A 2D Smoothed Particle Hydrodynamics method for ...

Journal

of Applied

Mechanics

Vol.8, pp.591-599

A 2D Smoothed

(August

2005)

Particle Hydrodynamics

JSCE

method for liquefaction

induced

lateral spreading analysis

Mounir NAILI*, Takashi MATSUSHIMA**, * *

Yasuo YAMADA***

PhD student, Dept of Eng Mechanics and Energy, University of Tsulcuba, Tennodai1-1-1,

Tsukuba 305-8573

Dr. of Eng., Associate Professor, Dept of Eng Mechanics and Energy, University of Tsukuba, Tennodai 1-1-1, Tsukuba 305-8573 ***

Dr. of Eng., Professor, Dept of Eng Mechanics and Energy, University ofTsukuba,

Tennodai 1-1-1, Tsukuba 305-8573

This paper investigatesliquefactioninduced lateralspreadingusing a 2D Smoothed particle hydrodynamics(SPH) based numericalmethod in the frameworkof fluid dynamics.Owing to the factthat the induceddisplacementduring lateralspreadmay reach several meters and the shear strain develops till 100%, a method which is essentially a Langrangian meshfree particle was developed to deal with large displacement.By assumingthe liquefiedsubsoilto behave as a viscousfluid during earthquakeshaking, a Bingham type model is used to model the liquefiedsoil. The ability of the method is checked using results from shaking table experimentsto reproducefree surfaceshape,flow velocityand to investigatethe relationshipbetween timehistoryof flowvelocity,liquefiedsoilthicknessand surfacegroundslope. Key Words:lateralspread SPH,Binghammodel,flow velocity

1. Introduction

monotonic shear tests, shaking table and centrifuge experiments have revealed that the liquefied soil behave either as a solid or a

Liquefactionof loose, saturated,cohesionlesssoils and other granularmaterialdue to earthquakeshaking constitutea major source of damage to constructed facilities. For instance, liquefactioninduced lateral spreading represents a potential source of distressto piles foundation,waterfrontstructuresand lifelines.DuringNiigataearthquake(1964)and Kobe earthquake

viscousfluidduringthe courseof the loading(Kawakamiet al2) Towhata3),Yasuda4)).Thus, many numerical methods were developedby consideringthe liquefiedsoil eitheras a solid or a fluid.For example,Aydan5) used a FEM formulationto assessthe inducedpost-liquefactiondisplacementby assumingthe soil as a visco-elasticmedium. Yasuda et al.6)proposed a simplified

(1995), severe damageswere inflictedto pile foundationsand waterfrontstructures. Lateral spreads at the margins of Port and Rokko Islands caused widespread seaward displacement of caisson walls that severely disrupted port operations. According to Doi and Hamada1),the induced displacement reachedseveralmeters in Ohgataarea in Niigatacity, wherethe maximumdisplacementreachedover8 m. Hence,in order to designstructuresbeing able to withstand lateral spreading,the assessmentof the induced displacement formsthe cornerstoneto any seismicbaseddesignmethod. Numericalmethodsto assesspost-liquefaction displacements rely on our understandingof the mechanicalbehavior of the liquefied soil during shaking. In this respect, undrained

procedurefor the analysisof the permanentdisplacementusing the FEM by consideringa reductionof the soil stiffnessbefore and after the occurrence of the liquefaction.Assuming the liquefiedsoil as a Bingham fluid,Uzuoka7) used the volume of fluid method to predict both the flow process and lateral spreadingload.Usingthe same hypothesisfor the liquefiedsoil, Towhata et al.8) proposed a solution for the permanent displacementon the basis of the minimum energy principle. Using the OP (Cubic Interpolatedpseudopartcile)method, Hadush et al9) proposed a numericalmethod for liquefaction induced lateral spreading analysis in the frameworkof fluid dynamicsand were able to simulatelateralspreadof a liquefied groundwith an overlyingnon-liquefiedlayer. Because induced displacementsare large as reported from

― 591―

post-earthquakeobservations,gridbasednumericalmethodssuch as FEM face sometimesmesh distortionproblemas it is difficult to handle largedeformation.In this case, specialtreatmentof the mesh is necessaryto overcomenumericalinstabilityand this may lead to time-consumingcomputation.Similarly,VOF based on a staggeredmeshneedsspecialtreatmenttotrackmovingboundary and interface. Recently, Maeda and Sakai10) developeda procedurefor seepage failureanalysisof granularground using smoothedparticlehydrodynamics(SPH). In this paper,the SPH is investigatedin the frameworkof fluid dynamics to assess ground displacementsinduced by lateral spreading.The soil is consideredas a Non-Newtonianfluidby means of Binghamfluidmodel.Underthis hypothesis,the soil is capableof resistingany shearstressbelow the yield definedby the residualundrainedshearstrength.The abilityof the methodto reproducefree surfaceshape,timehistoryof flow velocitiesand its relationto liquefiedsoil thicknessand surfacegroundslope is investigated.

gravity. ƒÐƒ¿ƒÀ sum

is the

of an isotropic

total

stress

tensor

component

p and

which

can

a viscous

be written

as the

component ƒÑƒ¿ƒÀ

(5)

SubstitutingEq. (5) into the momentum Eq. (4) leads to the equationof motionexpressedin the formof (6)

Where:

v =ƒÊ/ƒÏ

represents

2.2 SPH formulation

the kinetic

viscosity.

for Navier-Stockes

In SPH a function representing

equations

a physical quantity can be

written in an integral form as:

(7) 2. SPH formulation

h representsthe smoothinglengthdefiningthe influencedomain The SPH is essentially a Lagrangian meshfree particle method,

and W(x-x',h)

the kernelfunction,which shouldsatisfythe

where the continuous medium is represented by a set of particles that follow the fluid motion and advect its physical quantities such as mass and momentum.

followingnormalizationcondition:

The SPH was originally developed to

(8)

deal with astrophysical problem and is nowadays applied to treat a variety of problem including fluid flow and solid mechanics. The derivative of the kemel function is expressed as: 2.1 Governing

equation

(9) For a Newtonian strain rate

incompressible fluid, the shear stress-shear

relationship

can be expressed

by the

following

equation:

In this study, we adopted the quintic spline kernel function having the following expression:

(1) Where ƒÊ

eƒ¿ƒÀ:

: Dynamic

viscosity

Deviatoric

strain

rate

tensor.

(2) (10) Where v : Velocity vector The motion of an incompressible,

Where r is the distance between the particles, s is equal to r/h and uniform density fluid is

govemed by the conservation of mass and momentum principles. It is expressed by the following two equations.

is a normalization

αD

7/4787ƒÎh2

and

respectively. 1.1 times

(3)

The using

3/359ƒÎh3 In

this

the initial SPH

in one, study,

spacing

formulation

Eq. (7) and

constant having the values of 1/120h, two

the

between for

Eq. (9). After

and

three-dimensional

smoothing

Eq.

length

h

space is taken

as

particles. (3)

and

transformation

Eq.

(6) can we

be achieved

get:

(4) (11)

In Eq. (3) and (4),p is the densityand g is the accelerationof

― 592―

3.1 SPH formulationfor a non- Newtonianfluid (12)

From

Eq.

particles

(11),

subscripts

identification;

express

the

particle

while

coordinate

Wij represents

Using

Roman

the

Greek

system.

smoothing

ij are

used

to

indices ƒ¿ƒÀ

Therefore, function

in Eq. of

denote are

(11)

particle

the

used and

to (12),

i evaluated

at

j. Eq. (5) for the total

stress,

Eq. (12)

can

be expressed

as:

The generalequationof the momentuminEq. (13) is usedto modela Non-Newtonianfluid.The mainidea consistsin defining an equivalent viscosity to be incorporated in the SPH approximationof the viscousforces. Accordingto the conceptof the Binghammodel, the soil is capableof resistingany shearbelow the yield stressrepresented by the minimumundrainedshearstrengthandthen behaveslikea Newtonianfluidfor shearstressesgreaterthan the yield stress.A Binghamtypefluidhas the followingshearstress-shearstrainrate relationship. (15)

(13) Where Ąmin

In Eq. (13), the first part of the right hand side represents the SPH approximation for the pressure term, while the second part represents the viscous forces. The pressure terms are computed

represents

the minimum

undrained

shear

strength

and

μB itsviscosity

If we definean equivalentviscosityin the form of Eq. (16),a non Newtonianfluidcan be modeledas a Newtoniantype fluid.

using an appropriate equation of state which relates the (16)

hydrostatic pressure to local densities.Monaghan11)suggested the following equation of state while modeling free surfaceproblem. (14)

Where ƒÁ rate

Where ƒÁ represents for

the

density sound

is a constant the

initial

and density

it is taken while

maximum

change

fluctuation

to an acceptable

speed

should

of

the

be appropriately

equal

to 7 in this

B is a parameter density. value

In of

that order

1%, the

by

the

second

invariant

of the

deviatoric

strain

tensor.

study. ƒÏ0 sets

to

is given

a limit

limit

value

(17)

the

of the

4. Numerical simulations

chosen.

3. Extensionof SPH to model lateral spread

In this section,the SPH methodpreviouslydescribedwill be validated using results obtained from 1g shaking table In this study,the SPH methodis formulatedin the framework experiments.The firstpartof thenumericalsimulationattemptsto of fluiddynamics.Hence,the liquefiedsoil is assumedto behave reproduce the final free surface shape and the velocity flow as a viscousfluidduringshaking. obtained from a lateral spread experiment conducted by In orderto extendthe SPH methodtomodellateralspread,an Hamada et al15).The secondpart of the numericalsimulations appropriateviscositycoefficientshould be definedfor liquefied aimsto investigatethe relationshipbetweenthe flowvelocitywith soil. In this respect,viscosityof liquefiedsoil was investigated respectto the soilthicknessand the slopegradient and find out using shaking table experiments Towhata12),Hamada and how the shapeof the timehistoryof the flowvelocitiesdo change Wakamatsu13), droppingball methodMiyajimaet al.14), steelball withthe soillayerthickness. pull-uptest and viscometerKawakamiet al.2).According to the obtained experimentalresults, it was found that the viscosity 4.1 Lateral spreading experiment coefficientof the liquefiedsoil decreaseswith increasingshear strain rate. This suggeststhat the behaviorof the liquefiedsoil To investigatethe characteristicsof the ground movement couldbe assumedas a non-Newtonianfluid.Based on the results afterthe earthquakemotion ceases,Hamada et al.15)conducteda obtained from viscometer,Uzuoka7)proposed a relationship seriesof 1g shakingtable experimentsin which a model ground betweenthe shearstrainrateand the shearstressin the form of a with an initialplane surfacewas constructedin a rigid soil box. Binghammodel.Accordingly,this studyadoptsthe conceptof a The modelhas 3 m long, 1 m wide and 0.3 m high. The box was Binghammodelto extendthe SPH methodto a non Newtonian vibratedin the lateraldirectionuntil a total liquefactionoccurs, fluidas it will be discussedinthe nextsection. and then liftedwith a specificgradientusing a hydraulicjack. During the experimentthe final shape of the free surfacewas continuouslymonitored and the time history of the induced

―593―

displacements at 3 different depths in the central section of the

(1)

box was measured. Different tests were performed by varying the

Numerical A total

initial relative density and the inclination of the soil box. Figure 1

equally

illustrates the experimental setup.

ground

setup

number

the

with

a

Lennard-Jones boundary setup

Fig.

In

a

Outline

previous

simulations at

1

study,

to assess

three

locations

the results

In this surface

with

simulation,

and

flow

we

and According

displacement

of

reached

an

undrained soil

deposit

10

with

well

the

Pa•Es. The

as

an

was

undrained

slope

bottom.

results

on

shear

strength.

undrained

observation, when

the

the

the

slope

shear of

shear

from

into

0.01

the

=ƒÁh cos(ƒÆ)sin(ƒÆ)

, where

can

a sloped

ground

strength

of

represents Table during

the

soil

equal

the soil height 1 bellow

to

1800

the

model

Particles

with

2 shows

a non

process.

can

be

y

is

the

kN/m3

in

numerical

condition

at t=0s.

used

it

(Courant-Fredrich-Levy) adopted

XSPH

that

of

overcome

according

the

shortly

blowup To

technique

move

throughout

noticed

a sudden

condition,

the this

proposed to Eq.

calculation

after

the

particles

1

Numerical

parameters

used

during

process.

start

of

occurred

numerical

by

a time

the

in the

instability,

Monaghan17)

we

where

the

(18).

to Where states

c is a that

constant

each of

in

particle

the

the

range

moves

neighboring

0•¬ƒÃ•¬1

in a velocity particles.

_ This

closer By

technique

to the

average

conducting

several

unit

and

h

the

and

observed

acceptable parameters

was

domain.

the

particles

expressed

study

results

= 10-4 s was

calculation

minimum

saturated

this

Initial

comparing one,

and

a

comparable

value

the simulated of results

E equal as shown

free surface to

0.5

used

simulations

Fig. 3

―594―

shapes

seems

in figures

the simulations.

Table

slip

the numerical

(18)

at which Ąmin is assessed.

summarizes

represent

Figure

the

be

needed

the

step Ģt

simulations weight

to

distributed,

Pa•Es

strength

represents

2

the CFL

computation,

of the

of

Simulations To satisfy

However,

minimum

taking

varied

forces

This

a

velocity by Ąmin

employed

of the calculation

Fig.

(2)

constitute

characteristics

was

resistant

represent

soil box

at the middle

the

equilibrium

free

Moreover,

undrained

The

observed

minimum

computed

equilibrium

into

the

triggered

the

minimum

inclination Į. the

was

and

the

coefficient

coefficient

the

the

experimental

soil

at

viscosity

considering

maintain

this,

the

were

at the wall surface.

at the beginning

uniformly to

gradient Į=4.2%.

potential11)

conditions

used

numerical

section,

in which

Accordingly,

experimental

soil,

assessed

to liquefied

surface

particles

were

observations.

to reproduce

viscosity

box

displacement

cross

experiment

achieve

strength Ąmin

as

the

liquefied

central

material

soil

15)

out

of the induced

the

inclination Į=2%. shear

account

to

the

carried

history

attempt

the

prerequisite.

al.16)

at

of the To

strength

et

setup

the experimental

velocity

inclination Į=42%. shear

Naili

the time

different

compared

of the experimental

of 2250

spanning

Final free surface shape at t = 12s.

to

3 and

with give 4.

Fig. 6 Fig. 4

Time history of flow velocity for casel.

Simulated free surface for case 1 and 5. Figure 6 depicts the time history of the flow velocity during

The simulated free surface for case 1.and 5 as seen in figure 4 reproduced

clearly the trend of the observed free surface during

the experiment

By representing the vectors displacement of the

free surface, we notice that the upstream part subsided due to

the first 2 s of the simulation container

observations can be made: ・The

simulated maximum

flow velocity for case 1 is

smaller than the observed one during the experiment as

gravity while the downstream heaved due to the volume transfer from the upstream. The central part kept almost the same configuration as the initial ground surface. Figure 5 shows the

at the central part of the soil

at three different depths for case 1. The following

shown in Table 2. ・The

current observed feature of the free surface in case 1.

velocities decrease slowly after the steady state is

reached and do not vanish as in the experiment. This is due to the fact that the model

does not take into

consideration the pore water pressure dissipation and the recovery of the soil rigidity.

Fig. 5

Vector displacement at the top of the free surface

Fig. 7

for case 1. Table 2 below summarizes the maximum

Displacement

distribution along the height at the central

cross section for cases 1 and 4. flow velocities

observed during the experiment. Table 2

Experimental observed maximum flow velocities 15)

Fig. 8 Displacement distribution along the height at the central cross section for case 5.

―595―

Figures7 and 8 show the displacementdistributionalongthe heightat the centralcrosssectionof the liquefiedlayerfor case 1, 4 and 5 respectively.It canbe seenthatthe induceddisplacements are insensitiveto the yield viscosity coefficient This can be explainedby the factthatthe inducedshearstrainrate throughout the simulationis smallerthan 1/s,which accordingto Eq. (15)

The Bingham model is taken as the constitutive law for the liquefied ground considering a yield viscosity p. of 1 Pa•s. The minimum undrained shear strength was assessed for each case at the middle of the soil layer. Figure 9 shows the computed minimum

undrained

shear strength

for each liquefied

layer

thickness.

givesalmostthe same valuesof the equivalentviscosity,and thus same range of the induced displacement. However, the displacementsrespond stronglyto any change in the minimum undrained shear strength, as in case 5 where the induced displacementsare half of the simulatedone for cases 1and 4. In both cases 1 and 4, the inducedshearstrainis of the order of 20% which is in the range of the observedvalue during earthquake. 4. 2 Numerical simulation of flow velocity with respect to surfaceslope gradient and liquefiedlayer thickness In their investigationusing shakingtable test Hamada and Wakamatsu13) conductedexperimentsto clarifythe mechanismof liquefaction-induced lateralspreadingand find out the nature of the liquefiedsoil.They alsofocus theirresearchon predictingthe magnitudeof the inducedlateraldisplacements.They statedthat the liquefiedsoil behavesas a viscousfluidduringshakingand as a solid after the dissipation of the pore water pressure. Accordingly,the Reynoldssimilitudelaw encounteredin fluid dynamicscan be appliedbetweenmodelsand actualgrounds. By running several experimentsof a slopedground model withdifferentlayerthicknessrangingbetween 15and 38 cm and surfacegradientvaryingbetween1 and 5% preparedin a soilbox of 3m longand 1m wide,Hamada and Wakamatsushowedthat the velocityof the groundflow is proportionalto the squareroot of the layerthickness.By assuminga flowvelocitydistributionas one fourth of a sine wave, they proposeda relationshipfor the maximumvelocityat the groundsurfacewithrespectto soil layer thickness,groundslopeand viscosityof the liquefiedlayer. In this section,we do not intendto simulatethe experiment performed by Hamada and Wakamatsu, but rather use the experimentalsetup described in section (4.1) to investigate numericallythe relationshipbetween the shapes of the time historyof the flow velocity,the liquefiedlayerthicknessand the groundgradientslope. We conductednumericalsimulationsusing liquefiedlayer thicknessranging from 16 cm to 50 cm with a ground gradient slopeof 3% and 5%. We attemptto focus on how the maximum velocityand the requiredtime to reach this maximumdo very withlayerthicknessand groundgradientslope. (1) Numerical setup Total 10 cases of simulationswere performedconsideringa liquefiedgroundmodel of thicknessvaryingbetween 16cmand 50 cm with a ground slopeof 3% and 5% insidea soil box of a 3m length.

―596―

Fig. 9 Minimum undrained shear stress versus layer thickness (2) Simulations results Numerical simulations showed that the maximum flow velocity occurs within the first 2 s of the time calculation. Accordingly, for better resolution, the time history of flow velocity is plotted during this time span.

Fig.

10

Time

Figures velocities

history

10 for

and

of flow

11

a surface

velocity

depict

the

ground

for

slope

time

slope Į

gradient Į=3%

history equal

of to

the

3%

flow

and

5%

respectively. It can with

be seen

decreasing

increases

that liquefied

with

increasing

necessary

to reach

liquefied

soil layers.

Generally, velocity agreement

is

the

the

shape soil soil

larger with

curve

thickness

the

for

thick

liquefied

physical

and

experimental

the

However,

velocity

that

at its base

while

thickness.

maximum

it is observed

of the

is longer

induced soil

velocity

flow

the time

span

for

maximum layer

observations.

widens

which

shallow

flow is

in

Moreover, both graphs illustrate clearly that the peak of the

(19)

flow velocity for each curve lies on an envelope curve that decays in an exponential way with increasing time.

(20) In the same manner, the relationship between the maximum flow velocities and the liquefied layer thickness is illustrated in figure 13 for a slope gradient of 3% and 5%.

Fig.

11

(3)

Time

history

Regression The

for

slope

gradient Į=

5%

the

simulations the

showed

maximum

flow

that

is

there

velocity

and

relationship

the

exists

maximum

thickness.

flow

Figure

12

also

between

velocity

expresses

the time

and

this

the

required

to

liquefied

relationship

for

layer a

Fig. 13 Flowvelocityversusliquefiedlayerthickness

a the

soil thickness.

A similar reach

of

between

liquefied

velocity

analysis

results

relationship

of flow

Similarly,regressionanalysiswas performedto expressthe relationshipbetweenthe flow velocityand the squareroot of the liquefiedsoilthickness.Thisrelationshipis givenby Eq. (21).

slope

(21) gradient Į=

3%

and

5%.

(4)

Time The

the

liquefied

for

a slope

corresponding

Fig.

12

Time

to reach

maximum

velocity

versus

history

time

of surface

history layer

of the was

also

gradient Į=3% time

displacement induced

history

displacement

assessed and for

at the

5%. each

Figures liquefied

at the surface central

cross

14 and

15 depict

layer

of

section the

thickness.

liquefied

soil thickness

Regression time

span

needed

corresponding The (19)

and

analysis

best Eq.

to reach

liquefied fit that (20)

to express the

the

maximum

soil thickness expresses

respectively

relationship

this

flow

was

velocity

the

and

the

by

Eq.

derived.

relationship

for slope

between

is given

gradient Į=3%

and Į=

5%.

Fig.

14

Time

history

of the induced gradient Į=3%

―597―

displacement

for a slope

Fig.

15

Time

history

of the

induced

displacement

velocitycurveat its base narrowswith increasinglayerthickness. Moreover,the maximum peak velocitiesflows lay on a curve representingthe required time to reach the maximum velocity withrespectto the layerthickness. The relationshipbetween the required time for the flow velocityto reach its maximumwas expressedby a power type functionin term of liquefiedlayerthickness.We also expressed the relationshipbetweenthe flowvelocityand the squareroot of the liquefiedlayerthickness. It is worthto mentionthat this studyrepresentsa contribution to analyzeliquefactioninducedlateralspreadingwithLagrangian meshfree particle method using smoothed particle hydrodynamics.It is a promising tool able to handle large displacements.However, it needs improvementsto take into accountporewaterpressuredissipationandstrengthrecovery.

for a slope

gradient Į=5%

References 5. Conclusions

This

paper

investigates

a

method

using

smoothed

particle

induced

lateral

spreading

analysis.

Based

on

post

results,

the

during

earthquake

was

throughout soil.

results

obtained

study

proposed from

to

was

liquefaction

experimental fluid

constitutive the

first

table

particle

as a viscous

type

simulate

method

a shaking

and

to behave

A Bingham

this

The

free for

observations

soil is considered shaking.

mesh

hydrodynamics

earthquake

liquefied

used

liquefied

Lagrangian

model

behavior

of

checked

experiment

the

to verify of

a lateral

spread. The

simulated

confidence free

the

surface

was

almost

the

kept The

soil

due

to volume

was

also

surface

shape

one.

Moreover,

clearly

observed

same

subsided

locations was

free observed

transfer. at the

along

height.

the than

the

slowly

than

the

of the

constitutive

pore

water The

central

shear

amplitude

experiment that

and

displacement

was

displacements

also

but

sensitive

of

we

carried

the

velocity

This

does

not

different

flow

the

one.

velocities

flow

velocity

is mainly take

due

into

to

account

recovery. along

and

the

flow

at three

the

it was

to the yield to

the surface

configuration.

the

and

simulated

are insensitive

range,

of

strength

of

of the

in the downstream

section

one,

of the

section

simulated

The

law

initial

history cross

dissipation

distribution cross

induced the

pressure

the

the

degree trend

where

heaved

middle

observed

decreases nature

time

general

middle

in

and

The

simulated

smaller

as

upstream

a certain

the

at the

inclination

in the

fit with

height

at the

found

that

viscosity

minimum

the

within

undrained

strength. In

the

last

investigate the

liquefied

on

numerical

velocities gradient.

part,

the relationship soil

thickness

results, increase

We

also

out

between and we

with noticed

the

observed increasing that

numerical

simulation

the maximum ground that layer the

shape

flow

gradient the

of

slope.

maximum

thickness the

to

velocity,

and time

Based flow slope history

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