1. Introduction

Bender Element is a non-destructive test performed in soil specimens to determine the small-strain shear modulus (G_max) of the soil. G_max is an important soil property which helps us to understand the elastic behaviour of the soil and to evaluate its response to dynamic loading, such as earthquakes, passing vehicles and vibrations.

This post intents to give a summary on the testing procedure of bender elements in triaxial specimens of soil for the determination of the small-strain shear modulus, Gmax. It is advised to the reader to use also the Glossary of Terms at www.vjtech.co.uk for the definition of the terms used herein.

1.1 What is a Bender Element?

A bender element is a piezoelectric transducer which can convert electrical energy into mechanical energy and vice versa. Practically, this means that whenever these elements are supplied with a voltage they are deforming on a specific way. On the other hand, when they are deforming they produce a small voltage. As such, bender elements are used to examine the propagation of ground waves through a soil specimen and measure their velocity which depends on its elastic properties. To do so, a bender element set consists of two elements which are installed at the opposite sides of a soil specimen. One of the elements acts as a transmitter and the other as a receiver. In simple terms, each element consists of a pair of two ceramic plates which are separated by a thin metallic sheet. There is a specific way the two ceramic plates are wired to allow them to deform crosswise, as shown in Figure 2 (left), when they are supplied with a DC voltage. Usually, the DC voltage is supplied in the form of a sine wave, as shown in Figure 2 (right). The sine wave is causing the bender element to deform on the same way causing the soil particles to move as well. The movement of the particles is producing a shear wave (S) that propagates towards the other side of the specimen and arrives there after some time. When it arrives, the shear wave is causing the deformation of the receiving element (the one at the top cap) and this produces a corresponding electrical signal, many times smaller than the electrical signal than initiated the wave. By synchronising and comparing the two electrical signals, the time of travel is obtained. Therefore, by knowing that the distance of travel is the tip-to-tip distance between the two elements, the speed of the shear wave, Vs, can be determined as:

where t is the time of travel and L is the distance that the wave travels. L can be taken as the distance between the tips of the two bender elements (transmitter and receiver).

G_max=ρV_s²        (2)

where ρ is the bulk density of the soil.

Bender elements are encapsulated within the base pedestal or top cap of the triaxial cell and are sealed with an epoxy coating that insulates them. In this way, the received signal is not interfering with any electrical noise that might occur during the process. Apart from the two piezometric elements, two devices are needed to perform the Bender Element test.

• Function generator: This is needed in order to create the sinusoidal curve of the transmitting signal shown in Figure 2 that will produce the deformation of the transmitting element. The user selects the desirable period (or frequency) and amplitude of this signal. In most of the cases, frequency ranges between 1 kHz and 50 kHz and amplitude between 1V and 12 V. The selection of the amplitude depends on whether the captured signal is clear enough to be used for the determination of the travel time.
  
• Digital oscilloscope: This acts as a data logger and is recording the signal captured by the receiving element. Moreover, the oscilloscope synchronises the two electrical signals, i.e. transmitted and received, in order to be comparable. The received signal can be as small as 10^-4 times smaller than the transmitting signal; therefore the oscilloscope must be very sensitive with low amplitude resolution (10^-5 V). Also, its time resolution must again be very low (in the order of 10^-6 s) so that the signals are well defined. Finally, the oscilloscope can incorporate averaging functions which will average a number of received signals to eliminate the electrical noise.

1.2 Types of Bender Elements

  Most frequently, bender elements are installed along the vertical axis of triaxial specimens and are used to determine the shear modulus at the vertical direction; these are called Vertical Bender Elements and consist of the transmitting element (Transmitter) at the base, which produces the shear wave, and the receiving element (Receiver) at the top which captures the shear wave. When there is a need to determine the shear modulus in different planes, mostly to determine soil’s anisotropy, bender elements can also be installed at the mid-height of the sample, at opposite sides with the elements looking at each other. These are called Horizontal Bender Elements (Figure 2) and the shear wave is propagating on the horizontal direction. The installation of the horizontal bender elements is a very delicate process and needs to ensure the minimum disturbance of the soil. The horizontal bender elements are not embedded in the pedestal and top cap, as the vertical ones, but they bring a rubber sleeve that allows them to protrude the soil only by the minimum depth and avoid any leaks through the elastic membrane. Usually, the horizontal bender elements are installed together with on-sample transducers to provide comprehensive instrumentation of the triaxial specimen. Finally, two sets of horizontal benders can be installed on the sample at an angle of 90°.

The anisotropy of the soil structure in the small-strain shear modulus is defined as the ratio between the vertical shear modulus and the horizontal shear modulus. The horizontal small-strain shear modulus is given from Equation (2) if Vs is the shear wave velocity determined by the horizontal Bender Element test.

Apart from shear waves, bender elements can be used for the propagation of compression (P) waves through the specimens. To produce this kind of waves, the elements have to be wired differently than in the shear wave elements in order to deform parallel to the direction of the wave propagation. Compression waves are faster than shear waves and their use can be useful for indirect measurement of the saturation level of the sample. This can be done by comparing the speed of the compression wave Vp that propagates through the specimen with the speed of the compression waves when they propagate through water. Since it is known that Vp=1450 m/s for water, the speed of the compression waves through a saturated specimen should have an approximate value.

2. Standards

ASTM D8295 – Standard Test Method for Determination of Shear Wave Velocity and Initial Shear Modulus in Soil Specimens using Bender Elements

3. Procedure

 3.1 Apparatus preparation

 To perform a Bender Element test on a triaxial apparatus, the base pedestal and the top cap must have embedded a set of bender elements. Triaxial cells must be specifically designed for bender element testing so that they allow the cables of the elements to pass through them without any water leaks. Usually, pedestals and top caps in these cells are interchangeable and can be replaced any time the user desires to run a Bender Element test.

Before using them, the bender elements must be checked in order to achieve the correct polarity. If this is the case, then the transmitted and received signals will start on the same direction at the time graph, i.e. they will both move initially upwards. This can be checked by bringing the two element tips into contact and firing a signal; the results should be two signals moving on the same direction (Figure 4). If the polarity is reversed, the received signal will first move on the opposite direction from the transmitted signal. The reversed signal should be avoided because it can be mixed up with the reversed first arrival seen in received wave forms caused by the near-field effect (see section 3.3). When the correct polarity is found, the base pedestal and top cap must be marked so that it can be visible the correct orientation of the elements when installing them into the specimen.

Moreover, the bender elements must be checked for possible time delay between the transmitted and received signals. This delay, although it is very low, it could be critical for the correct calculation of the shear wave velocity, therefore it should be eliminated. The best way to determine the delay between the two signals is to bring the two elements in direct contact and firing a square-type signal. If no delay exists, the two signals should almost match. If a delay can be detected, it should be determined and extrapolated from the actual test results.

3.2 Specimen preparation

 The preparation of the triaxial specimen is exactly the same as for normal triaxial tests (see also Triaxial Testing - An Introduction support document by VJTech). The specimens are undisturbed, coming from a borehole and trimmed to the right size, or reconstituted to specific dimensions, density and moisture. If reconstituted, the specimens can be formed on the base pedestal to avoid any disturbance during the installation.

The user must ensure a very good contact between the element tip and the soil sample so that the deformation of the element produces a nice and clean shear wave. The sample placed on the base pedestal with care to avoid any damage that might be caused during the penetration of the element. Moreover, the size of the element must be small enough to prevent any cracks occurred. The donut-shaped porous stone that is placed around the bender element helps to leave exposed only the necessary length that will penetrate the sample. When the specimen sits on the pedestal it should not be rotated because this can cause the loss of the good contact. After the installation of the sample on the pedestal, the top cap is placed on the top surface of the specimen and is slightly pushed so that the element penetrates. A rubber membrane is placed around the sample to prevent any contact of the soil with the confining fluid. Two O-rings are placed on the pedestal and top cap to seal the membrane against the sample. Normally, the applied confining pressure will offer a better contact between the soil grains and the elements and therefore test results will be better during the consolidation stage.

Another important consideration during the sample installation is the alignment of the bender elements. As discussed in section 3.1, the pedestal and the top cap must be marked at the outside so that they can be aligned during the installation. The two elements, transmitter and receiver, must be parallel in order to achieve a good quality received signal.

In the case of horizontal bender elements, their installation happens after the sample has been formed, placed on the based pedestal and wrapped with the rubber membrane. Two small holes are formed on the membrane and the rubber sleeve with the element are passing through it and pushed against the sample to establish a good contact. When they are in place, the interface between the elastic membrane and the rubber sleeve is sealed with some appropriate substance (e.g. silicone).

3.3 Test procedure

From a geotechnical point of view, Bender Element is a very simple test that takes no more than a few seconds to be performed. However, care should be taken to the preparation of the specimen and the correct installation of the elements which will ensure a good response from the receiver element and clear captures of the received wave, as described in the previous sections.

Initially, the triaxial sample is been prepared in the same way as in a normal triaxial test with the inclusion of the bender element set at the base pedestal and top cap. Additional sets can be installed in the specimen horizontally. The triaxial test starts by applying set values of cell and back pressure in order to saturate and then consolidate the sample. At any point, during this process, the user can select to fire a signal and measure the small-strain shear modulus, G_max, by using Equation 2. Normally, G_max has to be determined after the end of the consolidation process in order to evaluate the sample condition prior to its compression and failure.

The Bender Element test starts by providing a sine wave signal to the transmitting element, as shown in Figure 2. The element is then deforming towards one side, then to the other and finally comes back to the starting position. The magnitude of this deformation is controlled by the amplitude of the signal which usually ranges between 1 V and 12 V. The actual movement of the element is very small and is not causing a plastic deformation to the sample. If the received signal is not captured very clearly by the receiver, the user should increase the amplitude. It is found that an amplitude of 10 V works well for specimens with a length of 140 mm. On the other hand, the speed of the movement is controlled by the set frequency of the signal (= 1/T, where T is the period of the sine wave) which can range from 0.5 kHz and up to 50 kHz. In order to get comparable results, both transmitter and receiver should operate in the same frequency. Experimental results have shown that the received signal gets its maximum amplitude when the input frequency matches the resonant frequency of the bender elements. At higher frequencies, the received signal appeared to be weaker and more difficult to interpret. Therefore, the resonant frequency of the bender element set is considered as the optimum input frequency for the sine wave.

As soon as the shear wave reaches the receiver, the sine wave signal is captured by the element. Ideally, the received signal will look like the one in Figure 4, with the first wave being clearly identified and the subsequent ones gradually faded away. The first wave in the received signal corresponds to the first arrival of the shear wave and is the one that is used to determine the shear velocity. However, in reality, the received signal is not that clear but it contains a number of deficiencies which will make it difficult to be distinguished. The cause of these deficiencies might be one or more of the following:

• Near-field effect (Figure 5): This is identified by a minor “drop” in the received signal which could mistakenly be identified as a reversed first arrival of the shear wave. Also, can cause a faulty identification of the shear wave if the method of the first arrival, for the interpretation of the results, is adopted. In reality, though, the first arrival comes immediately after this small drop. The near-field effect is caused by waves that are travelling with different velocities through the sample and are coupled, producing this small “disturbance” in the received signal (Arulnathan et al, 1998). The near-field effect is common in cases where the two tips of the elements are very close, i.e. when the sample height is small. The property that shows the susceptibility of a testing system to be affected is the wavelength ratio which is defined as the ratio between the tip-to-tip distance and the wavelength of the shear wave. When this ratio is greater than 2, near-field effect is smaller.
• P-Waves: P-Waves travel much faster than S-Waves, therefore they might reach the receiver element first. In this case, the time of travel can be overestimated. The best way to reduce this effect is to place the two elements in parallel directions when setting up the triaxial specimen.
• Noise: Electrical noise, caused by poor insulation of the bender elements, can interfere with the transmitted signal and captured in the received signal. If the signal appears to be too noisy, then the Bender Element system must be checked for damages. Also, noise can be caused by mechanical movements, for example from the loading of the sample.

 Normally, the transmitted signal includes a number of signals fired at small time intervals (e.g. every 5 seconds). The final transmitted and received signals are then averaged in order to produce smooth signals that will be easier to interpret. The averaging is obtained with the use of applied filters with varied intensity. The strongest the filter the smoother the obtained signal. However, the user should select the least heavy filter, which can produce reasonable results.

3.4 Interpretation

The most difficult part in a Bender Element test is the interpretation of the results, i.e. the correct determination of the travel time for the shear wave. Most of the cases, the received signal contains noise and/or is affected by the arrival of other types of waves which makes the shear wave difficult to be distinguished. For this reason, several methods have been implemented to the interpretation of the Bender Element test. The most common methods are going to be presented in this document, Peak-to-Peak and Start-To-Start methods. Other methods can be found in the literature and are not covered in this document.

The transmitted and received signals must be synchronised, by the oscilloscope, and presented on a single plot so that they can be analyzed. The time difference between those two will define the time of travel for the shear wave. The transmitted signal is very well defined and its characteristics can be seen very clearly, i.e. its start, its first peak etc. But the received signal might not be very clear and so the user must make a judgement about what method of interpretation must be used.

 4. VJTech Bender Element testing system

4.1 Bender Element testing system

VJTech offers the Bender Element system that is shown in Figure 2. The system consists of a set of elements fitted on modified base pedestals and top caps; therefore the bender element test can be performed on specimens during triaxial tests. The diameter of the modified pedestal and top caps can range between 50 mm and 100 mm allowing tests on these diameter samples. Moreover, horizontal elements are available and can be installed at opposite sides of the sample and at various heights. The system, also, includes a Bender scope unit which incorporates the necessary function generator and digital oscilloscope. The procedure for the test is fully automated through the csBender module of Clisp studio software. Some of the system features include: 

• The piezometric elements are implemented into capsules made by titanium to reduce their weight;
• Both shear (S) and compression (P) waves can be generated;
• S and P waves can be transmitted either vertically or horizontally;
• Maximum sampling frequency: 50 kHz;
• Convenient USB interface to PC;
• Built-in signal generator with sine, square and triangular waveforms.

4.2 csBender software

 csBender is a very powerful and user-friendly module of Clisp studio software from VJTech. This module can be used in conjunction with the Bender Element testing system by VJTech that allows the determination of the small-strain shear modulus G_max in triaxial soil specimens. The great advantage of csBender is that it offer the automation of the test which is performed in just a few seconds. At first, the user must select the appropriate settings for the transmitted signals, i.e. the frequency (in kHz) and the amplitude (in V) (Figure 7). The waveform type can also be selected. Although for a normal Bender Element test, the sinusoidal waveform must be selected, other shapes like triangular, square and arbitrary can be fired. Moreover, properties for the visualisation of the received signal can be selected. Additionally, the number of the transmitted signals that will be averaged is selected, together with the time interval between them. Specimen details (current height and bulk density) can be added in the software or gathered by active triaxial tests. Finally, the averaging options can be modified in the initial setup.

After the initial setting, the user can initiate the test whenever G_max needs to be determined. This is done automatically by firing the signals and capturing the received ones from the set of the bender elements. Both signals are then synchronized and plotted on the same graph with the same x-axis (time) and different y-axes (transmitting signal in V and received signal in mV) (Figure 8). Signals can be sorted into different groups based on whether they are vertical or horizontal waves and also based on the type of waves that have been measuring (S or P). If the repeated signals option has been selected, the software will fire repeated signals and then combine them to produce an average waveform which will be clean of any random noise that is affecting it. The user can select from a range of Low Pass Filters the most appropriate to get the best result. Graphical tools offer a very handy way to obtain the correct arrival times and determine the time of travel for the shear waves. The Peak-to-Peak method of interpretation is implemented into the software, offering an automated way of determining the time of travel for the shear wave. However, by using the on-screen cursors, other methods of interpretation can be used to determine the time of travel. Once this is determined, csBender works out the shear wave velocity Vs, or compression wave velocity Vp, and the shear modulus G_max of the sample. Results can be exported into excel files in order to be processed further or transferred.

5. References

• Arulnathan, R., Boulanger, R. W., and Riemer, M. F. 1998. Analysis of Bender Element Tests. Geotechnical Testing Journal, GTJODJ, Vol. 21, No. 2, pp. 120-131.
• Poul V. Lade. 2016. Triaxial Testing of Soils. John Wiley & Sons, Ltd.