Weaker in one place, and better on every average

A pushover is run to find out where the deformation goes, and the reason it is worth the trouble is that every cheaper analysis reports an average.

The hero is the demonstration. One eight-storey frame, run twice: once regular, once with its ground storey at half the stiffness and half the strength of the others. The regular frame reaches 219 mm at roof level and spreads it evenly. The soft one reaches 266 mm and puts 5.58 per cent of storey drift into the ground storey against 0.53 next to it — a concentration of 5.9 where the regular frame’s is 1.7.

What makes this essay rather than a restatement is what happens to the numbers a report would contain.

Three global numbers, all of which improve #

Run the same two frames through the quantities an engineer actually writes down.

The ductility demand falls. The regular frame’s target of 219 mm against an idealised yield of 70 mm is a demand of 3.10. The soft frame’s 266 mm against a yield of 136 mm is 1.96. On the metric that is supposed to say how much inelastic deformation is being asked for, the dangerous frame asks for less.

The first mode carries more of the mass. The regular frame’s first mode holds 85.6 per cent and needs two modes to reach 90; the soft frame’s holds 91.1 per cent and reaches 90 in one. On the measurement that says whether a single-mode analysis is trustworthy, the dangerous frame scores better.

And the period lengthens. 0.93 s becomes 1.04 s, which on a typical design spectrum is a move down the falling branch and a lower spectral acceleration. On the input, the dangerous frame is asked for less force.

None of the three is wrong. They are all correct statements about quantities that do not distinguish the two buildings, and the reason is the same in each case: they are properties of the structure as a single oscillator, and a soft storey is a statement about how one oscillator’s deformation is distributed inside it.

The yield displacement is the quantity to watch there. It doubled because the frame now yields early and softens gradually rather than late and sharply, so the equal-area bilinear fit puts its corner further out. A ductility demand is a ratio, and this frame improved its ratio by making its denominator worse.

Position decides almost everything #

The reason is one line of statics and it is worth having in front of the arithmetic. The storey shear at any level is the sum of the inertia forces above it. At the top storey that sum is one floor’s worth; at the ground storey it is all eight.

So a storey’s demand-to-capacity ratio is set by how much building is above it, and the triangular load pattern makes the shear profile roughly parabolic — steep at the base, nearly flat at the top. Halving the capacity of a storey carrying one-eighth of the total shear leaves it comfortable. Halving the capacity of the storey carrying all of it moves the whole mechanism there.

This is why the famous failures are ground-floor failures and not because the ground floor is special in any other way. An open ground storey for parking, a shopfront, a lobby with tall glazing — each is an architectural requirement that happens to land on the one storey where a strength reduction cannot be absorbed.

It also says something useful about the reverse case. A strong ground storey buys very little, because the mechanism simply forms in the next weakest storey above it, and the drift concentration reappears one floor up.

The order the storeys give way in #

That order is worth a section on its own, because it is not what a strength-based intuition predicts.

Every storey in the regular frame has the same stiffness and the same strength; what differs is the shear each one carries, and the shear is largest at the base. So the ground storey ought to yield first, and it yields fourth.

The reason is the strength taper. The generator gives each storey a capacity that falls with height, in the way a real frame’s does — columns get lighter as the load above them falls — and the taper is applied as a smooth function while the shear demand is roughly parabolic. Where the two curves are closest is not at the base but a third of the way up, so the first hinge forms there.

Which means the regular frame’s mechanism is a distributed one and its first hinge is nowhere near its most heavily loaded storey. That is the healthy behaviour: seven storeys yield in turn over a long flat branch, each one shedding load to the others, and the roof travels 219 mm while no storey exceeds 1.33 per cent.

The soft frame’s order is 1, 3, 4, 2, 5, 6, 7 — the ground storey has jumped to the front, and once it is there the sequence behind it barely matters. The remaining storeys yield because the analysis keeps pushing, not because the building is asking them to; by the time storey 3 yields the ground storey is already at four times its yield deformation.

A yield sequence is therefore a better diagnostic than a yield load, and it is one of the few outputs of a pushover that no elastic analysis can approximate at all.

How weak is weak enough to matter #

Put the four points together and the shape of the relationship is the finding: 100 per cent gives 1.7, 80 gives 2.9, 65 gives 4.3, 50 gives 5.9. The concentration rises smoothly and steeply from the moment the storey is weaker at all.

That matters because a code irregularity limit is a threshold, and a threshold implies a cliff. The physical quantity has no cliff in it. A frame at 81 per cent is regular and a frame at 79 is irregular, and they behave within a few per cent of each other; the difference between the regular frame and the one at the limit is larger than the difference the limit is drawing.

The limit is doing something useful anyway, and it is worth saying what. It is not identifying the frames that will concentrate their drift — every non-uniform frame does that. It is identifying the frames for which the simplified analysis method is no longer good enough, and requiring a better one. That is a statement about the analysis, and reading it as a statement about the structure is the mistake this section exists for.

What the mass distribution says, and why it says the wrong thing #

That is the sharpest instance of the essay’s argument, because effective modal mass is the measurement the profession uses to decide whether a simple analysis is adequate.

The soft frame’s first mode does carry 91 per cent of the mass. A single-mode analysis of it is an accurate representation of its dynamics — of the displacement of the roof, of the base shear, of the period. Everything it computes is right.

And what it computes has nothing to say about the storey it will fail in, because a mode shape describes the ratio of the storey displacements and the analysis reports a single scalar multiplying it. The dangerous quantity is inside the mode shape, and the modal mass is a measure of how much of the structure the mode describes rather than of what it describes.

The mode shape itself carries the information — the soft frame’s first mode has almost all its curvature in the bottom storey — and it is discarded the moment the analysis multiplies it by a spectral value and reports a force.

What an elastic analysis reports instead #

Set beside the pushover the analysis that would actually have been run, and the omission is specific rather than general.

A response spectrum analysis takes the frame’s modes, reads a spectral acceleration at each mode’s own period, and produces a set of storey forces, storey shears and storey drifts. It reports storey drift — so the objection cannot be that the quantity is missing.

What it reports is the elastic drift, and the elastic drift of the soft frame is concentrated by a factor it can compute honestly: the ground storey is half as stiff, so it takes about twice the drift of the others. A factor of two, from a frame that will deliver 5.9.

The gap between 2 and 5.9 is the inelastic redistribution, and it runs the wrong way. In the elastic model a soft storey attracts drift in proportion to its compliance and nothing more; once it yields, its stiffness falls to the post-yield value — three per cent of the elastic one here — and every further increment of roof displacement goes almost entirely into it. The concentration is a runaway rather than a ratio, and no linear analysis contains the mechanism that produces it.

Codes patch this with a displacement amplification factor: multiply the elastic drift by something between 4 and 6 to estimate the inelastic one. Applied uniformly to every storey, which is precisely the assumption that fails here — the regular frame’s drifts want a factor near 3 and the soft frame’s ground storey wants a factor near 12.

So the elastic analysis is not blind to the soft storey. It sees it, prices it at a factor of two, and applies a correction that assumes the concentration does not grow. That is a more uncomfortable position than being blind, because the output looks like an answer.

What the drift is actually doing to the building #

The concentration is a ratio and the thing it is a ratio of is a rotation.

A storey drift of 5.58 per cent on a 3.5 m storey is 195 mm of horizontal movement across one floor, and every column in that storey has to accommodate it as a chord rotation of the same 5.58 per cent while carrying the whole weight of the building above.

Three things arrive together at that number and each is a different limit.

The columns’ plastic rotation capacity. A well-detailed reinforced concrete column can deliver perhaps 3 to 4 per cent chord rotation before its cover spalls, its bars buckle and its axial capacity is lost. 5.58 is past it.

The P-delta moment. The whole weight above, displaced by 195 mm, is an overturning moment on the storey that no analysis of first-order equilibrium contains — and it is the mechanism that turns a large drift into a collapse rather than a repair.

And the non-structural consequences, which arrive earlier than either: cladding connections, stair flights that become struts, and services that cross the storey.

None of the three has any dependence on the roof displacement. They are all statements about one storey, and one storey is the thing the global numbers averaged away.

What a retrofit is actually doing #

The obvious repair for a soft storey is to strengthen it, and the arithmetic here says why that is both right and easy to get wrong.

Adding strength to the weak storey moves the mechanism. Bring the ground storey back to parity and the frame’s yield sequence returns to 3, 4, 2, 1 — the distributed one — and the concentration returns to 1.7. That is the whole repair, and it is why jacketing the ground-floor columns is the standard intervention.

Adding stiffness without strength does not. A stiffer ground storey attracts more shear, and if its capacity has not risen with it the mechanism still forms there, now at a lower roof displacement. Stiffness is not strength is a distinction that costs nothing to state and is the difference between a retrofit and a redecoration.

And adding either to a storey that is not the weak one makes things worse. A new wall on the first floor increases the demand on the ground storey below it, because the shear that wall attracts has to reach the foundation through the storey underneath. This is the mechanism behind a whole class of failures in buildings extended or infilled after construction.

There is a fourth option that inverts the logic. Deliberately making a storey weaker and softer than everything above it is base isolation, and the difference is entirely in what that storey is made of: an isolator delivers 200 mm of displacement as a design property, with no strength loss and no degradation, while a column delivers it once. The soft storey and the isolation storey are the same idea built from a component that can and a component that cannot.

The comparison is worth holding, because it says that concentrating deformation into one level is not itself the fault. The fault is concentrating it into a level that was not designed to deliver it, and the pushover is what distinguishes the two.

What to carry away #

A soft storey improves the global metrics. Lower ductility demand, higher first-mode mass, longer period — all three, all correctly computed, all describing a building that concentrates its deformation into one storey.

Position is most of the danger. Half the strength removed from the top storey changes nothing; the same removal at the base multiplies the drift concentration by three and a half.

There is no cliff at the code limit. 80 per cent gives a concentration of 2.9 against a regular frame’s 1.7, and the relationship is smooth from 100 downwards.

And the quantity that matters is inside the mode shape. A modal analysis discards it at the moment it reports a force.

There is a last point about the pushover itself that this comparison makes better than any description of the method could. The analysis’s output is not a number, it is a distribution. A pushover that is read for its base shear, its yield displacement or its ductility demand has been reduced to exactly the summary quantities that failed to separate these two frames, and could have been replaced by a hand calculation. What it produces that nothing else does is the storey-by-storey deformation at the target, and the order the storeys arrive at their capacities in — and both of those are shapes rather than scalars. An analysis whose value is a shape and whose report is a scalar has been run and not used.

Where the model stops #

Stiffness and strength are reduced together here. The generator’s one parameter scales both, and a real soft storey usually has them decoupled — tall columns are less stiff without being weaker, while a discontinued wall removes both. A storey that is soft but not weak concentrates elastic drift and may not concentrate the mechanism.

The frame is a shear building. Each storey is a spring between two lumped masses, so the columns’ own bending is not modelled, no beam yields, and a real frame’s strong-column weak-beam hierarchy has nothing to act on here.

P-delta is discussed and not computed. Every capacity curve on this page is a first-order one, and the softening branch a real soft-storey frame shows — a curve that turns down rather than flattening — is exactly the second-order effect this model omits.

The load pattern is held fixed. The pattern is the analysis’s main assumption, and it is least defensible precisely here: once the ground storey has hinged, the inertia forces above it redistribute towards a rigid-body shape the triangular pattern does not describe.

And nothing here is a torsional statement. A soft storey that is soft on one side of the plan adds a rotation about a vertical axis, which is a plan irregularity on top of a vertical one and is what actually happened in most of the corner buildings that failed.

The ladder from here #

Later rungs on this anchor: adaptive pushover, where the load pattern is recomputed as the structure softens, and multi-modal procedures that run several patterns and combine them. The P-delta branch, and the negative post-yield stiffness that decides whether a soft storey is a repair or a collapse. Incremental dynamic analysis, which replaces the single target displacement with a suite of records and a distribution of answers. And the capacity design response to all of it — proportioning so that the mechanism forms where it was chosen rather than where the arithmetic happens to put it.

The soft-storey failure is the most photographed structural failure there is, and its diagnosis long predates any means of computing it. Buildings sitting intact on a crushed ground floor were catalogued after Caracas in 1967, San Fernando in 1971 and Mexico City in 1985, and the mechanism was obvious in every photograph. What was not available until the capacity-curve procedures of the 1990s was a way of showing that a frame would do it, on a drawing, before the building was built — and, less comfortably, a way of showing that all the numbers being reported about it would look fine.