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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
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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
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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
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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)
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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
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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
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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
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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)
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- 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?
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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
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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)
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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)
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-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
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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/
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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
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Transfer function SDOF
-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
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Transfer function SDOF
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.
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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.
8/10/2019 1 M. CoopN G Alvarado Talk 17 March 2010
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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
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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
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
=
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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
)
Transfer function BES
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
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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
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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
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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.
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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.