Earthquake plus tsunami is not arithmetic

If you want to know how a building will fare in an earthquake, you compute a fragility curve: the probability of exceeding a damage state, plotted against shaking intensity. If you want to know how it will fare in a tsunami, you compute another one against inundation depth. Both are standard practice. Both are defensible.

The problem is what happens when you need both.

The instinct is to treat them as independent and combine the probabilities. If the earthquake carries a 20% chance of collapse and the tsunami 15%, the union gives 1 − (0.80)(0.85) = 32%. That arithmetic is correct — for two independent hazards acting on two independent structures.

It is wrong for a subduction event.

The sequence is the point

In a subduction event the ground motion arrives first, and lasts seconds to a couple of minutes. The tsunami arrives minutes to tens of minutes later. By the time the water reaches the structure the shaking has stopped — and the frame the water meets is not the frame the earthquake met.

Beam ends have hinged. Column bases have yielded. The structure carries a residual drift and a lower lateral stiffness than it had an hour earlier. Its capacity against hydrodynamic loading is not its design capacity; it is whatever survived the shaking.

So the second hazard does not act on the original structure. It acts on a damaged one:

θ_ts,eff = θ_ts · (1 − k · P_eq)

The tsunami capacity is reduced in proportion to the earthquake failure probability, which stands in for the damage the shaking has already caused. The two events are sequential and coupled, not simultaneous and independent. Treating them as independent does not merely lose precision — it systematically understates risk, because it credits the structure with capacity it no longer has.

Why one curve is not enough

Once capacity depends on the other hazard, a single curve stops being sufficient. You cannot plot failure probability against shaking intensity alone, because the answer also depends on inundation depth. You cannot plot it against depth alone, for the same reason.

What you need is a surface: failure probability across the full space of both intensities. Every point on it answers a specific question — this much shaking, then this much water, gives this probability of exceeding this damage state.

Computing that surface is what my dissertation set out to do. A trial realisation goes out to a nonlinear structural analysis, a response comes back to a reliability search, the search updates, and the loop cycles until the design point converges. Run that across a discretised parameter space — every combination of intensity and configuration — and the individual curves join into one continuous surface.

What it costs, and what it buys

The cost is computation. A single fragility curve might take a few hundred nonlinear time-history analyses. A surface takes that at every point in the grid. The cases are independent, though, which makes the problem embarrassingly parallel: weeks of sequential analysis become hours when distributed.

What it buys is a comparison you could not otherwise make. A surface reduces to a single index, so two structural systems can be ranked against combined hazard rather than against each hazard separately. That is the question an owner or a planner actually asks — not "how does this building do in an earthquake", but "how does this building do here."

The honest limits

FERUM and ZEUS-NL are established analysis codes. The contribution is the coupling, the multi-hazard extension, and the parallelised run across a parameter space. Tsunami forces follow ASCE 7 Chapter 6; the components modelled here are hydrostatic and drag, with the Froude number setting the flow velocity. The capacity-degradation coefficient k is calibrated rather than derived; it is the weakest link in the chain and the one most worth improving.

And the method is not specific to earthquakes and tsunamis. Those are the cases shown because they are the ones I studied. Any two demands that can be varied, where the first alters the capacity against the second, fit the same method.