4.2 Soil Liquefaction Triggering & Post-Liquefaction Assessment

4.2 Soil Liquefaction Triggering & Post-Liquefaction Assessment

1. Physics of Soil Liquefaction

Soil liquefaction is one of the most destructive geotechnical hazards caused by strong earthquake ground shaking. Under undrained cyclic shear loading, loose to medium-dense saturated cohesionless soils (sands, non-plastic silts, and gravelly sands) tend to contract. Because pore water cannot drain rapidly during brief earthquake shaking, contractive volumetric tendencies transfer stress from the soil skeleton to the pore fluid, generating excess pore water pressure ($\Delta u$).

As excess pore pressure accumulates, the excess pore pressure ratio $r_u$ approaches unity:

$$r_u = \frac{\Delta u}{\sigma'_{vo}} \rightarrow 1.0$$

According to Terzaghi's effective stress principle ($\sigma' = \sigma - u$), when $\Delta u \approx \sigma'_{vo}$, the effective stress drops to zero ($\sigma' \to 0$). Consequently, frictional shear strength ($\tau = \sigma' \tan\phi'$) vanishes, causing the soil deposit to transform transiently into a dense fluid suspension. Two primary phenomena result:

1. Flow Liquefaction: Occurs in loose soils where static shear stresses (such as on slopes or under footing edges) exceed the residual strength of the liquefied soil ($s_r$), triggering catastrophic, unlimited flow slides.
2. Cyclic Mobility: Occurs in medium-dense soils under level or gently sloping ground where cyclic stress reversals produce large incremental shear deformations, but static equilibrium is maintained due to strain-hardening upon dilation at larger strains.

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2. Liquefaction Susceptibility Criteria

Not all soils are susceptible to liquefaction. Geotechnical screening evaluates grain size distribution, density, groundwater depth, and soil plasticity:

• Cohesionless Soils: Clean sands ($FC < 5\%$), silty sands, and non-plastic silts with high relative density ($D_r > 75\%$) or high blow counts ($N_{1,60cs} \ge 30$) are non-liquefiable under typical design earthquakes.
• Fine-Grained Soil Susceptibility (Bray & Sancio 2006 / Boulanger & Idriss 2006):
  • Susceptible to Liquefaction (Sand-like Behavior): Plasticity Index $PI < 12$, Liquid Limit $LL < 37$, and in-situ water content ratio $w_c / LL > 0.85$.
  • Moderately Susceptible: $12 \le PI \le 18$ and $LL < 45$.
  • Clay-like Cyclic Softening (Non-Liquefiable by Classical Sand Rules): $PI > 18$. These soils undergo cyclic softening rather than classical liquefaction and must be evaluated using fine-grained dynamic strength reduction procedures.

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3. Simplified Triggering Procedure (Seed & Idriss Methodology)

The industry standard for evaluating liquefaction triggering is the Simplified Procedure (Seed & Idriss 1971; Youd et al. 2001; Boulanger & Idriss 2014). It involves comparing the seismic demand (Cyclic Stress Ratio, $CSR$) against the soil capacity (Cyclic Resistance Ratio, $CRR$).

Step A: Compute Seismic Demand ($CSR$)

The average cyclic shear stress ratio induced by an earthquake at depth $z$ is calculated as:

$$CSR = 0.65 \cdot \left(\frac{a_{max}}{g}\right) \cdot \left(\frac{\sigma_{vo}}{\sigma'_{vo}}\right) \cdot r_d$$

where:

• $a_{max}$ = peak horizontal ground surface acceleration.
• $g$ = acceleration of gravity ($9.81\text{ m/s}^2$ or $32.2\text{ ft/s}^2$).
• $\sigma_{vo}$ = total vertical overburden stress at depth $z$.
• $\sigma'_{vo}$ = effective vertical overburden stress at depth $z$.
• $r_d$ = stress reduction coefficient accounting for soil column flexibility (Boulanger & Idriss 2014): $$r_d = \exp\left( -1.012 - 0.00518 z + 0.0024 z^{1.5} + (0.0158 + 0.0006 z) M_w \right)$$ Or for simplified hand calculations ($z \le 9.15\text{ m}$ / $30\text{ ft}$): $$r_d \approx 1.0 - 0.00765 z \quad (z \text{ in meters})$$

Step B: Compute Soil Capacity ($CRR_{7.5}$)

Field penetration resistance is first corrected to an equivalent clean-sand blow count ($N_{1,60cs}$):

$$N_{1,60} = N_{60} \cdot C_N \cdot C_B \cdot C_R \cdot C_S$$

where $C_N = \sqrt{\frac{P_a}{\sigma'_{vo}}} \le 1.7$ is the overburden correction factor ($P_a = 100\text{ kPa} \approx 1\text{ atm}$).
The fine-content correction yields $N_{1,60cs}$:

$$N_{1,60cs} = N_{1,60} + \Delta N_{1,60}$$ $$\Delta N_{1,60} = \exp\left( 1.63 + \frac{9.7}{FC + 0.01} - \left(\frac{15.7}{FC + 0.01}\right)^2 \right)$$

For an $M_w = 7.5$ earthquake and effective overburden stress $\sigma'_{vo} = 1\text{ atm}$, $CRR_{M7.5, 1atm}$ is estimated per Boulanger & Idriss (2014):

$$CRR_{7.5} = \exp\left( \frac{N_{1,60cs}}{14.1} + \left(\frac{N_{1,60cs}}{126}\right)^2 - \left(\frac{N_{1,60cs}}{23.6}\right)^3 + \left(\frac{N_{1,60cs}}{25.4}\right)^4 - 3 \right)$$

Step C: Compute Factor of Safety ($FS_{liq}$)

To adjust $CRR_{7.5}$ for different earthquake magnitudes ($M_w$) and overburden stresses ($\sigma'_{vo} \neq 1\text{ atm}$):

$$FS_{liq} = \frac{CRR_{7.5} \cdot MSF \cdot K\sigma}{CSR}$$

where:

• Magnitude Scaling Factor ($MSF$): Corrects for duration of shaking relative to $M_w = 7.5$: $$MSF = 6.9 \cdot \exp\left(-\frac{M_w}{4}\right) - 0.058 \le 1.8$$
• Overburden Correction Factor ($K_\sigma$): Accounts for non-linear reduction in cyclic strength ratio at high effective stresses: $$K\sigma = 1 - C\sigma \ln\left(\frac{\sigma'_{vo}}{P_a}\right) \le 1.1$$ where $C_\sigma = \frac{1}{18.9 - 2.55 \sqrt{N_{1,60cs}}} \le 0.3$.

A soil layer is predicted to liquefy if $FS_{liq} < 1.0$ (or $FS_{liq} < 1.2 - 1.5$ per engineering design standards).

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4. Post-Liquefaction Assessment: Settlement & Lateral Spreading

When liquefaction is triggered ($FS_{liq} < 1.0$), geotechnical engineers must quantify post-liquefaction deformations:

1. Post-Liquefaction Volumetric Settlement (Reconsolidation):

   • Excess pore pressures dissipate upward after shaking, causing volumetric contraction strain ($\varepsilon_v$) in liquefied layers.
   • $\varepsilon_v$ is evaluated as a function of $FS_{liq}$ and $N_{1,60cs}$ (Ishihara & Yoshimine 1992; Zhang et al. 2002).
   • Total ground surface settlement $\Delta S$ is the integral of volumetric strain across layer thickness $H$: $$\Delta S = \sum_{i=1}^{m} \varepsilon_{v,i} \cdot \Delta z_i$$

2. Residual Shear Strength ($s_r$):

   • Liquefied soil retains a small residual undrained shear strength $s_r$ due to soil fabric jamming and friction at large strains.
   • $s_r$ (or normalized ratio $s_r / \sigma'_{vo}$) is estimated from empirical SPT correlations (Olson & Stark 2002; Idriss & Boulanger 2008): $$s_r = P_a \cdot \exp\left( \frac{N_{1,60cs}}{16} + \left(\frac{N_{1,60cs}}{16}\right)^3 - 3.0 \right) \quad \text{or} \quad \frac{s_r}{\sigma'_{vo}} = 0.02 - 0.12$$

3. Lateral Spreading Displacement:

   • Gentle slopes ($0.1\% - 5\%$) or free-face margins (riverbanks, seawalls) undergo lateral movement driven by static gravity stresses operating on liquefied soil.
   • Lateral Displacement Index ($LDI$): Calculated by integrating maximum shear strains ($\gamma_{max}$) over depth: $$LDI = \int_{0}^{z_{max}} \gamma_{max} \cdot dz$$
   • Empirical models (Youd, Hansen, & Bartlett 2002 Multi-Linear Regression) predict horizontal displacement ($D_H$) as a function of slope steepness ($W\%$ or $S\%$), earthquake magnitude ($M_w$), distance to source ($R$), and cumulative liquefied thickness ($T_{15}$).

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5. Liquefaction Mitigation Strategies

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6. Comprehensive Step-by-Step Worked Example

Problem Statement:

A level site proposed for a commercial building has a saturated clean sand layer ($FC = 3\%$) located at a depth of $z = 6.0\text{ m}$.

• Water table depth: $z_{w} = 1.5\text{ m}$
• Total unit weight above water table: $\gamma_{m} = 18.0\text{ kN/m}^3$
• Saturated unit weight below water table: $\gamma_{sat} = 19.5\text{ kN/m}^3$
• Measured field blow count at $z = 6.0\text{ m}$: $N_{60} = 10$
• Design Earthquake: Magnitude $M_w = 7.5$, Peak Ground Acceleration $a_{max} = 0.30g$
• Overburden correction exponent $C_N = 1.15$, hammer efficiency ratio $C_B C_R C_S = 1.0$.
• Simplified stress reduction coefficient at $z = 6\text{ m}$: $r_d = 1.0 - 0.00765(6.0) = 0.954$.
• Volumetric strain correlation for this $N_{1,60cs}$ and $CSR$: $\varepsilon_v = 2.5\%$.
• Thickness of liquefied sand layer: $H = 4.0\text{ m}$.

Calculate:

1. Initial total stress $\sigma_{vo}$ and effective stress $\sigma'_{vo}$ at $z = 6.0\text{ m}$.
2. The Cyclic Stress Ratio ($CSR$).
3. The normalized clean sand blow count ($N_{1,60cs}$) and Cyclic Resistance Ratio ($CRR_{7.5}$).
4. Factor of Safety against Liquefaction ($FS_{liq}$).
5. Estimated post-liquefaction reconsolidation settlement ($\Delta S$).

Solution:

Step 1: Overburden Stresses at $z = 6.0\text{ m}$

• Total overburden stress: $$\sigma_{vo} = (1.5\text{ m} \times 18.0\text{ kN/m}^3) + (4.5\text{ m} \times 19.5\text{ kN/m}^3) = 27.0 + 87.75 = \mathbf{114.75\text{ kPa}}$$
• Pore water pressure: $$u_o = 4.5\text{ m} \times 9.81\text{ kN/m}^3 = \mathbf{44.15\text{ kPa}}$$
• Initial effective stress: $$\sigma'_{vo} = \sigma_{vo} - u_o = 114.75 - 44.15 = \mathbf{70.60\text{ kPa}}$$

Step 2: Compute Cyclic Stress Ratio ($CSR$)

$$CSR = 0.65 \times \left(\frac{a_{max}}{g}\right) \times \left(\frac{\sigma_{vo}}{\sigma'_{vo}}\right) \times r_d$$ $$CSR = 0.65 \times 0.30 \times \left(\frac{114.75}{70.60}\right) \times 0.954 = 0.195 \times 1.6253 \times 0.954 = \mathbf{0.3023}$$

Step 3: Compute $N_{1,60cs}$ and $CRR_{7.5}$

• Normalized blow count: $$N_{1,60} = N_{60} \times C_N = 10 \times 1.15 = 11.5$$ Since fine content $FC = 3\% < 5\%$, fine content correction $\Delta N_{1,60} = 0$, so $N_{1,60cs} = 11.5$.
• Cyclic Resistance Ratio ($CRR_{7.5}$) using Boulanger & Idriss formula for $N_{1,60cs} = 11.5$: $$CRR_{7.5} = \exp\left( \frac{11.5}{14.1} + \left(\frac{11.5}{126}\right)^2 - \left(\frac{11.5}{23.6}\right)^3 + \left(\frac{11.5}{25.4}\right)^4 - 3 \right)$$ $$CRR_{7.5} = \exp\left( 0.8156 + 0.0083 - 0.1158 + 0.0421 - 3.0 \right) = \exp(-2.2498) = \mathbf{0.1054}$$

Step 4: Factor of Safety against Liquefaction ($FS_{liq}$)
Since $M_w = 7.5$, $MSF = 1.0$. For $\sigma'_{vo} = 70.6\text{ kPa}$, $K_\sigma \approx 1.05$.

$$FS_{liq} = \frac{CRR_{7.5} \times MSF \times K\sigma}{CSR} = \frac{0.1054 \times 1.0 \times 1.05}{0.3023} = \frac{0.1107}{0.3023} = \mathbf{0.366}$$

Since $FS_{liq} = 0.366 \ll 1.0$, liquefaction is triggered in this layer.

Step 5: Post-Liquefaction Settlement ($\Delta S$)
With volumetric strain $\varepsilon_v = 2.5\%$ across $H = 4.0\text{ m}$ ($4000\text{ mm}$):

$$\mathbf{\Delta S} = \varepsilon_v \times H = 0.025 \times 4000\text{ mm} = \mathbf{100\text{ mm}} \quad (3.94\text{ inches})$$