4.1 Seismic Site Characterization & Ground Motion Hazards

4.1 Seismic Site Characterization & Ground Motion Hazards

1. Introduction & Wave Propagation Mechanics

Earthquake engineering in geotechnical practice begins with understanding how seismic waves originate from fault ruptures and propagate through the Earth's crust to reach engineering structures. Fault ruptures release stored strain energy in the form of seismic waves, broadly categorized into body waves and surface waves:

1. Body Waves (propagate through the interior of the earth):

   • Compressional or Primary Waves (P-waves): Longitudinal waves where particle motion is parallel to the direction of propagation. P-waves travel fastest through solid rock and soil, according to the velocity relation: $$V_p = \sqrt{\frac{M}{\rho}} = \sqrt{\frac{K + \frac{4}{3}G}{\rho}}$$ where $M$ is the constrained modulus, $K$ is the bulk modulus, $G$ is the shear modulus, and $\rho$ is total mass density.
   • Shear or Secondary Waves (S-waves): Transverse waves where particle motion is perpendicular to the direction of wave propagation. S-waves cannot propagate through fluids ($G = 0$) and are the primary driver of cyclic shear stresses in soil deposits during earthquake shaking. Shear wave velocity is expressed as: $$V_s = \sqrt{\frac{G}{\rho}}$$

2. Surface Waves (propagate along the earth's surface or boundary layer):

   • Rayleigh Waves (R-waves): Exhibit retrograde elliptical particle motion in a vertical plane aligned with propagation. Rayleigh waves decay slowly with depth and often dominate ground motion amplitude at long periods.
   • Love Waves (L-waves): Transverse horizontal motion confined to surface layers, requiring a velocity gradient (softer layer over stiffer substrate).

Seismic Wave Taxonomy:
├── Body Waves
│   ├── P-Waves (Compressional, parallel particle motion, fastest)
│   └── S-Waves (Shear, perpendicular particle motion, key for geotechnical shear stress)
└── Surface Waves
    ├── Rayleigh Waves (Retrograde elliptical motion in vertical plane)
    └── Love Waves (Transverse horizontal motion in surface layer)

---

2. Ground Motion Intensity Measures

To quantify earthquake ground motions for geotechnical analysis and structural design, several intensity measures are utilized:

• Peak Ground Acceleration ($PGA$): The maximum absolute amplitude of horizontal acceleration recorded on an accelerogram. It serves as the baseline parameter for simplified liquefaction and pseudostatic slope analyses.
• Peak Ground Velocity ($PGV$): Reflects ground motion energy in the intermediate period range ($0.5\text{ s} \le T \le 2.0\text{ s}$) and strongly correlates with structural drift and pipeline damage.
• Peak Ground Displacement ($PGD$): Dominates long-period structural response and permanent ground deformation.
• Spectral Acceleration ($S_a$): The peak response acceleration of a single-degree-of-freedom (SDOF) system with a specified natural period $T$ and critical damping ratio $\zeta$ (typically 5%).
• Arias Intensity ($I_a$): A cumulative measure of total seismic energy density per unit mass, integrated over the duration of shaking: $$I_a = \frac{\pi}{2g} \int_{0}^{t_d} [a(t)]^2 dt$$ where $a(t)$ is ground acceleration and $t_d$ is total motion duration. $I_a$ strongly correlates with seismic slope displacement and liquefaction triggering potential.

---

3. ASCE 7 / IBC Site Classification Framework

Local soil conditions dramatically alter the amplitude, frequency content, and duration of bedrock ground motions. ASCE 7-16 and ASCE 7-22 classify sites into six primary categories (Site Classes A through F) based on average soil properties within the top 30 meters (100 feet) of the subsurface profile.

---

4. Calculation of 30-Meter Average Shear Wave Velocity ($V_{s30}$)

$V_{s30}$ is defined as the time-weighted harmonic mean shear wave velocity across the top 30 meters of the site. It represents the total travel time of an S-wave propagating vertically through the upper 30 m:

$$V_{s30} = \frac{\sum_{i=1}^{n} d_i}{\sum_{i=1}^{n} \frac{d_i}{V_{si}}} = \frac{30\text{ m}}{\sum_{i=1}^{n} \frac{d_i}{V_{si}}}$$

where:

• $d_i$ = thickness of layer $i$ (in meters), such that $\sum_{i=1}^{n} d_i = 30\text{ m}$.
• $V_{si}$ = shear wave velocity of layer $i$ (in m/s).

Similarly, average equivalent SPT blow count ($\bar{N}_{60}$) for cohesionless layers is calculated as:

$$\bar{N}_{60} = \frac{\sum d_i}{\sum \frac{d_i}{N_{60,i}}}$$

---

5. Construction of Design Response Spectrum

Under ASCE 7 procedure, short-period ($S_s$ at $T = 0.2\text{ s}$) and 1-second period ($S_1$ at $T = 1.0\text{ s}$) Risk-Targeted Maximum Considered Earthquake ($MCE_R$) spectral accelerations mapped for bedrock (Site Class B/C boundary) are modified by site coefficients $F_a$ and $F_v$ to determine maximum site accelerations:

$$S_{MS} = F_a S_s$$ $$S_{M1} = F_v S_1$$

The Design Spectral Acceleration parameters ($S_{DS}$ and $S_{D1}$) are taken as two-thirds of the $MCE_R$ values:

$$S_{DS} = \frac{2}{3} S_{MS} = \frac{2}{3} F_a S_s$$ $$S_{D1} = \frac{2}{3} S_{M1} = \frac{2}{3} F_v S_1$$

Key transition periods for building the design curve are defined as:

$$T_s = \frac{S_{D1}}{S_{DS}}$$ $$T_0 = 0.2 T_s = 0.2 \frac{S_{D1}}{S_{DS}}$$

The design spectral response acceleration $S_a(T)$ across period $T$ is constructed in four period ranges:

1. For $0 \le T < T_0$: $$S_a = S_{DS} \left(0.4 + 0.6 \frac{T}{T_0}\right)$$
2. For $T_0 \le T \le T_s$: $$S_a = S_{DS}$$
3. For $T_s < T \le T_L$ (where $T_L$ is the mapped long-period transition period, typically 4–12 s): $$S_a = \frac{S_{D1}}{T}$$
4. For $T > T_L$: $$S_a = \frac{S_{D1} T_L}{T^2}$$

---

6. Seismic Hazard Analysis: PSHA vs. DSHA

Evaluation of ground motion hazards is performed using two alternative paradigms:

1. Deterministic Seismic Hazard Analysis (DSHA):

   • Selects a specific controlling seismic source scenario (controlling fault, maximum credible earthquake magnitude $M_{max}$, and minimum distance $R$).
   • Uses Ground Motion Models (GMMs / attenuation relations) to compute median or 84th-percentile ground motion intensity parameters.
   • Produces a single controlling worst-case ground motion scenario but ignores earthquake recurrence frequencies.

2. Probabilistic Seismic Hazard Analysis (PSHA):

   • Accounts for all potential seismic sources, rupture probabilities, magnitude recurrence statistics (Gutenberg-Richter relation $\log N = a - bM$), and ground motion variability.
   • Integrates probability distributions over magnitude, distance, and ground motion attenuation to construct a site seismic hazard curve (annual rate of exceedance $\lambda$ vs. peak parameter).
   • Standard PE exam design return periods:
     • Design Earthquake (DE): 10% probability of exceedance in 50 years $\rightarrow$ Return Period $T_R \approx 475\text{ years}$.
     • Maximum Considered Earthquake ($MCE_R$): 2% probability of exceedance in 50 years $\rightarrow$ Return Period $T_R \approx 2,475\text{ years}$.
     $$\lambda = -\frac{\ln(1 - P_e)}{t} \implies T_R = \frac{1}{\lambda}$$
   • Deaggregation: Identifies the principal contributing earthquake magnitude ($M^*$) and source distance ($R^*$) that govern the probabilistic hazard at a specific design period.

---

7. Field & Geophysical Testing Techniques

Determination of dynamic subsurface profile properties relies on specialized field methods:

• Crosshole Seismic Survey (ASTM D4428): Involves two or more cased boreholes. S-waves are generated in one hole and recorded at identical depths in adjacent holes. Provides highly accurate, direct layer-by-layer $V_s$ measurements.
• Downhole Seismic Survey: Uses a single borehole. Surface impact source generates waves detected by a downhole geophone receiver array at incremental depths.
• Multichannel Analysis of Surface Waves (MASW): Non-destructive surface wave method measuring Rayleigh wave dispersion to invert $V_s$ profile without drilling.
• Seismic CPT (sCPT): Geophone embedded inside CPT cone allows simultaneous measurement of tip resistance $q_c$, sleeve friction $f_s$, pore pressure $u_2$, and interval shear wave velocity $V_s$.

---

8. Comprehensive Step-by-Step Worked Example

Problem Statement:

A geotechnical site investigation for a high-rise foundation in Seattle, WA provides the following subsurface stratigraphy down to 30 meters depth:

• Layer 1 ($0 - 4\text{ m}$): Loose silty sand, $V_{s1} = 140\text{ m/s}$
• Layer 2 ($4 - 12\text{ m}$): Dense sand, $V_{s2} = 260\text{ m/s}$
• Layer 3 ($12 - 22\text{ m}$): Very dense gravelly sand, $V_{s3} = 420\text{ m/s}$
• Layer 4 ($22 - 30\text{ m}$): Weathered bedrock, $V_{s4} = 650\text{ m/s}$

Mapped short-period acceleration $S_s = 1.50g$ and 1-second period acceleration $S_1 = 0.60g$. Site amplification coefficients from ASCE 7 for this Site Class are $F_a = 1.00$ and $F_v = 1.50$.

Calculate:

1. The 30-meter average shear wave velocity ($V_{s30}$).
2. Determine the ASCE 7 Site Class.
3. Compute design spectral parameters $S_{DS}$, $S_{D1}$, and transition periods $T_0$ and $T_s$.
4. Calculate the Design Spectral Acceleration $S_a$ at a structural period $T = 0.50\text{ s}$.

Solution:

Step 1: Compute $V_{s30}$
Layer thicknesses ($d_i$):

• Layer 1: $d_1 = 4\text{ m}$, $V_{s1} = 140\text{ m/s} \implies \frac{d_1}{V_{s1}} = \frac{4}{140} = 0.02857\text{ s}$
• Layer 2: $d_2 = 8\text{ m}$, $V_{s2} = 260\text{ m/s} \implies \frac{d_2}{V_{s2}} = \frac{8}{260} = 0.03077\text{ s}$
• Layer 3: $d_3 = 10\text{ m}$, $V_{s3} = 420\text{ m/s} \implies \frac{d_3}{V_{s3}} = \frac{10}{420} = 0.02381\text{ s}$
• Layer 4: $d_4 = 8\text{ m}$, $V_{s4} = 650\text{ m/s} \implies \frac{d_4}{V_{s4}} = \frac{8}{650} = 0.01231\text{ s}$

Total travel time through upper 30 meters:

$$\sum_{i=1}^{4} \frac{d_i}{V_{si}} = 0.02857 + 0.03077 + 0.02381 + 0.01231 = 0.09546\text{ s}$$ $$\mathbf{V_{s30}} = \frac{30\text{ m}}{0.09546\text{ s}} = \mathbf{314.3\text{ m/s}}$$

Step 2: ASCE 7 Site Class Determination
Since $180\text{ m/s} \le V_{s30} = 314.3\text{ m/s} \le 360\text{ m/s}$, the site is classified as Site Class D (Stiff Soil).

Step 3: Calculate Design Spectral Parameters

• $S_{MS} = F_a S_s = 1.00 \times 1.50g = 1.50g$
• $S_{M1} = F_v S_1 = 1.50 \times 0.60g = 0.90g$
• $S_{DS} = \frac{2}{3} S_{MS} = \frac{2}{3} \times 1.50g = \mathbf{1.00g}$
• $S_{D1} = \frac{2}{3} S_{M1} = \frac{2}{3} \times 0.90g = \mathbf{0.60g}$

Transition periods:

• $T_s = \frac{S_{D1}}{S_{DS}} = \frac{0.60g}{1.00g} = \mathbf{0.60\text{ s}}$
• $T_0 = 0.20 \times T_s = 0.20 \times 0.60\text{ s} = \mathbf{0.12\text{ s}}$

Step 4: Calculate Design Acceleration $S_a(T = 0.50\text{ s})$
Since $T_0 (0.12\text{ s}) \le T (0.50\text{ s}) \le T_s (0.60\text{ s})$, the building period falls within the plateau region of the design spectrum:

$$\mathbf{S_a(0.50\text{ s})} = S_{DS} = \mathbf{1.00g}$$