earthquake deconvolution

8/11/2019 earthquake deconvolution

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4th International FLAC Symposium on Numerical Modeling in Geomechanics 2006 Hart & Varona (eds.) Paper: 04-10 2006 Itasca Consulting Group, Inc., Minneapolis, ISBN 0-9767577-0-2

1 INTRODUCTION

Design earthquake ground motions developed forseismic analyses are usually provided as outcropmotions often rock outcrop motions. However, for

FLAC analyses, seismic input must be applied at the base of the model rather than at the ground surfaceas illustrated in Figure 1. The question then arises:what input motion should be applied at the base of a

FLAC model in order to properly simulate the designmotion?

The appropriate input motion at depth can becomputed through a deconvolution analysis using a1-D wave propagation code such as the equivalent-linear program SHAKE. This seemingly simpleanalysis is often the subject of considerable confu-sion resulting in improper ground motion input for

FLAC models. In this paper the application ofSHAKE for adapting design earthquake motions for

FLAC input is described. Numerical examples are presented illustrating two typical cases:1 A rigid base, where an acceleration-time history

is specified at the base of the FLAC mesh.2 A compliant base, where a quiet (absorbing)

boundary is used at the base of the FLAC mesh.

2 SEISMIC INPUT TO FLAC

Input of an earthquake motion into FLAC is typi-cally done using either a rigid base or a compliant

base. For a rigid base, a time-history of accelera-tion (or velocity or displacement) is specified for

grid points along the base of the mesh. While simpleto use, a potential drawback of a rigid base is thatthe motion at the base of the model is completely

prescribed. Hence, the base acts as if it were a fixed-displacement boundary reflecting downward propa-gating waves back into the model. Thus, a rigid baseis not an appropriate boundary for general applica-tion unless a large dynamic impedance contrast ismeant to be simulated at the base (e.g. low velocity

sediments over high velocity bedrock).

Soil

Target earthquake givenas outcrop motion

Bedrock

InputMotion?

Figure 1. Seismic input to FLAC.

For a compliant base simulation, a quiet (also re-ferred to as absorbing) boundary is specified alongthe base of the FLAC mesh. FLAC uses the viscous

boundary scheme developed by Lysmer & Kuhle-meyer (1969), consisting of two sets of dashpots at-tached independently to the mesh in the normal andshear directions. Limiting discussion to the shear di-

Earthquake deconvolution for FLAC

L.H. MejiaURS Corporation, Oakland, CA, USA

E.M. DawsonURS Corporation, Los Angeles, CA, USA

ABSTRACT: Design earthquake ground motions for dynamic analyses are typically specified as outcrop mo-tions, which may have to be modified for input at the base of a FLAC model. Often a deconvolution analysisusing a 1-D wave propagation code, such as the program SHAKE, is performed to obtain the appropriate in-

put motion at depth. This seemingly simple analysis is often the subject of considerable confusion. In this paper the theory and operation of the program SHAKE and input requirements of FLAC are reviewed, and theapplication of SHAKE for adapting design earthquake motions for FLAC input is described. Numerical ex-

amples illustrating typical cases are presented, and several questions that commonly arise are addressed.

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rection only, the dashpots provide a viscous sheartraction given by:

s s s vC = (1)

where and C S are the density and shear wave veloc-ity of the base material, and vS is the shear-component of particle velocity at the boundary. Notethat equation (1) is simply the relation between shearstress and particle velocity in an elastic shear wave(Kolsky 1963). The viscous dashpots of the quiet

boundary absorb downward propagating waves sothat they are not reflected back into the model.At a quiet boundary, an acceleration time history

cannot be input directly because the boundary must be able to move freely to absorb incoming waves.Instead the acceleration-time history is transformedinto a stress-time history for input. First the accel-eration is integrated to obtain velocity and then the

proportionality of stress to velocity in an elasticwave is used, as in Equation (1).

FLAC input requires that a factor of two be addedto this relation because of the stress is absorbed

by the viscous dashpots. FLAC does not take care ofthis numerical detail internally, but instead requiresthe user to add the factor of two. Thus for applica-tion of a stress-time history through a quiet base, theshear stress is given by

su s s vC 2= (2)

where v su is the particle velocity of the upward propagating motion. Note that if a history of accel-eration is recorded at a grid point on the quiet base,it will not necessarily match the input history. The

input stress-time history specifies the upward propa-gating wave motion into the FLAC model, but theactual motion at the base will be the superposition ofthe upward motion and the downward motion re-flected back from the FLAC model.

3 THEORY AND OPERATION OF SHAKE

SHAKE (Schnabel et al. 1972) is a widely used 1-Dwave propagation code for site response analysis.

SHAKE computes the vertical propagation of shearwaves through a profile of horizontal visco-elasticlayers. Within each layer, the solution to the waveequation can be expressed as the sum of an upward

propagating wave train and a downward propagatingwave train. The SHAKE solution is formulated interms of these upward and downward propagatingmotions within each layer as illustrated in Figure 2.

The relation between waves in one layer andwaves in an adjacent layer can be solved by enforc-ing the continuity of stresses and displacements atthe interface between the layers. These well knownrelations for reflected and transmitted waves at the

interface between two elastic materials (Kolsky1963) can be expressed in terms of recursion formu-las. In this way, the upward and downward propa-gating motions in one layer can be computed fromthe upward and downward motions in a neighboringlayer.

To satisfy the zero shear stress condition at thefree surface, the upward and downward propagatingmotions in the top layer must be equal. Starting atthe top layer, repeated use of the recursion formulas

allows the determination of a transfer function be-tween the motions in any two layers of the system.Thus, if the motion is specified at one layer in thesystem, the motion at any other layer can be com-

puted.SHAKE input and output is not in terms of the

upward and downward propagating wave trains, butin terms of the motions at: a) the boundary betweentwo layers, referred to as a within motion, or b) ata free surface, referred to as an outcrop motion.The within motion is the superposition of the up-ward and downward propagating wave trains. The

outcrop motion is the motion that would occur at afree surface at that location. Hence the outcrop mo-tion is simply twice the upward propagating wavetrain motion. If needed, the upward propagating mo-tion can be computed by taking half the outcrop mo-tion. At any point, the downward propagating mo-tion can then be computed by subtracting the upward

propagating motion from the within motion.This SHAKE solution is in the frequency domain,

with conversion to and from the time-domain per-formed with a Fourier transform. Although this pa-

per is only concerned with the use of SHAKE for thelinear elastic case, SHAKE can address non-linearsoil behavior approximately through the equivalent-linear approach. Analyses are run iteratively to ob-tain shear modulus and damping values for eachlayer that are compatible with the computed effec-tive strain for the layer.

G1 1 1Layer 1

Layer 2

Layer n(halfspace)

G2 2 2

G n n n

Upward propagating

Downward propagating

Free surface

Figure 2. Layered system analyzed by SHAKE. Layer proper-ties are shear modulus, G; density ; and damping fraction, .

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4 DECONVOLUTION FOR RIGID ANDCOMPLIANT BASE

The deconvolution procedure for a rigid base is il-lustrated in Figure 3. The goal of the exercise is todetermine the appropriate base input motion to

FLAC such that the target design motion is recov-ered at the top surface of the FLAC model. The pro-file modeled consists of three 20-m thick elastic lay-ers with shear wave velocities and densities as

shown in the figure. A nominal material damping of0.1% is used for all layers in order to minimize ap- proximations introduced by the Rayleigh dampingmodel employed in the FLAC analysis (Rayleighdamping is frequency dependent). The target accel-eration-time history, shown in Figure 4, is a modi-fied Kobe Earthquake recording, scaled to a peakground acceleration (PGA) of 1.0 g.

The SHAKE model includes the three elastic lay-ers and an elastic halfspace with the same propertiesas the bottom layer. The FLAC model consists of acolumn of 120 linear elastic elements. The targetearthquake is input at the top of the SHAKE columnas an outcrop motion. Then, the motion at the top ofthe halfspace is extracted as a within motion(shown in Fig. 5) and is applied as an acceleration-time history to the base of the FLAC model. The re-sulting acceleration at the surface of the FLAC

model is shown to be virtually identical to the targetmotion in Figure 6. The SHAKE within motion isappropriate for rigid base input because, as de-scribed above, the within motion is the actual mo-tion at that location, the superposition of the upwardand downward propagating waves.

vs = 150 m/sec = 18 kN/m 3

+0 m

-20 m

-40 m

-60 m

vs = 225 m/sec = 19 kN/m 3

vs = 350 m/sec = 22 kN/m 3

Halfspacevs = 350 m/sec = 22 kN/m 3

Target earthquakeapplied at surface asoutcrop motion

SHAKE FLAC

SHAKE within motion appliedas acceleration time history tobase of FLAC model

Computed accelerationrecord at surface

vs = 150 m/sec = 18 kN/m 3

+0 m

-20 m

-40 m

-60 m

vs = 225 m/sec = 19 kN/m 3

vs = 350 m/sec = 22 kN/m 3



SHAKE FLAC

SHAKE within motion appliedas acceleration time history tobase of FLAC model


Figure 3. Deconvolution procedure for rigid base.

0 5 10 15 20 25time (sec)

-1.0

-0.5

0.0

0.5

1.0

a c c e

l e r a

t i o n

( g

)

Modified Kobe EarthquakeScaled to PGA = 1.0 g

Figure 4. Target earthquake acceleration-time history.

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0 5 10 15 20 25

time (sec)

-0.5

0.0

0.5

a c c e

l e r a

t i o n

( g

) Within Motion

Figure 5. Computed within motion from SHAKE.

0 5 10 15 20 25time (sec.)

-1.0

-0.5

0.0

0.5

1.0

a c c e

l e r a

t i o n

( g

) Target MotionFLAC

Rigid Base

Figure 6. Computed and target motions: rigid base.

The deconvolution procedure for a compliant base is illustrated in Figure 7. The SHAKE and FLAC models are identical to those for the rigid body exercise, except that a quiet boundary is applied to the base of the FLAC mesh. For applicationthrough a quiet base, the upward propagating wave

motion ( the outcrop motion) is extracted fromSHAKE at the top of the halfspace. This accelera-tion-time history (shown in Fig. 8) is integrated toobtain a velocity, which is then converted to a stresshistory using Equation (2). Again, the resulting ac-celeration at the surface of the FLAC model is virtu-ally identical to the target motion (Fig. 9).

As an additional check of the computed accelera-tions, the response spectra for both the compliant

base and rigid base cases are shown in Figure 10.These closely match the response spectra of the tar-get motion.

5 DECONVOLUTION FOR TYPICAL CASESENCOUNTERED IN PRACTICE

Although useful for illustrating the basic ideas be-hind deconvolution, the example presented above inSection 4 is not the typical case encountered in prac-tice. More common is the situation shown in Figure11, where one or more soil layers (expected to be-have non-linearly) overly bedrock (assumed to be-have linearly). A FLAC model for this case will usu-ally include the soil layers and an elastic base of

bedrock. To compute the correct FLAC compliant base input, a SHAKE model is constructed as shownin the Figure. The SHAKE model includes a bed-rock layer equal in thickness to the elastic base ofthe FLAC mesh, and an underlying elastic half-spacewith bedrock properties. The target motion is input

to the SHAKE model as an outcrop motion at the topof the bedrock (point A). Designating this motion asoutcrop means that the upward propagating wavemotion in the layer directly below point A will be setequal to the target motion. The upward propagat-ing motion for input to FLAC is extracted at Point Bas the outcrop motion.

For the compliant base case there is actually noneed to include the soil layers in the SHAKE modelas these will have no effect on the upward propagat-ing wave train between points A and B. In fact, forthis simple case, it is not really necessary to perform

a formal deconvolution analysis, as the upward propagating motion at point B will be almost identi-cal to that at point A. Apart from an offset in time,the only differences will be due to material damping

between the two points, which will generally besmall for bedrock. Thus, for this very common situa-tion, the correct input motion for FLAC is simply of the target motion. (Note that the upward propa-gating wave motion must be converted to a stresstime history using equation (2) which includes a fac-tor of 2 to account for the stress absorbed by the vis-cous dashpots. Alternatively, the target motion can

be directly used with Equation (1).)4


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vs = 150 m/sec = 18 kN/m 3

+0 m

-20 m

-40 m

-60 m

vs = 225 m/sec = 19 kN/m 3

vs = 350 m/sec = 22 kN/m 3



SHAKE FLAC

SHAKE upward propagatingmotion applied to base ofFLAC Model


Downward propagatingmotion absorbed byquiet base

Figure 7. Deconvolution procedure for compliant base.

0 5 10 15 20 25time (sec)

-0.5

0.0

0.5

a c c e

l e r a

t i o n

( g

) Upward Propagating

Figure 8. Upward propagating motion from SHAKE.

0 5 10 15 20 25time (sec)

-1.0

-0.5

0.0

0.5

1.0

a c c e

l e r a

t i o n

( g

)Target MotionFLAC

Compliant Base

Figure 9. Computed and target motions: compliant base.

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0.01 0.1 1 10period (sec)

0.0

0.5

1.0

1.5

2.0

2.5

p s e u

d o - s p e c

t r a

l a c c e

l . (

g )

Target MotionFLAC: Compliant BaseFLAC: Rigid Base

5% damped

Figure 10. Computed and target response spectra.

A

Bedrock

Target earthquakeapplied at top of bedrockas outcrop motion



Non - linear soil Non-linear soil

Bedrock

B

A

Bedrock


SHAKE FLAC



Non - linear soil Non-linear soil

Bedrock

B

Figure 11. Compliant base deconvolution procedure for a typical case encountered in practice.

For a rigid base analysis, the within motion at point B is required. Since this within motion incor- porates downward propagating waves reflected offthe ground surface, the non-linear soil layers must

be included in the SHAKE model. However, soilnon-linearity will be modeled quite differently in

FLAC and SHAKE. Thus, it is difficult to computethe appropriate FLAC input motion for a rigid baseanalysis.

Another typical case encountered in practice is il-lustrated in Figure 12. Here, the soil profile is deep,and rather than extending the FLAC mesh all theway down to bedrock, the base of the model ends

within the soil profile. Note that the mesh must beextended to a depth below which the soil response is

essentially linear. Again the design motion is inputat the top of the bedrock (point A) as an outcrop mo-tion, and the upward propagating motion for input to

FLAC is extracted at point B. As in the previous ex-ample, for a compliant base analysis there is no needto include the soil layers above point B in theSHAKE model. These layers have no effect on theupward propagating motion between points A and B.Unlike the previous case, the upward propagatingmotion can be quite different at points A and B, de-

pending on the impedance contrast between the bed-rock and linear soil layer. Thus, it is not appropriateto skip the deconvolution analysis and use the target

motion directly.6


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A

Bedrock


SHAKE FLAC

SHAKE upward propagatingmotion applied to base ofFLAC Model Downward propagating

motion absorbed byquiet base

Non-linear soil Non-linear soil

BLinear soil

Linear soil

A

Bedrock


SHAKE FLAC

SHAKE upward propagatingmotion applied to base ofFLAC Model Downward propagating

motion absorbed byquiet base

Non-linear soil Non-linear soil

BLinear soil

Linear soil

Figure 12. Compliant base deconvolution procedure for another common case encountered in practice.

6 REFLECTIONS OFF RIGID BASE

One of the main disadvantages of an assumed rigid base boundary is that downward propagating wavesare reflected back into the model rather than radiat-ing out through the base. These reflections are oftennot readily apparent in complex non-linear FLAC analyses, as they can be masked by the high damp-ing at larger strains in non-linear soil models. Re-flections off the base are clearly observed in elasticsystems with very low damping.

6.1 1-D Column

To illustrate how easily unwanted reflections can beinduced off a rigid base, the example shown in Fig-ure 3 is modified slightly. For simplicity all layersare assigned a uniform shear velocity of 250 m/secand uniform density. A SHAKE analysis is then per-formed to compute the appropriate within motion forapplication at the base of the FLAC mesh. Nowimagine that this within motion is applied to a FLAC model that has a shear wave velocity 5% lower thanthe 250 m/sec used in the SHAKE analysis. Thissituation might occur, for example, in a 2-D FLAC analysis where the rock stiffness is slightly lower atone end of the mesh.

Applying this mismatched rigid-base input mo-tion results in the surface acceleration-time historyshown in Figure 13. Clearly large amplitude periodicvibrations develop for the rigid-base boundary.These are due to the excitation of standing waveswithin the model. The period of these standingwaves can be seen in the response spectrum (Fig.

14) which has prominent peaks at 0.20 seconds and0.34 seconds, corresponding to the standing waves

modes shown in the figure. Note that the boundaryconditions for these waves are fixed displacement atthe base of the model and zero shear stress at theground surface.

Also shown in Figures 13 & 14 are the surfaceacceleration and response spectra for the corre-sponding compliant base analysis (velocity in the

FLAC model reduced by 5%). As expected, thesediffer only slightly from the target motion with nosigns of standing waves.

6.2 Embankment

As a further illustration of the spurious reflectionsthat can be caused by a rigid base, consider the em-

bankment shown in Figure 15. Acceleration input isapplied through a rigid base, using a within motioncomputed so that the target earthquake motion is re-covered at point A, the free-field. The computed ac-celeration at the crest of the embankment (Fig. 16)and the response spectrum of this motion (Fig. 17)again show large amplitude periodic vibrations. Thecorresponding compliant base analysis does notshow these vibrations.

7 CONCLUSIONS

Input of an earthquake motion into FLAC is typi-cally done through either a rigid or compliant base.For a rigid base, a time-history of acceleration isspecified at the base of the FLAC mesh. For a com-

pliant base, a quiet (or absorbing) boundary is speci-

fied along the base of the FLAC mesh and the inputmotion is applied as a stress-time history.7


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0 5 10 15 20 25time (s)

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

a c c e

l e r a

t i o n

( g

)Rigid BaseCompliant Base

Figure 13. Computed acceleration at top of column for rigid base with 5% velocity mismatch.

0.01 0.1 1 10period (sec)

0

1

2

3

4

5

6

7

p s e u

d o - s p e c

t r a

l a c c e

l . ( g )

Target MotionRigid BaseCompliant Base

5% damped

3/4 = H

5/4 = H

H

Figure 14. Response spectrum of surface motion for model with rigid base and 5% velocity mismatch.

Input motion deconvolvedso as to recover targetmotion at free field

vs = 350 m/sec

vs = 350 m/sec A

Acceleration measured atcrest of embankment

B

Dynamic input applied at rigid base or compliant base

1.5

1 100 m

100 m

Figure 15. Embankment analyzed with rigid and compliant base.

0 5 10 15 20 25

time (s)

-3

-2

-1

0

1

2

3

a c c e

l e r a

t i o n

( g

)Rigid BaseCompliant Base

Figure 16. Computed accelerations at crest of embankment.8


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