Earthquake Response Analysis

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.