wave energy in surface layers for energy-based damage ... · takaji kokusho , ryuichi motoyama,...

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Soil Dynamics and Earthquake Engineering 27 (2007) 354–366

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Wave energy in surface layers for energy-based damage evaluation

Takaji Kokusho�, Ryuichi Motoyama, Hiroshi Motoyama

Faculty of Science and Engineering, Chuo University, Tokyo

Received 17 April 2006; received in revised form 5 August 2006; accepted 7 August 2006

Abstract

Seismic wave energy in surface layers is calculated based on vertical array records at four sites during the 1995 Hyogo-ken Nambu

earthquake by assuming vertical propagation of SH waves. The upward energy generally tends to decrease as it goes up from the base

layer to the ground surface particularly in soft soil sites. Theoretical study on 1D multi-layers model to investigate the basic energy flow

mechanism indicates that the energy at the ground surface can be smaller on softer soils due to high soil damping during strong shaking

even if resonance effect is considered. A simple calculation for a shear-vibrating structure resting on foundation ground shows that

induced strain in the structure is directly related to the energy or the energy flux of surface layers. Hence, a general perception that soft

soil sites tend to suffer heavier damage than stiff sites should be explained not by greater incident energy but by other reasons such as

degree of resonance. Furthermore, it is recommended that not only acceleration or velocity but also S-wave velocity should be specified

at a layer where a design seismic motion is given, so that the seismic wave energy can clearly be quantified in seismic design practice.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Seismic wave energy; Performance-based design; Resonance; Impedance ratio; Damping

1. Introduction

Conventional seismic design is based on inertia forcesgiven by acceleration (e.g. maximum acceleration) orseismic coefficients. Though the force-based design methodhas long been used to date, it is recognized increasingly thatacceleration may not be an appropriate parameter forseismic damage evaluation. More and more strong accel-erograms with a maximum value exceeding 1G have beenobtained in recent years without any significant damage,e.g. in Tarzana, California during the 1994 Northridgeearthquake, in Kushiro, Hokkaido during the 2003Tokachi-oki earthquake, in Toka-machi, Niigata during2004 Niigata-ken Chuetsu earthquake, etc. Velocity isincreasingly used in place of acceleration because it isbelieved to be closely related to seismic energy. Then, whydon’t we quantify the wave energy along with accelerationor velocity in seismic damage evaluation or seismic design?

The wave energy was investigated by seismologists;Gutenberg and Richter [1,2] in order to evaluate the total

e front matter r 2006 Elsevier Ltd. All rights reserved.

ildyn.2006.08.002

ing author. Tel.: +81338171798; fax: +81338171803.

ess: [email protected] (T. Kokusho).

seismic wave energy released from a seismic source basedon observed earthquake records assuming spherical energyradiation for body waves or cylindrical radiation forsurface waves from a source. The total seismic energywas also discussed by Tsuboi [3] from a viewpoint of anequivalent rock volume capable to store the energy.However, few researchers investigated wave energy froma viewpoint of engineering design. In Japan, the energyconcept has already been proposed or implicitly used inseismic designs in buildings [4] or road bridge design [5],although it is still limited within superstructures, withoutexplicitly considering energy flow from foundation groundto superstructures. Kokusho and Motoyama [6], as apreliminary research on energy flow in surface layers,investigated energy dissipation mechanisms based onvertical array seismic records, although the energy flowup to ground surface and superstructures was notdiscussed.As will be explained later, the S-wave energy is basically

calculated as a product of the square of particle velocity ofpropagating seismic wave times soil impedance. Thisindicates that, even if the same particle velocity ismeasured, at A and B-site for example, the energy is twice

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ARTICLE IN PRESS

A

B

C

Eu

Ed

Eu

Ed

Es Es

Base layer

Surfacelayer

1

2

m

n

Seismogragh

Groundsurface

z

Layer No.

Am Bm

AnBn

h1

h2

hm

Fig. 1. Energy flow in vertical array system.

T. Kokusho et al. / Soil Dynamics and Earthquake Engineering 27 (2007) 354–366 355

larger at A if the impedance is twice larger at A than B. Itmaybe said that this effect is not clearly identified in thepresent seismic design practice, provided that the energyreally serves as a decisive parameter for evaluating seismicdamage.

In the first part of this paper, the energy flow of seismicwaves during the 1995 Hyogo-ken Nambu earthquake(sometimes called as the Kobe earthquake) in 4 verticalarray sites is calculated. Accumulated wave energy andenergy flux (flow rate) in surface layers and at the groundsurface are calculated from vertical array records byassuming vertical propagation of SH waves in surfacelayers. Then, idealized 1D multi-layers linear models withvariable S-wave velocities and damping ratios of soils arestudied to understand energy flow and dissipation mechan-isms in surface layers. Finally, a simplified model of asuperstructure resting on foundation ground is discussed toconsider the effect of seismic wave energy in surface layerson induced strain and seismic damage of superstructures.

2. Wave energy evaluation by vertical array records

Based on a postulate that dominant seismic motionspropagate in the vertical direction by SH waves [7], theenergy increment DE transported by the SH-wave througha unit area in a time increment Dt is expressed [6] as

DE ¼ DEk þ DEe ¼ rV sDtdu

dt

� �2

, (1)

where r is the soil density, Vs the wave velocity and du/dt

the particle velocity of the soil. Note that the wave energyDE is transmitted 50% by the kinetic energy DEk and 50%by the strain energy DEe. Let us define the time derivativeof the energy;

E ¼DE

Dt¼

dE

dt¼ rV s

du

dt

� �2

(2)

as energy flux or energy flow rate. If a time interval for aseismic motion to go through a certain depth t ¼ t1�t2 isconsidered, the accumulated energy is expressed as

Eðt ¼ t1�t2Þ ¼

Z t2

t1

E dt ¼ rV s

Z t2

t1

du

dt

� �2

dt. (3)

Note that du/dt in Eqs. (1) and (2) is the particle velocitynot directly of recorded motions but of traveling waves ineither upward or downward direction. Therefore, it isessential to separate a measured motion at a point intoupward and downward waves in order to evaluate theindividual energies.

If a site consists of a set of horizontal soil layers and theybehave as linear materials, upward and downward waves atany point can be calculated from a surface record based onthe multiple reflection theory [8] from which the flow of theenergy there is readily evaluated. During strong earth-quakes, seismic motions at the ground surface are verymuch influenced by the soil nonlinearity. However, the

deeper the soil is, the more linearly soil behaves evenduring strong earthquakes [9]. If vertical array records areavailable, the energy flow in deeper ground can beevaluated by using earthquake records at deeper levelswhere seismic wave is less contaminated by soil nonlinear-ity. The separation of upward and downward waves frommeasured motions at two different underground levels ispossible based on the multiple reflection theory asexplained in another literature [6]. Wave energies or energyflux can be calculated from the velocity time histories byEq. (2) or Eq. (3). On the other hand, the upward energy ata ground surface (Point A in Fig. 1) can be calculated bysubstituting a half of particle velocity there into du/dt inEq. (2) or (3).

3. Energy calculations at 4 sites during the Kobe earthquake

Four vertical array sites used in the energy calculationfor the Kobe earthquake are Port Island in Kobe city (PI),Research Institute of Kansai Electric Power Company[KEPCO] in Amagasaki city (SGK), KEPCO power plantin Takasago city (TKS) and KEPCO transformer station atKainan-ko in Wakayama city (KNK). PI is just next to thecausative fault, SGK and TKS are about 20 km far andKNK is about 65 km far from the epicenter. In Fig. 2, thesoil profiles of the 4 sites are shown together with theseismograph installation levels. Three to four seismographs

ARTICLE IN PRESS

0 200 400 600

0

20

40

60

80

100

S-wave velocity Vs (m/s)

Dep

th (

m)

WL

GL-0m

GL-16.4m

GL-32.4m

GL-83.4m

PI

GL-4.0mSG

CH

S

GS

G

CH

G

Seismograph

A

B

C

Damping ratio D (%)

0 20 40 60

0 100 200 300 400 500

0

20

40

60

80

100


Dep

th (

m)

SFMGSF

G

CH

S

CH

CH

CH

WL

SGK

GL-2.0m

GL-0m

GL-25m

GL-97m

Seimograph

CH

C

B

A

Damping ratio D (%)

0 4 8 12 16 20

0 200 400 600

0

20

40

60

80

100


Dep

th (

m)

SF

M

G

G

S

CH

G

M

TKS

WLGL-2.5m

GGL-0m

GL-25m

GL-100m

Seismogragh

C

B

A

Damping ratio D (%)

0 5 10 15

0 1000 2000

0

20

40

60

80

100

Vs-initial

Vs-inv.NS

Vs-inv.EW

D-inv.NS

D-inv.EW


Dep

th (

m)

C

S

M

WL

S

SF

M

M

Rock

KNK

GL-2.0m

GL-0m

GL-100m

Seimograph

SGM

M

M

G

GFG

GL-25m

A

B

C

Damping ratio D (%)0 5 10 15 20

(a) (b)

(c) (d)

Fig. 2. Borehole log and profiles of measured or back-calculated vs and damping ratio at 4 vertical array sites utilized in this research.

T. Kokusho et al. / Soil Dynamics and Earthquake Engineering 27 (2007) 354–366356

are installed between the ground surface and the deepestlevel of 84–100m. In the same charts, profiles of S-wavevelocities (Vs) measured by S-wave logging tests are showntogether with Vs and damping ratios (D) back-calculatedfrom the main shock records. Details on the back-calculation are available in other literatures [9].

Acceleration records in two horizontal directions at thedeepest level (Point C) and the second deepest level (PointB) shown in Fig. 2 are utilized to evaluate the energy flows

at the two levels. Surface acceleration records are used toevaluate the energy at surface levels (Point A). In theenergy evaluations based on Eq. (2) or Eq. (3), the back-calculated S-wave velocities and damping ratios shown inFig. 2 are assigned as the soil properties between Points Band C, and also at Point A. In PI, the horizontal motionsconverted to the major principal axis (in which the largestacceleration occurred) and the minor principal axisperpendicular to it are used, while, in the other three sites,

ARTICLE IN PRESS

0 5 10 15 20 25 30 35 40-0.2

-0.1

0.0

0.1

0.2

Up , Down

GL-97mV

el. (

m/s

)-0.2

-0.1

0.0

0.1

0.2GL-24.9m

Vel

. (m

/s)

Up , Down

-0.2

-0.1

0.0

0.1

0.2GL-0m

Vel

. (m

/s)

Up , Down

0

20

40

60

80

Ene

rgy

(kJ

/m2 )

Eu (Es), Ed, Eu-EdGL-0m GL-24.9m , , GL-97m , ,


motions in NS and EW directions are used. Accelerationrecords at the ground surface and the two deeper levels aretransformed into frequency spectra by the FFT technique.The low-frequency portion of the spectra (frequency:fo0:1Hz) is then cut off to remove a long period driftbased on the assumption that the energy contribution forthe frequency lower than 0.1Hz may be negligible.

Figs. 3–6 show analytical results in the major principaldirection in PI and in NS direction in other 3 sites. In eachfigure, the top chart indicates time-histories of energies andthe second to fourth charts depict velocity time histories ofupward and downward waves at Points A, B and C,respectively. The energy time histories are drawn for theupward energies at Points A, B and C, the downwardenergies at Points B and C and the differences between theupward and downward energies at Points B and C. Allenergies are expressed in KJ/m2 corresponding to theamount of seismic wave energy passing through a unit areaof 1m2.

Energy flux or accumulated energy calculated by Eq. (2)or Eq. (3) is dependent on impedance ratios of soil layersrVs. Here, the soil density r is estimated with a reasonableaccuracy from the soil type in conjunction with in situS-wave velocity and the ground water level shown in thesoil profile. As for the S-wave velocity, it is assumed that insitu low-strain values by S-wave logging tests are reliable.The value of Vs actually used in the energy calculation is

0 5 10 15 20 25 30 35 40-0.9-0.6-0.30.00.30.60.9

Up , Down

GL-83.4m

Vel

. (m

/s)

Vel

. (m

/s)

Vel

. (m

/s)

Time (s)

-0.9-0.6-0.30.00.30.60.9

GL-32.4m

Up , Down

-0.9-0.6-0.30.00.30.60.9

GL-0m

Up , Down

0

100

200

300

400

Ene

rgy

(kJ/

m2 )

GL-0m GL-32.4m , , GL-83.4m , ,

Eu (Es), Ed, Eu - Ed

Fig. 3. Time-histories of energy (top) and particle velocities (bottom) at PI

site in major principal axis.

Time (s)

Fig. 4. Time-histories of energy (top) and particle velocities (bottom) at

SGK site in NS direction.

strain-dependent degraded value which may involve acertain amount of uncertainties due to the back-calcula-tion. Uncertainty in the energy calculated based on Vs

may not be so large at Point B or C, because the soildegradation is relatively limited there. The uncertaintytends to increase near the ground surface, Point A, wherethe soil modulus degrades considerably due to exerted highstrain and/or pore-pressure development particularly insites near the fault such as PI and SGK. It should be notedtherefore to take this trend of uncertainty in mind to assessthe following results of energy evaluation.As shown in Fig. 3 for the major principal axis of PI, the

upward energies Eu remarkably increase within the firsttwo cycles of strong acceleration until t ¼ 6:3 s. The finalvalue of Eu at the deepest level (Point C; GL-83.4m)amounts to 305 kJ/m2 as a scalar sum of the energies in themajor and minor principal axes. At GL-32.4m (Point B),the upward energy Eu, which shows similar time-dependentchange, is about 80% of the energy at Point C. At theground surface (Point A), the upward energy Eu is about20% of Point C, indicating a clear decreasing trend ofupward energy with decreasing depth. The value (Eu–Ed) atPoint B or C reaches to a final value at t ¼ 17 s, whichcorresponds to dissipated energy, Ew in the surface layersabove that point. This amounts to 155 kJ/m2 at GL-32.4m

ARTICLE IN PRESS

0 5 10 15 20-0.10

-0.05

0.00

0.05

0.10

Up, Down

GL-100m

Vel

.(m

/s)

Time (s)

-0.10

-0.05

0.00

0.05

0.10GL-25m

Vel

.(m

/s)

Up, Down

-0.10

-0.05

0.00

0.05

0.10GL-0m

Vel

.(m

/s)

Up , Down

0

2

4

6

8

Ene

rgy

(kJ/

m2 )

GL-0m GL-25m , , GL-100m , ,

Eu (Es), Ed, Eu - Ed


TKS site in NS direction.

0 5 10 15 20 25 30 35 40

-0.04-0.020.000.020.04

Up , Down

GL-100mV

el. (

m/s

)

Time (s)

-0.04-0.020.000.020.04 GL-25m

Vel

. (m

/s)

Up , Down

-0.04-0.020.000.020.04 GL-0m

Vel

. (m

/s)

Up , Down

0.0

0.4

0.8

1.2

1.6

Ene

rgy

(kJ/

m2 )

Eu (Es), Ed, Eu - EdGL-0m GL-25m , , GL-100m , ,


KNK site in NS direction.


as the total energy dissipated in the two directions, andabout 65% of the corresponding total upward energy, Eu

at that level [6]. Near this vertical array station, extensivesand eruptions by liquefaction were observed during theearthquake despite that the station was neighboring to astorage building, the foundation ground of which had beenlightly compacted. Back-calculations using the samevertical array records also demonstrated that the fill layer16m thick from the surface extensively liquefied duringthe earthquake (e.g. [9,10]). This high percentage ofthe dissipated energy hence indicates that the surfacesoil served as an energy absorber for the strong seismicmotion [6].

Similar results for SGK site in the NS direction areshown in Fig. 4. The upward energy at GL-97m totaling inthe NS and EW directions is 83 kJ/m2 at t ¼ 15 s, only 27%of that at GL-83.4m in PI site. The upward energy Eu atGL-25m (Point B) and at the surface (Point A) evaluatesabout 90% and less than 20%, respectively, of Eu at GL-97m (Point C), indicating again the clear decreasing trendof upward energy with decreasing depth. The value (Eu–Ed)shows rapid increase until t ¼ 15 s and stays almostconstant only with slight fluctuations after that. Consider-ing a low possibility of liquefaction in this site judging fromthe soil condition, this increase seems to reflect thehysteretic energy dissipation due to a nonlinear stress–

strain relationship in non-liquefied soils during strongshaking.The results of TKS site in NS directions are shown in

Fig. 5. At this site, Eu at the surface (Point A) is againmuch smaller than the deeper levels at GL-25m (Point B)and GL-100m (Point C) despite that the amplitude of thevelocity time history is evidently larger at the surface. Thevalue (Eu–Ed) at GL-25m shows a negative value inthe latter part of the time history probably due to errorsinvolved in soil modeling.The results of KNK site in the NS direction are shown in

Fig. 6. A remarkable difference in the velocity amplitudebetween GL-25m and GL-100m exists in this site onaccount of the big difference in the impedance between thebase rock of V s ¼ 1630m=s and the overlying soil layer asshown in Fig. 2. In a good contrast with the previous 3sites, the upward energies Eu at GL-100m (Point C),GL-25m (Point B) and GL-0m (Point A) are not sodifferent to each other despite the aforementioned differ-ence in wave amplitude. The dissipated energy Ew is muchlower than Eu, indicating minimal soil nonlinearity even inthe upper layer in this site.In most of the energy evaluations described above, the

increasing trend in (Eu–Ed) almost stops in the middle ofthe records despite that Eu and Ed are still increasing. Themoment when (Eu–Ed) seems to stop its increase is pointedout by the arrow mark in Figs. 3–6. After that, the wave

ARTICLE IN PRESST. Kokusho et al. / Soil Dynamics and Earthquake Engineering 27 (2007) 354–366 359

energy is likely to propagate not only in the verticaldirection but also in the horizontal directions by surfacewaves. The accumulated energy up to this point will beused in the later energy analysis.

Fig. 7 shows the energy flux per second at three depths inthe principal axis of PI site. The upward energy flux Eu ¼

dEu=dt are very variable with time comprising multiplepeaks which correspond to the steeper gradients of the timehistory of cumulative energies shown in Fig. 3 and hencereflect wave form characteristics. The greater the depths,the higher the flow rate peaks and the earlier they appear.Fig. 8 indicates the relationships between the cumulativevalue or the maximum flow rate of upward energy versusthe depth obtained by the similar calculations at the 4 sites.The energy values in the graph are the sums in the twohorizontal directions and expressed in the logarithmicscale. Tremendous differences exist in the upward energyamong the 4 sites depending on the differences in the focaldistance. It should be noted that both the cumulativeenergy and the energy flux reduce considerably as theyapproach to the ground surface except in KNK where thesoil responded almost linearly.

0

50

100

150

200

250

4 5 6 7 8 9 10 11 12

Ene

rgy

flow

rat

e (k

J/s)

Time (s)

Es

Eu 32.4

Eu 82.4

Fig. 7. Time-histories of energy flux at PI site in major principal axis.

1 10 100 1000

0

20

40

60

80

100

PI (dEu/dt) max

PI Eu

SGK (dEu/dt) max

SGK Eu

TKS (dEu/dt) max

TKS Eu

KNK (dEu/dt) max

KNK Eu

Max energy flow rate (dEu/dt) max (kJ/m2/s)

or Accumulated energy Eu (kJ/m2)

Dep

th (

m)

Fig. 8. Distributions of accumulated upward energy and maximum energy

flux along depth at 4 sites (total values in the two horizontal directions).

Let us then compare the energies Es evaluated at theground surface (Point A) with the damping ratios in thecorresponding sites. Here, the damping ratios in individualsublayers, which were back-calculated in the separateinvestigation by Kokusho et al. [9], are averaged bymultiplying the weight of the thickness of each sublayershallower than Point B in Fig. 2. In Fig. 9, the energy ratiosEs/Eu are plotted versus damping ratios D with opensymbols in the two directions at the 4 sites. It may well beassumed despite some data scatters that the ratio of surfaceenergy Es to the upward energy Eu at Point B decreaseswith increasing averaged damping ratio as approximatedby the thin curve. Almost the same trend can be recognizedfor the ratios of the maximum values of the energy fluxðEsÞmax=ðEuÞmax ¼ ðdEs=dtÞmax= dEu=dt

� �max

plotted withclosed symbols in the same figure.In Fig. 10, the energy ratios Ew/Eu are compared with

the averaged damping ratios, where Ew and Eu are thedissipated energy in the surface layer shallower than PointB and the upward energy at Point B, respectively. Theincreasing trend of Ew/Eu with increasing damping ratiomay be assumed as the thin line. The dissipated energy in

0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40 50

PI Es /EuPI (dEs /dt) max/(dEu/dt) maxSGK Es/EuSGK (dEs /dt) max/(dEu/dt) maxTKS Es /EuTKS (dEs/dt) max/(dEu /dt) maxKNK Es/EwKNK (dEs/dt) max /(dEu/dt ) max

Rat

io o

f ene

rgy

or m

ax. f

low

rat

e be

twee

n E

s an

d E

u

Damping ratio D (%)

Fig. 9. Accumulated energy and energy flux versus optimized damping

ratio at 4 sites.

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50

PISGKTKSKNK

Ene

rgy

ratio

Ew

/Eu

Damping ratio D (%)

Fig. 10. Dissipated energy versus optimized damping ratio at 4 sites.

ARTICLE IN PRESS

�1 Vs1

�2 Vs2

EdEu

Ep Layer boundaryA1

A2 B2

z

0

0.25

0.5

0.75

1

0

0.5

1

1.5

2

0.1 1 10

Ene

rgy

ratio

Ep/E

u or

Ed

/Eu

Am

plitu

de r

atio

Impedance ratio � = �1Vs1 /�2Vs2

Energy flow ratio

Amplitude ratioA1 /A2

Ep/Eu

Ed /Eu

(a)

(b)

Fig. 11. One-dimensional propagation of SH wave through a layer

boundary (a) and amplitude or energy ratios of waves versus impedance

ratio (b).


the surface layer amounts to 70–50% of the upward energyat the base in the near-fault site PI, while it is around 20%or less in the remote site KNK, indicating that the rest ofthe upward energy returned to deeper earth again. Thecombination of Figs. 9 and 10 indicates that, in a site withsmaller shaking, dissipated energy is small and a largepercentage of energy comes up to the ground surface. Incontrast, more than a half of the upward energy is lost bysoil liquefaction or damping in the surface layer and only asmall portion arrives at the surface in a near-fault site withstrong shaking.

4. Energy flow mechanism in layered ground

4.1. Stationary response to harmonic waves

In order to understand the mechanism controlling theenergy flow in layered soil deposits under the assumptionof 1D propagation of the SH wave, let us go back to a basicmodel on a harmonic waves traveling in the verticaldirection. The first model is an infinite media consistingof two layers with a horizontal boundary as shown inFig. 11(a). z-axis is taken upward from the boundary. S-wave velocities are V s1 and V s2 , and soil densities are r1and r2 in the upper and lower layers, respectively. Wavedisplacements at the lower layer u1 and at the upper layeru2 are expressed as

u1 ¼ A1eiðot�k1zÞ; u2 ¼ A2e

iðot�k2zÞ þ B2eiðotþk2zÞ, (4)

where i ¼ffiffiffiffiffiffiffi�1p

, t is the time, o the angular frequency, k1and k2 are the wave numbers in the upper and lower layers,respectively, defined by k1 ¼ o=V s1 , k2 ¼ o=V s2 . A1, A2

and B2 are the wave amplitudes for the input upward wavein the lower layer, the propagating upward wave in theupper layer, and the reflecting downward wave in the lowerlayer, respectively. If the impedance ratio a is defined bya ¼ r1V s1=r2V s2 , then, from the continuity of deformationand stress at the boundary, the constants A1, A2, B2 arecorrelated as

A1=A2 ¼ 2=ð1þ aÞ; B2=A2 ¼ ð1� aÞ=ð1þ aÞ. (5)

Using Eq. (2), corresponding energy ratios are obtainedas

Ep=Eu ¼ 4a=ð1þ aÞ2; Ed=Eu ¼ ð1� aÞ2=ð1þ aÞ2, (6)

where Eu, Ep and Ed are the energy flux per unit timefor the input upward wave in the lower layer, thepropagating wave in the upper layer, and the reflectingwave in the lower layer, respectively. In Fig. 11(b), theamplitude ratio and the energy ratio of waves are takenalong the two vertical axes versus the logarithm of theimpedance ratio in the horizontal axis. Note that theenergy ratio Ep=Eu decreases symmetrically as the im-pedance ratio a is distant from unity, whereas theamplitude ratio monotonically decrease with increasingimpedance ratio. This indicates that, if a layer boundaryexists, the energy Ep always decreases from Eu irrespective

of relative stiffness of the two layers because a partof the energy Ed is inevitably transferred to the reflectingwave.The second model is a two-layers system with a free

surface at the top. A surface layer with a height H is restingon an infinitely thick base layer as shown in Fig. 12. z-axisis taken upward from the boundary, again. In this case, thewave numbers k�1, k�2 are taken as complex values in orderto take the soil damping into consideration. If a harmonicwave is given in the base layer, a stationary response of thesystem can be written as

u1 ¼ A1eiðot�k�1zÞ þ B1e

iðotþk�1zÞ,

u2 ¼ A2eiðot�k�2zÞ þ B2e

iðotþk�2zÞ, ð7Þ

where A1, B1 are wave amplitudes of propagating upwardwave and reflecting downward wave at the bottom of theupper layer, while A2, B2 are those for input upward waveand reflecting downward wave at the top of the lower layer,

ARTICLE IN PRESS

Surface layer

Base layer

H

z

A1

A2

B1

B2

As As

Eu Ed

Es Es

�2, Vs2

�1, Vs1

Fig. 12. Two-layers model for 1D propagation of SH wave.

0

2

4

6

8

10

0.1 1 10

Ene

rgy

ratio

l (d

Es

/dt)

/ (dE

u/d

t) l

Ene

rgy

ratio

l (d

Es

/dt)

/ (dE

u/d

t) l


f/f0 =1.0 2.0

0.8, 1.2

0.6, 1.4

0.2, 1.8

0.4, 1.6

D1 = D2 = 0

0

1

2

3

0.1 1


D1 = 5%, D2 = 0

f /f0 = 1.0

1.40.6

0.81.2

(a)

(b)

Fig. 13. Energy ratios between surface and base in 2-layers systems for

D1 ¼ 0% (a) and D1 ¼ 5% (b).


respectively. From the stress free condition at the surface

B1=A1 ¼ e�2ik�1H (8)

and also from the continuity of deformation and stress atthe layer boundary, the amplitude ratios can be decided as

A1

A2¼

2

ð1þ a�Þ þ ð1� a�Þe�2ik�1H

,

B2

A2¼ð1� a�Þ þ ð1þ a�Þe�2ik

�1H


, ð9Þ

where k�1 and a* are the complex wave number and thecomplex impedance ratio, respectively, defined by

k�1 ¼ k11

1þ 2iD1

� �1=2

; a� ¼r1Vs�1r2Vs�2

¼ a1þ 2iD1

1þ 2iD2

� �1=2

(10)

in which, complex S-wave velocities V�s1 , V�s2 areV�s1 ¼ V s1ð1þ 2D1Þ

1=2, V�s2 ¼ V s2 ð1þ 2D2Þ1=2, and D1,

D2 ¼ damping ratios in the lower and upper layers,respectively. If the amplitude at the ground surface As istaken in place of A1, the ratio of As to A2 is written as

As

A2¼

2

ð1þ a�Þeik�1H þ ð1� a�Þe�ik

�1H

. (11)

Using Eq. (2), the corresponding energy ratios areobtained as

Es=Eu

�� ¼ ja�j As=A2

�� 2

¼ ja�j2

ð1þ a�Þeik�1H þ ð1� a�Þe�ik�1H

��2

, ð12Þ

Ed=Eu

�� ¼ B2=A2

�� 2 ¼ ð1� a�Þ þ ð1þ a�Þe�2ik�1H


��2

, (13)

where Eu and Ed are the energy flux for the upward anddownward waves, respectively, at the top of the lower layerand Es is the energy flux for the incident wave at the

ground surface. Quite obviously, the energy lost in thesurface layer per unit time Ew can be calculated as

Ew=Eu

�� ¼ 1� Ed=Eu

�� (14)

because the energy gap between Eu and Ed has to bedissipated in the surface layer, if the vibration is assumedstationary.Based on Eq. (12), the energy ratio Es=Eu

�� ¼ jðdEs=dtÞ=ðdEu=dtÞj is calculated for various frequencies and thecurves are drawn against the impedance ratio a from 0.1 to10 on a semi-log graph in Figs. 13(a) and (b). Here, thefrequency f is normalized by the first resonant frequencyf 0 ¼ V s=4H. In Fig. 13(a) for zero damping (D1 ¼ 0) in the

ARTICLE IN PRESS

0

50

100

150

200

(b)

(a)

Input wave

Fou

rier

spec

. (ga

l Ese

c)

0.1 1 100

3

6

9

(c) Frequency (Hz)

Impedance ratio �1.0 0.727 0.3640.182 0.091

Spe

c.ra

tio

0 5 10 15 20 25 30 35 40-6

-3

0

3

6

Acc

. (m

/s2 )

Time (sec)

PI GL-83.4m (Incident wave)

Fig. 14. Acceleration (a), Fourier spectrum (b) and spectrum ratio (c)

used in the analysis.

10.10.0

0.5

1.0

1.5

2.0

Ene

rgy

ratio

E

s/E

u


D1 = 0%

2.5%

5.0%

10 %

20 %

40 %

Fig. 15. Energy ratio versus impedance ratio relationships for different

damping ratios.


surface layer, jEs=Euj is unity for a ¼ 1 and the curvessystematically shift in a symmetrical way as f/f0 changesfrom zero to 2. For normal geotechnical conditions inwhich ao1:0, resonance occurs at f/f0 ¼ 1.0 with the wavelength of 4H, while, a41:0, another resonance occurs at f/f0 ¼ 2.0 with the wave length of 2H. During the resonance,substituting f ¼ f 0 ¼ V s=4H into Eq. (12) and assumingzero damping ratio in the upper layer,

jEs=Euj ¼ ðdEs=dtÞ=ðdEu=dtÞ�� ¼ 1=a (15)

indicating that the softer the surface layer, the larger theenergy at the ground surface, as it is reversely proportionalto the impedance ratio a. However, it is obvious inFig. 13(a) that, if the frequency is off-resonance and f/f0is less than 0.6 or larger than 1.4, the energy ratio jEs=Euj

becomes smaller than unity even for zero damping. Thistrend is more evident as a gets smaller or the surface layerbecomes softer. In Fig. 13(b), similar curves for 5%damping (D1 ¼ 5%) in the surface layer are drawn againstthe impedance ratio a from 0.1 to 2. Here again, the energyratio jEs=Euj becomes smaller than unity if the frequency isoff-resonance with f/f0 less than 0.6 or larger than 1.4.

4.2. Transient flow by earthquake waves in a 2-layers system

Next, energy flow by a transient wave in 2-layers systemof Fig. 12 is calculated. The thickness of the surface layer isH ¼ 30m and the soil density in the surface and base layersare r1 ¼ r2 ¼ 2:0 t=m3. These values are chosen so that thetwo-layers model can roughly represent the surface layerresponse in the vertical array sites explained before. S-wavevelocity in the base layer is kept constant as V s2 ¼ 330m=swhile that in the surface layer V s1 is parametrically variedfrom 330 to 30m/s. The impedance ratio a ¼ r1V s1=r2V s2

correspondingly varies from unity to 0.0909. The dampingratio in the base layer is assumed D2 ¼ 0 while that in thesurface layer is varied as D1 ¼ 0240%. Input accelerationmotion shown in Fig. 14(a), which is identical to theupward wave at Point C in the principal axis in PI site, isgiven at the top of the base layer. The Fourier spectrum ofthe input motion shown in Fig. 14(b) is compared withthe transfer functions of the two-layers system shown inFig. 14(c), calculated for various V s1 with the dampingratio D1 ¼ 5%. Note that the dominant frequency of theinput motion is about 0.78Hz although the spectrum hasseveral peak frequencies around there.

Fig. 15 shows the ratio of surface energy Es to theupward energy Eu at the base layer in the vertical axisversus the impedance ratio a in the horizontal axiscalculated from the parametric study described above.For D1 ¼ 0, Es takes a maximum value when a ¼ 0:182and the peak frequency of the spectrum ratio coincideswith dominant frequencies included in the input motionas shown in Figs. 14(b) and (c). However, the resonanteffect diminishes as D1 becomes larger, and Es mono-tonically decreases as a decreases for D1410%. In Fig. 16,the ratio of maximum flow rates ðEsÞmax=ðEuÞmax ¼

ðdEs=dtÞmax=ðdEu=dtÞmax between surface and base is takenin the vertical axis in place of the ratio of the accumulatedenergy. In this case, too, the flow rate shows a monotonic

ARTICLE IN PRESS

1

1.0

1.5

2.0

(dE

s/d

t)m

ax/(

dE

u/d

t)m

ax

D1

= 0%

2.5%

5.0%

10%

20%

40%

0.0

0.5

0.1


Fig. 16. Relationships between ratio of maximum energy flux and

impedance ratio for various damping ratios.

10.0

0.1

0.5

1.0

Ene

rgy

ratio

Ew

/Eu

D1 = 0%

2.5%

5.0%

10 %

20 %

40 %


Fig. 17. Dissipated energy ratio versus impedance ratio relationships for

different damping ratios.

100 200 300 400 500 600 700-120

-100

-80

-60

-40

-20

0

Dep

th (

m)


2-layer model

3-layer model

4-layer model

5-layer model

Fig. 18. S-wave velocity distributions along depth in multi-layers models

having the same resonant frequency.

0 10 20 30 400.0

0.2

0.4

0.6

0.8

1.0

1.2

Ene

rgy

ratio

(E

s/E

u)

Damping ratio D (%)

2-layer model

3-layer model

4-layer model

5-layer model

Fig. 19. Energy ratio versus damping ratio relationship for multi-layers

models having the same resonant frequency.


decrease with decreasing impedance ratio for D1X10%,indicating that the softer the surface soil, the smaller theenergy or its flow rate for higher soil damping in thesurface layer. Fig. 17 depicts the ratio of dissipated energyEw to the upward energy Eu versus a obtained from thesame parametric study. While Ew ¼ 0 quite obviously forD1 ¼ 0, Ew takes the maximum value when a ¼ 0.182under non-zero damping ratios because the more waveenergy is trapped and dissipated in the surface soil due tothe resonance effect.

4.3. Transient flow by earthquake waves in multi-layers

system

In the 2-layers model above, the surface layer has beenidealized as a single uniform layer. In order to examine theeffect of this idealization, the surface layer of 100mthickness resting on a infinitely thick base layer is divided

into 2–4 sublayers of equal thicknesses to make 3–5 multi-layers systems as shown in Fig. 18. The S-wave velocity ofthe base layer is set as 700m/s while the velocities of surfacesublayers are adjusted properly so that the first resonantfrequency keeps the constant value of 0.78Hz in eachmulti-layers system. The damping ratio of each sublayer isvaried in concert from 0% to 40% while that in the baselayer of infinite thickness is assumed as zero. The sameinput motion as illustrated in Figs. 14(a) and (b) having thedominant frequency of 0.78Hz is given as an upward waveat the base layer.In Fig. 19, the ratio of the energy at the ground surface

to that at the base layer Es/Eu is taken against the dampingratio D1 of the surface sublayers. Quite reasonably, thewave energy arriving at the surface decreases withincreasing damping ratio in the surface layer in all themulti-layers model. The decreasing curve is concave andqualitatively similar to that in Fig. 9, indicating that the

ARTICLE IN PRESS

0 10 20 30 400.0

0.1

0.2

0.3

0.4

0.5

Ene

rgy

ratio

(E

w/E

u)

Damping ratio D (%)

2-layer model

3-layer model

4-layer model

5-layer model

Fig. 20. Dissipated energy ratio versus damping ratio relationship for

multi-layers models having the same resonant frequency.


similar mechanism dominates both in the actual groundand in the linear multi-layers model. Also noted is that thewave energy arriving at the surface tends to decreasemonotonically with the increasing number of sublayers.

In Fig. 20, the ratio of the dissipated energy in thesurface layer to the upward energy in the base layer, Ew/Eu,is taken versus the damping ratio D1. This relationshipqualitatively resembles to the corresponding curve shownin Fig. 10 obtained in the actual ground. It should also bepointed out in Fig. 20 that, in contrast to Fig. 19, thedissipated energy shows little difference among the variousmulti-layers models. By combining Figs. 19 and 20, it maybe said that, with increasing subdivision in the surfacelayer, more wave reflection occurs at the sublayerboundaries, taking back more wave energy into deeperground before arriving at the surface, while the energydissipation in the surface layer is almost insensitive to thenumber of subdivisions.

Shear-vibrating structure

Foundation ground

�st

, Vsst

�1, Vs

1

Hst

z

Est

Est

Es E

d

Fig. 21. Two-layers shear-mode system idealizing a superstructure resting

on soil.

5. Energy-based seismic response evaluation of structures

Through the above energy flow analyses in actualground and in the idealized 1D models, it has been foundthat various parameters are controlling the amount ofenergy at the ground surface, namely impedance ratio,nonlinear soil properties, soil damping, resonance effect,number of soil layers, etc. It has also been recognized thatthe seismic wave energy tends to be smaller in soft soil sitesthan in stiff soil sites because, unlike the wave amplitude, italways decreases in passing through a layer boundary andalso because surface soft soil layers cannot store so muchenergy even if resonance occurs due to high energydissipation by soil damping. This finding may not becompatible with a widely accepted perception that soft soilsites tend to supply greater seismic energy leading toheavier damage than stiff sites, provided that the seismic

wave energy is actually a key for evaluating damage ofstructures.During the 1923 Kanto earthquake, larger number of

wooden houses are said to have collapsed in down-townsoft soil areas than Pleistocene stiff soil areas in Tokyo andtriggered fires killing a great number of people. The sametrend seems to hold in the 1987 Loma Prieta earthquake,when major damage of wooden houses and geotechnicaldamage of life lines was concentrated in soft ground alongthe Bay area.However, during 1995 Kobe earthquake on the contrary,

buildings and civil engineering structures were heavilydamaged in areas of competent soils. In contrast, structuraldamage due to seismic inertia effect was not so serious insoft liquefied soil areas along the seashore less than a fewkilometers apart from the heavily damaged areas, althoughgeotechnical damage was prevalent there. In what follows,some discussions will be made how the seismic wave energyat ground surface is related to shaking-induced structuraldamage.Obviously, the degree of structural damage is strongly

dependent on induced strain in components of super-structures in comparison with threshold yield strains. Inperformance-based design, increasingly employed recently,structural performance and structural damage duringstrong earthquakes are mainly evaluated through inducedstrain levels. Consequently, it is meaningful to go back to abasic relationship between the induced strain and theseismic wave energy by a simple model.If a superstructure is idealized as a shear beam of a

large lateral dimension resting on foundation ground,then, the interaction between the two may be approximatedby a 2-layers shear-vibration system as depicted in Fig. 21which is basically the same as the 2-layers ground shown inFig. 12. The shear strain in the surface layer in Fig. 12 isgiven by differentiating the first of Eq. (7) by z and usingEq. (8),

g ¼ 2k�1A1 sin k�1ðH � zÞeiðot�k�1HÞ. (16)

ARTICLE IN PRESS

Table 1

An example of ratios for pertinent values between Site-A and B under the

same design velocity

Site A:B

Design velocity: du/dt 1:1

Impedance of ground: rVs 2:1

Impedance ratio (structure/ground): ast 1:2

Energy flux at ground surface: Es 2:1

Max. structural strain at resonance: gmax 2:1


Then, resorting to the first of Eq. (9) and Eq. (12), andreplacing H ! Hst, r! rst, Vs! Vsst, dEs=dt!

ðdEst=dtÞ, dEu=dt! dEs=dt, A1! Ast, A2! As,a� ! a�st, k�1 ! k�st, the shear strain in the superstructureis expressed as

g ¼4 sin k�stðHst � zÞ

ð1þ a�stÞeik�stHst þ ð1� a�stÞe�ik�stHst

�� a�stEs

rstVs�3st

��1=2

, (17)

where Hst is the height of the structure, k�st the complexwave number of the structure, a�st the complex impedanceratio, rst the equivalent density of the structure and V�sst thecomplex S-wave velocity of the structure. The first absolutevalue on the right-hand side of Eq. (17) indicates asignificant contribution of the resonance effect to inducedstrain in the structure. In the first resonance; the angularfrequency is o ¼ jpV�sst=ð2HstÞj, then the maximum shearstrain in resonance is expressed as

gmax ¼ 2E1=2s

�a�strstVs�3sst

�� 1=2. (18)

Because Es ¼ rV sðdu=dtÞ2 from Eq. (2), Eq. (18) can bealso written in terms of particle velocity, du/dt, as

gmax ¼ 2rV sðdu=dtÞ=rstV2sst. (19)

This equation clearly indicates that the maximum strainof the structure depends not only on the particle velocity ofthe ground but also on its impedance.

In reality, superstructures are not so simple as idealizedby uniform shear beams. They behave more like compli-cated mass-spring systems with limited lateral dimensionsand vibrate in shear-bending modes. However, it may bepossible to find equivalent parameters for the idealizationwhich basically satisfies Eq. (17). The induced strain thusevaluated from the energy flux can be compared with yieldstrain and correlated with different levels of structuralbehavior to be used for the performance-based design.

For brittle structures with small ductility, such asconcrete buildings with insufficient reinforcements, ma-sonry or brick buildings, etc., the maximum shear straininduced cycle by cycle is decisive for the failure of thestructure. Consequently, the energy flux becomes a keyvariable as indicated in Eq. (17) for the performance baseddesign. In contrast, for structures with higher ductilityfactors and higher damping ratios such as well-designedconcrete structures with enough shear walls, soil structures,dams, etc., damage by repeated strain cycles is essential forstructural performance. Therefore, the accumulated energyshould be used in place of the energy flux in designing suchstructures.

Based on the above considerations, it may be said thatthe energy, the degree of resonance and the impedanceratio between the structure and the ground are the threekey factors influencing the induced strain and the degree ofdamage in a given superstructure. Considering that theseismic wave energy is relatively small at the groundsurface in soft soil sites, it is unlikely that structural

damage due to inertia effect is always higher there than instiff soil sites. Instead, damage concentration seems to bedifferent from one earthquake to another depending on theenergy and other factors such as foundation soils andstructures including the degree of resonance.In the above discussions, the wave energy is defined at a

ground surface on which the superstructure is resting. Indesign practice, the earthquake input is sometimes definedat a base layer of stiffer soils. However, it is clear that thesurface energy can be easily correlated with the waveenergy at the base layer based on the 1D multi-reflectiontheory if soil linearity is assumed. Thus, no matter wherethe seismic wave energy is defined, either at the groundsurface or at the base layer, it can be related to the energygiven to a superstructure.Considering that the energy is connected to induced

strain or damage in superstructures, it is desirable to definea design seismic input not only by acceleration or velocitybut also by energy. According to Eq. (2) or Eq. (3), it canbe done simply by specifying the seismic impedance rVs ofa layer where the design wave is given. For instance,suppose the same velocity time history is given at two sites,A and B, where the same structure is resting. If theimpedance of A-site is twice as large as B-site, the surfaceenergy and the maximum strain of the structure during theresonance is twice larger in A-site than B-site assummarized in Table 1 based on Eqs. (18) and (19).One may say that, even current design practices specify a

couple of different design spectra corresponding todifferent soil conditions considering the energy conceptimplicitly. However, it will be more preferable to quantifydesign seismic input not only by acceleration or velocitybut also by energy. It can be done simply by specifying thesoil impedance where the motion is given. Furthermore, thequantification of energy enables different analytical resultsusing different design motions with different dominantfrequency, different duration, etc. to compare on the samescale of energy.

6. Conclusions

Energy flow analyses on strong motions of the 1995Kobe earthquake at 4 vertical array sites yielded thefollowing major findings:

ARTICLE IN PRESST. Kokusho et al. / Soil Dynamics and Earthquake Engineering 27 (2007) 354–366366

(1)
The ratio of the energy at the ground surface to theupward energy at the base is much lower than unity insoft soil sites with strong shaking. The dissipatedenergy in the surface layer amounts to 50–75% of theupward energy at the base in near-fault sites withstrong input motions, while it is around 20% or less ina distant site. The energy dissipation is particularlylarge in a site with extensive liquefaction.
(2)
In a site with a small shaking amplitude, a largepercentage of energy comes up to the ground surfacebecause the dissipated energy is small. In a near-faultsite with strong shaking, more than a half of theupward energy is dissipated by soil liquefaction andlarge damping in the surface layer and only a smallportion arrives at the surface, indicating that thesurface energy tends to be lower than the energycoming up from the base layer in soft soil sites withstrong shaking. This trend is particularly pronouncedwhen liquefaction occurs, indicating a significanteffect of base-isolation on liquefied ground.
Basic studies on stationary and transient response of a2-layers or multi-layers linear system disclosed;

(3)
If a wave propagates through layer boundaries in aninfinitely large medium, the wave energy alwaysdecreases because a part of the energy is transferredto the reflecting wave.
(4)
With increasing damping ratio in the surface layer, theenergy at ground surface decreases while the dis-sipated energy increases. The trends are very similar tothose actually observed in the vertical array sites.
(5)
The energy or energy flux at a ground surface which isexpected to increase due to resonance effect in asurface layer cannot become so large if large soildamping is exerted in soft soils. For more than 10%damping, the surface energy tends to be smaller thanthat at a base layer and decrease monotonically withdecreasing impedance ratio.
(6)
Based on 5) above, a general perception that soft soilsites are more prone to seismic damage than stiff soilsites may not always be true if seismic wave energy is adecisive factor for structural damage by seismic inertiaeffect.
(7)
In a multi-layers system, the energy coming up toground surface decreases with increasing number ofsubdivision of a surface layer because of wavereflections at sublayer boundaries.
Finally, a simple calculation on 2-layers modelsimulating a structure resting on foundation groundindicated:

(8)
Energy, resonance effect and impedance ratio betweenstructure and ground are the three key factors
controlling the induced strain in a given super-structure.

(9)
The current practice to define design seismic motionwithout clear specification of the impedance of the soillayer where the design motion is given is not adequate,because, for instance, the same velocity time historyresults in twice as large energy if it is designated at astiffer layer with its impedance twice as large as that ofa softer layer.
(10)
It is hence desirable to define design seismic inputs notonly by acceleration or velocity (represented either bya maximum value or by time histories) but also byenergy, or instead, simply to specify the impedanceratio rVs of a layer where the motion is given.
Acknowledgments

The Kansai Electric Power Company and the KobeMunicipal Office who successfully recorded the verticalarray data and CEORKA (The Committee for EarthquakeObservation Research in Kansai Area) who generouslydistributed them are gratefully acknowledged.

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wave energy in surface layers for energy-based damage ... · takaji kokusho , ryuichi motoyama,...

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