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Ŕ Periodica Polytechnica
Civil Engineering
60(2), pp. 145–158, 2016
DOI: 10.3311/PPci.7952
Creative Commons Attribution
RESEARCH ARTICLE
Suitable Waves for Bender Element
Tests: Interpretations, Errors and
Modelling Aspects
Muhammad E. Rahman, Vikram Pakrashi, Subhadeep Banerjee, Trevor Orr
Received 04-02-2015, revised 23-04-2015, accepted 22-06-2015
Abstract
Extensive research on bender element tests has been carried
out by many researchers, but precise guidelines for carrying out
such tests have not yet been established. It is often recommended
that, when using a particular bender element test for the first
time on a particular soil to determine its small strain dynamic
properties, several methods should be tried and the results com-
pared in order to improve confidence in the results obtained.
Demonstrated use of relatively easy analytical models for inves-
tigating different scenarios of bender element testing is another
aspect that should be further looked into. This paper presents
laboratory experiments and dynamic finite element analyses to
determine a suitable wave for use in bender element tests in the
laboratory to measure small strain shear stiffness (Gmax). The
suitability of a distorted sine wave over a continuous sine wave
for tests is observed from laboratory experiments and dynamic
finite element analyses. The use of simple finite element models
for assessing a number of aspects in relation to bender element
testing is demonstrated.
Keywords
Bender Element Test ·Numerical Analysis ·Sine Wave ·Small
Strain Shear Stiffness
Muhammad E. Rahman
Faculty of Engineering and Science, Curtin University Sarawak, Malaysia
e-mail: merahman@curtin.edu.my
Vikram Pakrashi
Dynamical Systems and Risk Laboratory, Civil and Environmental Engineering,
School of Engineering, University College Cork, Cork, Ireland
e-mail: V.Pakrashi@ucc.ie
Subhadeep Banerjee
Department of Civil Engineering, Indian Institute of Technology, Madras, India
e-mail: subhadeep@iitm.ac.in
Trevor Orr
Department of Civil, Structural and Environmental Engineering, Trinity College,
Dublin, Ireland
e-mail: torr@tcd
1 Introduction
Dynamic analyses to evaluate the small strain stiffness of soil
and the response of earth structures to dynamic stress applica-
tions are finding increased popularity in civil engineering prac-
tice. Idealised models and analytical techniques may be used to
represent a soil deposit and its response in this regard. Estima-
tion of the small strain stiffness and the dynamic properties of
the soil are important and challenging problems. Precise mea-
surement of the small strain stiffness and dynamic soil prop-
erties are difficult tasks when analysing dynamic geotechnical
engineering problems [17]. Several field and laboratory tech-
niques are available to measure the dynamic properties, many
of which involve measurements at small-strain [24,28] or large
strain levels [21]. The choice of a particular technique depends
on the specific problem to be solved. The existing tests provide
insights into correlation with other tests methods, with types of
specimens or the methods, but the requirement of more data and
the approaches towards rapid modelling of scenarios still remain
a topical subject.
The key soil properties that influence wave propagation and
other low-strain phenomena include stiffness, damping, Pois-
son’s ratio and density. Of these, stiffness and damping are
the most important since the others usually have less influence
and tend to fall within relatively narrow ranges [17]. Labora-
tory tests are available to measure dynamic properties of soils
at small strain levels. The resonant column test [29], ultrasonic
pulse test [27], and bender element test [13] are the commonly
employed techniques to measure small strain stiffness and dy-
namic properties.
Extensive research on bender elements test has been carried
out by many researchers in last few decades [2,4, 15] and [3] but
precise guidelines for carrying out such tests are not completely
established. It is usually recommended to try and compare sev-
eral methods when using a particular test for the first time on
a particular soil to determine its small strain stiffness and dy-
namic properties, in order to improve confidence in the results
obtained [15] and [4]. Theoretical analysis and experimental
validation in the frequency domain have been recently carried
out [1] using a transfer function to characterise different soils
Suitable Waves for Bender Element Tests: Interpretations, Errors and Modelling Aspects 1452016 60 2
with linear, but dispersive characteristics of soil in relation to
the waves. A wavelet-based approach for singularity detection
has also been proposed recently by [5] in connection with an
assessment of shear wave arrival time. Essentially, the use of
bender elements to predict shear modulus is a system identifi-
cation problem fraught with different levels of variability, noise,
method of assessment, inherent uncertainties in soil character-
istics, type of instrument used and the level of analytical effort.
Investigations into these aspects remain topical and important
[12] in this regard.
This paper considers the suitability of different waves for ben-
der element tests for various situations and also assesses the
variability of such results acknowledging the typical resources
available for experimental and analytical studies. Initially, ex-
periments are carried out on marine clay in Chennai, India with
sinusoidal excitation to obtain the variability of estimated shear
modulus. The study is augmented with an investigation on boul-
der clay, Dublin, Ireland and the suitability of a distorted sine
wave as an excitation in investigated. Traditional theoretical
modelling is taken up next to demonstrate that even a relatively
simple finite element model address a number of issues related
to the estimation, capturing a number of observations observed
experimentally. The study adds to the on-going understanding
of the different approaches towards the estimation of shear mod-
ulus using bender elements in the presence of varied methods,
equipment and analytical rigour.
2 Bender Elements
Bender elements have been used for the measurement of elas-
tic small strain stiffness and damping ratio in a triaxial cell. The
bender element technique has undergone significant develop-
ment in the last few decades [10]. In the early stage, piezo-
ceramics were mainly used to generate and receive compression
P-waves. Since little information about the soil structure can
be obtained from P-waves and since the P-wave velocities are
highly influenced by the pore fluid, the piezoceramics have been
combined in different forms to generate and receive shear waves.
Such combined forms of piezoceramic are used in gauges for
measuring vibrations known as bender elements [8] and [16].
Bender elements consist of two thin piezoceramic plates
rigidly bonded to a central metallic plate. Two thin conduc-
tive layers, which serve as electrodes, are glued externally to the
bender element. The polarization of the ceramic material in each
plate and the electrical connections are such that when a driving
voltage is applied to the element, one plate elongates and the
other shortens. When the measurement of the shear wave veloc-
ity is made using bender elements in the triaxial test apparatus,
one bender element is fixed in place in the top cap and the other
in the pedestal. The elements are of 1mm thickness, 12 mm
width and about 15mm length. In the set-up in the triaxial cell,
the bender elements at both ends protrude into the specimen as
cantilevers. When the bender element at the top is set into mo-
tion, the soil surrounding the bender element is forced to move
back and forth horizontally and its motion initiates the propaga-
tion of a shear wave through the soil sample. When the shear
wave reaches the other bender element at the other end of the
specimen in the triaxial apparatus, it causes it to bend and thus
produce a voltage. This output signal can be captured through
an oscilloscope and the travel time determined by measuring the
time difference between the input and the output signals. The
shear wave velocity can be found by dividing the travel distance,
L, by the travel time, t where the travel distance of the wave is
taken to be equal to the specimen’s length minus the protrusion
of the two-bender elements at the both ends. After determining
the propagation shear wave velocity, it is possible to calculate
the small strain shear modulus, Gmax,by the elastic continuum
mechanics relationship.
Gmax =ρv2
s(1)
Where ρis the soil density and vsis the shear wave velocity.
The damping ratio, ξ, can be determined from the frequency
response curve using the half–power bandwidth method. The
half-power bandwidth method is a very common method for
measuring damping [9] and uses the relative width of the re-
sponse spectrum.
The advantages of bender elements are that they can also be
incorporated into the oedometer test apparatus and simple shear
test device. However, normally they are incorporated into the
triaxial apparatus [14,16] and [17]. Anisotropy of the soil stiff-
ness can also be investigated by locating bender elements on
two vertical and opposite sides of a sample. The disadvantage
of the bender element is that the element must be waterproofed
to prevent short circuits, which is often difficult to achieve when
testing dense or hard saturated materials. Insertion of the bender
element into such hard materials can easily damage the sealing
for waterproofing [14].
The shear modulus of soil is related to the shear wave veloc-
ity determined using a bender element test. The bender element
induces very small strains which lie typically within the elastic
limit. Dyvik & Madshus (1985) estimated the maximum shear
strain induced by bender elements to be less than 0.001%, so
that Gmax is relevant for very small strain [7,11] and [16]. Jovi-
cic (1997) validated the assumption of elasticity experimentally
by finding that there was no volume change in specimens when
drained bender element tests were performed. No pore water
pressure was generated when undrained tests were carried out
[15].
Although the use of the bender element apparatus is simple,
the application of bender element test for the measurement of
small strain stiffness and the damping ratio may not be straight-
forward as it is difficult to measure the exact travel time between
the input and output signals. The strain level induced by the
bender element test is directly proportional to the displacement
at the tip of the bender element so that it is difficult to measure.
Determination of the shear wave velocity is a key element
Period. Polytech. Civil Eng.146 Muhammad E. Rahman, Vikram Pakrashi, Subhadeep Banerjee, Trevor Orr
for establishing Gmax in bender element tests. The shear wave
velocity is directly related to the shear wave travel distance and
the travel time and is calculated by dividing the travel distance
by the travel time. Initially, there was some doubt as to what
should be taken as the true travel distance. Intuitively one would
take the distance between the bender element tips although some
researchers thought it might be the full height of the sample.
Viggiani and Atkinson (1995a) carried out some laboratory tests
on a set of reconstituted samples of Speswhite kaolin of different
lengths to investigate what should be taken as the travel distance.
Travel times are plotted against the overall length of the sample
for different confining stress state conditions [25]. The test data
fall on straight lines, each with an intercept of about 6mm on
the vertical axis where the bender elements used for those tests
were 3 mm long. From these tests, it has been concluded that the
travel wave distance should be between the tips of the elements
rather than the full length of the sample. This is in agreement
with previous experimental work by Dyvik and Madshus (1985)
[11].
Consequently, the most important parameter to be determined
in a bender element test is the exact travel time of the shear wave
between the transmitter and the receiver. Actually the principal
problem with the bender element test has always been the sub-
jectivity of the determination of the arrival time used to calculate
shear wave velocity [15] and [19]. The travel time is dependent
on the shape of the wave transmitted through the soil sample.
The procedure commonly used in bender element tests is to gen-
erate a square wave and to determine the time of first arrivals
[15] and [25] although there is considerable distortion of the
output signal using a square wave. The problem with the square
wave is that it is composed of a wide spectrum of frequencies
[15]. From the received signal of the square wave alone, it is
uncertain whether the shear wave arrival is at the point of first
deflection, the reversal point, or some other point.
To reduce the degree of subjectivity in the interpretation, and
to avoid the difficulty in interpreting the square wave response,
Viggiani and Atkinson, (1995a) suggested using a sine pulse as
the input signal [25]. Consisting of a single frequency, the out-
put wave is generally of a similar shape to the input signal, but
it is still very difficult to determine the travel time. The problem
arises due to a phenomenon called the near field effect. The near
field effect is caused by the very rapid P-waves that mask the
first arrivals of the slower, S-waves. The near field effect may
mask the arrival of a shear wave when the distance between the
source and the receiver is within the range of 1/4 to 4 wave-
lengths. This is the situation in bender element tests where the
distance between the transmitter and the receiver is relatively
small and is about 2 to 3 wavelengths [25]. Brignoli & Gotti
(1992) also found the existence of near field effects in bender el-
ement tests and these have been further investigated by Jovicic
et al.(1996) [7,15]. The wavelength can be estimated from:
λ=vs
f(2)
Where, fis the frequency of the input signal in Hertz.
Jovicic et al., (1996) derived an analytical solution for the
time record at a point resulting from the equation for the excita-
tion by a transverse sine pulse of a point source within an infinite
isotropic elastic medium, obtained by Sanches-Salinero et. al.,
1986 [15,22].
The fundamental solution for the transverse motion S [22] is
given in the form
S=1
4πρv2
s
Γ(3)
where the function Γis given by:
Γ = 1
de−i($d/vs)+
1
i$d2
vp−
1
$2d3
v2
p
e−i($d/vs)−
− vs
vp!2
1
i$d2
vp−
1
$2d3
v2
p
e−i($d/vp)
(4)
where dis the distance between the transmitter and receiver
of bender element, vsis shear wave velocity, vpis primary wave
velocity and ωthe angular velocity.
From Eq. (4), it can be seen that there are three coupled com-
ponents to the transverse motion, which comprise both the near
field and far field effects. This is because the motions of the ele-
ments are not pure compressive or shear motions [7] and gener-
ally some compressive or shear motion occurs. All three terms
represent transverse motion. While they propagate with differ-
ent velocities, the first two travel with the velocity of a shear
wave, and the third travels with the velocity of a compression
wave. The attenuation occurs at different rates with geometrical
damping for the three components, the second and third terms
attenuating at a rate an order of magnitude faster than the first
term. The coefficient of the first term is proportional to the in-
verse of the distance and the coefficients of second and third
terms are proportional to the inverse of square of the distance
and the cube of the distance, which implies that the last two
terms are only significant for small distances and are called near
field terms and first term is significant for larger distances and
is called the far field term. In bender element tests, the effect is
amplified by the S-wave source and substantial P-wave energy
that is often developed.
Sanches-Salinero et. al., (1986) expressed their results in
terms of the ratio d/λ, where λis the wavelength of the input
signal [22]. Later Jovicic et. al., (1996) denoted this ratio as
Rd[15]. The value of Rdrepresents the number of wave lengths
that occur between the bender element transmitter and receiver
and which control the shape of the received signal through the
Suitable Waves for Bender Element Tests: Interpretations, Errors and Modelling Aspects 1472016 60 2
degree of attenuation of each term of Eq. (4) that occurs as the
wave travels through the sample. Rdis calculated from:
Rd=d
λ=df
vs(5)
For low values of Rdthere is an initial downward deflection
of the trace before the shear wave arrives, representing the near
field effect given by the third component of Eq. (4). For high
values of Rdthe near field effect is almost absent.
Jovicic et. al., (1996) carried out a series of experiments
using bender elements on Speswhite kaolin specimens with an
isotropic effective stress of 200kPa and values of Rdof 1.1 and
8.1 [15]. The results confirmed that for low Rdvalues the near
field effect dominates while for high Rdvalues the near field ef-
fect is almost insignificant. The authors recommended selecting
a high enough frequency so that the soil being tested has a high
Rdratio. In practice, this cannot be always achieved, because at
the high frequency required for stiffer materials, overshooting of
the transmitting element can occur. These experiments were car-
ried out on cemented granular soft rock, with Gmax of 2.5 GPa at
an effective stress of 200kPa. At 2.96 kHz the element follows
perfectly the input wave while at 29.6kHz it does not and over-
shooting occurs. The limiting frequency at which overshooting
starts to occur depends on the relative impedances of the soil and
the element and overshooting is found to be severe for stiffer soil
as well as being pronounced for square waves.
3 Bender Element Tests
The following section details the results of bender element
tests which would be later used for performance evaluation of
numerical simulations. The first series of tests were conducted
on Chennai marine clay. Some basic properties of Chennai ma-
rine clay for all the tests are shown in the Table 1.
The bender element tests were carried in the triaxial cham-
ber to simulate the field confining pressure. The transmitter el-
ement is mounted at the pedestal of the triaxial chamber. The
receiver element is inserted on the top of the soil sample. Elec-
tric pulses are applied to the transmitter through the wave-form
generator. The continuous shear waves, thus produced, would
travel through the soil sample before being recorded by the re-
ceiver at the other end. By measuring the travel time (t) and the
distance (L) between the tips of the bender elements, the shear
wave velocity (vs) can be obtained as,
vs=L/t(6)
The maximum shear moduluscan thus be computed as,
Gmax =ρv2
s(7)
Where ρis the density of the soil.
The estimation of maximum shear modulus obtained in this
regard is shown in Table 2.
The results presented in Table 2 establish the repeatability of
the estimates and the control of the test. The observed shear
wave velocities and corresponding maximum shear modulus are
in accordance with the results obtained from field tests on Chen-
nai Marine clay, as reported by Boominathan et. al., (2008) [6].
Moreover, the range of maximum shear modulus obtained for
Chennai marine clay was found to be comparable with the pre-
dictions computed from empirical correlations proposed by Vig-
giani and Atkinson (1995) for clays with different mineralogy
[25,26].
The typical input signal and receiver output from a bender el-
ement test on Chennai marine clay is shown in the Fig. 1. It
should be noted that the raw data of the receiver output gener-
ally contain noises of high frequencies. As it is well-understood
that the resonant frequency of marine cannot have a resonant
frequency greater than 10Hz, a low pass filter with cut-offfre-
quency of 10 Hz is used to remove the spurious signals. It should
also be noted that the travel time and thus the shear wave ve-
locity are computed from the first few signals, beyond which
reflection of the waves may incur uncertainty in the measure-
ments [14]. While this is associated with some subjectivity in
the analysis, it also opens up the possibility of investigating the
suitability of relatively rapidly created axisymmetric models for
estimation of maximum shear modulus using bender elements
and further analyses related to the suitability of the input waves.
It is well-established that the soil type, experimental tech-
niques and conditions influence the measurement of maximum
shear modulus [14,25, 26]. Viggiani and Atkinson (1995b) in-
dicated that the maximum shear modulus of the natural clay
may substantially vary from that of the reconstituted clay [26].
Consequently, it was deemed important to compare with tests
on other types of clay in relation to the suitability of the input
waves.
Bender element tests using a Wykeham-Farrance 100mm tri-
axial cell were carried out on Dublin boulder clay with cylindri-
cal specimens of 200mm length and 100 mm diameter bender
elements in the top cap and base pedestal. A Thurlby Thander
TGA 1240 function generator, a Pico ADC-212 high-resolution
oscilloscope and shielded output cables were also used. The ex-
citation signal, produced by a function generator was amplified
and sent to the bender element in the top cap with maximum
peak-to-peak amplitude of 20V. The waves transmitted through
the soil specimen from the top were recorded at the base by the
receiver and displayed on an oscilloscope. Calibration was car-
ried out by placing the two platens in direct contact and mea-
suring the time interval between the initiation of the electrical
impulse sent to the transmitter and the initial arrival of the wave-
form recorded at the receiver. The top platen was marked with
respect to the base platen in order to avoid ambiguity in test
interpretation [18]. The measured waveforms and calibration
times obtained for the bender elements were checked to ensure
that there was no time lag. The absence of wave transmission
paths, other than through the soil specimen, was checked as well
by placing the two platens in the triaxial cell without any soil and
without contact and ensuring that no wave arrival was recorded
Period. Polytech. Civil Eng.148 Muhammad E. Rahman, Vikram Pakrashi, Subhadeep Banerjee, Trevor Orr
Tab. 1. Basic Properties of Chennai Marine Clay
Liquid limit 54%
Plastic limit 30%
Sand 10%
Silt 34%
Clay 56%
Tab. 2. Summarizes the Shear Wave Velocity and Maximum Shear Modulus Obtained for Chennai Marine Clay.
Maximum shear modulus (MPa)
Sl.no. Shear wave velocity
(m/sec) From Bender Element tests
(MPa)
Field data from
Boominathan et al. (2008)
(MPa)
From Viggiani and
Atkinson (1995) (MPa)
BET-1 91 15.39
17
Range: 15–40
BET-2 85 13.43 (Depending on plasticity
BET-3 93 16.08 and mineralogy of
BET-4 87 14.07 the cohesive soil)
by the receiver when pulses were generated by the transmitter.
Average properties of Dublin boulder clay are shown in the Ta-
ble 3. Bender element measurements were carried out by ex-
citing a transmitter with a standard sine waves and arbitrarily
distorted sine waves, followed by detection of the first arrival of
the waves at the receiver. Signals were sent top down, from the
top cap to the base. Lings and Greening (2001) found that there
was no difference between the signals when they were sent top
down or bottom up [20]. Input and output traces of a standard
sine wave (Fig. 2) and an arbitrary distorted sine wave (Fig. 3)
indicate the presence of a near-field effect for a sine wave and
the absence of such an effect for a distorted sine wave.
As discussed in this section, there remains an interesting pos-
sibility in investigating the use of traditional axisymmetric mod-
els for assessing shear modulus from bender elements and a
number of observations were made from such simpler mod-
elling, in relation to the experimental evidences.
4 Investigations into the Usefulness of Traditional Ax-
isymmetric Finite Element Models
4.1 Finite Element Method and Model
In this paper, a Finite Element (FE) model is developed to
calculate the time gap between the transmitted wave and the re-
ceived wave in the cylindrical triaxial cell sample. The prob-
lem is three-dimensional in nature, but a simplified model con-
sisting of two-dimensional plane strain, linear-elastic finite el-
ement analyses were deemed sufficient and carried out using
the commercially available general-purpose finite element pack-
age PLAXIS. The soil was modelled using a 15-noded plane
strain triangular element. For a 15-node triangle, the order of
interpolation for deflections is four and the integration involves
12 stress points. The dimensions of the finite element mesh
(Fig. 4) used to model the bender element were 225mm high
and 100mm wide. The behaviour of the bender elements is
modelled by use of a 5 noded beam element that represents real
plates in the out-of-plane direction and the interface between the
soil and the bender elements was characterized as an interface
element. The beam elements are based on Mindlin’s beam the-
ory, which allows for beam deflections due to shearing as well
as bending. The bender element top cap was modelled by use
of a 15-noded triangular element and the interface between the
soil and the top cap was characterized as an interface element.
The boundary conditions were applied by selecting the standard
fixities, which meant that the horizontal top and bottom bound-
aries (Fig. 4) were fully fixed and the side boundaries were kept
stress free to replicate the actual test condition [4]. The load
and boundary conditions and triangular mesh are also shown in
Fig. 4.
In the analyses, the input voltage signal applied to the trans-
mitting bender element was modelled by means of a transverse
sinusoidal motion with an amplitude of 1.0x 10−3mm, which
acted at a point representing the tip of the transmitter. Based
on the assumption of plane wave fronts and the absence of any
reflected or refracted waves, the output voltage signal from the
receiving bender element was taken from the tip of the receiver
and the travel time of a shear wave between the transmitter and
the receiver were taken as the time between characteristic points
in the signals recorded at these two points. The most commonly
used characteristic points are the first peak, first trough or zero
crossing of the input and output signals. The travel time between
two points can be taken as the time shift that produces the peak
cross-correlation between signals recorded at these two points
([4]).
4.2 Model Properties
The Poisson’s ratio (ν) chosen for the soil was 0.495, replicat-
ing undrained condition of the soil. The shear wave velocity of
soil was calculated as 386.2m/s using:
vs=√Emax
p2ρ(1 +ν)=sGmax
ρ(8)
Where ρ, the soil density is equal to 2242 kg/m3and Emax,
Suitable Waves for Bender Element Tests: Interpretations, Errors and Modelling Aspects 1492016 60 2
Fig. 1. An experimental trace for a standard Sine wave input for Chennai marine clay
Fig. 2. An experimental trace with near-field event for a standard Sine wave input.
Fig. 3. An experimental trace without near-field event for a distorted Sine wave input.
Period. Polytech. Civil Eng.150 Muhammad E. Rahman, Vikram Pakrashi, Subhadeep Banerjee, Trevor Orr
Tab. 3. Summary of average basic material properties (Skipper et al., 2005)
Parameter Upper Brown Upper Black Lower Brown Lower Black
Moisture content,
%13.1 9.7 11.5 11.3
Bulk density,
Mgm−32.228 2.337 2.283 2.284
Liquid limit, % 29.3 28.3 30.0 29.5
Plastic limit, % 15.9 15.1 14.9 17.8
Plasticity index, % 13.4 13.2 15.1 11.8
Clay content, % 11.7 14.8 17.8 17.5
Silt content, % 17.0 24.7 28.3 30.5
Sand content, % 25.0 24.7 25.7 34.0
Gravel content, % 46.3 35.8 28.0 35.5
the small strain undrained Young’s modulus is equal to 1GPa.
This value is a typical Emax value obtained from laboratory tests
on Dublin Boulder Clay. The bender elements and top cap
were made of lead zirconate and aluminium with small strain
Young’s moduli of 70GPa and 65GPa, densities of 2700kg/m3
and 7500kg/m3and Poisson’s ratios of 0.33 and 0.3 respec-
tively.
4.3 Comparative Study between Plane Strain Elements
and Axisymmetric Elements
A comparison between the results of the FE analyses using
plane strain elements and axisymmetric elements help determine
the type of element more suitable for the purpose since the real
situation is neither a purely plane strain problem nor a purely
axi-symmetric problem. Sine waves of 2098 Hz and 10489Hz,
along with an assumed input Gmax of 334.45MPa were used.
The shear wave velocity was calculated using vs=L/t. The
Gmax values were calculated using Eq. (1). The percentage er-
rors in Gmax were calculated with respect to the input Gmax value
(Table 4) and the lower percentage errors for the plane strain el-
ements indicated that these were more suitable.
4.4 Effects of Mesh Dimension
An investigation of the Gmax values obtained using different
mesh dimensions indicated that the error in the predicted wave
velocities increases exponentially with the increase in the mesh
dimensions. The percentage of errors in Gmax due to using very
fine, fine, medium and coarse meshes (corresponding to approx-
imately 1000, 500, 250 and 100 elements respectively) were
0.55%, 0.97%, 2.7% and 5.5% respectively. For the subsequent
analyses, very fine mesh densities were selected. However, even
medium mesh is observed to give reasonable results.
4.5 Estimation using Single Sine Wave
A single sine wave (Fig. 5) in the transverse direction at the
tip of the transmitter bender element was used as an input signal
to cause a deflection of the bender elements and to generate a
wave through the soil to the receiver. A series of finite element
analyses was carried out for different values of Rd=d
λ=df
vs
between 1 and 8 The frequency range was 2.1kHz to 16.78 kHz.
The experimental output signals (Fig. 6) from the receiver were
analysed to obtain the Gmax deviation percentages for different
Rdvalues (Fig. 7) by comparing the predicted Gmax values ob-
tained from the predicted shear wave velocity (vs) at different Rd
values with an assumed Gmax value of 334.45MPa. The Gmax
deviations were obtained using a number of methods to predict
the shear wave arrival time. The first inflexion method yields
deviations of the order of 4.57%, 3.8% and 2.6% for the Rdra-
tios of 1, 2 and 3 respectively, while for the remaining Rdvalues
the percentage deviations are less than 2%. The percentage de-
viations obtained using the first peak-to-peak input and output
waves method are 6.2% and 3.89% for the Rdratios of 1, and
2 respectively, while for the remaining Rdvalues the percentage
errors are less than or around 3%. The percentage deviations
obtained using the cross correlation method are 5% and 9% for
the Rdratios of 1, and 2 respectively, while for the remaining
Rdvalues the percentage deviations are less than or around 2%.
In summary, there is a significant similarity between the Gmax
values obtained using the three interpretation methods and when
the Rdvalues are 1 and 2, the Gmax percentage deviation is higher
than 3.5% while it is less than 3.5% for Rdvalues from 3 to 8.
This suggests that the true shear wave arrival time is masked by
deviations due to near-field effects, as the predicted and theoret-
ical arrival times are not the same. The potential sources of de-
viations reduce as Rdincreases. The predicted Gmax percentage
deviations are presented in Table 5. The distance between the
tips of the transmitter and the receiver was 184mm and the shear
wave velocity of the soil was 386.6m/s. The predicted shear
wave arrival time corresponds to the theoretical arrival time of
0.476ms. The theoretical shear wave arrival time was not ob-
tained using the output from the receiver because there was an
initial downward deflection of the output signal due to the near-
field effect. The input frequency has a significant influence on
the near-field effect and this effect is not discernible when fre-
quency is increased. Jovicic et. al., (1996) has also found similar
trends. These results demonstrate the need to carry out a series
of tests with different input frequencies in order to eliminate the
near-field effects [15].
Suitable Waves for Bender Element Tests: Interpretations, Errors and Modelling Aspects 1512016 60 2
Fig. 4. A representative mesh of the finite element model.
Tab. 4. A Comparative Study of Plane Strain and Axisymmetric Elements
Frequency, Hz %Gmax error
Plane strain elements Axi-symmetry elements
2098 +4.57 -19.5
10489 +1.32 -25.2
Fig. 5. Input for estimation of shear wave velocity through a single Sine wave.
Period. Polytech. Civil Eng.152 Muhammad E. Rahman, Vikram Pakrashi, Subhadeep Banerjee, Trevor Orr
Fig. 6. Horizontal deflection of the receiving tip bender element for a single Sine wave input.
Tab. 5. Percentage error in Gmax predicted from shear wave velocity.
Frequency,
kHz Rd
Predicted
arrival time,
ms vs, m/s Predicted
Gmax, MPa
Calculated
Gmax, MPa % deviation
2.1 1 0.492 373.98 313.57 334.45 6.24
4.2 2 0.486 378.60 321.37 334.45 3.91
6.3 3 0.480 383.33 329.45 334.45 1.49
8.4 4 0.483 380.95 325.37 334.45 2.71
10.5 5 0.483 380.95 325.37 334.45 2.71
12.6 6 0.484 380.17 324.03 334.45 3.12
14.7 7 0.480 383.33 329.45 334.45 1.49
16.8 8 0.479 384.19 330.92 334.45 1.06
Suitable Waves for Bender Element Tests: Interpretations, Errors and Modelling Aspects 1532016 60 2
Fig. 7. Percentage error for the estimation of small strain shear stiffness using a single Sine wave input.
4.6 Estimation using a train of five Sinusoidal Waves
With unchanged mesh, boundary and material properties, a
train of five sinusoidal waves (Fig. 8) was used next in the trans-
verse direction at the tip of the transmitter, in FE analyses with
Rdvalues between 1 and 8. The shear wave outputs (Fig. 9) and
the Gmax errors (Fig. 10) indicate that the Gmax values obtained
using the first inflexion interpretation method are 17.4%, 4.6%,
3.74% and 3.3% for Rdratios of 1, 2, 3 and 4 respectively while
for remaining Rdvalues the percentage errors are less than 2%.
The results of the analyses using the first peak-to-peak input and
output wave method with an Rdratio of 1 give higher percent-
age errors while the remaining results give reasonably consistent
errors of between 1.37 and 2.6 percent for the calculated Gmax.
The single sine and the continuous sine signals behave very sim-
ilarly with respect to the initial downward deflection. The results
obtained using the continuous sine wave indicate that errors due
to (i) wave interference at the boundary, (ii) the near-field effect
and (iii) a non-one-dimensional wave travel mask the true shear
wave arrival time. The second and third sources of error reduce
as the input frequency, i.e. Rd, increases.
4.7 Estimation using a Distorted Sine Wave
A single distorted sine wave (Fig. 11), instead of a single stan-
dard sine wave in the transverse direction at the tip of the trans-
mitter was examined next as an input signal. The calculated
Gmax value is strongly dependent on the amplitude ratio (Arroyo
et al. (2003), which is the ratio between the amplitude of the
first upward cycle to the amplitude of the second downward cy-
cle. Jovicic et. al., (1996) recommended that the amplitude of
the first upward cycle of the wave should be reduced so as to
cancel out the near-field effect. A parametric study was carried
out to examine the effect of this ratio on the initial deflection.
Analyses carried out using amplitude ratios of 1/1.5, 1 /2, 1/3,
and 1/4 and the initial deflection values were normalised with
respect to the initial deflection value for an amplitude ratios of
1/3. The output waves (Fig. 12) indicate that the near-field ef-
fect reduces as the amplitude of the first upward cycle of the
wave decreases and is zero when the cycle ratio is 1/3. The
normalised deflection decreases with reducing cycle ratio up to
0.33 before increasing. The amplitude ratio of first upward cy-
cle to the first downward cycle of the waves can be chosen so
as to significantly cancel out the near-field effect. Consequently,
the distorted sine wave is favourable for avoiding the problem
of the near-field effects and determining the first arrival time as
compared to a single sine wave or a continuous sine wave.
5 Conclusions
Extensive research on bender elements test has been carried
out by many researchers in last few decades, but precise guide-
lines for carrying out such tests have not yet been established.
It is usually recommended to try and compare several methods
when using a particular test for the first time on a particular soil
to determine its small strain dynamic properties, in order to im-
prove confidence in the results obtained [15] and [4].
The investigations presented in this paper suggest that in ad-
dition to the soil types, the choice of input signal in bender ele-
ment tests may influence the determination of Gmax. The effects
of the input signal type on the Gmax for a particular soil can be
determined using finite element analysis and laboratory experi-
ment. However it is expensive, time consuming and need more
Period. Polytech. Civil Eng.154 Muhammad E. Rahman, Vikram Pakrashi, Subhadeep Banerjee, Trevor Orr
Fig. 8. Input for estimation of shear wave velocity through a continuous Sine wave.
Fig. 9. Horizontal deflection of the receiving tip bender element for a continuous Sine wave input.
Suitable Waves for Bender Element Tests: Interpretations, Errors and Modelling Aspects 1552016 60 2
Fig. 10. Percentage error for the estimation of small strain shear stiffness using a continuous Sine wave input.
Fig. 11. Input for estimation of shear wave velocity through a distorted Sine wave.
Fig. 12. Horizontal deflection of the receiving tip bender element for a distorted Sine wave input.
Period. Polytech. Civil Eng.156 Muhammad E. Rahman, Vikram Pakrashi, Subhadeep Banerjee, Trevor Orr
sample to determine the suitable wave for a particular soil us-
ing laboratory experiment than finite element analysis. Based
on this study it is concluded that for a particular soil it better to
conduct finite element analysis to select suitable wave and then
conduct laboratory experiment using best wave to determine the
Gmax.
A distorted sine wave is observed to be more favourable than
a single standard sine wave or a train of sine waves due to its
superior ability for cancelling out the near-field effect. The near-
field effect is dependent on the ratio between the amplitudes of
the first upward wave to the first downward wave of the input
signal (Rd). The Gmax values obtained from single standard sine
and continuous sine waves presented in this study suggest that
if the Rdratio is 3 or above, the near-field effect does not sig-
nificantly influence the measured travel time. However, it was
noted that for low values of Rdthere is an initial downward de-
flection of the trace before the shear wave arrives, representing
the near-field effect.
A comparison between the results of FE analyses using plane
strain elements and axisymmetric elements have been carried
out and it is found that the plane strain elements yield better
results compared to axisymmetric elements. When using rela-
tively simple FE models, employing fine mesh sizes were ob-
served to yield the best results, although acceptable results can
be obtained rapidly to investigate different testing scenarios even
when a medium mesh size is considered.
Acknowledgements
The authors gratefully acknowledge the involvement and help
of the following:
Department of Civil Engineering, University of Dublin, Trin-
ity College,
Nishimatsu Construction Co., Ltd,
The Geotechnical Trust Fund of the Institution of Engineers
of Ireland.
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