The earthquake asks for a displacement

The last essay ended with an uncomfortable number. A structure of one second period, on the record used there, attracts an elastic force of 294 kN per hundred tonnes of mass — around 30% of its own weight, applied sideways. Designing an ordinary building to resist that elastically is possible and nobody does it, because it would cost several times what the building is worth.

What is done instead looks reckless and is not.

That is the equal-displacement rule, and the whole of modern seismic design rests on it.

Why weaker is not further #

The reasoning is a energy argument and it is worth having in words before it is measured.

An elastic structure stores everything the earthquake gives it and gives it all back. A yielding structure stores what it can and converts the rest to heat, permanently, in the plastic deformation of its members. The energy the earthquake delivers is roughly fixed by the ground motion and the mass; if the structure can dispose of some of it, less has to be stored, and storing less means deflecting less than the strength reduction would suggest.

There is a second reason, and it is the one that makes the rule as sharp as it is. A structure that yields becomes softer, and a softer structure has a longer period, and a longer period moves it to a part of the spectrum where the demand is smaller. The yielding structure is not merely enduring the same earthquake with less strength; it is having a different and gentler earthquake, because it changed its own period while the shaking was going on.

Measured, over five records #

A rule this convenient deserves suspicion, so here it is measured rather than quoted. Five different records, a strength reduction factor of four in every case, and the ductility actually demanded:

From half a second upwards the ductility demanded is very nearly the strength reduction taken — 4.3 against 4 — and the displacement is within a tenth of the elastic one. That is the rule, and it holds well.

Below half a second it fails, and it fails in the unsafe direction. At 0.15 s a structure given a quarter of the elastic strength is asked to deform eleven times its yield displacement, and reaches 2.77 times the elastic displacement. A short-period structure has no time to soften its way out of trouble: each excursion is over before the period lengthening can help, and the reduced strength buys nothing but larger plastic demands.

That is why every code applies a reduction to the reduction at short periods, and it is one of the places where a rule of thumb has a genuinely dangerous failure region rather than merely an inaccurate one.

Two rules, and the period that separates them #

The pattern in the table has a name at each end, and both are Newmark and Hall’s, from work in the 1960s and 70s that turned a set of observations about integrated records into design rules.

Equal displacement, for long periods: the inelastic structure reaches the same displacement as the elastic one, so the ductility demanded equals the strength reduction taken, $\mu = R$.

Equal energy, for short periods: the area under the force–displacement curve is preserved instead, which gives $\mu = (R^2+1)/2$. At $R = 4$ that is 8.5 rather than 4 — a very much larger demand for the same reduction. Measured at 0.15 s over the five records here, the actual mean was 11.09, so even the equal-energy rule understates it.

The same integration run at 0.15 s shows what that means as a picture rather than as a coefficient.

Nothing about the structure changed between that figure and the first one except its period, and the rule reversed.

The transition between them sits somewhere near the period at which the spectrum stops rising, and the physical reason is the one already given: period lengthening only helps a structure whose period is on the falling part of the curve. A structure to the left of the spectral peak lengthens its period and moves towards the peak, which is the opposite of a rescue.

So the rule that makes seismic design affordable is a rule about long-period structures, and short, stiff, strong-looking buildings are the ones it does not protect. That is a poor match for intuition, and it is the reason squat masonry structures do so badly in earthquakes while tall flexible frames do well.

How much reduction is too much #

Compare the three reductions on the same structure and the same record:

Two things in that table are worth pausing on.

The R = 2 case deflects less than the elastic structure — 55.8 mm against 74.4. That is not an error and it is common: a little yielding at the right moment detunes the structure from the shaking and dissipates energy, and the net effect can be a smaller peak than the elastic response. A structure allowed to yield slightly is genuinely better off, which is a strange sentence and a well-established result.

The residual displacement is not monotone: 4.0 mm at R = 2, 11.8 mm at R = 4 and 0.8 mm at R = 6. Where a structure stops depends on the accident of which excursion was last and in which direction, and it is essentially unpredictable for a given record. Residual drift is a statistical quantity even when everything else about the analysis is deterministic.

The loop is the mechanism #

Two features of that figure are the whole of the argument.

The area is enormous compared with a damper’s. A viscous mechanism dissipates in proportion to the frequency and the square of the amplitude, and at the amplitudes and frequencies a building reaches, it cannot compete with a hysteresis loop that is as tall as the yield force and as wide as the excursion.

The area does not depend on the frequency. A viscous damper does less work when the motion is slow; a yielding member does not care. That matters because the periods that matter for buildings are long, which is precisely where viscous mechanisms are weakest.

What it costs #

The bargain is not free, and the price is in the same figure.

Permanent displacement. The 1.0 s structure finishes 11.8 mm off plumb. A building left out of plumb after an earthquake may be perfectly safe and impossible to insure, impossible to re-let, and uneconomic to straighten. Residual drift has become a design quantity in its own right for exactly this reason: the structure that survives and cannot be reoccupied has not really succeeded.

Damage that has to be found. Energy dissipated in a plastic hinge is energy that broke something at a scale of grains and grain boundaries. The member is still standing and its remaining capacity is not what it was.

Ductility that has to be delivered. The whole calculation assumes that when the member is asked to deform four times its yield displacement, it can. That is a detailing question rather than an analysis question — confinement reinforcement in a concrete column, compact sections and restrained flanges in a steel one, connections stronger than the members they join — and every one of those requirements exists to cash a cheque this calculation wrote. It is the same obligation the plastic hinge in a beam carries in the static case, with the difference that a static hinge rotates once and a seismic one rotates back and forth for thirty seconds.

Capacity design: choosing where the yielding happens #

The most important consequence of all this is not a number.

If a structure is going to yield, the designer gets to choose where. Making some members deliberately weaker than others forces the plastic deformation into them, and everything else can then be designed to resist the force those members can actually deliver rather than the force the earthquake might have applied. That is capacity design, and it converts an uncertain demand into a known one.

The idea has a static ancestor. A fixed-ended beam pushed to collapse forms its hinges in a definite order and fails in a definite mechanism, and the collapse load is a property of that mechanism rather than of the loading history, which is the whole content of plastic analysis. Capacity design is that argument used as an instruction rather than as a prediction: decide the mechanism first, then make every other part strong enough that no other mechanism can form.

The rule that comes out of it is strong-column, weak-beam: beams are allowed to hinge, columns are not, because a beam hinge damages a bay and a column hinge damages a storey. That is the same reasoning as the hinge put in on purpose, applied to a structure that will be pushed in both directions many times rather than loaded once.

Isolation: the other way to reduce the demand #

Yielding reduces the demand by making the structure softer during the earthquake. There is a way to do the same thing before the earthquake arrives, and it is worth putting beside ductility because it is the same physics used deliberately.

Mount the building on bearings that are very flexible horizontally and stiff vertically, and its period goes from perhaps 0.5 s to 2.5 or 3 s. On the spectrum, that moves it from near the peak to well down the falling tail — a reduction in acceleration of three or four times — and the price is paid in displacement, which is where a long-period structure suffers. A base-isolated building may move 200 mm at its bearings, which is why it needs a moat around it.

What the building above the bearings then experiences is a gentle, slow motion, and it can be designed to stay elastic. That is the significant difference: a ductile building trades damage for cost and an isolated one trades displacement at one plane for damage everywhere else. Isolation is chosen where the contents matter more than the structure — hospitals, data centres, museums — and where a building that survives and cannot be used would be a failure.

The connection to the rest of this site is exact: isolation is a deliberate soft storey, the thing the mode-shapes essay identified as a defect, engineered so that the storey doing the deforming is a row of bearings designed for the job rather than a set of columns that happened to be short of restraint.

The gravity load, which is what turns drift into collapse #

Nothing above collapses. The structure yields, drifts, dissipates and comes back, and the worst outcome is a building 11.8 mm off plumb. That is because the calculation has left out the weight of the building acting on its own sideways displacement, and putting it back changes the character of the problem rather than the size of the answer.

The measure is the stability coefficient

$$\theta = \frac{P\Delta}{Vh}$$

and it has a neat reading: the P-delta effect subtracts $\theta$ times the elastic lateral stiffness from whatever stiffness the structure has. Elastically that is a modest correction — $\theta$ is typically 0.05 to 0.15, and the response is amplified by $1/(1-\theta)$, which is the ordinary second-order amplification.

After yielding it is not modest, because there is very little stiffness left for it to subtract from. A yielded member’s tangent stiffness is its strain-hardening slope — a few per cent of the elastic value, call it $\alpha$. So the structure’s post-yield tangent stiffness, as a fraction of its elastic one, is

$$\alpha - \theta$$

and for a frame with $\alpha = 0.03$ and $\theta = 0.10$ that is negative: $-0.07$.

A negative post-yield stiffness is a different animal from a small positive one. It means that once the structure has yielded, pushing it further sideways requires less force, not more. Each excursion leaves it easier to push in the same direction than it was before, drift accumulates rather than oscillating about zero, and the structure walks steadily over until it falls.

That is sidesway collapse, and it is how most nominally ductile frames that have collapsed in earthquakes actually collapsed. Not by running out of strength, and not by exhausting their ductility in a single excursion, but by losing their restoring force to gravity and ratcheting.

Which reframes what the code limit on $\theta$ is for. It is usually presented as a limit on second-order amplification, and the elastic amplification at $\theta = 0.1$ is eleven per cent, which nobody would legislate about. The real requirement is the inequality above: $\theta$ has to stay below the strain-hardening ratio for the post-yield response to have any restoring force at all, and the strain-hardening ratio of a real frame is small.

It also sharpens the short-period failure this essay has already shown. The 0.3 s structure wandered away from zero and accumulated drift in one direction, and that was attributed to having no time to soften its way out of trouble. Gravity does the same thing by a different route and the two compound: a short-period structure ratchets because its excursions are over too quickly, and a heavily loaded one ratchets because it has lost its restoring force, and a building that is both has no mechanism left to bring it back.

There is a design response that follows directly and is worth naming, because it is the one thing that reliably prevents the ratchet. Give the structure a back-up system that stays elastic — a stiff spine, a rocking wall, a continuous column running the full height — whose job is not to carry load but to supply a restoring force after everything else has yielded. It contributes almost nothing before the earthquake and it is what stops the frame walking, and it is the reason a continuous core through a ductile frame is worth more than its strength suggests.

So the equal-displacement rule’s comfortable conclusion — a quarter of the strength, the same displacement — holds for a structure whose post-yield stiffness is still positive, and says nothing at all about one whose is not. The rule is about how far a structure goes. Whether it comes back is a separate question with a different inequality in it.

What the analysis cannot see #

Modes stop existing. Once part of a structure yields, its stiffness distribution changes while the motion is happening, so mode shapes and periods are no longer constants and the whole modal decomposition stops being available. The single-degree-of-freedom results above are used for insight and a real inelastic analysis integrates the whole structure step by step.

Cycles accumulate. The spectrum records the largest excursion; a yielding structure is damaged by the number of large excursions too, and two records with identical spectra can demand five plastic cycles or fifty.

Degradation is not in the model. The loops drawn here are the same size on the tenth cycle as on the first. Real members lose strength and stiffness as they are cycled — concrete spalls, reinforcement buckles, bolts elongate — and the honest models for that are empirical.

The strength is not known either. A structure’s actual yield strength exceeds its design value by the overstrength of the material, the reinforcement provided over the reinforcement required, and the contribution of everything that was not counted — so the real reduction factor is smaller than the one taken, and in the safe direction for the demand while being in the unsafe direction for everything the yielding members are supposed to protect. That is why capacity design multiplies the protected members’ demands by an overstrength factor rather than trusting the nominal strengths.

The record matters more than the model. The scatter across the five records used above is larger than the effect of any refinement to the structural model. A structure at 2.0 s came out at a mean ductility of 6.15 against the 4 the rule predicts, entirely because of which records happened to be in the set.

The elastic demand every strength in this essay is a fraction of comes from a response spectrum, and the fraction is chosen rather than computed — a decision about how much damage is acceptable, dressed as a coefficient. The scatter above is why that decision is made against a set of records rather than one.

Where the ladder goes #

The behaviour factor — the number a code divides the elastic force by — is the compressed form of everything above: a claim about how much ductility a particular structural system can deliver, and therefore how much strength it may be excused. It ranges from about 1.5 for an unreinforced masonry structure to 6 or 8 for a well-detailed frame, and the whole difference is detailing.

Two directions lead out. The first is the observation that ductility is not the only way to reduce the demand: a structure can be isolated from the ground instead, given a long period deliberately by mounting it on bearings, so that the earthquake it is asked to resist is the gentle far end of the spectrum. The second is that everything here has been about a structure damaged on purpose — the deliberate opposite of the redistribution nobody chose, where yielding rearranges a structure’s internal forces without anyone deciding it should — and there are structures for which that is not acceptable, and which have to buy their energy dissipation from a device instead of from their own members.