r4.05.06 Equivalent linear method for the propagation of 1D waves

Summary

This document describes the equivalent linear method and its implementation in*Code_Aster* in the DEFI_SOL_EQUI [U4.81.31] command.

The equivalent linear method makes it possible to calculate the response of a one-dimensional (1D — one component) or multidimensional (1D — three components) soil column, stratified horizontally to the vertical propagation of shear waves (SV). It provides an approximate account of the behavior of soils under cyclic loading. The method for resolving the equivalent linear is an iterative procedure, where equivalent linear viscoelastic characteristics are evaluated at each iteration for each soil layer on the basis of the degradation curves of the shear modulus \(G\) and the increase in hysteretic damping measured experimentally.

The simplicity of implementation, the quality of the results and the speed of the calculations make it a widely used method in engineering. The one-dimensional equivalent linear method is thus implemented in numerous calculation codes (SHAKE, FLUSH, CYBERQUAKE, DEEPSOIL, EERA,…).

The application of the equivalent linear method remains valid as long as the volume deformation of the ground can be neglected. Otherwise, it is necessary to turn to non-linear models integrating the shear-mean pressure coupling (or deviatorial-dilatance deformations) such as the cyclic elasto-plastic model of Hujeux [R7.01.23] (Hujeux’s law of cyclic behavior for soils).

Traditionally, the equivalent linear model is accepted up to free field distortion deformations of the order of \({10}^{-3}\) (Guide ASN /2/01). This remains an order of magnitude of validity, which it is necessary to adapt to the situation and to the phenomena that one wishes to model.

• 1. Introduction
• 2. One-dimensional viscoelastic stress-strain relationship
• 3. Stress-strain relationship in three dimensions
• 4. Formulations of equivalent linear behavior
  • 4.1. Schnabel formulation
  • 4.2. Lysmer formulation
  • 4.3. Comparison of the energies dissipated by the two formulations
• 5. Simplified consideration of water pressure using the Byrne model
• 6. Implementation of the equivalent linear in code_aster
  • 6.1. Boundary conditions of the model
  • 6.2. Equivalent linear model formulations
  • 6.3. Resolution method
• 7. Choice of the weighting coefficient of the effective deformation
• 8. Choice of the weighting factor for the increase in effective water pressure
• 9. Verification and validation
• 10. Bibliography