Structural Engineering & Statics
where the load goes
Dynamics

The earthquake asks for a displacement

A structure a quarter as strong as the elastic demand does not deflect four times as far. It deflects almost exactly as far, yields on the way, and survives — which is why no ordinary building is designed for the force an earthquake would apply if it stayed elastic.

Assumes The spectrum is not a load and The only thing that stops it.

16 min read 8 figures The load that will not hold stillWhich failure arrives first

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.

A ground motion, on a structure of 1.00 s periodDisplacement against time for a single-degree-of-freedom structure of natural period 1.00 s and 5.0% damping, under a ground motion of 3.5 m/s² peak. The elastic peak is 57.71 mm, and the same frame given a 4th of that strength peaks at 65.46 mm and comes to rest 14.13 mm from where it started.02468101214161820-60-40-20204060time (s)displacement (mm)a ground motion of 3.5 m/s² peakelastic, and the same frame at a 4th of its strengthelastic: peak 57.71 mmyielding: peak 65.46 mmleft 14.13 mm off plumb
Fig. 1 The same structure twice through the same record: once elastic, and once with a yield strength of a quarter of the elastic demand. The elastic one peaks at 74.4 mm. The one that yields peaks at 69.9 mm — slightly less — and finishes 11.8 mm from where it started. A quarter of the strength, and essentially the same displacement.

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:

period ductility demanded displacement ÷ elastic
0.15 s 11.09 2.77
0.30 s 6.52 1.63
0.50 s 4.41 1.10
1.00 s 4.31 1.08
1.50 s 4.34 1.08
2.00 s 6.15 1.54

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.

A ground motion, on a structure of 0.300 s periodDisplacement against time for a single-degree-of-freedom structure of natural period 0.300 s and 5.0% damping, under a ground motion of 3.5 m/s² peak. The elastic peak is 13.1 mm, and the same frame given a 4th of that strength peaks at 29.18 mm and comes to rest 0.62 mm from where it started.02468101214161820-30-20-10102030time (s)displacement (mm)a ground motion of 3.5 m/s² peakelastic, and the same frame at a 4th of its strengthelastic: peak 13.1 mmyielding: peak 29.18 mmleft 0.62 mm off plumb
Fig. 2 A short-period structure — 0.3 s — with the same quarter-strength reduction. The yielding response no longer tracks the elastic one: it wanders away from zero, accumulating plastic deformation in one direction, and ends the record a long way from where it began. This is the failure mode the equal-displacement rule cannot see.

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, μ=R\mu = R.

Equal energy, for short periods: the area under the force–displacement curve is preserved instead, which gives μ=(R2+1)/2\mu = (R^2+1)/2. At R=4R = 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 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

A ground motion, on a structure of 1.00 s periodDisplacement against time for a single-degree-of-freedom structure of natural period 1.00 s and 5.0% damping, under a ground motion of 3.5 m/s² peak. The elastic peak is 57.71 mm, and the same frame given a 6th of that strength peaks at 92.48 mm and comes to rest 6.85 mm from where it started.02468101214161820-100-5050100time (s)displacement (mm)a ground motion of 3.5 m/s² peakelastic, and the same frame at a 6th of its strengthelastic: peak 57.71 mmyielding: peak 92.48 mmleft 6.85 mm off plumb
Fig. 3 The same one-second structure given a sixth of the elastic strength. It peaks at 74.5 mm — indistinguishable from the elastic 74.4 — and demands a ductility of 6.01. The displacement has stopped responding to strength almost entirely, which is the equal-displacement rule at its most extreme.

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

strength reduction yield force peak displacement ductility demanded residual
elastic 294 kN 74.4 mm 1.00 0
R = 2 147 kN 55.8 mm 1.50 4.0 mm
R = 4 73 kN 69.9 mm 3.76 11.8 mm
R = 6 49 kN 74.5 mm 6.01 0.8 mm

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.

Where the energy goes: one loop in force against displacementThe force the supports feel — the spring's and the damper's together — against the displacement, for one mechanism. yielding at 273.79 kN, enclosing 25.97 kJ over the record drawn. The yielding loop is a parallelogram whose area does not depend on how fast it is traced, and every circuit leaves the structure displaced from where it began.-60-40-20204060-300-200-100100200300displacement (mm)restoring force (kN)yielding at 273.79 kN — 25.97 kJ
Fig. 4 The same structure at half the elastic strength rather than a quarter. The loops are taller and narrower — the structure yields at a higher force and travels less far past it — and there are fewer of them. Halving the reduction roughly halves the ductility demanded and considerably more than halves the damage, because damage accumulates with the number of large excursions as well as their size.

The loop is the mechanism

Where the energy goes: one loop in force against displacementThe force the supports feel — the spring's and the damper's together — against the displacement, for one mechanism. yielding at 136.89 kN, enclosing 23.85 kJ over the record drawn. The yielding loop is a parallelogram whose area does not depend on how fast it is traced, and every circuit leaves the structure displaced from where it began.-40-202040-150-100-5050100150displacement (mm)restoring force (kN)yielding at 136.89 kN — 23.85 kJ
Fig. 5 Force against displacement for a yielding structure through a whole record. Each excursion past the yield force traces a parallelogram, and the enclosed area is energy that has left the structure for good. The structure never returns to the origin: every plastic excursion moves the loop sideways, and where it stops is where the building ends up.

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.

Where the energy goes: two loops in force against displacementThe force the supports feel — the spring's and the damper's together — against the displacement, for two mechanisms. viscous, 5% of critical, enclosing 34.02 kJ over the record drawn; yielding at 136.89 kN, enclosing 23.85 kJ over the record drawn. The yielding loop is a parallelogram whose area does not depend on how fast it is traced, and every circuit leaves the structure displaced from where it began.-40-202040-600-400-200200400600displacement (mm)restoring force (kN)viscous, 5% of critical — 34.02 kJyielding at 136.89 kN — 23.85 kJ
Fig. 6 The two mechanisms on one plot: the viscous ellipse and the yielding parallelogram, at comparable displacements. The comparison is why ductility is designed for rather than damping.

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 collapse mechanism of a fixed-ended beamA collapse mechanism, with the hinge position found by searching rather than quoted. Every position gives an upper bound on the collapse load; the lowest is 60.00, at a hinge 50.0 per cent along, which is a coefficient of 16.000 times Mp over the square of the span.sagging hinge at 4.00hinge at the fixed endand herelowest upper bound: 60.00every hinge position gives an upper bound on the collapse loadassumed position of the sagging hingecoefficient 16.00 Mp ÷ L²
Fig. 7 The static ancestor of the idea: a fixed-ended beam forms hinges in a definite order and collapses in a definite mechanism, and the collapse load is a property of the mechanism rather than of the loading history. Seismic capacity design is that argument used as an instruction — 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.

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 response spectrum of that record, at one damping ratioThe peak pseudo-acceleration of a single-degree-of-freedom structure against its natural period, for the record above, at 5% damping. Each curve is 44 complete time integrations — one structure per point. At zero period the structure is rigid and rides the ground, so the curve starts at the peak ground acceleration of 3.5 m/s²; it peaks at 7.07 m/s² at a period of 0.37 s, an amplification of 2.02.,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,00.511.522.5302468natural period (s)spectral acceleration (m/s²)the ground's own peak, 3.5 m/s²5% damping — peak 7.07 m/s² at 0.37 s
Fig. 8 The elastic demand the whole argument starts from. Every strength in this essay is a fraction of what this curve asks for, and the fraction is chosen rather than computed — a decision about how much damage is acceptable, dressed as a coefficient.

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.

Where this goes next

three structures · one recordT = 0.3 speak 12.54 mmT = 0.8 speak 46.2 mmT = 1.8 speak 108.95 mmpeak displacementperiod The spectrum is not a load A response spectrum looks like a load curve and is not one. Every point on it is the peak of a complete time integration of one particular structure, and the curve is what you get by doing that again for every structure there could be. sagging hinge at 4.69hinge at the fixed endlowest upper bound: 7.29every hinge position gives an upper bound on the collapse loadassumed position of the sagging hingecoefficient 11.66 Mp ÷ L² After the first yield, which is not the end A steel beam whose extreme fibre has reached yield has not failed. It has started forming a hinge, and collapse waits until there are enough hinges to make a mechanism. crown hinge — no moment here, by constructionH = 23.4H = 23.427.027.0thrust line and axis coincide — the definition of funicular The hinge put in on purpose An arch with two pinned feet cannot be solved by statics. Add a third hinge at the crown — deliberately weakening it — and the whole structure falls out of one moment equation. 00.511.522.530510152025forcing frequency ÷ natural frequencyamplitude ÷ static deflection1% damping — 50× at the peak2% damping — 25.01× at the peak5% damping — 10.01× at the peak10% damping — 5.03× at the peak The only thing that stops it Drive a structure at its own frequency and the amplitude grows without limit unless something takes energy out. What takes it out is damping, and damping is the one structural property that is never designed, never drawn, and never known until the thing is built. 02000040000600008000010000012000014000016000000.20.40.60.81joint rotational stiffness, kN·m/radend moment ÷ wL²/126.04%30%62.16%rigid boundarysemi-rigidfixed ended The redistribution nobody chose A beam designed as simply supported, on connections that are not pins, has end moments the analysis never predicted and a mid-span moment smaller than it was sized for. Usually that is safe. It is never intentional, and there is one direction in which it is not safe at all.

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Behaviour factorCapacity designDuctilityEnergy dissipationEqual displacement ruleHysteresisPlastic hingeResidual displacement