4.3 Dynamic Soil Properties & Ground Response Analysis

4.3 Dynamic Soil Properties & Ground Response Analysis

1. Small-Strain Dynamic Soil Properties

Under dynamic loading (earthquakes, machine vibrations, wave action), soil behavior is non-linear, inelastic, and strain-dependent. At very small shear strains ($\gamma \le 10^{-6}$ or $10^{-4}\%$), soil behaves in a linear-elastic manner without acoustic energy loss or permanent fabric rearrangement. The maximum shear modulus, designated as $G_{max}$ (or $G_0$), is a fundamental parameter in dynamic geotechnical engineering:

$$G_{max} = \rho V_s^2 = \left(\frac{\gamma_t}{g}\right) V_s^2$$

where:

• $\rho$ = mass density of soil ($\text{kg/m}^3$ or $\text{slugs/ft}^3$).
• $\gamma_t$ = total unit weight of soil ($\text{kN/m}^3$ or $\text{pcf}$).
• $g$ = acceleration of gravity ($9.81\text{ m/s}^2$ or $32.2\text{ ft/s}^2$).
• $V_s$ = low-strain shear wave velocity ($\text{m/s}$ or $\text{ft/s}$).

Empirical correlations express $G_{max}$ for clean sands as a function of mean effective stress ($\sigma'_m$) and void ratio ($e$) (Hardin & Drnevich 1972):

$$G_{max} = \frac{625}{0.3 + 0.7 e^2} \cdot P_a^{1-n} \cdot (\sigma'_m)^n$$

where $n \approx 0.5$ for granular soils and $P_a$ is atmospheric pressure.

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2. Modulus Reduction ($G/G_{max}$) and Damping Ratio ($D$) Curves

As cyclic shear strain amplitude ($\gamma$) increases beyond the elastic threshold, two key developments occur:

1. Modulus Reduction: Soil stiffness degrades due to micro-slippage at particle contacts. The secant shear modulus $G$ decreases below $G_{max}$, defining the normalized modulus reduction curve $G/G_{max}$ vs. $\gamma$.
2. Hysteretic Damping: Energy is dissipated per loading cycle through mechanical friction and plastic deformation. The equivalent hysteretic damping ratio $D$ increases from a minimum small-strain damping $D_{min}$ (typically $1\% - 3\%$) to large-strain values exceeding $20\% - 25\%$.

Strain Threshold Boundaries:
1. γ <= 10^-6 (10^-4 %): Linear Elastic Region (G = Gmax, D = Dmin, no permanent strain)
2. 10^-6 < γ <= 10^-4: Elastic-Plastic Non-Linear Region (G begins to decrease, D increases)
3. γ > 10^-4 (10^-2 %): Volumetric Strain Threshold γ_tv (Pore water pressure accumulation begins)

Influencing factors (Vucetic & Dobry 1991; Darendeli 2001):

• Plasticity Index ($PI$): High plasticity clays retain linear behavior up to higher shear strains compared to non-plastic sands. Consequently, high $PI$ soils exhibit less modulus degradation (higher $G/G_{max}$) and lower damping ratios ($D$) at a given shear strain level.
• Effective Overburden Stress ($\sigma'_m$): Higher effective confining stress increases $G/G_{max}$ curves and decreases damping $D$ in cohesionless soils.

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3. One-Dimensional Ground Response Wave Equation

Ground response analysis computes how vertically propagating body shear waves (S-waves) travel from underlying bedrock through soil layers to the ground surface. For a viscoelastic horizontal soil layer of mass density $\rho$, the 1D shear wave propagation equation is expressed as:

$$\rho \frac{\partial^2 u}{\partial t^2} = G \frac{\partial^2 u}{\partial z^2} + \eta \frac{\partial^3 u}{\partial z^2 \partial t}$$

where $u(z,t)$ is horizontal displacement, $G$ is shear modulus, and $\eta$ is viscous damping coefficient (where viscosity $\eta = \frac{2 G D}{\omega}$).
In frequency-domain complex notation, the dynamic shear modulus $G^*$ is written as:

$$G^* = G (1 + 2i D)$$

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4. Equivalent Linear (EQL) vs. Fully Nonlinear (NL) Analysis

Two principal numerical frameworks are used in geotechnical engineering practice to solve 1D ground response:

1. Equivalent Linear Method (EQL - SHAKE Approach)

• Concept: Approximates non-linear hysteretic soil response using linear viscoelastic wave propagation theory, solved iteratively in the frequency domain.
• Procedure:
  1. Assume initial estimates of $G = G_{max}$ and $D = D_{min}$ for each soil layer.
  2. Perform frequency-domain wave propagation to compute horizontal acceleration and shear strain time histories.
  3. Determine the maximum shear strain $\gamma_{max}$ in each layer and compute an effective cyclic shear strain: $$\gamma_{eff} = R\gamma \cdot \gamma_{max}$$ where $R_\gamma \approx 0.65$ (or $R_\gamma = \frac{M_w - 1}{10}$).
  4. Read updated values of $G_{new}$ and $D_{new}$ from the strain-dependent soil property curves ($G/G_{max}$ and $D$ vs. $\gamma_{eff}$).
  5. Check convergence: If $|G_{new} - G_{old}| / G_{old} > \text{tolerance}$ (typically $1\%$), repeat steps 2–4.
• Limitations: Cannot model permanent shear deformations or transient pore pressure buildup ($r_u$), and tends to over-damp high frequencies during strong shaking ($\gamma_{max} > 1\%$).

2. Fully Nonlinear Method (NL - DEEPSOIL / PM4Sand / PDMY)

• Direct time-domain integration of constitutive stress-strain relationships.
• Models hysteresis loops directly, including cyclic pore pressure generation, strain-hardening, dilatancy, and residual shear deformation.

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5. Site Period, Resonance, and Impedance Ratio

The fundamental period of a soil deposit ($T_s$) governs how the ground filters bedrock motions. For a uniform single soil layer of thickness $H$ over rigid bedrock:

$$T_s = \frac{4H}{V_s}$$ $$f_s = \frac{1}{T_s} = \frac{V_s}{4H}$$

For a multi-layered soil profile, the effective fundamental period is computed using travel-time summation:

$$T_s = 4 \sum_{i=1}^{n} \frac{d_i}{V_{si}}$$

Impedance Ratio ($\alpha$):

The contrast in dynamic stiffness between the underlying rock (base) and overlying soil layer 1 determines energy transmission and reflection across the boundary:

$$\alpha = \frac{\rho_1 V_{s1}}{\rho_{rock} V_{s,rock}}$$

At the fundamental frequency ($f_s$), the theoretical amplification ratio $A_{max}$ of ground motion from rigid bedrock to surface (with soil damping $D$) is:

$$A_{max} \approx \frac{1}{\alpha + \frac{\pi}{2} D}$$

When the natural period of an overlying building structure ($T_{struct}$) matches the fundamental site period ($T_s$), double resonance occurs, inducing high structural accelerations and catastrophic damage.

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6. Comprehensive Step-by-Step Worked Example

Problem Statement:

A foundation site consists of a $20.0\text{-meter}$ thick uniform stiff clay deposit over bedrock.

• Clay unit weight: $\gamma_t = 19.0\text{ kN/m}^3$
• Measured shear wave velocity: $V_s = 250\text{ m/s}$
• Bedrock properties: $\gamma_{rock} = 24.0\text{ kN/m}^3$, $V_{s,rock} = 1200\text{ m/s}$
• Soil damping ratio at small strains: $D = 4.0\% = 0.04$
• Structure proposed for the site: A 5-story reinforced concrete building with an estimated natural period $T_{struct} = 0.50\text{ s}$.

Calculate:

1. Small-strain shear modulus $G_{max}$ of the clay layer.
2. Fundamental period ($T_s$) and fundamental frequency ($f_s$) of the soil profile.
3. Impedance ratio ($\alpha$) between the clay layer and underlying bedrock.
4. Theoretical peak amplifications ratio ($A_{max}$) at the site's fundamental period.
5. Assess whether double resonance between the structure and ground site is a design concern.

Solution:

Step 1: Compute Small-Strain Shear Modulus ($G_{max}$)
Mass density of clay:

$$\rho = \frac{\gamma_t}{g} = \frac{19,000\text{ N/m}^3}{9.81\text{ m/s}^2} = 1,936.8\text{ kg/m}^3$$ $$G_{max} = \rho V_s^2 = 1,936.8\text{ kg/m}^3 \times (250\text{ m/s})^2 = 121,050,000\text{ Pa} = \mathbf{121.05\text{ MPa}}$$

Step 2: Fundamental Site Period ($T_s$) and Frequency ($f_s$)

$$T_s = \frac{4H}{V_s} = \frac{4 \times 20.0\text{ m}}{250\text{ m/s}} = \mathbf{0.320\text{ s}}$$ $$f_s = \frac{1}{T_s} = \frac{1}{0.320\text{ s}} = \mathbf{3.125\text{ Hz}}$$

Step 3: Compute Impedance Ratio ($\alpha$)

$$\rho_{rock} = \frac{24,000}{9.81} = 2,446.5\text{ kg/m}^3$$ $$\alpha = \frac{\rho_{clay} V_{s,clay}}{\rho_{rock} V_{s,rock}} = \frac{1,936.8 \times 250}{2,446.5 \times 1200} = \frac{484,200}{2,935,800} = \mathbf{0.1649}$$

Step 4: Peak Theoretical Site Amplification Ratio ($A_{max}$)

$$A_{max} \approx \frac{1}{\alpha + \frac{\pi}{2} D} = \frac{1}{0.1649 + \left(\frac{\pi}{2} \times 0.04\right)} = \frac{1}{0.1649 + 0.0628} = \frac{1}{0.2277} = \mathbf{4.39}$$

Step 5: Assessment of Building Resonance

• Soil site period $T_s = 0.32\text{ s}$
• Building natural period $T_{struct} = 0.50\text{ s}$
• Ratio $T_{struct} / T_s = 0.50 / 0.32 = 1.56$.
  Since $T_{struct}$ does not equal $T_s$ (they differ by over $50\%$), direct resonance is mitigated. However, non-linear softening during strong shaking will lengthen the soil period $T_s$ toward $0.45 - 0.50\text{ s}$, which could bring the degraded site period into resonance with the building during a major seismic event.