different shape, and for a deep member it is most of the answer.
Which member moved the roof
A beam sags because it curves. A truss has no curvature anywhere — it comes down because every one of its members changes length, and the sum of those changes, weighted member by member, is a ranking that names which ones are worth stiffening. Usually not the ones a designer worries about.
The area of a diagram is a rotation
A deflection is the double integral of a bending moment, and the two constants of integration are the whole difficulty. Mohr's theorems replace them with two pictures — an area, and where that area's centre of gravity sits.
Built to the wrong shape on purpose
A cambered beam is fabricated curved upward so that load bends it down to something like straight. Nothing in the analysis changes, no stress anywhere is altered, and almost every mistake made with it is a bookkeeping mistake about which loads count.
The hole that costs nothing, and everything
A service opening removes 30% of a beam's second moment and 0.7% of its deflection. What it costs is not that. Across the opening the shear has nowhere to go but through the two tees, and a tee carrying shear over a length bends.
The deck is not there to carry the load
A cable takes the shape of whatever is on it, which is exactly the problem — under a point load its shape is a kink, and a kink is not a road. The stiffening girder exists to spread the load until what reaches the cable is something the cable's own shape is right for.
The section that changes along the span
A prismatic beam is checked where the moment is largest, and everyone knows where that is. A tapered one is not, because the capacity is moving too — and for a cantilever with a load at its tip the governing station is exactly where the depth has doubled, with no length, no load and no material in the answer.
The deflection that belongs to the support
A beam calculation answers a question about a beam sitting on things that do not move. Real ones sit on bearings, on other beams and on columns that shorten, and every one of those is a spring in series with the member — so a deflection is the sum of two things and only one of them is a property of the beam.
The deflection that is a derivative
A structure's strain energy is one number. Differentiate it with respect to a load and out comes the displacement under that load — and the trick that makes it a method rather than an identity is that the load does not have to be there.
Half the studs, and most of the beam
Bonding two layers together quadruples the stiffness of the pair. A real connection is a row of studs that deform, so the layers slide a little and the beam sits between the two answers — but not halfway between, and the number that decides where is a single dimensionless group.
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.
Stiffer than its cracked section says
At a crack the concrete below the neutral axis has gone and the steel carries the tension alone. Between the cracks it has not gone — bond drags it back into tension, the steel strain drops, and the curvature averaged over a length of beam is neither section's.
Where a deflection comes from
The unit-load method gives a deflection as an integral, and this collection has treated that integral as a number to evaluate. It is not a number. It is a density, and it says which millimetres of the member produced the answer — which is not the same map as where the moment is largest.
A third of the load crosses sideways
A slab spanning both ways is usually explained as two beams sharing a load by a fourth power, and the explanation is not merely approximate — it is missing a mechanism. A real plate carries load three ways, and the third one has no beam strip in it: it is twisting, it accounts for a third of the load on a square panel, and it is why the corners lift.
The weight that has to be known before it can be found
Every other load arrives from outside and can be looked up. A structure's own weight depends on how big it is, and how big it is depends on the load — so the first calculation on any project is a fixed point, and the fraction of a member spent carrying itself turns out to be the square of its span as a fraction of a span it can never reach.
The curvature nobody applied
Concrete shrinks as it dries, by about half a millimetre in every metre. In a symmetrically reinforced member that is a shortening and nothing else. In a member with more steel in one face than the other — which is every beam and every slab — the steel holds one side back and the section bends, with no load on it at all.
The answer is continuous and the catalogue is not
Every optimisation in this subject returns a number with three significant figures in it, and nothing with three significant figures can be bought. What can be bought is a rolled series whose steps are a quarter to a half apart, so the member that goes on the drawing is on average a tenth stronger than the one that was calculated and can be a third stronger for no reason at all.
The gap nobody computed
A movement joint is sized by adding up everything the structure will do to it, and the deflection calculation — the only term anybody computes carefully — is usually the smallest one in the list. The largest is a construction tolerance, which is not a structural quantity at all, and the sum of the extremes is nearly twice what treating them as independent would ask for.
Stiffer than the model said
Measured natural frequencies of finished buildings come out between ten and sixty per cent above the values computed for them, consistently and in one direction only. Nothing on the list of reasons is a modelling error: every one is a real source of stiffness deliberately left out — and leaving stiffness out is conservative for deflection and unconservative for vibration.
The beam whose moment is a deflection
A beam's bending moment is the second integral of its load. Its deflection is the second integral of M/EI. They are the same problem, so a deflection can be found by loading a fictitious beam with M/EI and asking a statics question — and the only thing to remember is the supports, which are not remembered but derived, one boundary condition at a time.
The pile that is too short to bend
A laterally loaded pile is long or short in its own characteristic length, and the two are different structures. The slender pile bends and the ground below four characteristic lengths never hears about it; the monopile is two lengths deep, rotates about a point, and every tabulated coefficient written for the first case is wrong for the second.
The connection is busiest where the beam is not
A composite beam's studs are spaced evenly along it and the demand on them is not even at all. It peaks at the supports, where the bending stress is nothing, and falls to zero at mid-span, where the section is working hardest — so the connection is designed from a diagram nobody looks at.
An influence line is a deflected shape
Finding where a load has to stand to be worst means solving the structure once for every position it could stand in. Reciprocity says the answer is a single deflected shape — release the quantity being asked about, move it by a unit, and the shape the structure takes is the influence line.
The member that is not worth stiffening
A truss's deflection is a sum of one term per member, and a term is zero whenever either force in its product is. A vertical carrying the whole of a panel load can contribute nothing at all to the movement — which a total can never show and a per-member sum shows nothing else.
The stiffness that belongs to the span
A section's flexural rigidity is a property of the section, and a member's is not. Once shear deformation is counted the effective stiffness contains the span, so the same panel is a different member at three metres and at six — and for a sandwich the correction is not a correction.
Equivalent in work, not in resultant
Two force systems with the same resultant and the same moment about every point are interchangeable — for a rigid body. A finite element is not a rigid body, and substituting one for the other on a beam element leaves the tip of a cantilever a third too low with no warning of any kind.
The drawing that is right except for a rotation
Williot's construction finds every joint of a truss from its members' changes of length alone, in one drawing — and puts the roller six thousand units off its support. The error is one rigid rotation, Mohr's diagram takes it away, and a drawing started from the member symmetry holds still never makes it.
The shear that moves the moments
In a statically determinate beam, shear deformation adds movement and changes no force. In a redundant one the forces come from how the beam deforms, so a stubby beam soft in shear carries less moment at its wall and more in its span — and the carry-over factor, the one half every hand method passes along, falls to nothing at φ = 2 and then changes sign.
The creep that belongs to the member
A creep coefficient is quoted for a concrete, and half of it is not a property of the concrete. It is the member drying out, and drying goes through the surface — so one mix creeps a quarter more in a thin slab than in a deep beam, gets there years sooner, and in saturated air forgets its size altogether.
The calculus answer, rounded, is the worst one
A real truss is built from a catalogue, and every member is rounded up to the next section. That rounding costs its stiffness almost nothing: the few per cent that separate the strength design from the stiffest survive it. What does cost is the next step. When a deflection limit governs, the obvious move — take the continuous optimum and round it up — needs more steel than any other way of stiffening the truss, and the exact discrete answer is within one per cent of a bound no catalogue can beat.
The truss whose forces follow its sections
In a determinate truss each member's force is fixed before its section is chosen, so sizing for strength and sizing for stiffness can be done in either order. Put a counter-diagonal in every panel and they cannot. The fully stressed design becomes an iteration that starves some counters to nothing and keeps others, lands on a different truss from every start, and — whichever it lands on — weighs the same and deflects the same. Then enlarge one group of members to stiffen it, and a vertical nobody touched is overloaded by 58 per cent.
Named alongside it
The objects these essays reach for when they reach for this one.
ServiceabilityStiffnessVirtual workCreepCurvatureFree bodySecond moment of areaCompatibilityTrussUnit load methodComposite actionFlexural rigidity