The angle is not a detail

Bridges are skewed for reasons that have nothing to do with structural behaviour. A road crosses a river at an angle, or a rail line, or another road, and the alignment is fixed long before anyone sizes a pier. The skew angle is handed to the structural engineer as a constraint from the geometry of the site.

It is easy to treat it as a detail — the same bridge, set at an angle. It is not.

What skew does

In a straight bridge, the deck and the piers share a coordinate system. Seismic demand in the longitudinal direction is resisted longitudinally; transverse demand is resisted transversely. The response decomposes cleanly, which is why the standard analysis procedures work as well as they do.

Introduce skew and that decomposition stops holding. The deck's principal axes no longer align with the piers'. Longitudinal excitation now produces transverse response and vice versa, and the deck tends to rotate in plan under seismic loading — it wants to turn about a vertical axis in a way a straight deck does not.

That rotation concentrates demand unevenly. The obtuse corners of the deck and the abutments they bear on take more than their share, and pier demands become asymmetric even under symmetric excitation. None of this is exotic; it has been observed in real earthquakes for decades. The question is what it does to vulnerability in probabilistic terms.

Assumed versus derived

Here is the practice that motivated the work: fragility curves for skewed bridges are often taken from studies of regular geometry, on the reasoning that skew is a modest perturbation.

That is an assumption, and it is testable. So rather than adopting curves derived from straight bridges, we derived them directly — nonlinear time-history analysis across a range of skew angles, with fragility curves computed for each configuration.

The finding that matters is not a single coefficient. It is that skew is not a small perturbation to be absorbed by conservatism elsewhere. The fragility curves shift with angle, they shift differently for different damage states, and the shift is large enough that carrying a straight-bridge curve across to a skewed structure is not a defensible simplification.

The comparison problem

Deriving a curve per configuration creates its own difficulty. Once you have fragility curves for a family of skew angles, you have to compare them — and fragility curves are awkward to compare. Each is defined by a median and a dispersion, and two curves can cross. One structure can look better at low intensity and worse at high intensity. "Which is more vulnerable" then has no single answer.

This is a general problem, not one specific to bridges. It is why a lot of comparative fragility work quietly reduces to comparing medians and ignoring dispersion, which discards exactly the information that made the curve worth computing.

The response is to define an index: a single number derived from the whole curve, so that structures become rankable without throwing away its shape. That makes questions like "how much does a 45-degree skew cost us relative to a straight crossing" answerable with one figure rather than two plots and a judgement call.

Why this generalises

The skew work and the multi-hazard work are the same problem seen twice.

In both, the standard approach handles one variable well and then gets applied outside the conditions it was derived for — a straight-bridge curve used for a skewed bridge; a single-hazard curve used where two hazards act in sequence. In both, the fix is to stop assuming and compute across the parameter space instead. And in both, once you have a family of curves rather than one, you need a way to reduce them to something comparable.

That is the thread: vulnerability is specific to a structure's actual configuration, not to the idealised one the reference curves came from. Skew angle is one such configuration parameter. Mass distribution after a change of use is another. The hazard sequence is a third.

The limits

These results are for the bridge classes and ground motion set studied. The direction and magnitude of the effect are what transfer; the specific curves are not a lookup table for an arbitrary skewed bridge. And the index is a comparison tool, not a replacement for the curve — it is deliberately lossy, and what it discards should be understood before it is relied on.