Ground Motions, Intensities, and Building Damage

By the end of this lecture, you will be able to:

• [LO-17.1] Distinguish intensity (felt-effect) from magnitude (source) from peak ground motion (waveform measurement).
• [LO-17.2] Define PGA, PGV, PGD, and 5%-damped $S_a(T)$, and explain physically which carries the dominant energy at each band.
• [LO-17.3] Predict three site/source effects — distance attenuation, soft-soil amplification, soil liquefaction — and apply $T_{\text{building}} \approx N/10$ s.
• [LO-17.4] Critique an AI-generated explanation of expected shaking by separating source, path, and site contributions.

The framing question

The 2001 Nisqually earthquake was $M_W$ 6.8 — one number for the source.

But people in Olympia and Seattle did not experience $M_W$ 6.8. They experienced:

• A time history of acceleration, lasting ~30 s
• With a PGA of ~0.16 g downtown
• With MMI VI–VII (felt by all, slight damage)
• All depending on soil class, building height, and location

Magnitude, ground motion, and intensity are three different things. Confusing them is the most common single error in earthquake reporting.

Three observables, three time derivatives

For a record $\boldsymbol{u}(\boldsymbol{x}, t)$ at a station:

$$\boldsymbol{u} \xrightarrow{\partial_t} \boldsymbol{v} = \dfrac{\partial \boldsymbol{u}}{\partial t} \xrightarrow{\partial_t} \boldsymbol{a} = \dfrac{\partial^2 \boldsymbol{u}}{\partial t^2}$$

A factor of $\omega$ in the spectrum at each derivative.

PGA, PGV, PGD, and $S_a(T)$

A real building responds to a band of frequencies, not just to PGA.

$S_a(T)$ = peak acceleration of a 5%-damped oscillator of period $T$.

Why $S_a(T)$ matters: the building rule of thumb

$$T_{\text{building}} \;\approx\; \frac{N}{10}\,\text{s} \qquad (N = \text{storeys})$$

As cities build taller, the relevant period band lengthens — exactly where megathrust earthquakes radiate.

Buildings × Source types

The forward problem: GMPE

A Ground-Motion Prediction Equation (GMPE) relates ground motion to source, path, and site:

$$\ln Y \;=\; f_{\text{source}}(M)\;+\;f_{\text{path}}(M, R)\;+\;f_{\text{site}}(V_{S30})\;+\;\varepsilon$$

• $Y$: PGA, PGV, or $S_a(T)$
• $\sigma(\varepsilon) \approx 0.6$ in natural log → factor of ~1.8 in linear scatter

Even the best modern GMPE disagrees with observation by a factor of two routinely.

USGS ShakeMap (Worden et al., 2020) implements this prediction in real time.

Site amplification: impedance contrast

For an SH wave going from rock ($Z_1$) into soft soil ($Z_2 \ll Z_1$):

$$\frac{A_{\text{surface}}}{A_{\text{rock}}} \;=\; \frac{2\,Z_1}{Z_1 + Z_2} \;\approx\; 2$$

Plus resonance of a layer of thickness $H$ over rigid rock:

$$f_0 \;=\; \frac{\beta_2}{4 H}$$

Duwamish flats: $H \approx 200$ m, $\beta_2 \approx 350$ m/s → $T_0 \approx 2.3$ s — close to mid-rise resonance!

(1985 Mexico City: $T_0 \approx 2$ s, amplification ~50.)

Site amplification — visualised

The inverse problem: Modified Mercalli Intensity (MMI)

For a historical earthquake (1556 Shaanxi, 1700 Cascadia) — no instrumental record exists.

Only the felt and damage descriptions survive: the MMI scale (I–XII).

Modern Ground Motion / Intensity Conversion Equations (GMICE, Worden 2012) convert PGA MMI statistically.

Three caveats on intensity

1. Ordinal, not interval. The step VI → VII is not the same physical jump as VII → VIII.

2. Mixes source, path, and site. Soft soil + URM construction reads higher MMI for the same source.

3. Depends on the building stock. A modern wood-frame house fails at higher PGA than a 1900 brick building. The same shaking is reported as different MMI in different decades.

Intensity is not a measure of an earthquake — it is a measure of an earthquake at a place.

Same magnitude, different felt area: the East–West contrast

The 2011 M5.8 Virginia event was felt by 20× more people than the M6.0 Parkfield 2004. Reason: eastern crust has higher $Q$ (less attenuation), not bigger source.

Liquefaction

Saturated, loose sand + sustained shaking → pore pressure rises until the granular skeleton fails.

Three conditions:

1. Loose sandy soil below the water table
2. PGA $\gtrsim 0.1\,g$ for ~10+ cycles
3. Shallow ground water (within a few metres of the surface)

Examples:

• 1964 Niigata (buildings tilted on intact foundations)
• 2011 Christchurch (sand boils throughout the city)
• 2001 Nisqually (Duwamish flats and Harbor Island)

In Seattle: Washington Geological Survey Liquefaction Susceptibility maps show extensive risk in fill areas.

Building failure modes

Each failure mode has an engineering counterpart:

• Soft-storey → shear walls, especially at ground level
• Frame collapse → cross-bracing, gussets
• Foundation failure → ground improvement, deep piles

Earthquake-resistant retrofit strategies

Three classical approaches plus modern dampers:

1. Shear walls — vertical reinforced-concrete plates between window openings absorb lateral shear.
2. Cross-bracing / gussets — diagonal members add shear strength to existing frames.
3. Base isolation — laminated rubber-and-steel bearings decouple building from ground; shifts $T$ longer to escape the high-frequency band.
4. Viscous dampers — convert kinetic energy to heat (the big "X" piston in retrofits).

Cost of base isolation: ~5–10% of building cost. Used in hospitals, City Halls, the LA Civic Centre.

Cascadia: three source types, three signatures

A Cascadia $M_W$ 9 will produce modest PGA in Seattle but record-setting $S_a(3\text{ s})$ — exactly the band that excites the city's high-rise inventory.

Research Horizon

Site-specific GMPEs from machine learning
$V_{S30}$ alone misses ~50% of the residual; full $V_S$ profiles + microtremor H/V + deep learning halve the residual at well-instrumented stations (Bahrampouri 2024).

Earthquake Early Warning (ShakeAlert)
Deployed across WA/OR/CA 2021–2023. The first few seconds of the P-wave forecast the much larger S- and surface-wave shaking.

Physics-based Cascadia simulation
No instrumental record of a Cascadia $M_W$ 9 exists; 3D dynamic-rupture simulations (Frankel 2018, Wirth 2018) anchored to paleoseismic constraints provide the forecast.

Societal Relevance — Cascadia building codes

The 2018 Washington State Building Code adopted Cascadia-specific $S_a(1\text{ s})$ design values for coastal counties.

• WGS 2024 CSZ Tsunami Loss Estimate Study
• USGS ShakeAlert deployment (statewide 2023)
• WA DNR HAZUS loss-estimation maps

Every new mid-rise apartment from Bellingham to Astoria is being designed today against an explicit Cascadia long-period shaking forecast — a forecast that did not exist a generation ago.

AI Literacy — Critique a generated shaking forecast

Try this prompt with Claude or ChatGPT:

"I live in Seattle in a wood-frame house. A magnitude 9.0 earthquake happens on the Cascadia subduction zone, 200 km west of me. How strong will the shaking feel?"

Then evaluate the response for:

1. Source / path / site separation — does it acknowledge the question is underspecified?
2. Quantitative uncertainty — does it give a range, or a confident number?
3. Period dependence — does it note that wood-frame at $T = 0.1$ s is not the band that megathrusts excite most?
4. Non-uniqueness flagging — does it ask about soil class, building age, distance precision?

Submit a 250-word critique. Lab 4 has the rubric.

Concept Checks

1. A 30-storey building and a wood-frame house stand on the same lot in downtown Seattle. A crustal $M_W$ 6 earthquake gives PGA = 0.20 g, $S_a(3\,\text{s}) = 0.05\,g$. Which is at greater risk? How does the answer change for a Cascadia $M_W$ 9 with PGA = 0.15 g, $S_a(3\,\text{s}) = 0.4\,g$?

2. Sketch (qualitatively) PGA vs distance for two earthquakes: identical $M_W$ 6, both at 30 km from downtown Seattle, one on basement rock and one in the Duwamish fill. Which is higher at the surface, and why?

3. Why is using "intensity" to compare the 1556 Shaanxi (830,000 deaths) and the 2010 Maule, Chile ($M_W$ 8.8, 521 deaths) more informative than using "magnitude" — and why is the inverse also true?

Summary

• Three quantities, not interchangeable: magnitude (source), ground motion (waveform), intensity (felt effect).
• PGA dominates short-period band; $S_a(T)$ is the engineering quantity that matches a building of period $T$.
• $T_{\text{building}} \approx N/10$ s links architecture to seismology.
• Site effects are first-order: impedance + resonance can amplify by 2–10×.
• GMPEs predict mean and scatter; ShakeMap conditions on observations.
• Liquefaction and four failure modes explain most damage; engineering retrofits address each.
• Cascadia long-period forecasts are now embedded in WA building codes.

Next lecture: the same Cascadia rupture, viewed as an oceanic forcing — the tsunami.