The pattern that stopped describing the building

A pushover analysis applies a lateral load pattern, increases it, and records what the structure does. The pattern is an input. It is usually triangular, or the elastic first mode, on the reasoning that a building’s seismic response is dominated by its first mode and the forces should resemble it.

The reasoning is sound and it is about the elastic first mode. A pushover is run to see what happens after that.

What the pattern is standing for #

A load pattern is not the earthquake. It is a stand-in for the inertia forces a structure develops while it moves, and inertia forces are mass times acceleration — so the pattern that is right is the one proportional to $m_i \phi_i$, mass times the shape the structure is moving in.

At the start of a push, that shape is the elastic first mode and the standard patterns approximate it well. What happens next is that a storey yields, and a yielded storey’s stiffness drops to a few per cent of what it was.

A structure with a softened storey has a different first mode, and the difference is not subtle: the soft storey deforms and everything above it moves as a nearly rigid block. A block moving as one has an almost uniform mode shape, so the inertia forces on it are almost uniform, and the pattern that describes them is nearly flat.

A fixed-pattern analysis uses the first of those curves for all of it.

Why the answers separate where they do #

The storey shear at any level is the sum of the forces above it. A rising pattern puts most of its force high up, so the shears stay large all the way to the roof; a flat pattern puts more of it low down, so the shears fall away faster with height.

That is the whole of the disagreement, and it says exactly where the two answers should differ.

Read the two profiles as shapes rather than as numbers. The adaptive one is a mechanism: a large deformation at one level and almost nothing above it. The fixed one is a mechanism with a tail — the upper storeys still deforming, because forces are still being applied to them in proportions that assumed they were participating.

The tail is an artefact of the pattern. Nothing in the structure is producing it.

The quantity that agrees, and the one that does not #

The pair of numbers is the point of this essay and it is worth setting out plainly.

At the same roof displacement, the two analyses report base shears 8.7 per cent apart and fourth-floor drifts 376 per cent apart.

The base shear and the roof displacement are the two coordinates of the capacity curve, and the capacity curve is what a capacity-spectrum or coefficient procedure operates on to find a target displacement. So the procedure that consumes the pushover sees the quantities the two methods agree about, and reports a target displacement that is nearly the same either way.

What the engineer then reads off, at that target displacement, is the storey drift profile — which is the thing the two methods disagree about by a factor.

That is the same shape as the mass-participation rule in modal analysis, and the two are worth putting side by side because the mechanism is identical. A criterion or a procedure is written in a quantity that converges or agrees easily; the answer the analysis exists to produce is a different quantity; and the correlation between them is weakest exactly on the irregular structures where the analysis is most needed.

Watching the pattern change, step by step #

The evolution has a structure to it, and following it once makes the whole thing concrete.

Nothing happens for the first fortieth of a per cent of roof drift. The frame is elastic, the tangent stiffness is the elastic stiffness, and the recomputed pattern is the pattern it started with — 0.062 at the ground floor rising to 0.165 at the roof.

Then the ground storey yields, at 0.05 per cent of roof drift, which is very early: it is the weak storey and it reaches its capacity while everything above it is barely stressed. Its tangent stiffness falls to three per cent of what it was, the first mode of the tangent structure becomes a rigid block on a soft spring, and the pattern goes almost flat — 0.122 to 0.127 across the eight floors.

Then the upper storeys begin to catch up. At 1.09 per cent the second storey yields, at 1.22 the third and at 1.45 the fourth, and each time the pattern moves a little back toward a rising shape, because the storeys still elastic are now the stiff ones and the mode leans toward them.

By the end the pattern is 0.071 to 0.143 — rising again, but not the shape it started at. Five changes over six hundred increments, each one a discrete event, each one produced by a storey reaching its capacity.

That sequence is also the answer to a reasonable objection. If the pattern is recomputed continuously, is the analysis still doing anything a designer can follow? It is: the pattern changes when the structure changes, at moments a designer can name, and the changes are few enough to list. An adaptive push is not a black box with a changing pattern in it. It is five pushes with four events between them.

What the adaptive run is not #

It is not more conservative, and this is worth being clear about because “adaptive” sounds like “refined” and “refined” sounds like “safer”.

On the frame here the adaptive run gives a larger base shear (by 8.7 per cent) and a larger ground-storey drift (358.5 mm against 316.1), and smaller drifts at every floor from the third upward. Design on the fixed pattern and the ground storey is under-checked while the fourth floor is over-checked; design on the adaptive one and the reverse.

Neither is a bound. Both are approximations to a nonlinear dynamic analysis, and which is closer depends on the structure and on the record — which is the honest reason incremental dynamic analysis exists and the reason it is expensive.

The frame where it matters least, and the one where it matters most #

The size of the disagreement tracks how much the mode shape changes, which tracks how concentrated the yielding is.

That inversion is worth a paragraph. On a frame with a very soft ground storey the yielding is so concentrated that both analyses find the same mechanism almost immediately, and the pattern above the mechanism hardly matters because nothing above it is deforming either way — the two runs come out 1 per cent apart.

On a frame with a mildly soft storey the yielding spreads, the mode goes on changing throughout the push, and the pattern that was chosen at the start is wrong for the longest stretch of the analysis. The disagreement is largest on the ordinary building, not on the pathological one.

The sequence the analysis was run to find #

There is a second output a pushover produces, and the two runs disagree about it too.

A pushover reports the order in which things yield, and that order is what capacity design is checked against: a frame proportioned so that its beams hinge before its columns should show beam hinges arriving first, and a pushover is how that is demonstrated.

On the adaptive run here the ground storey yields at 0.05 per cent of roof drift, the second at 1.09, the third at 1.22 and the fourth at 1.45 — four events in a definite order. The fixed-pattern run finds a different order, because it is pushing the upper storeys harder throughout and brings them to their capacities sooner relative to the lower ones.

A method that gets the yield sequence wrong is failing at the thing capacity design is checked by, and it is failing at it silently, because a sequence is not a number anybody plots against anything. It appears in the output as a list of hinge formations with load steps beside them, and two analyses of the same frame produce two lists that both look plausible.

The connection to a plastic collapse mechanism is worth drawing, because the sequence is precisely what a limit analysis does not care about. A collapse load is a property of the mechanism and not of the route to it, so a plastic analysis is indifferent to the order and gets the right answer anyway. A pushover is not a limit analysis — it is a route — and the route is what it is being asked about.

What it costs to do #

Almost nothing, and that is worth stating because the method’s reputation is that it is elaborate.

Each step of an adaptive push needs one eigenvalue solve on the tangent stiffness matrix — the same solve that produced the elastic modes, on a matrix that has had a few diagonal entries reduced. On the frames here the pattern only changes when a storey yields, which happened five times over six hundred increments, so five eigenvalue solves buy the whole of the difference.

The reason it is not universal has more to do with what the procedure downstream expects. A capacity-spectrum method converts the pushover curve into an equivalent single-degree-of-freedom system using a participation factor computed from the mode shape — and when the shape changes during the push, that conversion has to change with it, which most implementations do not do. The obstacle is the bookkeeping around the analysis rather than the analysis, which is a familiar shape in this collection.

What to do with a fixed-pattern result #

Most analyses in most offices will go on being run with a fixed pattern, and there is a useful thing to do with one that does not require rerunning it.

Look at where the drift profile stops being a mechanism. A yielded structure’s drift profile should be concentrated: large where the hinges are, small everywhere else. A fixed-pattern profile with a substantial tail above the mechanism — the 26.3 and 16.6 mm at the fourth and fifth floors here, against a ground storey at 316 — is showing deformation the pattern is supplying rather than deformation the structure wants. The tail is the artefact and its size is the diagnostic.

Compare two patterns rather than one. Running the same frame with a triangular and a uniform pattern brackets a good deal of what an adaptive run would have found, because the uniform pattern is roughly what the post-yield mode looks like. Codes that require two patterns are asking for exactly this, and the envelope of the two is a defensible answer where a single pattern is not.

And treat the ground-storey drift as the number that carries the check. It is the quantity the two methods agree about best on a strongly irregular frame and the quantity a soft-storey failure is actually about. The drifts higher up, on the same analysis, are the ones to distrust.

None of the three needs any new machinery. What they need is knowing which parts of a familiar output are the reliable ones, which is the same discipline a truncated modal analysis asks for and is not automatic in either case.

The same failure of correlation, four times over #

This collection has now met the same shape often enough to state it as a rule rather than as four coincidences.

A method reports several quantities. One of them is easy to check, cheap to compute and prominent in the output. Another is what the method exists to produce. A criterion, a validation or a habit fixes on the first — and the two decorrelate on exactly the structures that made the method necessary.

A truncated modal analysis counts mass, gets the base shear right and the roof force wrong, and asks for fewest modes on the most irregular frame.

The portal method gets the storey shears right and puts the point of contraflexure in the wrong fifth of the column.

A mesh refined until the displacements settle has settled in the quantity that converges first; the stress at the re-entrant corner has not.

And a pushover agrees with itself on base shear and disagrees by a factor on drift.

In none of the four is the criterion wrong. Each measures something real and each was chosen because it is cheap and unambiguous. What none of them can do is know which output the design depends on, and that is a question about the building rather than about the analysis — which is why it stays with the engineer no matter how good the software gets.

The practical version is one sentence, and it is the same sentence in all four cases. Check the number the design is going to use, not the number the method prints first.

Storeys yield at once and frames do not #

A shear building yields storey by storey and a frame does not. Every model here concentrates the nonlinearity into storey shear-drift laws, so a “yielded storey” is a single event. A real frame yields hinge by hinge — beam ends first if it was designed properly — and the softening is gradual rather than stepped, which makes the mode evolve smoothly rather than in five jumps.

The push is monotonic and an earthquake is not. Nothing here cycles, so nothing degrades: the storey laws are bilinear with hardening and they retrace themselves. A real structure’s backbone under cycling is lower than its monotonic push, and by an amount that depends on the detailing.

The pattern is derived from one mode. Multi-modal pushover procedures run a separate push for each significant mode and combine the results, which is a different and larger correction than the one here — and combining nonlinear results is not something superposition permits, so the combination is a convention rather than a derivation.

And there is no P-delta. The vertical load acting through the drift adds moment and, at the drifts above, would give the capacity curve a falling branch. A soft storey with a negative post-yield stiffness is the difference between a repair and a collapse, and it is absent from every curve here.

One roof displacement, two structures #

The capacity curves are drawn against roof displacement, which is one number standing for a structure that is moving in a shape. Two structures at the same roof displacement can be in completely different states, and the whole of this essay is one instance of that — the same 426 mm of roof movement, distributed two ways, with the drift at the fifth floor differing by nearly five times.

The other absence is time. A pushover has none; it is a sequence of equilibrium states, ordered by load rather than by clock. The mode changing during the push is drawn as four snapshots, and in a real event the structure passes through those states in a fraction of a second and passes back through them, which is a motion no static analysis contains.

The assumption underneath both runs #

Both analyses assume that a static push with some pattern reproduces the peak deformations of a dynamic response, and the whole argument above is about which pattern rather than about whether.

That assumption is the larger one and it is the one the method is criticised for. It holds well for a structure whose response is dominated by one mode that does not change much — a low-rise regular frame — and it holds worst for the tall, irregular structures that most need the check. Adaptive pushover improves the pattern and does not repair the assumption, and the honest summary is that it removes one of the two objections to the method.

Which is still worth doing. A method with two known errors in it, one of which has been removed, is in a better position than one with two — and the removed one costs five eigenvalue solves.

What a multi-modal procedure would still not fix #

Later rungs on this anchor: multi-modal pushover procedures and the combination rule they need. 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 one target displacement with a suite of records and a distribution of answers. Cyclic backbone curves and the degradation a monotonic push cannot see. 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 most useful thing to carry out of this is not about pushover at all. It is that a method is usually checked against the quantity it reports most prominently, and that quantity is usually the one it gets right. The check worth running on any approximate method is on the output it is actually used for, and on this one that is a storey drift and never a base shear.