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Compliant base condition

Setting the appropriate boundary condition at the base of a soil sediment for wave propagation problems is not a trivial task. Application of the earthquake motion using a 'rigid base' (Dirichlet) boundary condition (usually as a time-history of acceleration1 leads to reflections of downward propagating waves back into the model as depicted in Figure1. These reflections are often not readily apparent in complex non-linear analyses, as they can be masked by the damping at larger strains in non-linear soil models or by 'natural' reflections caused by dynamic impedance contrasts at layer boundaries. Therefore, a rigid base is only appropriate for cases with a large impedance contrast at the base of the model 1\(^,\)2.

Figure 1: Schematic representation of the half space subjected to a seismic input motion (left) and its corresponding numerical model using a rigid base (middle) or a compliant base (right) boundary condition)

Lysmer and Kuhlemeyer 6 proposed a viscous traction as boundary condition to absorb incident waves. It can be used without further modification when the source of excitation is within the model, e.g. simulations of vibratory pile driving or vibrating machinery. To simulate excitations originating from the outside of the model, such as incoming seismic waves, this boundary condition needs to be extended. To derive the boundary condition, the one-dimensional wave propagation of an harmonic wave through an elastic homogeneous solid is considered. Starting from the d'Alembert's general solution of the one-dimensional wave equation

\[ \begin{equation}\label{eq:dalembert} u(x,t) = f(x+ct) + g(x-ct) \end{equation} \]

where \(f\) and \(g\) are two waves with constant velocity \(c\) moving in opposite directions (i.e. \(f\) moves to the bottom and \(g\) to the top of the column depicted in Fig. \ref{fig:compliant_base_scheme}). Differentiation of Eq. (\(\ref{eq:dalembert}\)) leads to

\[ \begin{equation}\label{eq:dalembert_derivative} u_{,t}=v=c\left(f'-g'\right) ~~~\mbox{and}~~~ u_{,x}=\varepsilon=f'+g' \end{equation} \]

The traction needed to suppress any upwards reflections and, at the same time, to prescribe the input motion is found by substituting Eq. (\(\ref{eq:dalembert_derivative}\))\(_1\) in Eq. (\(\ref{eq:dalembert_derivative}\))\(_2\)

\[ \begin{equation} \varepsilon=v/c+2g' \end{equation} \]

Limiting the discussion to the shear direction only (\(v=v^s\), \(c=c^s\) and \(\varepsilon=\gamma\)), using \(\gamma=\tau/G\) and \(G=(c^s)^2 \rho\), and imposing the input motion as a prescribed velocity such that \(\hat{v}^s=\hat{g}=c^s\hat{g}'\), the equivalent traction \(\hat{\tau}\) is obtained as

\[ \begin{equation}\label{eq:compliant_base_stress} \hat{\tau} = c^s \rho v^s + 2 c^s \rho \hat{v}^s = c^s \rho \left( v^s + 2 \hat{v}^s \right) \end{equation} \]

Therein \(\rho\) is the density of the material, \(v^s\) is the particle velocity at the boundary and \(c^s\) is the shear wave velocity. The prescribed velocity \(\hat{v}\) corresponds to the upward propagating motion of the seismic input signal.

Since the absorption is independent of frequency, the boundary condition can absorb both harmonic and non-harmonic waves 6. It is worth noting that in addition to preventing reflections from the base of the model, a compliant base simplifies the computation of the appropriate input motion, since only the upward propagating motion is required.

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  2. M. Mánica, E. Ovando, and E. Botero. Assessment of damping models in flac. Computers and Geotechnics, 59:12 – 20, 2014. URL: http://www.sciencedirect.com/science/article/pii/S0266352X14000299, doi:https://doi.org/10.1016/j.compgeo.2014.02.007. ↩

  3. P. B. Schnabel, J. Lysmer, and Seed B.H. Shake: a computer program for earthquake response analysis of horizontally layered sites. EERC Report 72-12, University of California, Berkeley., 1972. ↩

  4. JP Bardet, K Ichii, and CH Lin. Eera. A computer program for equivalent-linear earthquake site response analyses of layered soil deposits. University of Southern California, Department of Civil Engineering, user's manual, 2000. ↩

  5. David Robinson, Trevor Dhu, and John Schneider. Sua: a computer program to compute regolith site-response and estimate uncertainty for probabilistic seismic hazard analyses. Computers & Geosciences, 32(1):109 – 123, 2006. URL: http://www.sciencedirect.com/science/article/pii/S0098300405001160, doi:https://doi.org/10.1016/j.cageo.2005.02.017. ↩

  6. John Lysmer and Roger L Kuhlemeyer. Finite dynamic model for infinite media. Journal of the Engineering Mechanics Division, 95(4):859–878, 1969. ↩↩