A Mathematical Framework for the Evaluation of Soil Liquefaction

Introduction

Soil liquefaction is a complex geotechnical phenomenon wherein a saturated, cohesionless soil deposit loses its shear strength and stiffness in response to an applied stress, usually dynamic earthquake shaking, causing it to behave as a viscous liquid. The catastrophic failure of foundations, earthworks, and critical infrastructure during major seismic events is frequently attributed to this mechanism. Consequently, the rigorous evaluation of liquefaction potential is a cornerstone of modern geotechnical earthquake engineering.

This document presents a comprehensive mathematical framework for the analysis of soil liquefaction. It progresses from the foundational deterministic methods based on the factor of safety to advanced probabilistic, strain-based, and numerical approaches. The primary objective is to detail the engineering and mathematical formalisms required to:

1. Quantify the seismic demand imposed on a soil deposit (Cyclic Stress Ratio).
2. Evaluate the intrinsic capacity of the soil to resist liquefaction (Cyclic Resistance Ratio) using a variety of in-situ testing methods.
3. Incorporate uncertainty through probabilistic and reliability-based frameworks.
4. Predict the physical consequences of liquefaction, such as settlement and lateral displacement.
5. Introduce alternative and advanced analytical paradigms, including energy-based methods and effective stress-based numerical modeling.

The formulations presented herein are grounded in empirical data from landmark case histories and extensive laboratory testing, providing the engineer with a robust toolkit for the assessment and mitigation of liquefaction hazards.

Evaluating Liquefaction

Introduction

To assess the liquefaction potential of soil, the following are required:

1. Field tests (in-situ)
2. Laboratory analysis
3. Empirical correlations
4. Engineering judgment

Process

Soil Sampling and Testing

• Field Investigation: Collect soil data from the site using methods like Standard Penetration Test (SPT), Cone Penetration Test (CPT), or Shear Wave Velocity $(V_s)$ measurements.

• Laboratory Testing: Determine soil index properties such as grain size distribution, relative density, water content, and Atterberg limits to supplement in-situ data.

Determination of Soil Properties

• Soil Type: Identify if the soil is liquefaction-susceptible, typically non-plastic, sandy, or silty soils.
• Saturation Level: Evaluate the groundwater level and degree of saturation, as fully saturated soils are a prerequisite for liquefaction.
• Density and Relative Density: Measure the soil’s state parameters, which are primary controls on liquefaction resistance.

Seismic Parameters

• Peak Ground Acceleration (PGA): Estimate the maximum ground acceleration $(a_{max})$ during an earthquake, usually from seismic hazard maps or site-specific Probabilistic Seismic Hazard Analysis (PSHA).
• Magnitude of Earthquake ($M_w$): Consider the moment magnitude of the design earthquake for the region, which influences the duration of shaking.

The Simplified Stress-Based Procedure

The foundational approach in liquefaction analysis is the comparison of the seismic loading on the soil with the soil’s capacity to resist that loading.

Cyclic Stress Ratio (CSR)

The seismic demand on the soil at a specific depth is quantified by the Cyclic Stress Ratio (CSR).

• where:
  • $a_{max}$ is the peak ground acceleration.
  • $g$ is the acceleration due to gravity.
  • $\sigma_v$ is the total vertical stress.
  • $\sigma_v^{\prime}$ is the effective vertical stress.
  • $r_d$ is the depth reduction factor, accounting for the non-rigid response of the soil column.

Cyclic Resistance Ratio (CRR)

The soil’s intrinsic capacity to resist liquefaction is quantified by the Cyclic Resistance Ratio (CRR), determined empirically from in-situ test data.

Factor of Safety (FS)

The potential for liquefaction is evaluated using a Factor of Safety, defined as the ratio of capacity to demand.

A value of $\mathrm{FS} \leq 1$ suggests that liquefaction is expected to be triggered.

Evaluating CRR from In-Situ Tests

SPT-Based Evaluation

Empirical Formula for $\mathrm{CRR}_{7.5}$

The baseline CRR, for a magnitude 7.5 earthquake, is correlated to the corrected SPT blow count, $(\mathrm{N1})_{60}$. A common expression is:

Final Calculation of CRR

The baseline value is adjusted for non-standard earthquake magnitudes and in-situ stress conditions:

• where:
  • $\mathrm{MSF}$ is the Magnitude Scaling Factor: $\mathrm{MSF} = (7.5/M_w)^{2.56}$ or a similar relation.
  • $\mathrm{K}_{\sigma}$ is the overburden correction factor:
    $$ \begin{aligned} \mathrm{K}_{\sigma} = (\sigma_{v0}^{\prime}/P_a)^{f-1} \end{aligned} $$
    where, $f$ depends on soil properties.
  • $\mathrm{K}_{\alpha}$ is the static shear stress correction factor for sloping ground conditions.

CPT-Based Evaluation

CPT Parameter Normalization

The cone resistance ($q_c$) and friction ($f_s$) are normalized to account for overburden pressure. The primary parameter is the normalized cone resistance, $q_{c1N}$:

The soil behavior is classified using the Soil Behavior Type Index, $I_c$, which helps adjust for fines content.

CPT-Based CRR Formula

CRR is calculated as a function of the clean-sand equivalent normalized cone resistance, $q_{c1Ncs}$:

Shear Wave Velocity ($V_s$)-Based Evaluation

$V_s$ Normalization

The shear wave velocity is corrected for overburden stress to $V_{s1}$:

$V_s$-Based CRR Formula

The CRR for a magnitude 7.5 earthquake is given by empirical correlations of the form:

• where $a$, $b$, and $V_{s1,cr}$ are empirically derived constants.

Pore Pressure Generation and Dissipation

Pore Pressure Generation Models

The build-up of excess pore water pressure ($\Delta u$) during an earthquake is often expressed as the pore pressure ratio, $r_u = \Delta u / \sigma_{v0}^{\prime}$. The empirical model by Seed et al. (1976) relates $r_u$ to the cycle ratio ($N/N_L$):

• where $N_L$ is the number of cycles required to trigger liquefaction.

Pore Pressure Dissipation (Consolidation)

The dissipation of excess pore pressure after shaking is governed by Terzaghi’s one-dimensional consolidation theory. The governing partial differential equation is:

• where $c_v$ is the coefficient of consolidation. Solving this equation predicts the time-dependent settlement of the post-liquefied soil.

Probabilistic and Reliability-Based Analysis

Probability of Liquefaction ($P_L$)

To account for uncertainty, the deterministic Factor of Safety is converted to a Probability of Liquefaction ($P_L$) using a log-normal cumulative distribution function:

• where $\Phi$ is the standard normal cumulative distribution function and $\sigma_{\ln(\text{model})}$ represents the total model uncertainty.

Liquefaction Potential Index (LPI)

The overall hazard for a soil profile is integrated using the Liquefaction Potential Index (LPI):

• where $F(z)$ is a severity function based on FS, and $W(z) = 10 - 0.5z$ is a depth-weighting function.

Reliability Index ($\beta$)

A more rigorous probabilistic assessment uses reliability methods. The state is defined by a performance function, $g = \text{CRR} - \text{CSR}$. The reliability index $\beta$ is:

The probability of liquefaction is then $P_L = \Phi(-\beta)$. This framework allows for systematic uncertainty propagation using methods like the First-Order Second-Moment (FOSM) approach.

Post-Liquefaction Consequence Analysis

Settlement

Post-liquefaction settlement ($S_v$) is calculated by integrating the reconsolidation volumetric strain ($\varepsilon_v$) over the depth of the liquefied layers.

• The volumetric strain, $\varepsilon_v$, is estimated from empirical charts relating it to factors like FS, relative density ($D_r$), or corrected SPT blow count ($(\mathrm{N1})_{60}$).

Lateral Spreading

Horizontal ground displacement ($D_H$) is estimated using multilinear regression models, which relate the displacement to seismic parameters ($M_w, R$) and site characteristics (slope $S$, free-face ratio $W$, thickness of liquefiable layer $T_{15}$).

Advanced Analytical Frameworks

Strain-Based Evaluation

This approach compares the earthquake-induced cyclic shear strain ($\gamma_{cyc}$) to the soil’s shear strain capacity ($\gamma_{L}$). The demand is calculated iteratively:

• where the modulus reduction $G/G_0$ is strain-dependent.

Energy-Based Approach

This paradigm postulates that liquefaction triggers when the seismic energy dissipated by a soil volume ($\Delta E$) reaches its energy capacity ($W$).

• Seismic Energy Demand: $\Delta E = \int \tau(t) , d\gamma(t)$
• Factor of Safety: $\mathrm{FS}_E = W / \Delta E$

Effective Stress-Based Numerical Modeling

For critical projects, direct numerical simulation using finite element or finite difference methods offers the highest fidelity. These models solve the coupled equations of motion and fluid flow. The core is an advanced plasticity-based constitutive model that mathematically describes the soil’s behavior under cyclic loading and captures the generation and dissipation of pore water pressure in time. The incremental elasto-plastic stress-strain relationship is expressed as:

• where $D^{ep}$ is the elasto-plastic constitutive matrix derived from a yield surface, flow rule, and hardening law.

Conclusion

The mathematical evaluation of soil liquefaction has evolved from a single, deterministic Factor of Safety to a sophisticated, multi-faceted discipline. As detailed in this document, a comprehensive analysis requires a hierarchical approach, beginning with the foundational stress-based method and escalating, as project demands dictate, to more complex probabilistic, deformation-based, and numerical techniques.

The accurate characterization of in-situ conditions through SPT, CPT, and $V_s$ testing forms the empirical bedrock of any analysis. The subsequent application of correction factors and carefully chosen correlations is essential for refining the soil’s Cyclic Resistance Ratio. However, acknowledging the inherent uncertainties in both seismic loading and soil capacity is critical. Probabilistic and reliability-based methods provide the necessary framework to move beyond a binary prediction of triggering to a more nuanced assessment of failure probability and risk.

Ultimately, the triggering of liquefaction is only the precursor to its engineering consequences. The analysis of post-liquefaction settlement and lateral spreading translates the geotechnical assessment into tangible metrics of structural performance and damage potential. For the most critical infrastructure, advanced numerical modeling provides the highest-fidelity tool for simulating the complex soil-structure-fluid interaction throughout a seismic event. The selection of an appropriate analytical methodology, from the simplified method to a full effective stress analysis, remains a key exercise of engineering judgment, balancing computational effort with the risk tolerance and performance objectives of the project at hand.