The crack is a strain, not a slope

The settlement that matters is the difference makes the case that a building does not care how far down it went, only how much it tilted — and measures the tilt as an angular distortion, the slope between neighbouring footings, against limits of one in five hundred and one in two hundred and fifty.

Those limits work. They are also a proxy, they were obtained by looking at damaged buildings rather than by any argument, and there is a mechanism underneath them that says when they will mislead.

Which free body produced the number #

The free body is the whole building, taken as a deep beam spanning over the settlement profile: a rectangle of length $L$ and height $H$, of some modulus, with the ground’s movement imposed on it as a deflected shape.

That is a large idealisation and it is the useful move. A building is not a beam, but a masonry facade is a plane element with a length and a height carrying its own weight, and what cracks it is a tensile strain in that plane — so a model that produces the strain field from the shape is answering the right question in a way an angular distortion cannot.

The beam deflects in two ways at once and each puts tension somewhere different.

Bending puts a tensile strain at the extreme fibre — the bottom of the wall in sagging, the top in hogging — and the strain for a given deflection is proportional to the distance from the neutral axis to that face.

Shear puts a tensile strain at 45°, which is a diagonal crack, and the strain for a given deflection has nothing to do with the neutral axis at all.

Setting each against a critical tensile strain gives two limiting deflection ratios, and the governing one is the smaller. Both are one line of algebra from the deep beam’s own deflection, split into a bending term and a shear term.

How it compares with the slope it replaces #

The two criteria answer different questions and it is worth putting numbers on how differently.

An angular distortion is a slope between two points; a deflection ratio is a sag over a length. For a settlement profile that is a smooth curve, the two are related — a parabolic sag of Δ over L has a maximum slope of about 4Δ/L — so a limit of 1 in 500 on the slope corresponds to a deflection ratio of about 0.5 per mille.

That is very close to the 0.41 per mille the deep beam gives for sagging at these proportions, and the closeness is not a coincidence: the angular distortion limits were calibrated on the same damaged buildings the critical strains were. Two descriptions of one dataset, and they agree where the dataset is.

Where they part company is everywhere else.

Hogging, where the deep beam says half and the slope criterion says nothing.

Proportions, where the deep beam has a minimum at L/H ≈ 1.6 and the slope criterion has one limit for every building.

And horizontal strain, which the slope criterion cannot contain because it is not a shape at all.

So the older criterion is a good summary of the cases it was drawn from and a poor extrapolation, which is exactly what an empirical limit should be expected to be. The settlement that matters is the difference remains the right first check; this is the one for the cases it was not calibrated on, and tunnelling under a terrace is the commonest of them.

Why hogging is exactly twice as bad #

The factor of two is the cleanest result in the model and it is geometry rather than a fit.

In sagging, the building’s neutral axis is near mid-height: the wall above the sag is in compression and the wall below is in tension, so the distance from the axis to the tension face is $H/2$.

In hogging, the neutral axis moves toward the bottom — a building hogging over a hard spot has its foundation acting as a stiff tie along the bottom, and masonry above cannot carry tension. The distance to the tension face becomes the full $H$.

Bending strain is proportional to that distance, so at the same deflection ratio the hogging building has twice the strain. It is exactly two, at any proportions, whenever bending governs.

That is worth carrying because it is invisible in an angular distortion. A settlement profile that hogs and one that sags can have identical slopes between footings, and one of them cracks the building at half the movement of the other.

The practical consequence is a rule about where a building is dangerous: the hogging zone is at the edge of a settlement trough, where the ground has already begun to flatten out, and it is a location a settlement contour plot makes look harmless. That is the same inversion the movement with no limit against it describes for twist: the quantity that is easy to draw is not the quantity that does the damage.

The minimum, and the two mechanisms #

The other feature of the first figure is that the limiting deflection ratio has a minimum rather than falling monotonically.

A squat building — L/H below about 1.5 — cracks diagonally. Its bending stiffness is enormous relative to its span, so almost all of the imposed deflection is taken as shear, and the diagonal tensile strain reaches the critical value first. The limit is high, because a deep beam is very stiff.

A long low building cracks in bending. Its shear stiffness is now the large one and the bending term dominates, and the limit rises again with the length. It is the same crossover a stubby beam has against a slender one, with cracking rather than deflection as the thing being decided.

Between them the limit is at its lowest, at L/H around 1.6, and that is where a building is least tolerant of a settlement profile. A three-storey terrace bay of 12 m width and 9 m height is squarely there.

That is a genuinely useful piece of guidance and it inverts the intuition. A long building is not more vulnerable than a short one; a building with proportions near square in elevation is, and the tall narrow one and the long low one are both better off.

What the horizontal strain does #

Everything above assumes the ground only settles. It does not.

Ground that moves down over a tunnel or an excavation also moves inward, so the surface stretches over the outer part of the trough and compresses over the middle. The stretch is a horizontal tensile strain in whatever is sitting on it.

It adds to the bending strain directly — both are tensile and in the same direction — so it comes off the top of the building’s capacity. Three millimetres over ten metres is 0.03 per cent, which takes the sagging limit from 0.41 per mille to 0.25.

And it coincides with the hogging. The outer part of a settlement trough is where the profile hogs and where the ground is in tension, so the two worst effects arrive at the same place. That is why the damage from tunnelling is concentrated at the edges of the trough rather than over the centreline, and it is a prediction the angular-distortion criterion cannot make at all.

The categories, which are strains #

The last piece is what the critical strain is, and the answer is that it is chosen rather than measured.

Burland’s damage categories are defined by what the damage looks like — hairline cracks under 0.1 mm are negligible, cracks up to 1 mm are very slight, up to 5 mm slight, up to 15 mm moderate — and each is associated with a limiting tensile strain: 0.05, 0.075, 0.15 and 0.3 per cent.

Those numbers are not material properties. Masonry cracks at a tensile strain of about 0.05 per cent, and the higher categories describe a wall that has already cracked and is being asked to open the cracks further. The category is a description of an outcome, and the strain is what has to be computed to predict it.

So the criterion is a chain: a ground movement gives a deflected shape, the shape and the proportions give a strain, and the strain is looked up in a table of what a building with that strain looks like. Every step is computable except the last, which is a judgement about acceptability — and that is the right place for the judgement to be, since a warehouse and a listed facade will accept different things.

Where it is used, which is under cities #

The criterion exists because of tunnelling, and the way it is used is worth describing because it is one of the few places in this collection where a structural calculation is done on thousands of buildings at once.

A tunnel under a city produces a settlement trough of known shape — a Gaussian, in the standard model, with a width set by the depth and the ground. Every building within it is assessed in three stages.

Stage one is the greenfield movement: compute the trough as though the buildings were not there, and take the deflection ratio and horizontal strain at each facade. Buildings below a threshold are cleared.

Stage two is this criterion: put those two numbers into the deep beam, get a strain, and read a damage category. Most of the remainder is cleared here, and the ones that are not go on.

Stage three is a detailed assessment: the building’s own stiffness, its condition, its openings, and often a monitoring regime and a protective measure.

The efficiency of that funnel is what makes a project like a metro line possible at all, and it depends entirely on stage two being both quick and defensible. A criterion that is one line of algebra and calibrated on real damage is what allows ten thousand buildings to be assessed and fifty to be underpinned, and it is why the deep-beam model is worth more than its accuracy on any single building would suggest.

There is one more thing the three-stage assessment does that is worth noticing, because it is a piece of engineering judgement dressed as a procedure. The greenfield movement used in stage one is deliberately the wrong number — it is the movement the ground would make with no building on it, and every building reduces it. Using it means the assessment is conservative by an amount that grows with the building’s stiffness, so the buildings most likely to be cleared early are the flexible ones that genuinely follow the ground, and the ones that survive to stage three are the stiff ones where the conservatism is largest. The funnel is arranged so that its errors push work toward the cases that deserve it.

Where the model stops #

A building is not a beam. It has openings, which reduce the shear stiffness enormously and concentrate strain at their corners — where the cracks actually appear. It has floors, which act as ties or do not depending on how they are connected. And its facade and its cross-walls behave differently.

The neutral axis is assumed rather than found. The hogging case’s “neutral axis at the bottom” is a statement about masonry not carrying tension, which is right for an old wall and wrong for a reinforced concrete frame with cladding.

The building is assumed to follow the ground. The relative stiffness argument says it does not: a stiff building bridges the trough and its deflection ratio is smaller than the ground’s. In practice both calculations are done — the ground’s profile gives an upper bound and the interaction gives the real one.

And the load path is not in it. The deep beam carries no load in this model; it is being deformed. A building whose weight is redistributing as it settles has forces changing too, and the strain from those is on top of everything here.

What the pictures cannot show #

Where the crack goes. The model gives a strain and a mechanism and says nothing about the crack’s location, which is decided by the openings, the bond pattern, the lintels and whatever the wall was repaired with. A prediction of “very slight damage” is a prediction about the total width of cracking rather than about any one crack.

They also cannot show time. Settlement arrives over months or years, and masonry creeps: a strain applied slowly is partly relieved before it cracks anything, which is why a building damaged by tunnelling in a fortnight and one settling the same amount over five years are not comparable — and why a crack width measured on site is a record of what has happened rather than of what was imposed. The criterion has no rate in it, and the conservatism that introduces is not quantified anywhere.

The assumption the figure rests on #

That the strain is uniform over the face it acts on.

It is not, in the one place the model is most used. A building with windows has piers between them, and the strain that the deep beam computes as a smooth field concentrates into those piers — by the ratio of the wall’s gross length to the sum of the piers, which for a Georgian terrace is a factor of two or three.

So the computed strain is an average and the strain that cracks a lintel is several times it. The model absorbs that by having its critical strains calibrated on real damaged buildings with real windows in them, which is what makes 0.075 per cent a useful number and not a masonry property.

Which is the general shape of the thing: a mechanical model with a calibrated constant in it is a good deal more useful than either a purely empirical rule or a purely mechanical one, and it inherits the limitations of both — it does not transfer to a building unlike the ones it was calibrated on, and it does not predict anything the mechanism cannot represent.

What to carry away #

Four things, and the last is the one that generalises past settlement.

A building damaged by ground movement is cracked by a tensile strain, and the deep beam is what turns a shape into one. Angular distortion is a proxy that works where it was calibrated.

Hogging is worse than sagging by exactly two, because the neutral axis moves to the bottom and the tension face becomes the whole wall. The hogging zone is at the edge of a trough, which is where a contour plot looks safest.

Horizontal strain subtracts directly from what is left, and it arrives at the same place as the hogging.

And a proxy is only as good as the range it was fitted over. The angular-distortion limits and the critical strains come from the same damaged buildings and agree on them; outside that range one of the two has a mechanism in it and the other does not, which is what decides which to trust. That is the same relationship a detail category has to a stress analysis, and the same one a bearing capacity has to a soil model.

The ladder from here #

Later rungs on this anchor: the interaction between the building’s stiffness and the ground’s, which decides the deflection ratio the building actually experiences rather than the one the greenfield profile offers. Openings and the strain concentration they produce, which is where the calibrated constants come from. Three-dimensional settlement troughs, where a building is skewed to the trough and has a twist as well as a sag. Protective measures — underpinning, compensation grouting, and the sequencing that keeps a trough away from a facade. And the same criterion applied to a frame rather than to masonry, where the tension face is a cladding panel and the mechanism is a joint rather than a crack.