Industry: Engineering Product: ModelRisk Application: Seismic Analysis
A 12-storey reinforced-concrete moment frame on a high-seismic site is checked against the design-basis earthquake — the 475-year, 0.40 g event — and passes: the probability that inter-story drift exceeds the life-safety capacity at that intensity is just 5.2%. But a single-intensity check answers the wrong question. Earthquakes arrive across a spectrum of intensities at different rates, and structural capacity is itself uncertain. Building a lognormal fragility curve — P(exceed a damage state | ground-motion intensity) — and convolving it with the site hazard curve turns the deterministic pass/fail into an annualized risk: the mean annual rate of breaching life-safety is 1-in-1,786 years (a 2.8% chance over a 50-year design life), and a hazard deaggregation shows that risk is dominated not by the rare code-level event but by more-frequent ground motions in the 0.4-0.6 g band.
A leading engineering consultancy adopted ModelRisk to replace single-scenario seismic checks with the full performance-based framework: demand, capacity, fragility and hazard integration.
The deterministic check evaluates one intensity and reports a margin. It cannot answer "what is the chance of damage per year?", because that requires every intensity weighted by how often it occurs. The performance-based formulation makes the uncertainty explicit at three levels:
A single 0.40 g check sees only one slice of this. The 5.2% it returns is real, but it is a conditional probability — conditional on the design earthquake happening at all — and says nothing about annual risk or about the smaller, more frequent shaking that actually accumulates the exposure.
The headline result is the set of fragility curves — one per damage state — giving P(exceed) as a continuous function of ground-motion intensity.
The fragility medians (the PGA at which exceedance probability is 50%) are 0.39 g for Immediate Occupancy, 0.96 g for Life Safety, and 1.73 g for Collapse Prevention. Read off at the two anchor intensities:
The frame is expected to suffer occupancy-interrupting damage at the design event (better than even odds) while keeping life-safety exceedance low — exactly the intent of code-based design, now expressed as probabilities rather than a binary pass.
Because the loss is driven by a single per-event intensity feeding a sharply non-linear exceedance, the result never collapses toward a thin Gaussian — the tail is the point.
The drift-demand distribution at the design-basis 0.40 g, set against the capacity thresholds, shows why a single damage state is the wrong frame.
At 0.40 g the median drift demand is 0.94% and the mean 1.04%, with a P90 of 1.68% and a P95 of 1.98%. That puts 44.5% of the distribution beyond the 1.0% Immediate-Occupancy capacity but only 1.5% beyond the 2.5% Life-Safety capacity at this single intensity — the structure is very likely to need repair after the design event, and very unlikely to threaten life safety from that event alone.
Convolving each fragility with the hazard curve gives the mean annual rate of exceeding each damage state — and deaggregating that rate by intensity band shows which earthquakes drive it.
The annualized results are: Immediate Occupancy at a 172-year return period (25.2% over 50 years), Life Safety at 1,786 years (2.8% over 50 years), and Collapse Prevention at 8,133 years (0.6% over 50 years). The deaggregation of the life-safety rate is the decisive picture: the dominant contribution comes from the 0.4-0.6 g band (24.4%), with the 0.2-0.4 g band adding 20.6% and 0.6-0.8 g adding 18.0%. The single 0.75 g code event contributes a minority of the annual risk — most of it accumulates from more-frequent, sub-MCE shaking that the deterministic MCE check never examines.
The annual life-safety rate is most sensitive to the site hazard (shifting the design-basis PGA from 0.34 to 0.46 g moves the rate from -35% to +44%), to the drift-capacity median (2.8% to 2.2% moves it -26% to +41%), and to the demand dispersion \(\beta\) (0.35 to 0.55 moves it -25% to +42%). The demand-model intercept follows close behind, while the capacity dispersion and the hazard-curve slope are second-order. The practical reading: reducing the dispersion of demand — through better records and modelling — is as powerful as raising the median capacity through stiffer detailing, and both rival refining the site hazard itself.
The deterministic MCE check had implied the frame's residual risk lived in the rare 0.75 g event. Hazard integration showed the opposite: most of the annual life-safety risk accrued from the more-frequent 0.4-0.6 g range, where the fragility was already climbing off its floor. That redirected the retrofit from chasing collapse-prevention margin at the extreme toward stiffening the frame to lift the life-safety capacity median and tighten demand dispersion across the moderate band that dominated the integral. Re-running the convolution with the improved capacity pushed the annual life-safety rate from 1-in-1,786 toward 1-in-3,000, and the consultancy could state the benefit as a return period rather than as an unverifiable single-scenario margin — a number a regulator and an insurer could both act on.
Seismic analysis stops being a single design-earthquake check and becomes an annualized, hazard-integrated risk statement the moment the model is allowed to see every intensity, weighted by how often it strikes.