Structural Engineering & Statics
where the load goes
Deflection

The settlement that matters is the difference

A building that goes down half a metre uniformly is undamaged and needs a new front step. One that goes down a twentieth as far, unevenly, has cracked. The superstructure can even the difference out — and the only way it can do so is by carrying the difference itself, as a force.

Assumes The deflection that belongs to the support, The beam that sits on the ground and Stiffness is not strength, and usually it is the one that governs.

21 min read 12 figures One support too manyWhich failure arrives firstThe load must go somewhere

Every building settles. The interesting question is not how far.

A structure that goes down uniformly is undamaged: no member changes length relative to its neighbours, no beam picks up a rotation it did not have, nothing cracks. It has a step at the front door that needs rebuilding and a drain that runs the wrong way, and structurally it is the building it was.

What damages it is the difference between one foundation and the next. And the measure that predicts damage is not the difference either — it is the difference divided by the distance over which it occurs.

β=δi+1−δiL\beta = \frac{\delta_{i+1} - \delta_i}{L}

the angular distortion, which is a rotation the structure has been given without being asked.

The building does not care how far it went down; it cares how much it tiltedFive footings on soil that is 35% as stiff under one of them, carrying 60 kN/m. They settle between 10 and 20 mm, and the number that matters is neither of those: it is the angular distortion between neighbours, 2.41 per thousand, or one in 415 — against a limit of one in 500 for cracking in finishes, which this does not. A building that went down half a metre uniformly would be undamaged and would need a new front step; this one has moved a twentieth as far and has cracked.1017201710the frame, before it settledsoft ground heresettlement in millimetres, mean 15worst angular distortion 1 in 415 between footings 0 and 1limit 1 in 500 — fails
Fig. 1 Five footings on soil that is 35% as stiff under one of them, carrying 60 kN/m. They settle between 12 and 54 mm, and the number that matters is neither of those: it is one in 231 between two neighbours, against a limit of one in 500 for cracking in finishes.

Which free body produced the number

The superstructure, taken as a beam sitting on springs.

The frame spans between its own foundations, so it is not a passenger. Each footing pushes up with kivik_i v_i, the frame bends between them, and the whole thing is an elastic beam on discrete elastic supports — which is the machinery this site already has for a beam on bearings, applied one level down.

Three quantities come out and they behave very differently.

The mean settlement is fixed by statics. Vertical equilibrium says ∑kivi=W\sum k_i v_i = W. If every kik_i is the same, the mean of the viv_i is W/nkW/nk and no arrangement of the frame’s stiffness can move it. Computed here: 24.0 mm for a frame at a thousandth of the soil’s stiffness and 24.0 mm for one at a thousand times it, agreeing to 3×10−103\times10^{-10}.

The differential settlement falls as the frame stiffens. The tilt across the building goes from 66.5 mm to 0.04 mm across the same range.

And the bending moment rises to meet it. 237 kNm to 1,665 kNm — a factor of seven, and still climbing at the stiff end.

How much of a deflection belongs to the beamThe share of the total deflection that is the beam's own bending, against the stiffness of what it sits on. A 24 m beam on five supports under a uniform load: on rigid supports every millimetre is the beam's, and the share falls away as the supports soften until almost none of it is. The beam drawn beside this figure sits at 1% — so 99% of what it does is happening somewhere a beam calculation never looks. The two flexibilities are in series, which means the softer one governs and stiffening the other buys nothing.4020080030002000000.20.40.60.81stiffness of each supportfraction of the deflection that is bendingrigid supportsare over here1%the beam drawn
Fig. 2 The same beam on springs asked the other question: how the load is shared rather than how far the supports move. The two are the same solve read at different outputs, and the sharing is what the evening-out costs.

The trade, drawn

Sweep the frame’s bending stiffness against the soil’s, holding everything else, and both curves are monotone in opposite directions:

EI/kL3EI/kL^3 tilt peak moment soft support’s share
0.001 66.5 mm 237 kNm 22.8%
0.03 50.8 269 18.1%
0.2 38.0 401 14.9%
1 20.1 963 11.6%
6 5.7 1,467 9.0%
200 0.2 1,659 8.1%

There is no stiffness at which both are small. The frame either accepts the movement or carries the force that prevents it, and the choice between them is what “designing for settlement” actually means.

The right-hand column shows the mechanism. A limp frame gives the soft support 22.8% of the total load — near its tributary share of 25% — while a rigid one gives it 8.1%. The frame is bridging over the soft patch, and the load it takes off that footing has gone onto the neighbours, along with the moment required to carry it there.

Evening the settlement out is a way of paying for it in forceTwo normalised curves against the superstructure's stiffness relative to the soil it stands on. The falling one is the differential settlement across the building — with a soft patch under one footing, a limp frame follows the ground and tilts by 33 mm, while a stiff one bridges over it and tilts by 0.0. The rising one is the bending moment in the frame, from 59 kNm to 416. There is no stiffness at which both are small: the frame either accepts the movement or carries the force that prevents it. The mean settlement, meanwhile, is fixed by statics — with uniform soil it does not move at all, to 3e-10 of itself, however stiff the building is made.10^-310^-210^-110^110^210^30%20%40%60%80%100%frame stiffness ÷ soil stiffness, EI ÷ k·L³share of its own largest valuemoment in the frameup to 416 kNmdifferential settlementfrom 33 mm
Fig. 3 The two curves normalised to their own largest values. The crossing region is where a real building sits, and it is broad — which is a mercy, because the soil stiffness is the least well known number in the whole calculation.
Two beams tied together, and the deeper one takes 89% of the loadTwo simply supported beams of 6 m, one twice as deep as the other, tied together at midspan so that they have to move as one. A load of 100 kN stands on the tie. Point stiffness is 48EI/L³, so the deeper beam is 8 times as stiff — depth cubed, nothing else — and the load divides in that ratio: 11.1 kN into the shallow beam and 88.9 kN into the deep one, 11% against 89%. Both midspan points move 50000.00 mm, which is the whole of the argument: the geometry of the load never entered it. The deflection is drawn 0 times full size — the real sag is 50000.00 mm on a 6 m span, about 1 in 0.P = 100 kNthe shallow beam takes 11.1 kN11% of it — one part of the stiffness in 9the deep beam takes 88.9 kN89% of it — 8 times the stiffness of its neighbourone load, two beams, one deflection: 50000.0 mm each
Fig. 4 The stiffest path takes the load, which is what the last column is. A stiff frame on soft ground routes load away from the soft support; the routing is free in the sense that nobody chose it, and expensive in the sense that a moment came with it.

Why the criterion is a rotation

Buildings do not crack because they moved. They crack because something in them was rotated past what it could take.

A partition sitting between two floors that differ in level by Δ\Delta over a length LL has been sheared by Δ/L\Delta/L. A brittle finish accommodates about one part in five hundred before it cracks visibly and about one in three hundred before the crack is structural. A frame’s members are more tolerant — cladding distortion and door frames generally complain first.

Which gives the numbers the whole subject is written in:

angular distortion what happens
1/1000 nothing
1/500 first cracking in brittle finishes
1/300 visible cracking, doors and windows binding
1/150 structural distress in the frame

The building drawn is at 1 in 231, which fails the first and is approaching the second. Its mean settlement of 31 mm is entirely unremarkable — a building can settle 100 mm uniformly and nobody will know.

Two buildings with the same settlement can be in completely different conditions, and the number that separates them is a slope.

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 1.69the limitstrengthdeflection
Fig. 5 Which limit arrives first. A settlement calculation and a strength calculation are two entirely separate checks with two entirely separate criteria, and on a building of ordinary proportions on ordinary ground it is the first that binds.

Sagging and hogging are not the same damage

The profile drawn dishes downward in the middle, which is the commonest shape and the more forgiving one. A building that hogs — settling more at its ends than at its centre, which happens over a stiff lens or where the ends are more heavily loaded — is in worse trouble for the same distortion.

The reason is where the tension is. A sagging profile puts the building’s top in compression and its bottom in tension, and the bottom of a building is generally a stiff, well-reinforced ground floor with a raft or a heavy beam in it. A hogging one reverses that: the tension is at the top, where there is nothing but the roof and the cladding, and a masonry facade in tension cracks at a much smaller distortion than one in compression.

The usual allowance is a factor of two: a hogging profile is treated as twice as damaging as a sagging one at the same angular distortion. That is a rule rather than a derivation, and what is worth carrying is why a rule is needed at all — the damage criterion depends on where the building’s own tension capacity is, which is a property of the structure and not of the ground.

The shear part is not a curve at allA 300 × 600 rectangle spanning 2,400 mm, a span-to-depth ratio of 4, with its bending deflection and its total drawn together. Bending gives 0.0254 mm and shear a further 0.0049 mm, so 16.3% of the movement is the term beam theory drops. The shear part is drawn separately beneath: two straight lines meeting under the load, each limb straight to 2e-16 of the peak. It is the shear diagram integrated once where the moment diagram above it is integrated twice, so it carries one degree of curvature less than the bending shape — none at all here. The real movement is 1 in 79,116 of the span; both panels are drawn at about 11,932 times it.100 kNthe gap is the shear: 16.3% of the totalbending alone, and what the beam really doesthe shear part alone, magnified 3 times furthertwo straight lines meeting under the loadshearγ = V/GAs is a slope the section is racked through, not a curvature —so this diagram is integrated once, where the moment diagram above it is integrated twice
Fig. 6 Two curvatures of opposite sign, which is the whole distinction. A settlement profile is a deflected shape imposed from below rather than caused by load, and everything this site knows about which fibre is in tension applies to it unchanged.

The timing matters as much as the magnitude

A settlement that happens before the finishes are installed does no damage at all. One that happens afterwards does all of it.

On sand, most of the settlement occurs as the load is applied — during construction, before the partitions are up — so the damaging fraction is small. On clay, consolidation takes years, and nearly all of it happens after the building is occupied.

That single distinction is worth more than any refinement of the calculation. The same 40 mm of differential settlement is harmless on one soil and cracks the building on the other, and the difference is entirely in the timing.

It is also the reason a construction sequence is a settlement variable. Building the heavy core first and the light perimeter later lets the core’s settlement happen before the connection between them is made — the same manoeuvre that delayed connections use for differential shortening, applied to the ground instead of to creep.

Two differences up the same building, peaking in different placesDifferential shortening between a perimeter column and the core of a 40-storey building, plotted up the height. The part driven by load peaks at level 20 — exactly half way up, because a floor near the top has almost nothing built above it to shorten what is beneath, and a floor near the bottom has almost nothing beneath it to shorten. The part driven by shrinkage does not care what is above it at all and accumulates all the way to the roof. Their sum is worst at level 40, at 43 mm, which across a 9 m bay is a floor out of level by one in 208.-40-30-20-10010020406080100120140column shorter than core (mm)height (m)from loadfrom shrinkagethe sumworst 43 mmat level 40one in 208
Fig. 7 The same problem one level up: two vertical elements shortening by different amounts, connected by floors that have to accommodate the difference. Settlement and shortening are the same geometry with the movement happening at opposite ends of the column.
Cambered against the wet loadA 24 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 404.4 mm of camber, it moves through -330.9, 0.0, 119.6, 202.7 mm as the four stages arrive — 0.0 mm on the day the slab is poured, and 202.7 mm at the end, which is one part in 118 of the span. The largest curvature it ever has is 404.4 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 14935 times the horizontal.levelas fabricated: 404.4 mm of camber-330.90.0119.6202.7self-weight of the steel: 73.5 mm on EI = 94wet concrete: 330.9 mm on EI = 94finishes and services: 119.6 mm on EI = 260imposed load: 83.1 mm on EI = 260
Fig. 8 And the sequence as a running total. What a partition sees is not the settlement but the settlement after the day it was built, which is a different curve with a different peak.

The raft, which changes the question

Everything above assumes separate footings. Put the building on a raft and the problem changes shape, because the raft is a continuous beam on a continuous elastic foundation rather than a frame on discrete springs.

The mechanism is the same and the governing number becomes a relative stiffness of the classic kind:

Kr=EsIsEsoilB3K_r = \frac{E_s I_s}{E_{soil} B^3}

with the structure’s own stiffness over the soil’s. Below about 0.01 the raft is flexible and follows the ground; above about 0.5 it is rigid and settles as a plane. In between it does both, and the moments in it are a function of that ratio rather than of the load.

The uncomfortable consequence is the same one: a rigid raft has almost no differential settlement and enormous bending moments in it, and a flexible one is the other way round. The raft’s thickness is chosen on that trade, not on the pressure under it.

The rigid calculation is wrong in both directions at onceWhat a rigid raft calculation gets, divided by what the beam on the ground gets, against the raft's length in characteristic lengths. Spreading 1000 kN uniformly over a length L gives a moment PL/8 and a pressure P/L, while the flexible answer is P/4β and Pβ/2 whatever L is — so the two ratios are βL/2 and 2/βL, which are reciprocals. They cross at βL = 2, that is L = 5.13 m, and they are the only pair of errors that vanish together. Past it the rigid calculation overstates the bending moment without limit, because PL/8 grows with the raft and P/4β does not, and understates the peak contact pressure in the same proportion. The raft drawn here is 24.00 m, or βL = 9.36: its moment comes out 4.68 times the truth and its pressure 0.21 times, whose product is 1.000. Hetényi's classification by βL calls a raft of that length long — behaves infinite.12345600.511.522.53βL — the raft's length in characteristic lengthsrigid answer ÷ flexible answerboth right at βL = 2, L = 5.13 mmoment: ×4.68overstated, without limitpressure: ×0.21understated, in proportiontheir product: 1.000
Fig. 9 The raft as a beam on a continuous foundation, which is the same equation with the springs smeared. Its characteristic length decides how far a column load spreads, and whether the raft behaves as one member or as a set of independent footings.
A range of 100 in the ground is a range of 3.2 in the answerThe characteristic length of the same 540 × 10³ kNm² strip on eight soils, each drawn as the band its subgrade modulus is quoted over rather than as a point. From 5 to 500 × 10³ kN/m³ is a factor of 100, and 1/β = (4EI/k)^¼ turns it into a factor of 3.16 — the fourth root, 3.16, exactly. So the softest ground here gives 4.56 m and the stiffest 1.44 m, and the design moment P/4β moves by the same 3.16 rather than by 100. Eight soils span 2.0 decades of stiffness and 0.50 decades of length. Arguing about the subgrade modulus to two figures is not where the uncertainty is.very soft clay4.56 mloose sand3.59 mfirm clay3.08 mstiff clay2.45 mdense sand2.11 mdense gravel1.64 mweak rock1.21 msound rock0.81 m012345characteristic length 1/β (m)
Fig. 10 And the range of soils the same raft might sit on — two orders of magnitude of subgrade stiffness, which moves KrK_r by the same two orders and takes a raft from flexible to rigid without anything about the structure changing.

A settlement is a support that moved

There is a shorter way to hold all of this, and it connects to something this site established long ago.

A support that moves in a determinate structure causes no forces at all — the structure simply takes up a new position, and every internal force is exactly what it was. In an indeterminate one it causes a complete field of moments in equilibrium with no applied load whatever, proportional to EIEI.

Differential settlement is precisely that, with the movement supplied by the ground rather than by a jack. So the whole trade above is a restatement of the redundancy result: a stiffer structure attracts more force from an imposed displacement, and a determinate one attracts none.

Which gives the design move the last section was circling. A building that cannot be made stiff enough to bridge the soft patch can instead be made less redundant — a movement joint through the full height, articulated foundations, a determinate arrangement of transfer beams — and it will then follow the ground with no force at all. The two strategies are the two ends of the same curve and there is nothing in between except compromise.

One support too manyThe same uniformly loaded beam with three sets of restraints, and the bending moment in each. Adding restraint moves moment from mid-span to the supports and lowers the peak — but only the first case can be solved by statics.simply supportedstatics alonesag 4320.0propped at one endneeds stiffnesssag 2430.0hog 4320.0built in at both endsneeds stiffnesssag 1440.0hog 2880.0the load never changes; only what is holding the endsthe built-in case peaks at two-thirds of the simple span's moment
Fig. 11 The redundant that a movement creates. Push one support of a two-span beam down and a complete moment diagram appears with no load anywhere on the beam — proportional to EI, invisible to statics, and exactly what a settling foundation does.
The same diagram, from two entirely different arithmeticsBending moments on a two-span beam under 60 kN/m. The curve is the stiffness solution; the marked points are what moment distribution reached after eight cycles of hand arithmetic — 0.0, 1080.0, 0.0 kNm at the supports against an exact 0.0, 1080.0, 0.0. The two share no code and no equations, which is the only arrangement in which agreement is evidence of anything.0.01080.0the largest disagreement is 1.6e-2 kNm
Fig. 12 And the redistribution that follows once the frame starts to crack. A settlement moment is an imposed-deformation effect, so the first hinge that forms relieves it rather than propagating it — which is why these effects are so much less dangerous in practice than a linear analysis suggests.

Where the model stops

The soil is a set of independent springs. It is not: pressing one footing down pulls its neighbours with it, because the soil is a continuum. That coupling makes the real settlement bowl wider and shallower than a Winkler model gives — so the model above overstates the local differential and understates the overall dish. Getting it right needs an elastic half-space rather than springs.

The spring stiffnesses are known. The subgrade modulus is the least reliable number in foundation engineering: it is not a soil property at all, it depends on the size of the loaded area, and quoting it to two significant figures is an act of optimism. Everything on this page should be read as a sensitivity study rather than a prediction.

And the frame is linear and elastic. A real frame cracks, its joints yield a little, and a settlement applied over ten years is resisted with a creep-reduced stiffness rather than a short-term one — all of which move it toward the flexible end of the trade and reduce the moments it actually carries. Settlement effects are self-relieving, which is why buildings survive differential settlements that a linear analysis says should have destroyed them.

What the pictures cannot show

The settlement profile is drawn as a smooth line through five points. Real ground varies over metres, so the true profile has features between the footings that no calculation on a five-spring model contains.

Nor can the figures show time. Every diagram here is a final state, and the thing that damages a building is a rate as much as a magnitude — a settlement delivered over a decade is accommodated by creep and one delivered in a week is not.

The assumption the figure rests on

The soft patch is one support at 35% of the others’ stiffness, and the answer depends on that entirely. Real variability is not a patch under one footing; it is a field, and the worst arrangement is rarely the one somebody guessed. What is drawn is a load case rather than a prediction, and the honest use of it is to run several — a soft patch under each footing in turn, and a smooth tilt across the whole plan — and design for the envelope.

The measurement that settles it

None of this is predictable to better than a factor of two, and the profession’s response is worth recording because it is unusual: it measures.

Levelling points are installed on the columns before the frame is complete and read at intervals for years. What comes back is a set of settlement histories, one per column, from which the differential and its rate are read directly — and if the readings are diverging from the predictions, the response is available while there is still something to be done about it.

That is the observational method, and it is one of the few places in structural engineering where a design is allowed to be provisional. The calculation sets the trigger levels; the measurements decide whether the contingency is needed. It exists here rather than elsewhere because the input — the ground — is the one material a designer cannot specify, test in advance, or replace.

The habit worth taking from it is narrower than the method. When the governing input has a factor of two of genuine uncertainty in it, the useful output is a trigger level rather than an answer, and a number quoted to three figures from such an input is a number pretending to be a different kind of number.

The ladder from here

Later rungs on this anchor: the elastic continuum in place of springs, and how much the coupling changes the answer. Consolidation settlement against immediate settlement, and the timing question made quantitative. The relative-stiffness parameter KrK_r derived rather than quoted, and the two thresholds that bound it. Damage criteria from the other direction — deflection ratio and horizontal strain, which predict cracking better than angular distortion does for a sagging profile. Piled rafts, where the piles and the raft share the load in a proportion that is itself a stiffness ratio. And the case this page keeps circling: a building whose stiffness is deliberately reduced — movement joints, articulated foundations — so that it follows the ground rather than fighting it.

Where this goes next

4020080030002000000.20.40.60.81stiffness of each supportfraction of the deflection that is bendingrigid supportsare over here1%the beam drawn 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. 1000 kN1/β = 2.56 mthe beam lifts off at 6.04 msettlement 3.90 mm · contact pressure 195 kN/m · moment 641 kNm, none of which contains the length of the beam The beam that sits on the ground Every other beam in this collection is held at points. A footing is held everywhere, by something that pushes back in proportion to how far it is pushed — and that single change hands the structure a length it did not choose. Two or three of those lengths from the column, nothing knows the load happened. this support 0.01 lowmoment19.6 sagging24.5 hogging30.6 if the spans were simplereactions 14.0 38.5 38.5 14.0 — the inner supports carry far more than a sharethe continuous case needed stiffness; the comparison did not The support that moved A redundant structure knows things statics cannot see. Settle one support by ten millimetres and a complete set of bending moments appears — in equilibrium with no load at all, and larger for a stiffer beam. 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.

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.

Angular distortionBeam on springsDamage criterionDifferential settlementFoundationLoad pathRaftRedistributionRelative stiffnessServiceabilitySoil structure interactionSubgrade modulusTilt