1600
Proceedings of the 18
th
International Conference on Soil Mechanics and Geotechnical Engineering, Paris 2013
=
(2)
The homogeneous form of equation (1) can be rewritten as
M
C
∗
K L
C
M M
(3)
L
L H G
C
Summarizing the system of simultaneous ordinary differential
equations (3) in matrix form yields
A
B
(4)
wherein,
,
A C
∗
M L
O
M O O O
L
O
O O
G O O O ,
B K
O O C
O M O O
L
O
C
O H O O O
(5)
Assuming that
,
，
and
, we see that
(6)
Here
(7)
If an
{ }
x
such as that expressed in equation (6) exists, we arrive
at the following general eigenvalue problem.
A
B
(8)
3 SIMULATION CONDITIONS
Calculations were performed under two-dimensional plane
strain conditions. We examined the case of an embankment
built on level ground. The finite element mesh and boundary
conditions used for the calculations are presented in Figure 1.
As boundary conditions for the ground, a viscous boundary
(
ρ
=2.0g/cm
3
,
V
s=1000m/s) was imposed in the horizontal
direction, a velocity boundary (fixed conditions) in the vertical
direction along the lower-end face and a periodic boundary
along the lateral faces of the ground. In addition, the lateral and
lower-end faces of the ground were assumed to be undrained
boundaries. The embankment, which was assumed to be
saturated, was established by progressively adding elasto-plastic
finite elements representing the two-phase soil-water in the
location shown in Figure 1, and the simulation was continued
until consolidation was completed. For the constitutive equation
for the soil skeleton, we employed the elasto-plastic constitutive
equation SYS Cam-clay model (Asaoka et al. 2002), which is
Figure 1. Finite element mesh and boundary condition
capable of describing the function of the skeleton structure. The
sand in the lower portion of the ground was assigned a material
constant corresponding to silica sand no. 6, and the overlying
clay in the upper portion of the ground was assigned a material
constant for
Tochi clay
. The embankment comprised a mixture
of silica sand no. 7 and
Tochi clay
and was assigned a material
constant for an intermediate soil (Noda et al. 2008b). The
natural frequency and seismic response analyses of the soil
structure-ground system after consolidation were analyzed
under the above conditions.
4 INITIAL NATURAL FREQUENCY AND NATURAL
FREQUENCY MODE OF THE SOIL STRUCTURE-
GROUND SYSTEM
The initial (the post consolidation) natural frequency modes
causing large-scale deformation of the ground (MODE 1) and
large-scale deformation of the embankment (MODE 2) yielded
by our calculations are presented in Figure 2. Although the
natural frequency mode in our analytical approach is expressed
as a linear combination of the real and imaginary parts of the
complex eigenvector, here we only show the mode expressed by
the imaginary part.
(a) MODE 1: Large scale deformation of ground
(b) MODE 2: Large scale deformation of embankment
Figure 2. Natural frequencies and natural frequency modes (imaginary
part)
5 SEISMIC RESPONSE ANALYSIS OF GROUND WITH
AN OVERLYING EMBANKMENT
Our analyses were performed on the post consolidation soil
structure-ground system described in the previous section. The
input seismic waves had dominant frequencies close to the
natural frequencies of the system as a whole, and their
magnitudes were adjusted to generate a maximum acceleration
of 200 gal. The acceleration history and Fourier amplitude
spectrum of the input waves are shown in Figure 3.
WAVE 1 represents a seismic wave with a dominant
frequency equal to the natural frequency of MODE 1, and
WAVE 2 represents a seismic wave with a dominant frequency
equal to the natural frequency of MODE 2. After inputting these
seismic waves into all the nodes at the bottom face of the
ground representing the horizontal viscous boundary, the
simulation was allowed to run until consolidation of the ground
stopped. The inputs of WAVE 1 and WAVE 2 are referred to
below as CASE 1 and CASE 2.
The changes in the shear stress distribution during and after
the simulated earthquake are presented in Figure 4. It is evident
that in both CASE 1 and CASE 2, delayed failure occurred after
the earthquake. Examining the progression of the shear stress, it
can be seen that in CASE 1, a slip surface (strain localized area)
extending from the top of the sand layer to the middle of the
ground developed during the earthquake. Thereafter, the slip
surface continued to expand over time from the ground to the
embankment, ultimately resulting in delayed failure.
Meanwhile, in CASE 2, although strain was localized at the top
of the sand layer, no large-scale deformation of either the
embankment or the ground occurred during the earthquake.
f
= 0.699 Hz
f
= 1.935 Hz
