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Introduction1

Seismic performance of earth systems requirescomputation of their dynamic response

Two General

Categories:

Equivalent-Linear

Cyclic Non-Linear

Frequency-domainvs.

Time-domain

Understand how

frequency-

vs. time-domainimplementation affects dynamic response

Objective

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Equivalent-Linear & Non-Linear 2

Equivalent-Linear Cyclic Non-Linear

G sec

DampingratioMasing (1926)

hypothesis2maxsec

loop

G

A

21

Secant shearmodulus G tanTangent shearmodulus

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3

Curve-Fitting Process

Equivalent-Linear & Non-Linear

Establish consistency with physically-meaningful

soil properties

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Dynamic Response Analyses4

AnalysesEquivalent-linear SHAKECyclic non-linear FLAC v.6 (Hysteretic)

SHAKE-FLAC Comparisons

1. Shear strain ( ) and shear modulus (G sec

, G tan

)time histories

2. Acceleration time histories

To understand, at a mechanistic level, how

time-varying stiffness affects dynamic behavior

To understand how time-varying stiffness

translates into differences in ground surfaceaccelerations and amplification response

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1D Soil Column5

Soil Column Properties:

Height of 30 m (60 zones/layers)

Shear wave velocity: 250 m/s Unit weight: 20 kN/m 3

Seed & Idriss

(1970) curves

SAND

FLAC fit using [default] function

Histories:

Shear strain ( ) and shear modulusat depth of 5 m

Acceleration at top of soil column

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Stiffness Reduction & Damping6

Empirical curves in SHAKEreplaced

with FLAC fits

Issue of excess dampingeliminated

Both codes have identical curves!!

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Input Ground Motions7

Sine motion (f m

= 1.25 Hz)(Ramped, 5 s intense shaking, 10 s quiet shaking)

Earthquake recording (f m

= 1.28 Hz)

(Hollister Hall, USGS 1028, from 1989 Loma Prieta, CA, USA)

Intense 0.2g

Quiet 0.05g

0.2g

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Strain & Modulus Time Histories8

Single, cycle duringintense

shaking

Single, cycle duringquiet

shaking

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Observation 19

Difference between FLAC and SHAKE most pronouncedduring interval of intense shaking

Discrepancy between frequency-domain and time-domainmodels is sensitive to the magnitude of the induced strains

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Observation 2

Intense Period

10

FLAC is softer

than SHAKE (phase shift, slower)

Greatest difference when G tan

is a minimum (points b , d )

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Observation 412

Steeper slope;more non-linear

Intense Period

Corresponds to shear modulus that isreducing at a rapid rate

From b to c , strains change rapidly Backbone curve

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Observation 413

More gentle slope;Less non-linear

Intense Period

Corresponds to shear modulus thatis reducing at a slower rate

From c to d , strains change slowly Backbone curve

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Observation 514

Quiet Period

FLAC is stiffer

than SHAKE

Lower induced strains, response is nearly identical(linear-elastic)

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Hysteresis Loops16

ShakingLevel

G sec

(MPa)

(%)

SHAKE FLAC SHAKE FLAC

Intense 94.1 78.7 5.4 7.4

Quiet 94.1 107.7 5.4 3.4

< Greater

Greater Response

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Ground Surface Accelerations17

FLAC Peak Accelerations

50% greater

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Conclusions18

When damping is the same, differences in dynamic

response are attributed to the coupled interactionbetween time-varying stiffness and shear strain.

Amplification response governed more by time-

varying stiffness than material damping. Time-domain implementation more effectively

captures peak points

of shaking (e.g. peak groundacceleration, PGA)

Non-linear models more appropriate for large strain situations (e.g. high intensity motions, soft sites,

resonance)

Frequency-

vs. Time Domain

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Thank you!

Second International FLAC/DEM Symposium on Numerical Modeling