1 M. CoopN G Alvarado Talk 17 March 2010

8/10/2019 1 M. CoopN G Alvarado Talk 17 March 2010

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The Use and Interpretation of Bender Elements

Matthew R. Coop

Giovanny Alvarado

Imperial College London

Edafos Engineering Consulting, Greece


2/27

Bender Elementselastic shear stiffness, G 0 from velocity of shear wave

InputOutput

D

G 0 = v s 2 ( = mass density)

(Dyvik & Madshus, 1985)

Piezoelectric Bender ElementsKramer (1996)

piezo-ceramic plates used for source & receiver

bend when subjected to voltage & generatea voltage when bent

elements need to be isolated from

pore water with epoxy coating


3/27

Platen Mounted Elements

Vstransmittedsignal

functiongenerator transmitter element

D

receivedsignaldigitaloscilloscopereceiver element

measure time delay between transmitted and received signal onoscilloscope T a

use tip to tip distance (D) between bender elements to calculate velocity V s


4/27

204.7

204.9

205.1

205.3

205.5

]

Comparisons Between Bender Element

Stiffnesses and Monotonic Measurements

LVDTs (Cuccovillo & Coop, 1997)

203.5

203.7

203.9

204.1

204.3

204.5

0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006

a [% ]

a

[ k P

loadunload

(Gasparre, 2005)


5/27

200

300

o d u

l u s

( M P a

)

elastic stiffnessfrom bender elementindicating confiningstress

650 kPa 650 kPa

Comparison between bender elements and G values from LVDTs

q

G tan = dq/3d s

1E-5 1E-4 1E-3 1E-2 1E-1shear strain (%)

0

100

u n

d r a

i n e

d s

h e a

r

63 kPa

150 kPa

250 kPa

good agreement depends on soilbeing isotropic (kaolincompressed to high isotropic

stresses)

(Jovicic & Coop, 1997)

s


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behaviour defined by:Ev = vertical Youngs modulusEH = horizontal Youngs modulus VH = Poissons ratio for influence of V on H

HV

= Poissons ratio for influence of

H

on V HH = Poissons ratio for influence of H1 on H2 or H2 on H1

GVH = shear modulus in vertical planeGHV = shear modulus in vertical planeGHH = shear modulus in horizontal plane

For a Cross-Anisotropic Soil:

5 independent parameterse.g. for a homogenous elastic

G VH

direction ofpropagation

plane of

polarisation

platen mounted elements measure G VH

ma er a VH= HV


7/27

Laterally Mounted T- Elements (Pennington et al., 1997)

samples can be cut horizontally and orientated vertically in apparatus to measureGHV and G HH with platen mounted elements

G HV

but laterally mounted T-elements are much easier

G HH

38mm specimen100mm specimen


8/27

K

G eq

q

p

Ev, VH

Eh , HV

0

0=>

r

a

Axial compression

0

0=>

a

r

Radial compression

Static probes using LVDTs used to find elastic parameters

(Gasparre et al., 2007)

v

vh

h

hvv

vv

vhh

h

hhh

E E v

E

v

E

v

+=

=

2

1

vhv

h

vv

v E

=

=

0=h

0=v

Lings et al. (2000)

Kuwano & Jardine (1998)

+

hhG hhhv

h E

,

FROM BENDERELEMENTS


9/27

20

10

0

e p

t h [ m ]

0 200 400Young's Modu li [MPa]

0 100 200Shear modu li [MPa]

0 100 200Bulk modu lus, K [MPa]

C

B 2(c)

B2(b)

B2(a)

Elastic Parameters for London Clay (Gasparre et al., 2007)

-0.5 0.0 0.5 1.0 1.5Poisson's ratios

vhhnvh

nhv

hh vh hv

40

30

D

Ev' (TX)

Ev' (HCA)

Eh' (TX)

Eh' (HCA)

Gvh (BE)

Ghh (BE)

Gvh (RC)

Gvh (Static)

A3(2)

B 1

good agreement between BE, resonant column and HCA values of G vh

(HCA)


10/27

- early work used square pulse:

Time Domain Analysis

near fieldeffect?

(Viggiani & Atkinson, 1995)

received signal does not resembletransmitted wave because of wide

range of frequencies in a square wave

- which feature should be used as theshear wave arrival?

early explanation of initial dip was near field effect due to complex natureof wave transmitted (not just a pure shear wave)

near field effect dissipates more quickly should not be a problem for:

D/>2 (Jovicic et al., 1996) or >0.6 (Arroyo et al., 2003) where is wavelength

but do reflections from boundaries of sample contribute to complex signal?


11/27

because of difficulties with square waves, other time domain methods developed

often trying to avoid/eliminate near field effect

Distorted sine pulse Sine burstSine pulse

Input Signals for Time Domain Analysis

(Jovicic et al., 1996)Sinusoids

(Pennington et al., 2001)

(Jovicic et al., 1996)

(Viggiani & Atkinson, 1995)

complex shapes may give apparently clearerarrival, but is it the correct arrival time?

either wave shape altered or more cycles added


12/27

Frequency Domain Methods

use of continuous single frequency ensures output same as input exceptfor phase shift e.g. point method (Greening & Nash, 2004)

identify frequencies (f) that

give input and output perfectlyin or out of phaseD=sample length =wavelength

o u

t p u

t

o u t p u

t

input

input

oscilloscope - plotinput against output,not both against time

1.0

1/t arr

input

0.5

D/

D/

f

output0.5 1 1.5 2

(Blewett et al., 1999)


13/27

point method is laborious more commonly done with acontinuous sweep of frequencies

Comparison of stackedphase from sine-sweepand -point data(Greening & Nash, 2004)

d i a n s

)

-points

p h a s e

( r

frequency (kHz)

sine-sweep

(Rio, 2006) a m p l i

t u d e

( v )

time (ms)


14/27

-20

020

40

60

l t a g e

( A r b

i t r a r y u n

i t s

)

2 kHz

3 kHz

4 kHz

5 kHz

6 kHz

Real Systems Time domain

-80

-60

-40

0 0.5 1 1.5

Time (ms)

O u

t p u

t V o

7 kHz8 kHz

9 kHZ

arrival time often apparently dependent on frequency

(Alvarado, 2007)Toyoura sand, p= 6.7MPa


15/27

Real Systems - Frequency Domain

40

50

60

70

80

90

f ( k H z )

which range is representative ofmaterial?

very stiff cemented sand

several phase shifts in the wholerange

0

10

20

30

0 1 2 3 4 5 6 7 8 9 10 11

n. 1

p=200 kPa

ta = 129 s

ta = 101 s

ta = 215 s

(Alvarado, 2005)

D/


16/27

Transfer function SDOF

)()()( f F f H f D =Mass-spring-damper system

Gain factor

222

21

1)(

+

=

nn f f

f f

k f H

Relative magnitudebetween input and output

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

0.0 0.2 0.4 0.6 0.8 1.0

Time (ms)

A m p l

i t u d e

( [ k N ] o r [ m m

* 1 2 0 ] ) Input

Output

Input frequency: 3kHz

Input: Force, Output:Displacement

= 2

1

1

2tan)(

n

n

f f

f f

f

Phase factor

Phase difference betweeninput and output


17/27


-1.5

0.0

1.5

g a i n

Input: 1 cycle

Input: 3 cycles

Transfer function

Output

Central input frequency

-15.5

-15.0

-14.5

-14.0

-13.5

-13.0

-12.5

0.0 0.2 0.4 0.6 0.8 1.0

Time (ms)

A m p l

i t u d e

( [ k N ] o r

[ m m

* 1 2 0 ] ) Input

Output

Input frequency: 3kHzEffect of input duration

The closer to a forced vibrationcondition the better matchbetween input and output, but thetransfer function still controls thephase .

-6.0

-4.5

-3.0

0.1 1.0 10.0Frequency (kHz)

N o r m a l i s e d

Input: 5 cycles

Input: 10 cycles

For a constant-parameter linearsystem the transfer function isindependent of the type andduration of the input


18/27


The frequency content of theoutput is controlled by theproximity of the central inputfrequency to the resonantfrequency-1.5

0.0

1.5

a i n

Input: 1 kHz

Input: 2 kHz

Transfer function

Output

Central input frequency

Effect of input frequency (sine pulse)

After the pulse input the systementers free vibration .

-6.0

-4.5

-3.0

0.1 1.0 10.0

Frequency (kHz)

N o r m a l

i s e d

Input: 3 kHz

Input: 5 kHz = f n

The Bender Element System is nota SDOF but it is constant-parameter and linear . The arehowever some similarities.

The closer the input frequency isto the resonant frequency thestronger the output and thecloser in frequency content to theinput.


19/27

Real BE Systems Time domain

Reconstituted Londonclay Input sine pulse

f= 4kHz, p =30kPa

-3

-2

-1

0

1

2

34

0.00 0.50 1.00 1.50 2.00

A m p l

i t u d e

( m V )

Output - R1PAsin(2p1.9t) - sin(3.3t)t1 (ms)

sin(2 1.9t-t1)-sin(2 3.3t-t2)

Most real output signals show at least tomain frequency components (f so & f b),

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

0.00 0.50 1.00 1.50 2.00

Time (ms)

A m p l

i t u d e

( m V )

3.3 kHz

1.9 kHz

t1t2

Wave components arrive at differenttimes. One of them should be ashear wave.

-5

-4

Time (ms)

none of them being necessarily in theinput .

Output signal duration is largerthan the input (extra cycles). Anindication of damping.


20/27

Multiple modes of vibration (system is not SDOF )

1.5

2.0

2.5

3.0

a r b . u

n i t s )

220 kPa -11 kHz 3480 kPa -11 kHz

6370 kPa -11 kHz 22400 kPa -13 kHz

Transfer function BES

fbf so

0.0

0.5

1.0

0 5 10 15 20 25

Frequency (kHz)

G a i n

(

Toyoura sand

( ) )sin( f t t B y =

( ) )sin( t A x t = ( ) cons A

H f == f f =

Gain factor Phase factor

Ideal System


21/27

1.5

2.0

2.5

3.0

a r b . u

n i t s )

220 kPa -11 kHz 3480 kPa -11 kHz

6370 kPa -11 kHz 22400 kPa -13 kHz


fb

f so

0.0

0.5

1.0

0 5 10 15 20 25

Frequency (kHz)

G a i n

(

Location of modes is dependent on stress-level

Toyoura sand

As stress increases input needs to have higher frequency. so

sob f f

f f N

=


22/27

0

500

1000

1500

0.0 2.0 4.0 6.0 8.0 10.0

Freq (kHz)

s e

f a c

t o r

( d e g

)


c

t o r

( d e g )

2000

2500

3000 S t a c

k e

d p

h

Thanet sand, p=100kPa

P h a s e

f

For the idealised model a straight line is expected (nodispersion)

Even if a straight line, only group velocity can be

measured which is not necessarily equal to shearwave velocity


23/27

0.901.001.101.201.301.401.50

t g / t a r r

Nf=25%

Nf=75%

System parameters

Parametric study using synthetic signals

arr s so t f Rd *0=

sob f

f f N

=

Number of waves at f so that fit

between input and output.System parameter

0.500.600.700.80

0.0 2.0 4.0 6.0 8.0 10.Rds0

=0.2, near field effect, Arroyo et al(2003b)

Proximity between modes of vibration may have a significanteffect on frequency domain interpretation.

System parameters would depend not only on the materialbut also on sample geometry and stress level.

so

Relative distance betweenmain frequencies


24/27

waveguidelower signal frequenciessignificant geometric dispersion

unboundedmedium

v e

l o c

i t y

( m

/ s )

correct V s

Geometry effects (Rio, 2006)

transition

near fieldeffect

H2/Diam

w a v e

height to diameter ratio affects apparent V s largely due to changingeffects of wave reflection on boundaries


25/27

The use of Bender Elements is a technique that has been used successfully forover 20 years. It now used in a number of commercial labs around the world.

Conclusions

Interpretation can be often difficult and therefore a number of interpretationmethods have appeared over the years.

There seems to be agreement that interpretation difficulties arise from thecomplexity of the BES .

,means that planar shear wave propagation can take place and the effects of othercomponents of the system could be filtered out should the transfer function beknown.

Careful selection of input frequencies and redundancy can make interpretationeasier and results more reliable.

There is a need for standards and perhaps the use of system parameters as away to make results cross-comparable.

The industry need to be aware of the advantages and limitations of the method.


26/27


27/27

G vh for Toyoura sand

1000

10000

G v h

( M P a )

Conventionalfrequency domaininterpretation ( groupvelocity ) produces thelowest values. At veryhigh pressureinconsistent results.

Time domaininterpretation lower

10

100

100 1000 10000 100000p' (kPa)

Time domainGroup velocityFrequency domainTatsuoka (2005)

values thanTatsoukas. Here,frequency effect used.

Corrected frequencydomain show

highest values.

Time and frequency domain show similar trends.

Documents

1 M. CoopN G Alvarado Talk 17 March 2010