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 for seismic analyses are usually provided as outcrop motions – often rock outcrop motions. However, for FLAC analyses, seismic input must be applied at the base of the model rather than at the ground surface as 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 design motion?’

The appropriate input motion at depth can be computed through a ‘deconvolution’ analysis using a 1-D wave propagation code such as the equivalent-linear program SHAKE. This seemingly simple analysis is often the subject of considerable confu-sion resulting in improper ground motion input for FLAC models. In this paper the application of SHAKE 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 simple to use, a potential drawback of a rigid base is that the 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 base is not an appropriate boundary for general applica-tion unless a large dynamic impedance contrast is meant to be simulated at the base (e.g. low velocity sediments over high velocity bedrock).

Soil

Target earthquake given as outcrop motion

Bedrock

Input Motion?

Figure 1. Seismic input to FLAC.

For a compliant base simulation, a quiet (also re-

ferred to as absorbing) boundary is specified along the 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 and shear directions. Limiting discussion to the shear di-

Earthquake deconvolution for FLAC

L.H. Mejia URS Corporation, Oakland, CA, USA

E.M. Dawson URS 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’ analysis using 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 pa-per the theory and operation of the program SHAKE and input requirements of FLAC are reviewed, and the application 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.

rection only, the dashpots provide a viscous shear traction given by:

sss vCρσ = (1)

where ρ and CS are the density and shear wave veloc-ity of the base material, and vS is the shear-component of particle velocity at the boundary. Note that equation (1) is simply the relation between shear stress and particle velocity in an elastic shear wave (Kolsky 1963). The viscous dashpots of the quiet boundary absorb downward propagating waves so that 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 transformed into 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 elastic wave is used, as in Equation (1).

FLAC input requires that a factor of two be added to this relation because ½ of the stress is absorbed by the viscous dashpots. FLAC does not take care of this numerical detail internally, but instead requires the user to add the factor of two. Thus for applica-tion of a stress-time history through a quiet base, the shear stress is given by

suss vCρσ 2= (2)

where vsu 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 the actual motion at the base will be the superposition of the 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-D wave propagation code for site response analysis. SHAKE computes the vertical propagation of shear waves through a profile of horizontal visco-elastic layers. Within each layer, the solution to the wave equation can be expressed as the sum of an upward propagating wave train and a downward propagating wave train. The SHAKE solution is formulated in terms of these upward and downward propagating motions within each layer as illustrated in Figure 2.

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

interface between two elastic materials (Kolsky 1963) 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 from the upward and downward motions in a neighboring layer.

To satisfy the zero shear stress condition at the free surface, the upward and downward propagating motions in the top layer must be equal. Starting at the 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 the system, 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, but in terms of the motions at: a) the boundary between two layers, referred to as a ‘within’ motion, or b) at a 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 a free surface at that location. Hence the outcrop mo-tion is simply twice the upward propagating wave train 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 the linear elastic case, SHAKE can address non-linear soil behavior approximately through the equivalent-linear approach. Analyses are run iteratively to ob-tain shear modulus and damping values for each layer that are compatible with the computed effec-tive strain for the layer.

G1 ρ1 ζ1Layer 1

Layer 2

Layer n(halfspace)

G2 ρ2 ζ2

Gn ρn ζn

Upward propagating

Downward propagating

Free surface

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

4 DECONVOLUTION FOR RIGID AND COMPLIANT BASE

The deconvolution procedure for a rigid base is il-lustrated in Figure 3. The goal of the exercise is to determine 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 of 0.1% is used for all layers in order to minimize ap-proximations introduced by the Rayleigh damping model employed in the FLAC analysis (Rayleigh damping is frequency dependent). The target accel-eration-time history, shown in Figure 4, is a modi-fied Kobe Earthquake recording, scaled to a peak ground acceleration (PGA) of 1.0 g.

The SHAKE model includes the three elastic lay-ers and an elastic halfspace with the same properties as the bottom layer. The FLAC model consists of a column of 120 linear elastic elements. The target earthquake is input at the top of the SHAKE column as an outcrop motion. Then, the motion at the top of the 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 target motion in Figure 6. The SHAKE ‘within’ motion is appropriate for rigid base input because, as de-scribed above, the ‘within’ motion is the actual mo-tion at that location, the superposition of the upward and downward propagating waves.

vs = 150 m/secγ = 18 kN/m3

+0 m

-20 m

-40 m

-60 m

vs = 225 m/secγ = 19 kN/m3

vs = 350 m/secγ = 22 kN/m3

Halfspacevs = 350 m/secγ = 22 kN/m3

Target earthquake applied at surface as ‘outcrop’ motion

SHAKE FLAC

SHAKE ‘within’ motion applied as acceleration time history to base of FLAC model

Computed acceleration record at surface

vs = 150 m/secγ = 18 kN/m3

+0 m

-20 m

-40 m

-60 m

vs = 225 m/secγ = 19 kN/m3

vs = 350 m/secγ = 22 kN/m3

Halfspacevs = 350 m/secγ = 22 kN/m3

Target earthquake applied at surface as ‘outcrop’ motion

SHAKE FLAC

SHAKE ‘within’ motion applied as acceleration time history to base of FLAC model

Computed acceleration record at surface

Figure 3. Deconvolution procedure for rigid base.

0 5 10 15 20 25time (sec)

-1.0

-0.5

0.0

0.5

1.0

acce

lera

tion

(g)

Modified Kobe EarthquakeScaled to PGA = 1.0 g

Figure 4. Target earthquake acceleration-time history.

0 5 10 15 20 25time (sec)

-0.5

0.0

0.5

acce

lera

tion

(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

acce

lera

tion

(g) Target Motion

FLACRigid 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 ap-plied to the base of the FLAC mesh. For application through a quiet base, the upward propagating wave motion (½ the outcrop motion) is extracted from SHAKE at the top of the halfspace. This accelera-tion-time history (shown in Fig. 8) is integrated to obtain a velocity, which is then converted to a stress history 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 CASES ENCOUNTERED IN PRACTICE

Although useful for illustrating the basic ideas be-hind deconvolution, the example presented above in Section 4 is not the typical case encountered in prac-tice. More common is the situation shown in Figure 11, 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 shown in the Figure. The SHAKE model includes a bed-rock layer equal in thickness to the elastic base of the FLAC mesh, and an underlying elastic half-space with bedrock properties. The target motion is input to the SHAKE model as an outcrop motion at the top of the bedrock (point A). Designating this motion as ‘outcrop’ means that the upward propagating wave motion in the layer directly below point A will be set equal to ½ the target motion. The upward propagat-ing motion for input to FLAC is extracted at Point B as ½ the outcrop motion.

For the compliant base case there is actually no need to include the soil layers in the SHAKE model as these will have no effect on the upward propagat-ing wave train between points A and B. In fact, for this 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 be small for bedrock. Thus, for this very common situa-tion, the correct input motion for FLAC is simply ½ of the target motion. (Note that the upward propa-gating wave motion must be converted to a stress time 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).)

vs = 150 m/secγ = 18 kN/m3

+0 m

-20 m

-40 m

-60 m

vs = 225 m/secγ = 19 kN/m3

vs = 350 m/secγ = 22 kN/m3

Halfspacevs = 350 m/secγ = 22 kN/m3

Target earthquake applied at surface as ‘outcrop’ motion

SHAKE FLAC

SHAKE upward propagating motion applied to base of FLAC Model

Computed acceleration record at surface

Downward propagating motion absorbed by quiet base

Figure 7. Deconvolution procedure for compliant base.

0 5 10 15 20 25time (sec)

-0.5

0.0

0.5

acce

lera

tion

(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

acce

lera

tion

(g) Target Motion

FLACCompliant Base

Figure 9. Computed and target motions: compliant base.

0.01 0.1 1 10period (sec)

0.0

0.5

1.0

1.5

2.0

2.5

pseu

do-s

pect

ral a

ccel

. (g)

Target MotionFLAC: Compliant BaseFLAC: Rigid Base

5% damped

Figure 10. Computed and target response spectra.

A

Bedrock

Target earthquake applied at top of bedrock as ‘ outcrop ’ motion

SHAKE upward propagating motion applied to base of FLAC Model

Downward propagating motion absorbed by quiet base

Non - linear soil Non-linear soil

Bedrock

B

A

Bedrock

Target earthquake applied at top of bedrock as ‘ outcrop ’ motion

SHAKE FLAC

SHAKE upward propagating motion applied to base of FLAC Model

Downward propagating motion absorbed by quiet base

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 off the ground surface, the non-linear soil layers must be included in the SHAKE model. However, soil non-linearity will be modeled quite differently in FLAC and SHAKE. Thus, it is difficult to compute the appropriate FLAC input motion for a rigid base analysis.

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 the way down to bedrock, the base of the model ends within the soil profile. Note that the mesh must be extended to a depth below which the soil response is

essentially linear. Again the design motion is input at 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 need to include the soil layers above point B in the SHAKE model. These layers have no effect on the upward propagating motion between points A and B. Unlike the previous case, the upward propagating motion 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 appropriate to skip the deconvolution analysis and use the target motion directly.

A

Bedrock

Target earthquake applied at top of bedrock as ‘outcrop’ motion

SHAKE FLAC

SHAKE upward propagating motion applied to base of FLAC Model Downward propagating

motion absorbed by quiet base

Non-linear soil Non-linear soil

BLinear soil

Linear soil

A

Bedrock

Target earthquake applied at top of bedrock as ‘outcrop’ motion

SHAKE FLAC

SHAKE upward propagating motion applied to base of FLAC Model Downward propagating

motion absorbed by quiet 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 waves are reflected back into the model rather than radiat-ing out through the base. These reflections are often not 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 elastic systems with very low damping.

6.1 1-D Column To illustrate how easily unwanted reflections can be induced off a rigid base, the example shown in Fig-ure 3 is modified slightly. For simplicity all layers are assigned a uniform shear velocity of 250 m/sec and uniform density. A SHAKE analysis is then per-formed to compute the appropriate within motion for application at the base of the FLAC mesh. Now imagine that this within motion is applied to a FLAC model that has a shear wave velocity 5% lower than the 250 m/sec used in the SHAKE analysis. This situation might occur, for example, in a 2-D FLAC analysis where the rock stiffness is slightly lower at one end of the mesh.

Applying this mismatched rigid-base input mo-tion results in the surface acceleration-time history shown in Figure 13. Clearly large amplitude periodic vibrations develop for the rigid-base boundary. These are due to the excitation of standing waves within the model. The period of these standing waves can be seen in the response spectrum (Fig. 14) which has prominent peaks at 0.20 seconds and 0.34 seconds, corresponding to the standing waves

modes shown in the figure. Note that the boundary conditions for these waves are fixed displacement at the base of the model and zero shear stress at the ground surface.

Also shown in Figures 13 & 14 are the surface acceleration and response spectra for the corre-sponding compliant base analysis (velocity in the FLAC model reduced by 5%). As expected, these differ only slightly from the target motion with no signs of standing waves.

6.2 Embankment As a further illustration of the spurious reflections that can be caused by a rigid base, consider the em-bankment shown in Figure 15. Acceleration input is applied through a rigid base, using a within motion computed 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. The corresponding compliant base analysis does not show 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 is specified 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 input motion is applied as a stress-time history.

0 5 10 15 20 25time (s)

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

acce

lera

tion

(g) Rigid Base

Compliant 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

pseu

do-s

pect

ral a

ccel

. (g)

Target MotionRigid BaseCompliant Base

5% damped

3/4 λ = H

5/4 λ = H

H

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

Input motion deconvolvedso as to recover target motion at free field

vs = 350 m/sec

vs = 350 m/secA

Acceleration measured at crest 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 25time (s)

-3

-2

-1

0

1

2

3

acce

lera

tion

(g) Rigid Base

Compliant Base

Figure 16. Computed accelerations at crest of embankment.

0.01 0.1 1 10period (sec)

0

2

4

6

8

10

12

pseu

do-s

pect

ral a

ccel

. (g)

Target MotionRigid BaseCompliant Base

Figure 17. Response spectra of motion at crest of embankment.

If the program SHAKE is used to compute the

input motion for application at the base of a FLAC model, the ‘within’ motion should be used for a rigid base, as this is the actual particle motion, the super-position of the upward and downward propagating wave trains. For a compliant base, the upward propagating wave train should be used. The upward propagating wave is ½ the SHAKE outcrop motion.

A rigid base is only appropriate for cases with a large impedance contrast at the base of the model. A compliant base is almost always the preferred option because downward propagating waves are absorbed, while for a rigid base these waves are reflected back into the model. Although the presence of these re-flections is not always obvious in complex non-linear FLAC analyses, the can have a major impact on analysis results, especially when cyclic degrada-tion or liquefaction soil models are employed.

In addition to preventing reflections off the base of the model, a compliant base greatly simplifies computation of the appropriate input motion. Only the upward propagating motion is required. This in-coming or incident wave is not affected by the mate-rial above. In contrast, a rigid base requires that the within motion be applied, which is a motion that de-pends not only on the incoming wave train, but also on the dynamic response of the model above.

REFERENCES

Itasca Consulting Group. 2005. FLAC – Fast Lagrangian Analysis of Continua, Ver. 5.0 User’s Guide. Minneapolis: Itasca.

Kolsky, H. 1963. Stress Waves in Solids. New York: Dover Publications.

Lysmer, J. and Kuhlemeyer, R.L. 1969. Finite Difference Model for Infinite Media. J. Eng. Mech., 95 (EMR), pp. 859-877.

Schnabel, P.B, Lysmer, J. and Seed, B.H. 1972. SHAKE, a computer program for earthquake response analysis of horizontally layered sites. Earthquake Engineering Re-search Center, University of California, Berkeley. Report No. EERC 72-12.