Page 158 - Numerical Analysis and Modelling in Geomechanics

MODELLING OF GROUND WAVES 139
Finite element and infinite element models
This section contains a preliminary verification of the capability of the chosen
elements to give a faithful representation of primary, P, shear, S, and Rayleigh, R,
waves. Infinite elements are then assessed for their performance in limiting or
eliminating reflected waves, and the simultaneous limit of P, S and R-wave
reflection at the model boundaries. The multi-stage models for impact and
for vibratory driving are then explained in detail. All computational analyses
were undertaken using the Abaqus (1999) suite.

Element verification

The normal choice for axisymmetric analysis is the 8-noded quadrilateral, which
is most effective when its aspect ratio is close to unity. For transient analysis, a
suitable mesh will have some 10 nodes per wavelength, and the time-step interval
should meet the stability criterion of ≥ x=c≥ t, where ≥ x is the mesh spacing, c is
the wave transmission velocity and ≥ t is the time step. Around the boundaries of
the FE mesh it is necessary to include infinite elements, which contain 5 nodes to
be compatible with the 8-noded quadrilateral, Figure 5.5. The normal mesh used
for the soil half-space comprised a 50×50 mesh of 1 m×1 m square axisymmetric
FEs, with a surround of IEs on the outer vertical and base-horizontal boundaries.
These elements were tested for purity of transmission of P, S and R waves and
for limitation of wave reflection (Ramshaw et al. 1998a). Consider the P-wave
test. A small mesh of FEs was set up in plane-strain, with constrained vertical
movement at top and bottom boundaries, and a compressive sine wave was
imposed at each end of the mesh with appropriate time delay, based on the
transmission velocity c of
p

(5.4)

where G is shear modulus, λ is a Lame constant and ρ is density.
A pure sine wave was observed. The right-hand boundary was then meshed
into infinite elements, and again a pure sine wave was observed, see Figure 5.6.
The geometric attenuation of the P-wave was studied by imposing a
compressive sine wave on a plane strain model, an axisymmetric mesh, and a
spherical expansion. The plane strain showed negligible attenuation, the
axisymmetric mesh gave attenuation of peak displacement proportional to 1/≥ r,
while amplitudes in spherical expansion were proportional to 1/r, in accordance
with standard wave theory (Ramshaw et al. 1998b), where r is distance from the
source.
Tests on S-wave generation and transmission were similar to those for the P-
wave, but with modified top and bottom boundary restraints, and with imposed
sinusoidal shear-waves, based on transmission velocity c of
s