1599
Effect of Seismic Waves with Different Dominant Frequencies on the Delayed
Failure Behavior of a Soil Structure-Ground System
Effets des ondes sismiques de fréquence dominante différente sur le comportement de rupture
retardée de structures en terre et de systèmes de sol
Shimizu R., Yamada S.
Nagoya University, Japan
ABSTRACT: In this research, a method of obtaining the natural frequencies and natural frequency modes was derived for a soil-water
2-phase system initial value/boundary value problem. Using this method, the natural frequencies and natural frequency modes of the
whole soil structure-ground system were calculated for the ground and an embankment constructed on it. The modes extracted
included (1) modes in which the ground deformed greatly and (2) modes in which the soil structure deformed greatly. In addition,
seismic response analysis was carried out by inputting two types of seismic wave having dominant frequencies close to the natural
frequencies corresponding to each mode. The results showed that delayed failure behavior progressing from the ground into the soil
structure was exhibited for the seismic waves that targeted (1) and that delayed failure behavior progressing from the soil structure to
the ground was exhibited for the seismic waves that targeted (2).
RÉSUMÉ : Cette étude se propose de dériver une méthode de calcul de la fréquence propre et du mode propre de vibration pour le
problème de la valeur initiale et de la valeur limite d'un système squelette eau-sol à deux phases. À l'aide de cette méthode appliquée à
des remblais créer sur un sol nous avons déterminé (1) un mode où le sol se déforme considérablement (2) un mode où les structures
de terre se déforment considérablement, en calculant la fréquence propre et le mode propre de vibration présismique du système
structures de terre-sol global. Nous avons par ailleurs analysé la réponse sismique en entrant deux types d'ondes sismiques possédant
une fréquence prédominante au voisinage de la fréquence propre pour chacun des modes pour amplifier chacun de ceux-ci. Les
résultats montrent l'apparition de comportements de rupture différée qui se propagent du sol à la structure de terre pour les ondes
sismiques en (1), et l'apparition de comportements de rupture différée qui se propagent de la structure en terre au sol pour les ondes
sismiques en (2).
KEYWORDS: natural frequency analysis, seismic response analysis, soil-water coupled finite deformation analysis.
1 INTRODUCTION
In this paper, we propose a method for evaluating the natural
frequency and natural frequency mode in the context of an
initial-boundary value problem of a two-phase soil-water
system. In addition, we calculate the natural frequency and
natural frequency mode of the entire soil structure-ground
system immediately before the simulated earthquake,
specifically that of an embankment and the ground on which it
is built, and determine the mode for which the ground and
embankment undergo large-scale deformation. Further, we input
two types of seismic waves with dominant frequencies close to
the natural frequencies of each mode, analyze the seismic
response using the soil-water coupled finite deformation
analysis code
GEOASIA
(Noda et al. 2008a), and thereby
demonstrate that differences in the dominant natural frequency
mode substantially impact the deformation/failure behavior of
the soil structure-ground system.
2 FORMULARIZATION OF A FINITE ELEMENT
DISCRETIZED RATE-TYPE EQUATION OF MOTION AND
SOIL-WATER COUPLED EQUATION AS AN
EIGENVALUE PROBLEM
In dynamic initial-boundary value problems for elasto-plastic
materials that can be described by a rate-type constitutive
equation, it is necessary to solve a rate-type equation of motion.
To this end, after deriving a weak form of the rate-type equation
of motion, we employ an elasto-plastic constitutive equation
formulated with effective stress as the constitutive equation for
the soil skeleton and perform finite element discretization.
Furthermore, we represent the pore water pressure
u
at the
center of each element in the soil-water coupled equation by
extending the physical models of Christian (1968) or Akai &
Tamura (1978) to the continuity equation for saturated soil or
the equation for the average flow velocity of pore water.
Moreover, when the viscous boundary or linear constraint
conditions (Asaoka et al. 1998) on velocity field of nodes are
applied, the final system of simultaneous ordinary differential
equations to be solved is as follows (Noda et al. 2008a):
M
+ C
∗
+ K − L
− C
=
L
− L + H + G
=
(1)
−C = 0
where
M
is the mass matrix,
K
is the tangential stiffness
matrix,
L
is a matrix for converting displacement velocities of
the soil skeleton to a volumetric change rate of the soil skeleton,
is a vector comprising the displacement velocities of nodes,
is the pore water pressure rate at the center of each element
as seen from the soil skeleton,
L′
is a matrix for the
acceleration term of the soil-water coupled equation derived
from
L
,
H
is the permeability matrix, and
G
is the matrix
for porosity and water compressibility,
is equivalent nodal
force vector,
is a vector for elevation head. Furthermore,
C
∗
is the damping matrix that results from imposition of the
viscous boundary, and represents non-proportional damping.
C
is a matrix imposing a linear constraint on node movement,
is the Lagrangian undetermined multiplier, and
−C
is a
term with physical significance that acts as a constraint imposed
on nodes.
Following the method of Foss (1958), we set
=
(2)
Effect of Seismic W ves with Different Dominan Frequencies on the Delayed
Failure Behavior of a Soil Structure-Ground System
Effets des ondes sismiques de fréquence dominante différente sur le comportement de rupture
retardée de structures en terre et de systèmes de sol
Shimizu R. & Yamada S.
Nagoya University, Japan
ABSTRACT: In this research, a method of obtaining the natural frequencies and natural frequency modes was deriv d f r a soil-wat r
2-phas syste initial value/boundary value problem. Using this meth , the natural fr quencies and natural frequency modes of the
whole soil structure-ground system were calculated for the ground and an embankment constructed on it. The modes extr cted
included (1) modes i which the ground deformed greatly and (2) mod s in which the soil structure def rmed greatly. I addition,
seismic response analysis was carried out by inputting two types of seismic wave having dominant frequencies close to the natural
frequencies corr sponding to each mode. The results showed that delayed failure behavior progressing from the ground into the soil
structure was exhibited for the seismic waves that targeted (1) and that delayed failure behavior progressing from the soil structure to
the ground was exhibit d for the seismic waves that targeted (2).
RÉSUMÉ : Cett ét de se propose de dériver une méthode de calcul de la fréquenc propre et du mode propre de vibration pour le
problème de la valeur initiale et de la valeur limite d'un système squelette eau-sol à deux hases. À l'aide de cette méthod appliquée à
des remblais créer ur un sol nous avons déterminé (1) un mode où le ol se déforme considérablement (2) un mode où l structures
de terre se déforment considérableme t, n calculant la fréquenc ropre et le mode propre de vibration présismique d système
structures de terre-sol global. Nous avons par ailleurs analysé la r ponse si mique en entrant deux types d'ondes sismiques po sédant
une fréquence prédominante au voisinage de la fréquence propre pour chacun des modes pour amplifier chacun de ceux-ci. L s
ré ultats montrent l'apparition de comportements de rupture différée qui se propagent du sol à la structure de terre pour les ondes
sismiques en (1), et l'apparition de comportements de rupture différée qui se propagent de la structure en terre au sol pour les ondes
sismiques en (2).
KEYWORDS: natural frequency analysis, seismic response analysis, soil-water coupled finite deformation analysis.
1
INTRODUCTION
I this paper, we propose a method for evaluating the natural
frequency an natural frequency mode in the context of an
i itial-boundary value pr blem of a two-phase soil-water
s st . In addition, we calculate the nat r l frequency nd
natural frequency mode of th entire soil structure-ground
system immediately before the simulated earthquake,
specifically that of an embankment and the ground on hich it
is built, and determine the mode for which th ground and
embankment undergo large-scale deformatio . Further, we input
two typ s of seismic waves with dominant frequ nci s close to
the natural frequencies of each mode, analyze the seismic
resp se using the soil-water coupled fi ite deformation
analysi code
GEOASIA
(Noda et al. 2008a), and thereby
demonstrate that difference in the dominant natural frequency
mode substantially impact the deformation/failure behavior of
the soil structure-ground system.
2 FORMULARIZATION OF A FINITE ELEMENT
DISCRETIZED RATE-TYPE EQUATION OF MOTION AND
SOIL-WATER COUPLED EQUATION AS AN
EIGENVALUE PROBLEM
In dynamic initial-boundary value problems for elasto-plastic
materials that can be described by a rate-type constitutive
equation, it is necessary to solve a rate-type equation of motion.
To this end, after deriving a weak form of the rate-type equation
of motion, we employ an elasto-plastic constitutive equation
formulated with effective stress as the constitutive equation for
the soil skeleton and perform finite element discretization.
Furthermore, we represe t the pore water press re
u
at the
center of each element in the soil- ater coupled equation by
extending the physical model of Christian (1968) or Akai &
Tamura (1978) to th continuity equati n for saturated soil or
the equation for the average flow velocity of pore water.
Moreover, when the viscous boundary or linear constraint
conditions (Asaoka et al. 1998) on velocity field of nodes are
applied, the final system of simultaneous ordinary differential
equations to be solved is as follows (Noda et al. 2008a):
M
+ C
∗
+ K − L
− C
=
L
− L + H + G
=
(1)
−C = 0
where
M
is the as matrix,
K
is the tangential stiffnes
matr x,
L
is a m rix for converting displacement velocities of
the oil skeleton o a v lumetric change rate of the so l skeleton,
is a vector comprising the displacement velocities of nodes,
is the pore wat r ressure rate at he ce ter of eac lement
as seen from the soil skel ton,
L′
is a matrix for the
acceleration term of the s il-wat r coupled equation derived
from
L
,
H
is the permeability matrix, and
G
i he matrix
for porosity and w ter compres ibility,
is equiv lent nodal
force vector,
is a vecto f r elevati n head. Furth rmore,
C
∗
is the dampi g matrix that res s from imposition of the
viscous boundary, and represents non-proporti al damping.
C
is a matrix imposing a linear constraint on node movement,
is the Lagrangian undetermined multiplier, and
−C
is a
term with physical significance that acts as a constraint imposed
on nodes.
Following the method of Foss (1958), we set
