dynamic response of the equivalent shear beam (esb) container

Dynamic Response of theEquivalent Shear Beam (ESB) Container

S.P.G. Madabhushi

CUED/D-SOILS/TR270 (1994)

Synopsis

It is important to simulate the semi-infinite extent of soil in a dynamic centrifuge model test. An

Equivalent Shear Beam box was designed and manufactured at the Cambridge University with

flexible end walls which simulate the shear deformation of a horizontal soil layer subjected to a

model earthquake.

The dynamic response of this ESB box at different ‘g’ levels is investigated by conducting a

centrifuge test. Several earthquakes were fired on the empty ESB box at different ‘g’ levels. The

data from this test are presented in this report. A detailed analysis of the data suggests that the

ESB box amplifies a higher frequency component during an earthquake fired at 40g. This

higher frequency component increases in strength along the height of the end wall. However the

magnitude of this higher frequency component is small when earthquakes are fired at other ‘g’

levels namely 5Og, 6Og, 70g and 80g. The higher frequencies of vertical accelerations are

amplified at all the ‘g’ levels at which earthquakes were fired. The amplification of the peak

horizontal acceleration increased with the increase of ‘g’ level. The peak vertical accelerations

however were attenuated even 1 hough the higher frequency components of the vertical

accelerations were amplified.

Contents

Synopsis

1 .O Introduction

2.0 Design Principles

3.0 Use of ESB box in centrifuge tests

4.0 Dynamic Response of the ESB box

5.0 Instrumentation

5.1 Accelerometers

5.2 Displacement measunng devices

6.0 Positioning of the instruments

7.0 Data from the centrifuge tests

8.0 Analysis of the data from the centrifuge tests

8.1 Earthquake I fired at 40 g

8.2 Earthquake 2jired at 50 g

8.3 Earthquake 3fired at 70 g

8.4 Earthquake lfired at 80 g

8.5 Earthquuke Sfired at 60 g

9.0 Amplification Factors

10.0 Conclusions

REFERENCES

DYNAMIC RESPONSE OF THE

EQUIVALENT SHEAR BEAM (ESB) CONTAINER

1.0 Introduction

Use of centrifuge modelling in understanding the dynamic behaviour of complex soil-s&ucture

systems has established itself as a powerful technique over the past decade. The advances made

in the study of earthquake related problems in geotechnical engineering were discussed by

Steedman (1991). The principles of dynamic centrifuge modelling are now well understood and

the data acquisition techniques and signal processing methods have been standardised. Data

from the model tests have been used to validate numerical codes, Madabhushi and Zeng

(1993). However there is a need to relate the data obtained from the centrifuge tests to the

problems encountered in the field. One of the major concern of a practising engineer about the

dynamic centrifuge tests is the boundary conditions created by the model containers. Use of a

model container with rigid-smooth end walls will lead to model conditions which are

significantly different from the semi-infiite conditions of a soil layer in the field. These

differences between a horizontal soil layer in a rigid container and a semi-infiite soil layer in

the field arise from the following aspects;

i) strain dissimilarity,

ii) stress dissimilarity and

iii) generation of pressure waves.

The dynamic response of a soil layer can be obtained by treating the soil layer as shear beams.

The properties of soil, in general, will vary with the depth of the layer. The dynamic response

of such a soil layer will be the same along any vertical plane as shown in Fig. la. However, in a

centrifuge model there is no scope for deformation of soil near the rigid end walls as shown in

Fig.lb. This results in a strain dissimilarity between the centrifuge model and the prototype soil

layer in the field.

The stress dissimilarity arises from the smoothness of the end walls. The stress state in any soil

element in the soil layer is shown in Fig.2. If the end walls were to be smooth the shear stresses

are not borne at the boundary resulting in a stress dissimilarity. However, if the end walls were

to be rough this dissimilarity will not arise.

Finally the use of rigid wall containers will result in generation of pressure waves from the

sides as illustrated in Fig.3. When the rigid container is subjected to base shaking the end walls

will cause compression and rarefaction of soil in the vicinity of the boundary. This will result

in horizontally travelling pressure waves in the model in addition to the vertically propagating

horizontal shear waves from the base. In a prototype soil layer there will be no horizontal

pressure waves.

These problems caused by using rigid-smooth end wall containers lead to the concept of using

a flexible and frictional end wall containers. An ideal boundary for dynamic centrifuge model

tests should satisfy the following criteria;

i) The boundary must have the same dynamic stiffness as the adjacent soil.

ii) It must have the same friction w the adjacent soil.

iii) The model container must have infinite lateral stiffness during the spin-up of the centrifuge

so that K. condition can be maintained.

iv) The frictional end wall must have the same vertical settlement as the soil layer so that there

are no initial shear stresses induced at the boundary.

Based on these principles an Equivalent Shear Beam (ESB) model container was developed at

the Cambridge University, Zeng (199 1).

II

/

II1IIII

II

II

I

-1

IIIIIIII

II

fI

dI d

I II I

I I

Fig. 1 a Soil deformation in a semi-infinite half space

7

IIIIII 8II

r i g i d w a l l II

I

III

‘III

II ?

II

soil

II

layer :III

Fig. lb Soil deformation in a centrifuge model with rigid end walls

PrOtOtyp8

III

: soil layerII

II aD

II

rigid wall

rigid smooth boundary

P V

A-T sll-Gh

flexible rough boundary

Fig.2 Stress conditions in a soil element in the half space and in thevicinity of the rigid smooth boundary in a centrifuge model

soil layer

S waves

rigid wallP waves

S waves

soil layer rigid wall

Fig.3 Stress waves in a centrifuge model and in the half space

Design print Jples 1) same friction at the boundary

2) same dynamic stiffness

a) same deflection under seismic loadingb) same natural frequency

aluminium section

I ,’I I

prototype soil layer/ I

/ /

rubber layer

flexible boundary in the model

Fig.4 End wall which simulates the soil in the centrifuge model

2.0 Design Principles

The interaction between the end walls of the model container and the soil will be a minimum if

the end walls have the same dynamic stiffness and friction as the adjacent soil column during

the earthquake loading. This is illustrated in Fig.4. The stiffness of a saturated soil layer

however is dependent on the void ratio, the effective confining stress, shear strain amplitude

and the magnitude of the excess pore pressures generated during the earthquake loading. It is

not possible to have end walls with such variable stiffness. Instead the end walls of the model

container can be designed to match the stiffness of a soil layer with specific void ratio subjected

to a specific range of shear strains and confining pressures.

Hardin and Dmevich (1972) have suggested a hyperbolic stress-strain relationship for sands

given as

z=G, Y- . . . . . . . . . . (1)1+x

where Gmax is the initial tangent modulus, y, is the reference shear strain given by

zmax” G

= - . . . . . . . . . . (2)max

where Oman is the shear stress at failure. The maximum shear modulus for sands can be

obtained from the following empirical formula obtained by Hardin and Dmevich (1972) based

on laboratory test data.

G max = loo[;l;ee:’ {o;}“.” . . . . . . . . . . (3)

where ‘e’ is the void ratio and bmis the mean confining pressure in MPa. The tangent modulus

of soil (Gt) may be obtained by using Eq.l as

& [ 12

Gt =-&

= Gmx 1-z‘tmax

. . . . . . . . . . (4)

The stiffness of the soil at different depths is calculated using the above equations and the end

wall of the container is designed to have the same overall stiffness by building a section with

rigid dural rings and flexible rubber rings placed alternatively. The section is designed such

that the end wall will have the same shear deflection as a column of sand. The shear strain

induced in such a column of sand may be expressed using Eq. 1 as

zy =G

z . . . . . . . . . . (5)m a - -

Y,

For a particular earthquake the shear stress z can be calculated as

z = K/iysz . . . . . . . . . . (6)

where y is the unit weight of sand, Kh is the peak amplitude of acceleration and z is the depth

at which we seek to calculate the shear stress. The maximum shear stress depends on the initial

& state of the soil and the shear stress due to the earthquake and can be obtained from the

Mohr’s circles for initial and fmal stress states shown in Fig.5 From this figure we can derive

the following relation for maximum shear stress.

z max = ~Oyl+Ko)O~sincpl~-IO.5~l-Ko)O~. . . . . . (7)

where <p is the internal friction angle and the 0, is the effective stress. KnowingZ- the

reference strain y, is calculated using Eq.2. Thus the shear strain y at any depth can be

calculated in a similar fashion. Now the total shear displacement relative to the base can be

obtained as

6(z)=j-y(z)dz=j- z %fIiz . . . . . . . . . (8)H H Gmr-

Fig.5 Mohr circle for K. state and for failure state

The Cambridge Equivalent Shear Beam (ESB) box was designed for a thickness of soil layer of

1Om with a void ratio of 0.75 and a dry density of 15 kN/m3 and a peak acceleration of 30%.

The three dimensional view of the ESB box is presented in Fig.6. The shear displacement of the

end walls of the ESB box under design conditions listed above is shown in Fig.7. This shear

displacement must match the shear displacement of the soil as shown in Fig.& Performance of

the ESB box was confirmed by using the data from a centrifuge test in which the horizontal soil

layer was tested under design conditions, Schofield and Zeng (1992). In Fig.9 the observed

shear displacement and the calculated (design) shear displacements are shown.

3.0 Use of Equivalent Shear Beam (ESB) box in centrifuge tests

Several centrifuge tests were carried out using the equivalent shear beam box as the mod4

container, Zeng (1992), Madabhushi (1993). The ESB box was designed for specific ‘g’ level,

soil density, void ratio and earthquake strength as explained in earlier section. In some of the

tests cited above not all of these conditions could be satisfied. It is important to determine the

dynamic response of the Equivalent Shear Beam box in order to delineate the effects of the

boundary on the soil model during earthquake loading. This is particularly important when the

ESB box is used at different ‘g’ levels and with soil properties and earthquake magnitudes

which are different from the design values.

The natural frequency of the end walls of the ESB container may be determined by using the

Rayliegh’s method which involves equating the peak kinetic energy to the peak potential energy

of the system. Assuming a deflected shape of the end wall as

Ii = G(z)sin ant . . . . . . . . . . . (9)

where a, is the natural frequency. Differentiating Eq.9 we obtain the velocity function as

v(z,t) = ciLs(z)costiLt . . . . . . . . . . . (10)

giving the maximum kinetic energy of

(KE)max= ]0

o.5y(z)&z . . . . ...*..... (11)

The maximum potential energy is

I

metal sheet alued with sandA greased rubber sheet

/

6 6 04 1iT dural base plate

rubberm dural

+ l/4’ BSP pipe fing

Fig.6 A three dimensional view of the CambridgeEquivalent Shear Beam (ESB) container

8

9

100.002 0.004 0.006 0.000 0.01 0.012

she(ir displacement (m)

Fig.7 Calculated soil deformation with depth

end wall

soi! layer

shear deformation of soil column

.--w deformation of the end walls

Fig.8 Deformation of end walls and the soil deformation

.

working depthDf possible model

soil near boundary

-1

- prediction of dt,formation of sand layer

m recorded deflection of end wall

4 6 8 10 12deflection (mm)

Fig.9 Observed and estimated shear deformation of the end walls

(PE)max = j 0.5(2-y,)dz . . . . . . . . . . . . (12)0

Equating 11 and 12 the natural frequency an of the end wall is obtained. For the design values

of the ESB box the natural frequency of the wall was computed to be 130 Hz.

4.0 Dynamic response of the ESB box

In this report the data from a series of earthquakes fired on the Equivalent Shear Beam box at

different ‘g’ levels will be presented. In this centrifuge test, designated as MG- 1, the empty ESB

box was subjected to earthquakes so that the dynamic response of the ESB box can be

identified. The ESB box was instrumented using accelerometers and LVDT’s. The dynamic

response of the ESB box at each ‘g’ level will be established using the data from the centrifuge

test.

A total of five earthquakes were fired during this centrifuge test and are designated as EQ-1 to

EQ-5. In Table 1 each of the earthquake, its strength and the ‘g’ level at which it was fired are

presented.

Table 1 Strength of earthquakes

EQ Num he-r G Level

I

Pleak acceleration ( % g)

14.8

25.5

19.3

25.6

5.0 Instrumentation

5. I Accelerometers

In the centrifuge tests reported here miniature piezoelectric accelerometers manufactured by

D.J. Birchall were used to measure acceleration in the soil, on the model container and on the

block building models during eanhquakes. The device has a resonant frequency of about 50

kHzandamaximurn error of 5%. The weight of the transducer is about 5 grams. Fig.10

shows the dimensions of the transducer.

5.2 Displacement measuring device

The displacement of the model container and the block building models during centrifuge tests

was measured by linear variable differential transformers (LVDTs) manufactured by Sangamo.

The dimensions of the transducer is shown in Fig. 11. Each transducer weighs about 36 grams.

The short body has a sensitive range of 10 VDC. Since the transducer has a comer frequency

of 57.7. Hz, the response of this type of LVDT to high frequency signals is limited by the

inertia characteristics of the devices. In the model earthquakes the magnitude and phase of

high frequency cyclic displacements recorded by a LVDT of this type does not correspond to

the magnitude and phase of the actual cyclic displacement. However the lower frequency

components of displacements are 1:orrectly recorded so that the record of a LVDT transducer

during a model earthquake shows the mean displacements reasonably well and the difference

between readings before and after a model earthquake correctly reflects the overall

displacements as a result of an earthquake.

6.0 Positioning of the instruments

The positioning of the accelerometers and the LVDT’s is shown in Fig.12. The accelerometers

were placed at different heights on either side to measure the horizontal accelerations. Vertical

accelerations were measured at the base and also at the top of the end walls to identify any

rocking of the box. ACC’s 3441 and 1225 were placed at the same height but on either side of

wilt mm mass: 0.005 kg

Fig. 10 Dimensions of a miniature accelerometer

DFand DFa series

Core assembly

D&in bearing litred

to DFg series only

unit: mm mass: 0.036 kg

Fig. 11 Dimensions of a small stroke LVDT

3466

34411225

1572

Equivalent Shear Beam (ESB)Container

o 3436

0 1258o ACC

0 L V D TI I

Fig. 12 Placement of instruments in centrifuge test MG- 1

Location of the Instruments

TG$fpqTy;)I Y( )17;

I z(mm)205

34773457570157543466344112251572

-33-33-33-33625625625625

115115115115115115-115115

1601157025208160160120

3492 (v) -20 40 2251900 (v) 610 40 225

the longitudinal axis of the ESB box so that they will detect any twisting of the ESB box during

earthquake loading.

7.0 Data from the Centrifuge tests

All the data presented in this section were subject to a low pass digital filter to remove the high

frequency components superposed on the signals due to the electrical noise in the slip rings of

the centrifuge. The cut off frequency of the filter was 2000.0 Hz. The filter characteristics are

presented in Fig. 13.

The data from the centrifuge tests are presented in Figs.14 to 28. In these figures the magnitude

of acceleration is expressed as a percentage of the centrifugal acceleration at which the model

is being tested. The magnitude of tiisplacement is in ‘mm’. The analysis of the data is presented

in the next section.

0

- 5 0

-1000 500 1000 1500 2000 2500

Fig. 13 Filter characteristics

1280 data points plotted per complete transducer record

10 Max=l5.6

A C C 3 4 5 7 - o E.-ID

Min=-16.7

Max=17.110

ACC1926

Max=l7.3w

10 Mox=16.3

ACCI 225 0 E-w

Min=-17.7-20

[m-cl

Scales : Model

TEST MG-1E Q - 1 ~

SHORT TERM G Level FIG.NO.MODEL DRY TIME RECORDS 4 0 i4FLIGHT - 1


ACC3436

ACC 1258

ACC 1900

ACC3492

4 4 0 4 6 0 4 6 0 5 0 0 520 540 560 5 6 0 600 6 2 0 6 4 0

[msec]

4 4 0 4 6 0 4 6 0 500 5 2 0 540 560 560 6&l 6;O 6;O

[msec]

[ msec]

5

0s

I II

4 4 0 4 6 0 4 8 0 so0

I 1 - 5

520 540 5 6 0 560 600 620 6 4 0

[ msec]

Scales : Model

Max=l5.1

Mln=-15.7

Max=7.6

Min=-9.5

Max=3.4

Min=-3.0

Max=4.7

Min=-4.8

I

TEST MG-1 G Level FIG.NO.MODEL DRYFLIGHT - 1

EQ-11~SHORT TERM

TIME RECORDS 4 0 1 5

I


ACC3436

ACC 1572

ACC5701

LVDT984

LVDT99 1

LVDT6257

I20 30 40 5 0 5n 7 0 60 90 loo 1 1 0 1 2 0 130 1 4 0 1 5 0 1 6 0 1 7 0 160 1 9 0 zoo 210

[msec]

1 0 Mox=14.7

0 E

-10Mln=-14.3

) 20

*fiw*,

Uax=16.6

.o E

Min=-18.5.-20

io ;o 4i io so io w SC1 1w 1 1 0 1 2 0 1 3 0 1 4 0 1x1 1w 170 1 5 0 1 9 0 200 210

[msec]

20 a0 40 50 50 70 60 So loo 1 1 0 120 130 140 154 150 170 150 190 2w 210

[msac]

Im 40 w &a 1 0 0 1 2 0 1 4 0 1 6 0 1 5 0 200

[msec]

m 40 50 50 loo lm 1 4 0 1 5 0 180 206

[msec]

0 . 0 5

Mox=O.l

0T

L

- 0 . 0 6

Ml=-0.1

I Jm 40 w 50 loo 1 2 0 1 4 0 1 5 0 180 200

[msec]

Scales : Model/ i

TEST MG-1EQ-11

SHORT TERM G Level FIG.NO.

MODEL DRY TIME RECORDS 4 0 1 6FLIGHT -1 I


ACC3436 .

.?D

ACC3457 .o z

.-a

b-1

ACC344

[m-cl

Max=25.5

Mill=-24.9

Mill=-13.0

Mox=27.9

Mln=-28.0

Mow=28.9

Min=-29.5

Max=30.0

Mill=-30.1

Max=32.2

Min=-31.7

Min=-29.8

Max=30.9

Mln=-33.0

uax=31.1

Mln=-27.7

Scales : Model

TEST MG-1 SHORT TERM G Level FIG.NO.MODEL DRY EQ-2;

TIME RECORDS 5 0 17FLIGHT - 1


AK3436 0 Yz

500 5 i o 5 4 0 560 5so 6 0 0 . - - 6 2 0

[ m s e c ]

ACC 1258

500 520 540 560 Se0 600 Go

[ m s e c ]

5

2.6

ACC 1900 r0 -

- 2 . 5

I4 0 0 4 2 0 4 4 0 4 6 0 4 6 0 500 520 5 4 0 5 6 0 560 600 620

[ m s e c ]

5

ACC3492 z

0-

I4 0 0 4 2 0 440 4 6 0 4 6 0 Jo0 520 5 4 0 5 6 0 560 600 620

[ m s e c ]

Mox=25.9

Mln=-24.4

MmG13.3

Min=-12.5

Max=5.0

Mb=-3.6

Max=6.6

Min=-3.9

Scales : Model

TEST MG-1 SHORT TERM G Level FIG.NO.MODEL DRY E Q - 2 ’

TIME RECORDS 5 0 18FLIGHT -1


20 Mox=25.4

ACC3436,* uz

I -20 MI=-24.7I J

20 M 40 50 6 0 70 60 90 loo 1 1 0 1 2 0 124 1 4 0 1 5 0 1 6 0 1 7 0 164 1 9 0 200 210[msec]

20 Max=33.4

ACC15720 E- 2 0

Mln=-35.7-40

20 30 40 5 0 60 7 0 60 so 100 1 1 0 1 2 0 130 I40 1 5 0 160 1 7 0 1 6 0 190 2w 210[msec]

‘20

*/yl$yM.Max=29.0

ACC570 1.o 3

-20 Mln=-27.6

20 30 40 54 60 70 80 90 loo 1 1 0 1 2 0 1.50 1 4 0 150 1 6 0 1 7 0 160 1 9 0 2 0 0 210[moec]

0.05 Max=O.l

LVDT9840 7

L

-0.06 Mill=-0.1I ,

20 40 60 60 loo 1 2 0 1 4 0 1 6 0 1 6 0 200[msec]

LVDT99

0.05 Mox=O.l

1 0 72

-0.05Min=-0.1

20 40 w 60 loo 1 2 0 1 4 0 1 6 0 1 6 0 200[msec]

Mox=O.l0.05

LVDT6257 70 z

Min=-0.1-0.05

20 40 6i so loo 1 2 0 1 4 0 1 6 0 160 200

[msec]

Scales : Model

TEST MG-11

MODEL DRY EQ-21SHORT TERM G Leve l FIG.NO.

TIME RECORDS 5 0 1 9FLIGHT -1

!!I


h-cl

ACC3457

A C C 3 4 7 7 .

ACC

A C C 3 4 6 6

.-7.0

10 IO s IO

[mrec]

Scales : Model

Min=-17.5

Max=25.4

Ml=-25.3

Max=25.7

Min=-27.5

Mar=24.3

Min=-25.5

Max=24.6

l

Mb=-26.9

Mm=30.6

Mln=-26.5

TEST MG-1 I SHORT TERM G Leve lEQ-31’

FIG.NO.MODEL DRY / TIME RECORDS 7 0 2 0

FLIGHT -1 I


ACC3436i

ACC1258

ACC1900

ACC3492

. - 1 0

3 4 0 3 6 0 36a 400 4 2 0 4 4 0 4 6 0 480 5 0 0 5 2 0

[msec]

I 1

340 S60 360 4 0 0 420 4 4 0 460 460 500 5 2 0

[msec]

2

027

- 2I I I

340 360 3 6 0 4 0 0 420 4 4 0 4 6 0 4&J 500 5 2 0

[msec]

I

3 4 0 3 6 0 360 400 4 2 0 4 4 0 4 6 0 4 6 0 5 0 0 5 2 0

[ msec]

2

OZ

- 2

Max=1 6.6

Ml=-17.0

Mox=B. 1

Min=-3.6

&x=2.7

MI=-2.6

t&x=2.9

Min=-3.2

Scales : Model

TEST MG-1E Q - 3 SHORT TERM G Level FIG.NO.

MODEL DRYI TIME RECORDS 70 2 1

FLIGHT - 1


.20

ACC3436 CM+Mox=19.5

1 0

.20 Mox=27.4

ACC1572.o E

. -20 MIn=-27.8

1 0 20 30 40 50 50 70 50 90 1 0 0 1 1 0 1 2 0 130 1 4 0 150

/

'20Max=Zl.4

ACC570 1

'-20 Min=-21.3

1 0 20 30 40 50 50 70 50 90 too 1 1 0 1 2 0 1w 140 150[msac]

0.05 Yax=O.l

LVDT9840 7

3

I . . Il0 20 30 40 ' 50 50 70 50 90 100 1 1 0 1 2 0 130 1 4 0 1 5 0

[msec]

- 0 . 0 5Ml=-0.1

0 . 0 5Uax=O.l

LVDT99 10 T

L- 0 . 0 5

Ml=-0.1

1 0 20 30 40 50 70 80 SO 100 ii0 Go li0 1 5 0[msac]

0 . 0 5 Mox=O.l

LVDT6257 Y0 L

-0.05 Mln=-0.1

1 0 2 0 so 40 / 5 0 So 70 80 PO 100 1 1 0 t20 130 MO 150[msec]

Scales : Model

TEST MG-1!

MODEL DRY EQ-31 SHORT TERM G Leve l FIG.NO.

FLIGHT -1 TIME RECORDS 7 0 22

II


ACC3436

Em-1

I

ACC1258 o E-0

ACC344

Max=25.5

Mb=-25.6

Max=7.6

Min=-9.6

Max=29.6

tan=-32.4

Mox=36.6

Min=-35.5

&x=43.6

Min=-37.6

Max=42.6

Min=-42.8

t&x=40.2

Min=-39.9

Max=41.5

urn=-41.2

Maw=44.1

MY=-45.6

Scales : Model

I ~

T E S T M G - 1MODEL DRY EO-41’

SHORT TERM G L e v e l FIG.NO.

I TIME RECORDS 8 0 ,23

FUGHT -1


ACC3436

ACC 1258

ACC 1900

ACC3492

2 0

0 g

- 2 0

I I3 0 0 3 2 0 3 4 0 360 360 400 4 2 0 440 4 6 0 4 6 0 500

[ m s e c ]

400

[msec]

400

[ msec]

- 2 . 5 z

-5

4 0 0

[ msec]

4 2 0 4 4 0 4 6 0 4 6 0 5 0 0

Scales : Model

Mox=26.3

Mb=-26.1

Mox=6.7

Min=-6.1

Mox=2.9

Mill=-3.0

Max=l.6

Mln=-6.5

TEST MG-1 SHORT TERM G Leve l FIG.NO.MODEL DRY E Q - 4 ’

TIME RECORDS 8 0 2 4

FLIGHT -1 I!I


20 MClX=ZS.S

ACC34360 z

-20 Mln=-25.5

1 0 20 30 40 50 5 0 70 I30 90 104 1 1 0 1 2 0 1 3 0 1 4 0 1 5 0[msec]

IiMox=43.5

'25ACC1572

.o E

-25Mln=-42.8

l

1 0 20 30 40 50 90 7 0 00 90 1 0 0 1 1 0 1 2 0 1% 1 4 0 1 5 0[msec]

.20 Max=33.1

ACC5701.o ur

. - 2 0Mln=-32.7

c tI1 0 20 so 40 50 90 70 w 90 1 0 0 1 1 0 1 2 0 1 3 0 1 4 0 1 5 0

[msec]

0 . 1Mox=O.l

LVDT9840 7

3

- 0 . 1 Mln=-0.1

1 0 20 30 40 50 50 70 110 90 loo 1 1 0 1 2 0 1 3 0 1 4 0 1 5 9[msec]

T E S T MG-I G Leve l FIG.NO.MODEL DRY EQ-4 1 SHORT TERM

TIME RECORDS 8 0 25FLIGHT - 11; 1I

LVDT99

0.1Yax=O.l

1 0 73

- 0 . 1 Mill=-0.1c

1 0 20 30 40 50 90 70 80 90 1 0 0 1 1 0 120 1 3 0 1 4 0 150

[msec]

0 . 0 5 Max=O.l

LVDT62570 -2

b

-0.05 Mln=-0.1

[msec]

I I 1Scales : Model

I I

1 1 5 2 d a t a p a i n t s p l a t t e d p e r c o m p l e t e t r a n s d u c e r r e c o r d

*.a

ACC 1258 0 z-*.o

to

ACC3477 0 z

.-to

‘0

ACC1225 o E.-lo

[m=l

Mill=-15.7

Maw=12.5

Mill=-13.6

Mox=13.4

Mln=-15.6

Mox=14.5

Min=-16.4

Max=13.4

Min=-15.2

Mm=l3.6

Mill=-15.7

Max=l3.3

MT=-16.3

Scales : Model

TEST MG-1EQ-5i

SHORT TERM G Level FIG.NO.MODEL DRY

I TIME RECORDS 6 0 26FLIGHT -1


. 10 Mox=lS.l

A C C 3 4 3 6 - .o z

v -. -10

t&l=-13.7

360 380 400 420 u o 460 460 500 520 540

[ msec]

5

2.5Max=4.6

ACC 1258 E0

-2.5WI=-3.6

360 380 400 420 440 460 480 500 520 540

[msec]

2

Mox=2.1

ACC 1900 0g

- 2 Mill=-2.0

360 300 400 420 440 480 460 500 520 640

[ msec]

2

Max=l.6

ACC34920

z

- 2

MI=-2.7

360 360 400 420 440 460 480 500 520 540

[msec]

Scales : Model

TEST MG-1 SHORT TERM G Level FIG.NO.MODEL DRY E Q - 5

/ TIME RECORDS 6 0 2 7

FLIGHT -1 I

1152 daita points p lot ted per complete t ransducer record

ACC3436

30 40 50 50 70 80 90 100 1 1 0 1 2 0 134 1 4 0 1 6 0 190 1 7 0 1 8 0 1 9 0 am

ACC 1572

ACC570

-10Ml=-14.1

[msac]

1 0 Max=lS.O

.OE

-10Mln=-16.7

I 130 40 so 90 70 60 90 1w 110 1 2 0 124 1 4 0 1 5 0 1 9 0 1 7 0 15x3 1 9 0 200

[msec]

. 10 Mox=12.5

1.O

z

-10Mln=-14.2

J

30 40 5 0 50 7@ 90 90 1 0 0 110 1 2 0 130 1 4 0 150 190 1 7 0 150 190 200

[msec]

0 . 0 5 Yax=O.l

LVDT984 70 3

kiln=-0.1- 0 . 0 6

LVDT99 1

I50 40 50 90 70 50 SO l o o 1 1 0 1 2 0 130 1 4 0 1w 1 9 0 (70 180 1 9 0 200

30 40 50 90 70 643 90 l o o 1 1 0 1 2 0 1 3 0 1 4 0 1 5 0 150 1 7 0 1 6 0 1 9 0 200

[msec]

r

LVDT6257

[ m s e c ]

I30 40 50 60 70 60 90 l o o 110 1 2 0 1 3 0 1 4 0 190 1 9 0 1 7 0 $60 1 9 0 200

[msec]

Scales : Model

TEST MG-1I I

MODEL DRY EQ-5’1SHORT TERM G Level FIG.NO.

TIME RECORDS 6 0 28FLIGHT - 1

/II

8.0 Analysis of the data from thy: centrifuge tests

As explained earlier each of the earthquakes were fved on the empty ESB box at a different ‘g’

levels so that the dynamic performance of the ESB box can be measured. The ESB box was

extensively instrumented using mmiature accelerometers and LVDT’s as explained earlier. In

this section the frequency analysels of the traces recorded during these centrifuge tests will be

presented. From these analyses the performance of the ESB box will be determined. In Table 1

the list of earthquakes and the ‘g’ level at which they were fired was presented. The analyses of

the data will be presented in the same order.

8.1 Earthquake 1 fired at 40 g

The data from this earthquake are presented in Figs.14 to 16. The maximum acceleration

recorded by the base accelerometer 3436 during this earthquake was 14.8 8. The strength of

the vertical acceleration was 12.2 % which is relatively large when compared with the

horizontal acceleration. The base shaking is predominantly at a single frequency with most of

the energy of the earthquake conce.ntrated at 87.5 Hz. This uniform base shaking is recorded by

all of the accelerometers located at the base of the ESB box, for example as indicated by the

traces recorded by ACC’s 3436, 5’154, 1572 and 5701 in Figs.14 and 16. The location of these

transducers is given in Fig.12. However there is a second higher frequency component in the

traces recorded by accelerometers placed at a distance above the base of the ESB box. For

example consider the traces in Fig.14 recorded by ACC’s 3457, 3477 and 1926 on the right

hand side end wall of the ESB box and ACC’s 3466, 3441 and 1225 on the left hand side end

wall of the ESB box (see Fig.12 for location of these transducers). By comparing the traces of

ACC’s 3457, 3477 and 1926 in the above figure it can be seen that the strength of the high

frequency component increases with the increase of the height at which the accelerometer is

placed. The frequency analyses of the base accelerometer 3436 and one of the accelerometers

at the top of the end wall ACC 1225 together with the time traces are presented in Fig.29. It

can be seen from this figure that the response of the ESB box at 265.0 HZ is significantly

amplified. It may be concluded from these observations that the ESB box has a resonant

frequency of 265 Hz when subjectfed to model earthquakes at 40 g.

In Fig.30 the vertical acceleration recorded at the base of the ESB box by ACC 1258 and at the

top of both the end walls by ACC 1900 and 3492 are presented (see Fig.12 for the location of

the transducers). The corresponding frequency analyses of each of these traces is presented on

the right hand side of this figure. The main vertical frequency of the model earthquake at this ‘g’

level is at 252.0 Hz. This frequency is reflected in the traces recorded at the top of the end

walls. The high frequency components are significantly amplified as indicated by the frequency

analysis of ACC’s 1900 and 3492. Also the frequency content of these accelerometers on either

of the end walls does not match suggesting the rocking of the ESB box.

The LVDTs did not record any significant movement of the end walls during this earthquake.

data paints platted per complete transducer record

20

E 0

-20

20 40 60 80 100 120 140 160 180

[ m s e c ]

1.5

iig 1

3$ 0.5x

0 100 200 300 400 500 600 700 800

[Hz1

1

20 - A C C 1 2 2 5

ij o-.

-20.

20 40 60 80 100 120 140 160 180

[msec]

2

1.5Gc5

1D.$s= 0.5

0 100 200 300 400 500 600 700 800

P-w

Scales : Model

TEST FIG-NO.MODEL

MG-1 1’m-1 i TIME RECORDS 29

FLIGHT __~. .-

data paints plotted per complete transducer record

0 . 7 5 -

n 3 0.5 -m 0 c

54go.25 -I I

- 1 0 , . ,, , .-5 0 0 600 0 0.25 0.5 0 . 7 5

[msec] [kHzl

0 . 2

92+j 0.12.$xI

4 5 0 5 0 0 5 5 0 6 0 0 0 0.25 . 0 . 7 5 1[msec] [k”H” ]Z

5 0 . 2

Gc5

7303-g 0.1I

- 5

4 5 0 5 0 0 5 5 0 6 0 0 0 0.25 . 0 . 7 5 1

[msec] [k”H” ]Z

Scales : Model

TEST ~_, FIG.NO.MODEL M G - 1

m-1 TIME RECORDS 3 0FLIGHT



recorded by the base accelerometer 3436 during this earthquake was 25.5 8. The peak strength

of the vertical acceleration was 19.4 % which is relatively large when compared with the

horizontal acceleration. The base ,&king is predominantly at a single frequency with most of

the energy of the earthquake concentrated at 96.9 Hz. This uniform base shaking is recorded by

all of the accelerometers located at the base of the ESB box, for example as indicated by the

traces recorded by ACC’s 3436, 5’154, 1572 and 5701 in Figs.17 and 19. The location of these

transducers is given in Fig.12. There is a second higher frequency component in the traces

recorded by accelerometers placed at a distance above the base of the ESB box. For example

consider the traces in Figs.17 recorded by ACC’s 3457, 3477 and 1926 on the right hand side

end wall of the ESB box and ACCs 3466,344l and 1225 on the left hand side end wall of the

ESB box (see Fig.12 for location of these transducers). By comparing the traces of ACc’s

3457, 3477 and 1926 in the above: figure it can be seen that the strength of the high frequency

component increases with the inc:rease of the height at which the accelerometer is placed.

However, the magnitude of this higher frequency component is much smaller than the one

recorded in the 40 g earthquake. The frequency analyses of the base accelerometer 3436 and

one of the accelerometers at the top of the end wall ACC 1225 together with the time traces are

presented in Fig.31. It can be seen from this figure that the response of the ESB box at 294.0

Hz has no significant amplification.






walls. The high frequency components are significantly amplified as indicated by the frequency

analysis of ACC’s 1900 and 3492. Also the frequency content of these accelerometers on either

of the end walls is similar suggesting there is very little rocking of the ESB box.

The LVDT’s did not record any significant movement of the end walls during this earthquake.

data points plotted per complete transducer record

20 A C C 3 4 3 6

-07 0

20 40 60 80 100 120 140 160 180 200

[msec]

0 100 200 300 400 500 600 700

[Hz]

20 -ACC 1225

-20 -

4-

3-

2-

1 -

20 40

I

$0 80 100 120 IdO 160 180 200

[ msec]

0 100 200 300 400 500 600 700

P-4

Scales : Model

TESTMG-1

FIG.NO.MODEL EQ-2FLIGHT TIME RECORDS 3’ :;.&,i

c

10

5

i--l0 0-


ACC 1258

[ msec]

I ACC 1900

[ m s e c ]

10 -

ACC 3492

tr: 0I

- 5

- 1 0

500

[ msec]

0.3

& 0.25d3Cii 0.1s

0

G 0.4UI2

3.$

H 0.2

[kw

0.75 1

Scales : ModelI

TEST--.

FIG.NO.MODEL

: MG-1 i’ EQ-2 I I TIME RECORDS 3 2

FLIGHT .



recorded by the base accelerometer 3436 during this earthquake wa.s lg.3 70. me p& s&en@

of the vertical acceleration was 8.8 8. This is relatively small compared to the horizontal

acceleration. The base shaking is predominantly at a single frequency with most of the energy

of the earthquake concentrated at 115.7 Hz. This uniform base shaking is recorded by all of the

accelerometers located at the base of the ESB box, for example as indicated by the traces

recorded by ACC’s 3436, 5754, 1572 and 5701 in Figs.20 and 22. The location of these

transducers is given in Fig.12. There is a second higher frequency component in the traces

recorded by accelerometers placed at a distance above the base of the ESB box. For example

consider the traces in Fig.20 recorded by ACC’s 3457, 3477 and 1926 on the right hand side

end wall of the ESB box and ACC’s 3466,344l and 1225 on the left hand side end wall of the

ESB box (see Fig.12 for location of these transducers). By comparing the traces of ACC’s

3457, 3477 and 1926 in the above figure it can be seen that the strength of the high frequency

component increases with the increase of the height at which the accelerometer is placed.

However, the magnitude of this higher frequency component is much smaller than the one

recorded in the 40 g earthquake. The frequency analyses of the base accelerometer 3436 and

one of the accelerometers at the top of the end wall ACC 1225 together with the time traces are

presented in Fig.33. It can be seen from this figure that the response of the ESB box at 350.0

Hz has no significant amplification.






walls. However the large component of vertical acceleration recorded by ACC 1258 at 240 Hz

is not reflected in the traces recorded by ACc’s 1900 and 3492. The high frequency

components are significantly amplified as indicated by the frequency analysis of ACC’s 1900

and 3492. Also the frequency content of these accelerometers on either of the end walls does

not match suggesting there is rocking of the ESB box.

The LVDTs 984 and 991 have recorded a small movement during the earthquake. However the

lowest LVDT 6257 did not move during this earthquake confkning that the lowest Dural ring

of the ESB box has no relative movement with respect to the base.


20 A C C 3 4 3 6

20 30 80 90 100 110 120 130 140 150

[msec]

0 100 200 300 400 500 600 700

[Hz]

20 A C C 1 2 2 5

z 0

-20

20 30 40 50 60 70 80 90 100 110 120 130 140 150

b0 100 200 300 400 500 600 700 800

[Hz]

c

Scales : Model

TEST M G - 1FIG.NO.

MODEL ; E Q - 3 TIME RECORDS3 3

FLIGHT -1

7s 0

- 1 0

5

EO

- 5

*


ACC 1258

400 500

[ m s e c ]

ACC 1900

350 400 450 500

[ m s e c ]

ACC 3492I

hi

0.4

';;ic53 0.2.zBI

r

0.3

F 0.2c53.$

f0. 1 -

l-

350 400 450 500

[ msec]

Scales

0

: Model

I I I

10.5

w-w

0.25

I I

TEST-_ FIG.NO.

MODELMG-1 ; ;2EQ-3 F TIME RECORDS 3 4

FLIGHT %I

8.4 Earthquake 4fired at 80 g

Many centrifuge tests in past were conducted at 80 g using the ESB box, &ng (1991),

Wdabhushi (1992), etc.,. The data from the earthquake fued at 80 g on the ESB box are

presented in Figs.23 to 25. The maximum acceleration recorded by the base accelerometer

3436 during this earthquake was 25.5 %. The peak strength of the vertical acceleration was 9.8

%. This is relatively small compared to the horizontal acceleration. The base shaking is

predominantly at a single frequency with most of the energy of the earthquake concentrated at

121.9 Hz. This uniform base shaking is recorded by all of the accelerometers located at the

base of the ESB box, for example as indicated by the traces recorded by ACC’s 3436, 5754,

1572 and 5701 in Figs.23 and 25. The location of these transducers is given in Fig.12. There is

a second higher frequency component in the traces recorded by accelerometers placed at a

distance above the base of the ESB box. For example consider the traces in Figs.23 as recorded

by ACC’s 3457, 3477 and 1926 on the right hand side end wall of the ESB box and ACC’s

3466, 3441 and 1225 on the left hand side end wall of the ESB box (see Fig.12 for location of

these transducers). By comparing the traces of ACC’s 3457,3477 and 1926 in the above figure

it can be seen that the strength of the high frequency component increases with the increase of

the height at which the accelerometer is placed. However, the magnitude of this higher

frequency component is much smaller than the one recorded in the 40 g earthquake. The

frequency analyses of the base accelerometer 3436 and one of the accelerometers at the top of

the end wall ACC 1225 together with the time traces are presented in Fig.35. It can be seen

from this figure that the response of the ESB box at 377.0 Hz has no significant amplification.





level is at 35 1.1 Hz. This frequency is reflected in the traces recorded at the top of the end

walls. There is a DC drag in the traces recorded by ACC’s 1258 and 3492. This may be due to

a drag on the amplifier bank and may not indicate a real event. However this downward drag

introduces a large component at rhe zero frequency mark in the frequency analyses of these

accelerometers. The high frequency components are significantly amplified as indicated by the

frequency analysis of ACC’s 1900 and 3492. Also the frequency content of these

accelerometers on either of the end walls does not match suggesting there is rocking of the ESB

box.

The LVDT’s 984 and 991 have recorded a movement of about 0.1 mm during this earthquake.

However the lowest LVDT 6257 did not move during this earthquake confirming that the

lowest Dural ring of the ESB box has no relative movement with respect to the base. The

displacement traces recorded by LVDT’s 984 and 991 are in phase suggesting that the end

walls are vibrating in their fundamental mode.


25 A C C 3 4 3 6

G 0

-25

20 30 40 50 60 70 80 90 100 110 120 130 140 150

[ msec]

6

B

2 ,

D.;ii

92I

o c

u-4

40

20

E 0

-20

-40 "

20 30 40 50 60 70 80 90 100 110 120 130 140 150

rmw1

10 -T

2

3.e 5.hs

0 100 200 300 400 500 600 700

[Hz]

Scales : Model

TEST e FIG.NO.MODEL

‘?hGml --:,

1 EQ-4, TIME RECORDS 3 5FLIGHT


0 . 6

[ m s e c ]

3 5 0 4 0 0 4 5 0

[ msec]

ACC 349:

5

0 . 2

i;ic53 0.1.$8I

1

~ 0 . 7 5Y)2

0 . 2 5

3 5 0 4 0 0 4 5 0

[ msec]

Scales : Model

0 . 2 5

r

0 0.25 0.5 0 . 7 5

lwl

TEST MG-1 3 FIG.NO.MODEL EQ-4 TIME RECORDS 36FLIGHT

8.5 Earthquake Sfired at 60 g

The bumpy road actuator produces a non uniform earthquake at 60 g. The edquake is

ch~mxis~ by small amplitude cycles preceded and followed by normal cycles. This feature

of the bumpy road is discussed in detail by Madabhushi (1991). The data from the e&quake

fired at 60 g on the ESB box are presented in Figs.26 to 28. The maximum acceleration

recorded by the base accelerometer 3436 during this earthquake was 14.0 %I. The peak strength

of the vertical acceleration was 4.5 8. This is relatively small compared to the horizontal

acceleration. The base shaking is predominantly at a single frequency with most of the energy

of the earthquake concentrated at 100.0 Hz. The base shaking at this frequency and the small

amplitude cycles in the middle of the earthquake are recorded by all of the accelerometers

located at the base of the ESB box, for example as indicated by the traces recorded by AK’s

3436, 5754, 1572 and 5701 in Figs.26 and 28. The location of these transducers is given in

Fig. 12. There is a second higher frequency component in the traces recorded by accelerometers

placed at a distance above the base of the ESB box. For example consider the traces in Fig.26

recorded by ACC’s 3457, 3477 and 1926 on the right hand side end wall of the ESB box and

ACC’s 3466, 3441 and 1225 on the left hand side end wall of the ESB box (see Fig. 12 for

location of these transducers). By comparing the traces of ACC’s 3457, 3477 and 1926 in the

above figure it can be seen that the strength of the high frequency component increases with the

increase of the height at which the accelerometer is placed. However, the magnitude of this

higher frequency component is much smaller than the one recorded in the 40 g earthquake. The

frequency analyses of the base accelerometer 3436 and one of the accelerometers at the top of

the end wall ACC 1225 together with the time traces are presented in Fig.37. It can be seen

from this figure that the response of the ESB box at 3 10.0 Hz has no significant amplification.




the right hand side of this figure. There is no clear main vertical frequency of the model

earthquake at this ‘g’ level apart from the one at driving frequency of the earthquake. The high

frequency components are signitkantly amplified as indicated by the frequency analysis of

ACC’s 1900 and 3492. Also the frequency content of these accelerometers on either of the end

walls does not match suggesting there is rocking of the ESB box.

The LVDT’s did not record any significant movement of the end walls during this earthquake.

9.0 Amplification of the input motion

The horizontal accelerations at the top of the end walls recorded during each of the earthquakes

had some amplification compared to the input acceleration. The amplification factor for each

earthquake is calculated as a ratio of the peak horizontal acceleration recorded at the top of the

end wall to the peak input horizontal acceleration. The accelerometers 1926 and 5754 placed at

the top and bottom of the end wall on the right hand side of the ESB box (see Fig.12) were

used in the above computation. In ‘Table 2 these amplification factors are presented.

Table 2 Amplification Factors for Horizontal Acceleration

Earthquake

EQ-1

EQ-2

EQ-3

G Level

40 g

5og

7og

Amplification Factor

1.1429

1.1541

1.2851

EQ-4 8Og 1.3210

EQ-5 60 g 1.0446

The amplification factors increase with the increase of ‘g’ level with the maximum amplification

occurring at 80 g.

A similar calculation is done for the vertical accelerations. In the case of vertical accelerations

there is an attenuation of the peak input vertical acceleration though the smaller high frequency

components are amplified. The attenuation factor is calculated as the ratio of peak vertical

20

E 0

-20


40 60 80 120

[ msec]

140 160 180

Gm2 1.5-

Ti l-.=sr" 0.5 -

-A-

100 200 300 400 500 600 700

[Hz10

20

s 0

-20

A C C 1 2 2 5

I2-

40 60 80 100 120

rmsnrl

140 160 180

Scales : Model

__--TEST

MODEL’ MG-I ’ FIG.NO.

EQ-5 1 TIME RECORDS37

FLIGHT


5

SO

-EiJ

n0I

10

5

0

-5

-10

ACC 1258

4 0 0 4 5 0 5 0 0 5 5 0

[ m s e c ]

ACC 1900

1 1 1 1

4 0 0 4 5 0 5 0 0 5 5 0

[ m s e c ]

ACC 3492

I I I I

4 0 0 4 5 0 5 0 0 5 5 0

[ msec]

Scales

0.2

GcL4cE+ 0.1ii

0.25

0 0.25 0.5 0.75 1

w-w

Model

0.25

TEST - -MG-1 I FIG-NO.

MODELEQ-5 TIME RECORDS 36

FLIGHT

acceleration recorded at the top of the end wall to the peak input vertical acceleration. The

accelerometers 3492 and 1258 placed at the top of the end wall and on the base of the ESB box

(see Fig.12) were used in the above computation. In Table 3 these attenuation factors are

presented.

Table 3 Attenuation Factors for Vertical Acceleration

Earthquake

EQ-1

EQ-2

EQ-3

EQ-4

EQ-5

G Level

4og

50 g

7og

80 g

60 g

Attenuation Factor

0.5053

0.4962

0.8421

0.8025

0.75

The vertical accelerations are attenuated the most in the 40g and 50g earthquakes. The

attenuation decreases at 70g and 80g. It is interesting to note that the attenuation at 80g is

greater than at 70g.

10.0 Conclusions

It is important to simulate the semi-infinite extent of soil in a dynamic centrifuge model test. An

Equivalent Shear Beam box was designed and manufactured at the Cambridge University with

flexible end walls which simulate the shear deformation of a horizontal soil layer subjected to a

model earthquake. The ESB box was designed to match the shear deformation of a 200 mm

soil layer subjected to a 30% earthquake at a centrifugal acceleration of 50g. However all these

design conditions may not be satisfied in a particular centrifuge test. The performance of the

ESB box at different ‘g’ levels must be known so that it can be used in any practical centrifuge

test series.

The dynamic response of this ESB box at different ‘g’ levels is investigated by conducting a

centrifuge test. Several earthquakes were fired on the empty ESB box at different ‘g’ levels. The

data from this test are presented in this report. A detailed analysis of the data suggests that the

ESB box amplifies a higher frequency component during an earthquake fired at 40g. This

higher frequency component increases in strength along the height of the end wall. However the

magnitude of this higher frequency component is small when earthquakes are fired at other ‘g’

levels namely 5Og, 6Og, 70g and 8Og. The higher frequencies of vertical accelerations are

amplified at all the ‘g’ levels at which earthquakes were fired. The amplification of the peak

horizontal acceleration increased with the increase of ‘g’ level. The peak vertical accelerations

however were attenuated even 1 hough the higher frequency components of the vertical

accelerations were amplified.

REFERENCES

Hardin, B.O. and Drnevich, V.P., ( 1972), Shear modulus and damping in soils: designequations and curves, Jnl. of Soil Mech. and Found. Eng.Div., ASCE, No.SM7, pp.667-692.

Madabhushi, S.P.G., (1991). Response of tower structures subjected to earthquake pertur-bations’, Ph.d thesis, Cambridge University, England.

Madabhushi, S.P.G. and Zeng, X., (1993), An analysis of the seismic behaviour ofquay walls, Proc. Verification of Lilquefaction Analyses by Centrifuge studies,(VELACS), Vol.II., K.Arulanandan and R.F.Scott (Editors), Davis, Caliiornia.

Schofield, A.N. and Zeng, X., (1992), Design and performance of an equivalent-shear-beam(ESB) container for earthquake centrifuge modelling, Technical Report, CUED/D-Soils/I’R245, Cambridge University, England.

Steedman, R.S., (1991), Centrifuge modelling for dynamic geotechnical studies, Second IrkConf. on Rec.Adv. in Geo. and Earthquake Eng. and Soil Dynamics, St.Louis, Missouri, USA.

Zeng, X., (1991), Design of the Cambridge Dynamic Shear Box, Technical Report, CUED/D-Soils/TR241, Cambridge University, England.

dynamic response of the equivalent shear beam (esb) container

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