Ductility, and what it buys

The two rules

Both relate the strength a yielding structure needs to the force an elastic one would attract. Both are observations from computed responses rather than derivations, and both are approximations that have survived because they are useful.

Equal displacement, R = μ. For structures of moderate to long period, the inelastic peak displacement is about the same as the elastic one would have been. If the displacements match, the strength can be reduced in direct proportion to the ductility available. A structure with μ = 4 needs a quarter of the elastic strength.

Equal energy, R = √(2μ − 1). For short-period structures, the areas under the elastic and inelastic force–displacement curves match instead. The reduction is smaller: μ = 4 buys R = 2.65, not 4.

The crossover is around the corner period of the spectrum — roughly 0.5 s, though it depends on the site. Below it, use equal energy. Above it, equal displacement. Very short-period structures get almost no benefit at all: as T → 0 the structure moves with the ground and ductility cannot help it.

The two rules disagree by a factor of about 1.5 at μ = 4. On a stiff building that is the difference between a design that works and one that does not.

What the displacement demand actually is

Under the equal-displacement rule the inelastic peak displacement equals the elastic one. That is easy to state and easy to misread.

It does NOT mean the yielding structure moves the same as the structure you designed. It means it moves the same as the ELASTIC structure of the same initial period would have — the one you did not build and would not have paid for. Since the real structure yields at u_y = f_y/k, and f_y has been reduced by R, the ductility demand is R, and the displacement is R times the yield displacement.

So halving the strength does not halve anything the occupant experiences. It doubles the ductility demand and leaves the displacement where it was.

Equivalent viscous damping, and why it flatters

Hysteretic dissipation is sometimes re-expressed as an equivalent viscous damping ratio, so that a linear analysis can be used with a higher ζ. Equating the energy dissipated per cycle gives, for an elastic–perfectly plastic system,

ζ_eq = (2/π)(1 − 1/μ)

At μ = 2 that is 31.8%, and at μ = 4 it is 47.7%. Those numbers should provoke suspicion, and they should: they are known to be too large, typically by a third or more.

The reason is in the derivation's assumption. The energy equivalence is set up for STEADY harmonic motion at the peak amplitude — many full loops, all of them at μ. A real earthquake response reaches its peak once or twice and spends the rest of the record on much smaller loops that dissipate far less. Design procedures that use equivalent damping therefore apply an empirical reduction, and the reduction is not a fudge — it is a correction for the difference between a harmonic test and an earthquake.

Residual displacement

When the shaking stops, a structure that yielded does not return to plumb. It stops at a residual displacement, and this is a demand that peak displacement does not describe.

It matters for a practical reason: a building with a large permanent lean may be structurally sound and still be uneconomic to repair. Residual drift has become the governing criterion for demolition decisions after real earthquakes, and it is the reason self-centring systems — rocking walls, post-tensioned frames — exist at all.

Residuals are also badly conditioned. A small change in the record, or in the post-yield stiffness, can change the residual by a large factor while barely moving the peak. A single-record residual is not a reliable number; a mean over a record set is.

Post-yield stiffness is the strongest lever on residual displacement. A system with α = 0 has no restoring tendency once yielded and can stop anywhere; a positive α pulls it back.

When a nonlinear analysis is actually needed

It is expensive, it needs decisions a linear analysis does not, and it can be wrong in ways that look plausible. It is worth it when:

• the structure is expected to yield significantly and the ductility demand needs to be quantified rather than assumed;
• yielding is concentrated, so a global R factor cannot represent it — a soft storey, a knee brace, an isolation system;
• residual displacement or damage sequence is part of the question;
• the structure is being assessed rather than designed, and its real capacity is what is at stake.

It is NOT worth it when the structure is intended to remain essentially elastic, when the question is a serviceability one, or when the model's material properties are not known well enough to justify the precision the method implies. A nonlinear analysis of a structure whose yield strength is known to ±25% reports its results to three figures and means none of them.