Theory and Methods

This section provides the mathematical and theoretical foundation for site response analysis methods implemented in PyStrata.

• Site Response Fundamentals
  • Physical Principles
  • Transfer Function Approach
  • Boundary Conditions
  • Time Domain vs. Frequency Domain
  • Validation and Verification
• Wave Propagation
  • Governing Equations
  • Solution Methods
  • Boundary Conditions
  • Numerical Implementation
  • Validation Examples
  • Implementation Details
• Nonlinear Soil Models
  • Equivalent Linear Approach
  • Empirical Models
  • Implementation in PyStrata
• Uncertainty Analysis
  • Sources of Uncertainty
  • Logic Tree Framework
  • Monte Carlo Simulation
  • Result Processing

Overview

Site response analysis relies on the fundamental principles of wave propagation in layered media. The one-dimensional assumption treats seismic waves as vertically propagating shear waves (SH waves) through horizontally layered soil deposits.

The governing physics include:

Wave Equation

The equation of motion for shear wave propagation in a continuous medium

Boundary Conditions

Stress and displacement continuity at layer interfaces

Material Constitutive Models

Relationships between stress, strain, and material properties

Damping Mechanisms

Energy dissipation through material and radiation damping

Mathematical Framework

The theoretical foundation is built on:

1. Linear Wave Theory - For small strain elastic wave propagation

2. Transfer Functions - Frequency domain representation of system response

3. Equivalent Linear Method - Iterative approach for strain-compatible properties

4. Random Vibration Theory - Statistical treatment of stochastic ground motion

5. Uncertainty Propagation - Monte Carlo and logic tree methods

Key Assumptions

Standard site response analysis makes several simplifying assumptions:

• One-dimensional propagation - Waves travel vertically through horizontal layers

• Linear viscoelastic behavior - For equivalent linear methods

• Uniform layer properties - Homogeneous properties within each layer

• Perfect layer bonding - No sliding at interfaces

• Infinite lateral extent - No boundary effects from finite dimensions

These assumptions are reasonable for most engineering applications but may require modification for complex site geometries or extreme ground motions.

Implementation Notes

PyStrata implements these theoretical methods with careful attention to:

• Numerical stability - Robust algorithms for wave propagation calculations

• Frequency resolution - Adequate sampling for accurate results

• Convergence criteria - Appropriate tolerances for iterative methods

• Validation - Comparison with analytical solutions and benchmark problems