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A Cyclic Constitutive Model forEarthquake Response Analysis
Kangwon National UniversityKIM, Yong-Seong
1. IntroductionThe 1995 Hyogo-ken Nambu Earthquake
The 1995 Hyogo-ken Nambu Earthquake
Railway
Wharf Facilities
sand
clay
claysand
liquefaction ?
amplification or damping of earthquake motion?earthquake
The 1995 Hyogo-ken Nambu Earthquake
Up to now
sand layer :
elasto-plastic model
clay layer :
elastic model
Cyclic Viscoelasto-viscoplastic model for clay besed on non-linear kinematic hardening rule
time dependent characteristics of clay in wide range of strain
Cyclic Model for Earthquake Response Analysis
1964 Niigata Earthquake
Three elements viscoelastic model
(Voigt model + linear elastic spring)
Kondner and Ho (1965) : dynamic behavior of clay Hori (1974) : Wave propagation test for sandy soil and clay Murayama (1983) : Distribution of relaxation time di Benedetto and Tatsuoka (1997) : Sand, soft rock
2. Review of Previous Studies
Viscoelastic - Viscoplastic Model
Small
viscoelastic
viscoplastic
Strain level
Large
cyclic viscoplastic model
viscoelastic3 parameter model
G1
G2
Infinitesimal
3. Cyclic Viscoelastic-Viscoplstic Model
Elastic Component
3 Elements Viscoelastic Component
Voigt Viscoelastic Component
Formulation of Viscoplastic Component
Overconsolidation Boundary Surface
Static Yield Function
Evolutional Law of *ij
Plastic Potential Function
Flow Rule
A Proposed Cyclic Viscoelastic and Viscoplastic Constitutive Model
Three elements viscoelastic model+
Cyclic viscoplastic model
Cyclic Triaxial Deformation Tests
0
4
8
12
16
20
Experiment(damping ratio)
Single amplitude axial strain, ( a)SA(%)
Hysteretic damping ratio, h(%)
1E- 3 0.01 0.10
40
80
120
160
200
Experiment(Young's modulus)
Equiv
alent
You
ng's
mod
ulus,
E eq(M
Pa)
T- 4, f=0.05(Hz)
Viscoelastic- viscoplastic Model
Elastic- viscoplastic Model
The results of the Cyclic Triaxial Deformation Test
- 10 - 5 0 5
- 100
- 50
0
50
100
- 10 - 5 0 5
- 100
- 50
0
50
100
0 5 10 15- 150
- 100
- 50
0
50
100
150
- 15 - 10 - 5 0 5- 150
- 100
- 50
0
50
100
150
0 50 100 150 200 250- 150
- 100
- 50
0
50
100
150
- 5 0 5 10 15- 150
- 100
- 50
0
50
100
150
0 50 100 150 200 250- 150
- 100
- 50
0
50
100
150
- 15 - 10 - 5 0 5- 150
- 100
- 50
0
50
100
150
0 50 100 150 200 250- 150
- 100
- 50
0
50
100
150
0 50 100 150 200 250
- 100
- 50
0
50
100
0 50 100 150 200 250
- 100
- 50
0
50
100
0 50 100 150 200 250- 150
- 100
- 50
0
50
100
150
Nc : 22Simulation
Devia
tor s
tress
, q(k
Pa)
Axial strain, a(%)
No. of cycle : 22Experiment
Devia
tor s
tress
, q(k
Pa)
Axial strain, a(%)
Nc : 23Experiment
Devia
tor s
tress
,q(kP
a)
Axial strain, a(%)
Nc : 5Experiment
Devia
tor s
tress
, q(k
Pa)
Axial strain, a(%)
Nc : 5Experiment
Devia
tor s
tress
, q(k
Pa)
Mean effective stress, p'(kPa)
Nc : 23Simulation
Devia
tor s
tress
, q(k
Pa)
Axial strain, a(%)
Nc : 23Simulation
Devia
tor s
tress
, q(k
Pa)
Mean effective stress, p'(kPa)
Nc : 5Simulation
Devia
tor s
tress
, q(k
Pa)
Axial strain, a(%)
(b) Effective stess path
(b) Effective stess path
(b) Effective stess path
(a) Stress- strain relation
(a) Stress- strain relation
(a) Stress- strain relation
Nc : 5Simulation
Devia
tor s
tress
, q(k
Pa)
Mean effective stress, p'(kPa)
Nc : 22Experiment
Devia
tor s
tress
,q(kP
a)
Mean effective stress, p'(kPa)
Nc : 22Simulation
Devia
tor s
tress
, q(k
Pa)
Mean effective stress, p'(kPa)
Nc : 23Experiment
Devia
tor s
tress
, q(k
Pa)
Mean effective stress, p'(kPa)
T-1 (d/2c=0.268)
T-2 (d/2c=0.332)
T-3 (d/2c=0.324)
4. Earthquake Response Analysis
LIQCA-2D(VE-VP) Effective stress analysis
based on infinitesimal strain theory u-p formulation FEM and FDM for the spatial discretization Newmark’s method for the time discretization A cyclic elasto-plastic model for sand A cyclic viscoelastic-viscoplastic model for clay
Port Island 1995/01/18 Liquefied Area
Kyoto
KobeOsaka
Port Island
Rokko Island
Soil profile Finite element meshes
Observed records during main shock at Port Island
(a) G.L. 0.0m (b) G.L. -83.0m
Input seismic wave
Governing Equations Using u-p Formulation
0 5 10 15 20
-400
0
400
-400
0
400
Time (sec)
Acc
eler
atio
n (g
al)
VE-VP Model NS 32.0m Acc. Max. -437(gal)
Observed Record NS 32.0m Acc. Max. 543.594(gal)
E-VP Model NS 32.0m Acc. Max. -389(gal)
(c) Acceleration vs. time relations at 32.0m sand layer
0 5 10 15 20
-400
0
400
-400
0
400
P
VE-VP Model NS 16.0m Acc. Max. -340(gal)
P
Time (sec)
Acc
eler
atio
n (g
al)
Observed Record NS 16.0m Acc. Max. 564.875(gal)
E-VP Model NS 16.0m Acc. Max. -351(gal)
(b) Acceleration vs. time relations at 16.0m sand layer
0 5 10 15 20-400
-200
0
200
400-400
-200
0
200
400
Acc
eler
atio
n (g
al)
Time (sec)
VE-VP Model NS 0.0m Acc. Max. -242(gal)
Observed Record NS 0.0m Acc. Max. -341.219(gal)
E-VP Model NS 0.0m Acc. Max. -244(gal)
(a) Acceleration vs. time relations at 0.0m sand layer
Maximum Value Distributions of Earthquake Response Analysis
Excess Pore Water Pressure Ratio vs. Depth Relations
Acceleration Response Calculated by E-VP and VE-VP Model
1995/01/17 05:53
After-Shock
G.L 0.0m sand layer
G.L -16.0m sand layer
G.L -32.0m sand layer
0.0 2.5 5.0 7.5 10.0
-40
0
40
-40
0
40
NS component
VE-VP Model; Max. 59.9(gal)
Time (sec)
Acc
eler
atio
n (g
al)
NS component
Obtained record; Max. 63.625(gal)
Obtained record; Max. 63.625(gal)
E-VP Model; Max. 60.7(gal)
(c) Acceleration vs. time relations at 32.0m sand layer
0.0 2.5 5.0 7.5 10.0
-30
0
30
-30
0
30
NS component
VE-VP Model; Max.-38.4(gal)
NS component
Obtained record; Max.-45.5(gal)
Acc
eler
atio
n (g
al)
Time (sec)
Obtained record; Max.-45.5(gal)
E-VP Model; Max.-51.5(gal)
(b) Acceleration vs. time relations at 16.0m sand layer
0.0 2.5 5.0 7.5 10.0
-60
0
60
-60
0
60
NS component
VE-VP Model; Max. 63.7(gal)
Acc
eler
atio
n (g
al)
Time (sec)
NS component
Obtained record; Max. 79.188(gal)
Obtained record; Max. 79.188(gal)
E-VP Model; Max. 85.5(gal)
(a) Acceleration vs. time relations at 0.0m sand layer
A Cyclic viscoelastic-viscoplastic model based on the non-linear kinematic hardening rule and three parameter theory was proposed.
The viscoelastic behavior of clay at the small strain range is an important characteristic during dynamic motion and the proposed model very well describes the viscoelastic behavior characteristics of cohesive soils in element simulations.
5. Conclusions
The accelerations calculated from the proposed model were in close agreement with the recorded accelerations on the Port Island down-hole array.
This study reveals that the viscoelastic-viscoplastic model can describe the damping characteristics of clay accurately at small strain levels, whereas the elastic-viscoplastic model cannot do so.