Pushed over until it will not stand

An earthquake asks a structure for a displacement rather than for a force, and the question that follows is whether the structure can supply one without falling down. Answering it properly means integrating the equations of motion for a non-linear structure under a set of ground motions, which is expensive and produces a different answer for every record.

A pushover answers something else, cheaply. Apply a lateral load pattern, increase it, and record the base shear against the roof displacement while the frame yields storey by storey. There is no time in it anywhere.

What comes out is a property of the structure: a curve that says how much it can carry against how far it can move. That curve is not the answer to the question. The answer is where the demand crosses it, and supplying the demand is the half of the method that carries all the assumptions.

Building the curve #

The frame is modelled as a shear building: each storey has a stiffness, a shear capacity and a post-yield stiffness, and the storey drifts add up to the roof displacement.

The load pattern is normalised so the base shear is one, and the storey shears follow as running totals from the top. Then the base shear is increased. A storey behaves elastically until its own shear reaches its capacity and follows a hardening slope afterwards:

$$\Delta_i = \frac{V s_i}{k_i} \quad\text{until } V s_i = V_{y,i}, \quad\text{then}\quad \Delta_i = \Delta_{y,i} + \frac{V s_i - V_{y,i}}{\eta k_i}.$$

The hardening $\eta$ is not a refinement, it is what makes the analysis possible. A shear building with no hardening cannot yield twice: the first storey to reach its capacity caps the base shear for ever, so the structure becomes a one-storey mechanism at first yield and the curve is flat from there.

That is a real behaviour — it is exactly what a soft storey does — and it is not what a designed frame does, because its columns are continuous and its members harden. Three per cent is enough to make the difference and is about what a bare steel frame shows.

The curve that results has three regions and they are worth naming, because the acceptance criteria attach to different ones. There is an elastic branch, whose slope is the frame’s lateral stiffness and whose end is first yield. There is a transition, over which storeys yield one after another and the slope falls in steps. And there is a plateau, on which the mechanism has formed and the frame is drifting at nearly constant shear. Most of a designed frame’s target displacement lands in the transition, which is the region least well represented by the bilinear idealisation everybody uses.

On the frame drawn the storeys yield in the order 3, 4, 2, 1, 5, 6, 7 — not from the bottom up, because the strength taper and the shear distribution do not have the same shape. The mechanism is a result of the analysis rather than a shape chosen in advance, and that is the single most useful thing a pushover produces.

Which yield, and why it matters #

The curve has no yield point on it. It has a first yield, a series of kinks, and a gradual flattening, and the ductility demand depends entirely on which of those is called the yield.

First yield is 28 mm. It is one storey reaching its capacity while every other one is still elastic, and it happens early because the storey capacities and the storey shears are not proportional. Quoting a ductility against it gives 7.7.

The equal-area bilinear yield is 70 mm. It is constructed so that the bilinear curve has the same area under it as the real one up to the target — that is, so that it absorbs the same energy — and it is the standard construction. The ductility against it is 3.1.

The same structure and the same earthquake, and a factor of two and a half between two defensible numbers. A ductility demand quoted without saying which yield it is measured from is not a quantity, and the difference is not a subtlety: acceptance criteria are written against the second and are sometimes checked against the first.

Where the demand comes from #

The curve is a property of a multi-storey frame and a spectrum is a property of a single mass on a single spring, so something has to connect them.

The connection is the equivalent single-degree system. The frame’s first mode shape is taken as the deformed shape, and the modal participation factor times the roof ordinate converts a spectral displacement into a roof displacement. For the frame drawn that product is 1.264.

The demand itself comes from the elastic spectrum at the building’s own period of 0.93 s, giving a spectral displacement of 173 mm — and the target roof displacement is 219 mm, which is 0.78 per cent of the building’s height.

The step that makes this legitimate is Newmark and Hall’s equal displacement observation: for structures with periods above about half a second, an inelastic system reaches roughly the displacement its elastic counterpart would have. It is an empirical result rather than a theorem, it fails for short-period structures, and the whole method rests on it.

Which free body produced the number #

There is no cut here that produces a force, because the analysis is incremental. The equivalent statement is that at every point on the curve the frame is in equilibrium under a set of static lateral forces, and the whole of the method’s content is in what those forces are.

The load pattern is a guess at the inertia forces the earthquake will produce, and inertia forces are mass times acceleration, distributed as the mode shape. So a triangular pattern is a first-mode assumption, a uniform one is a soft-storey assumption, and a modal one is the first mode computed properly.

The assumption fails at exactly the moment the analysis becomes interesting. A frame that has yielded in one storey no longer has the mode shape the pattern was drawn from; its deformation has localised, its effective period has lengthened, and the inertia forces have redistributed. The pattern is held fixed while the structure it describes changes underneath it.

That is not a small approximation and it is the main reason adaptive and multi-modal pushover procedures exist. It is also why the method is trusted for regular low-rise frames dominated by their first mode and distrusted for anything else.

What a soft storey does to the same curve #

The clearest demonstration of what a pushover is for is to run one on a frame that an elastic analysis would call adequate.

Take the same eight-storey frame and make its ground storey 45 per cent as stiff and as strong as the others — an open ground floor, which is an ordinary architectural requirement. The elastic period changes a little. The capacity curve’s base shear at the target falls from 2,543 kN to 2,136.

And the drift distribution changes completely. The regular frame at its target has drifts between 0.3 and 1.2 per cent, spread over eight storeys. The soft frame has 6.41 per cent in the ground storey and 0.39 in the one above it — a concentration of 6.5 against 1.7.

The roof moves nearly the same distance in both. One building has distributed that movement over eight storeys and the other has put it into one, and that is the whole difference between a frame that survives and a frame that does not.

The pattern is the assumption, and it can be tested #

The load pattern is the method’s weakest link and it is also the one that can be examined most cheaply: run the same frame with three of them and compare.

A triangular pattern is the classic first-mode approximation and gives the smallest base shear at the target. A uniform pattern loads the lower storeys harder, produces larger storey shears low down, and returns a larger base shear. A first-mode pattern computed from the eigenvector sits between them.

The spread between the three is the analysis’s own honest uncertainty band, and it is not small. What is worth doing with it is not averaging but reading: if the three patterns give the same mechanism, the mechanism is a property of the frame; if they give different ones, the frame’s weak link depends on how the earthquake happens to distribute its inertia, and that is a finding.

Overstrength, which is the other output #

The curve’s height above first yield is a quantity in its own right, and it is one of the few places where a designer gets to see it.

The frame drawn reaches 3,023 kN against a first yield at 1,994 — an overstrength of 2.29. That is not a safety factor anybody applied; it is the accumulated effect of members sized in standard increments, of minimum requirements, of the same section used over several storeys, and of the redistribution that follows each yield.

Overstrength is what makes seismic design work in practice and it is almost never computed. Codes contain an overstrength factor as a fixed number, applied to protect elements that must not yield; a pushover measures the real one for the structure in hand.

Two readings follow, and they point in opposite directions. A frame with large overstrength has more reserve than the design gave it credit for — and it also delivers larger forces to the elements it is supposed to protect, which is the whole argument of capacity design and is why the factor appears there as a demand rather than as a bonus.

The second reading is the one that catches people. A column, a foundation or a brace connection designed to remain elastic while something else yields has to be strong enough for the actual strength of the yielding element, not its design strength. If the frame is 2.29 times stronger than its first yield suggested, everything protecting it has to be 2.29 times stronger too, and a design that used the code’s fixed overstrength factor of, say, 1.5 has under-protected them.

The three things it cannot see #

The method’s limitations are specific, and each of them is a consequence of removing time.

Higher modes. A tall or irregular building responds in more than one mode, and the upper storeys of a tall frame are frequently governed by the second. A single fixed load pattern cannot represent a response with two shapes in it, and the error is largest exactly where the pattern is smallest.

Cumulative damage. A push is monotonic. A real earthquake cycles, and a member that has cycled has less capacity than one that has been pushed once — degrading, pinching, losing its compression side. A pushover uses a monotonic backbone curve and cannot know how many times it will be traversed.

Duration. Two ground motions with the same spectrum and very different durations produce very different damage, and a spectrum-based demand cannot tell them apart. That is a limitation the pushover inherits from the spectrum rather than one of its own, and it is the reason long-duration subduction motions have their own literature.

Reading it as a plastic analysis with a shape #

There is a way of placing the pushover among the things this collection already contains, and it makes both clearer.

Plastic collapse analysis finds the load factor at which a mechanism forms, and it is a strength calculation: the answer is a load, the mechanism is a shape, and no deformation appears anywhere. It has been available since the nineteen-thirties and it is exact within its assumptions.

A pushover is that calculation carried out incrementally so that the deformation is tracked alongside the load. The collapse load it ends at is the plastic collapse load for the given pattern; everything before that point is the extra information — how much the structure moved on the way, and which hinges formed when.

So the two are not competitors. A pushover is a plastic analysis with the displacement axis restored, and the restoration is what allows the result to be compared with a demand that is expressed as a displacement. That is the whole reason the method belongs to earthquake engineering rather than to plastic design.

What it is good for anyway #

The catalogue of limitations is long enough that it is worth being clear about why the method is used at all, and the answer is not that it is cheap.

It finds the mechanism. No elastic analysis names which storeys yield or in what order, and the mechanism is the most important qualitative fact about a structure’s seismic behaviour.

It finds the concentration. The soft-storey comparison above is the method’s central demonstration: two structures that an elastic analysis reports as similar behave completely differently, and the difference is visible only after yielding.

It produces a curve rather than a verdict. A pass/fail check tells a designer nothing about how close they were or which way to move. A capacity curve shows the whole trade between strength and deformation, and it is the only common analysis that does.

What the curve is checked against #

A capacity curve on its own is a picture. Turning it into an assessment requires acceptance criteria, and they are of two kinds that are worth keeping apart.

Global criteria are read off the curve: the roof drift at the target, the base shear, whether the target lies before or after the curve turns over. They are easy to compute and they say very little, because a structure can satisfy every global criterion while concentrating all of its deformation in one place — which is exactly what the soft-storey comparison above shows.

Local criteria are read off the deformed state at the target: the rotation demanded of each hinge, the drift of each storey, the strain in each critical element. They are where the assessment actually happens, and they are what the analysis had to become non-linear to produce.

The relationship between the two is the reason the method is worth its cost. A pushover is run to obtain a global number — the target displacement — and its value is almost entirely in the local distribution that comes with it. A report that quotes the performance point and not the storey drifts has thrown away the analysis and kept its summary.

Where the model stops #

The shear building is a crude frame. Real frames have flexible beams, so the storey drifts are not independent and the mechanism can be a beam-sway one rather than a storey one.

The hardening is a single number. Real members harden, degrade and lose stiffness at different rates, and the backbone curve is a composite of all of them.

The equal-displacement rule fails at short periods. Below about half a second an inelastic system displaces considerably more than its elastic counterpart, and a correction factor is needed that this arithmetic does not have.

The equivalent single-degree substitution is a first-mode one. Everything the frame does that is not first-mode — torsion in plan, higher modes in elevation — is outside the construction rather than approximated by it.

And the target is one number from one spectrum. The real demand is a distribution over records, and a single target displacement is its central estimate with the scatter removed.

Where the ladder goes #

Later rungs on this anchor: the capacity-spectrum method, where both curves are drawn in acceleration–displacement space. Adaptive pushover, where the load pattern is updated as the structure softens. Multi-modal procedures for tall and irregular buildings. The short-period correction to the equal-displacement rule. Cyclic backbone curves and the degradation a monotonic push cannot see. Incremental dynamic analysis, which is the honest and expensive alternative. Pushover for assessment of existing structures, which is where it is most used and where the uncertainties are largest. And the question underneath the whole method: what it means to answer a dynamic question with a static analysis, and when the answer is good enough.