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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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