Electronic Journal of Polish Agricultural Universities (EJPAU) founded by all Polish Agriculture Universities presents original papers and review articles relevant to all aspects of agricultural sciences. It is target for persons working both in science and industry,regulatory agencies or teaching in agricultural sector. Covered by IFIS Publishing (Food Science and Technology Abstracts), ELSEVIER Science - Food Science and Technology Program, CAS USA (Chemical Abstracts), CABI Publishing UK and ALPSP (Association of Learned and Professional Society Publisher - full membership). Presented in the Master List of Thomson ISI.
Volume 17
Issue 3
Civil Engineering
Available Online: http://www.ejpau.media.pl/volume17/issue3/art-07.html


Wojciech Sas1, Katarzyna Gabryś1, Alojzy Szymański2
1 Laboratory - Water Centre, Warsaw University of Life Sciences - SGGW, Poland
2 Department of Geotechnical Engineering, Warsaw University of Life Sciences - SGGW, Poland



In this paper the stiffness at small strains of selected cohesive soils from Warsaw area was evaluated on the basis of two varied laboratory techniques, namely, resonant column (RC) test and bender element (BE)  test. Both methods have different advantages and limitations  and  they are both increasingly popular. They are powerful and common laboratory tool for determining the shear wave velocity (VS) and hence the small-strain shear stiffness (Gmax) in soil. In this respect, special emphasis was placed on a brief presentation of these two techniques, together with some attempt to compare them. The values of Gmax obtained by the RC and BE tests were compared to study the effect of test methods. The main intention of the authors was to provide some contribution to the routine laboratory practice using resonant column and bender elements, with some insight in the interpretation of the received results. 

Key words: resonant column tests, bender elements tests, natural cohesive soils.


A crucial element in the solution process of geotechnical earthquake engineering problems is the measurement of soil’s dynamic properties [2]. Determination of soil response to e.g. earthquake or traffic vibrations requires some knowledge of the shear modulus (G) and the material damping (D) of the soil as well as how these properties can vary with shear strain amplitude.

At small strains, when shearing strain is less than about 0,001%, the relationship between the shear modulus and the shear wave velocity is the following:


VS – small-strain shear wave velocity,
ρ – unit
Gmax – small-strain shear modulus [4].

Among various methods for laboratory evaluation of shear modulus, the resonant column technique (RC) is recognized to be the most reliable [9]. In the resonant column method a cylindrical soil specimen is attached to the bottom and dynamically excited at its top. The torsional force operated at the top is produced with the use of an electrical motor consistings of four drive coils encircling four magnets fixed to a drive plate. The generated frequency can be up to 300 Hz. The vibration’s fundamental mode is found through the maximum amplitude of motion. Subsequently, from the resonant frequency the shear wave velocity and then the shear modulus, formula (1), are calculated using theory of elasticity. Material damping can be calculated based on two methods: half power bandwidth or a free-vibration decay curve created by shutting off the driving power.

Resonant column tests enable to determine the equivalent shear modulus of soil specimen for a strain level ranging from 10-4% to 0,5%. Basically RC tests can be carried out by the following three different procedures:

According to the above mentioned procedures a certain number of loading cycles is applied to the specimen, unfortunately precisely unknown.

Another disadvantage of RC technique is impossibility to control shear strain rate (). Generally, the average shear strain increases with growing strain levels (γSA) [10], according to the relationship:


Four orders of magnitude rise of the shear strain level causes a drop in the resonance frequency (fn) to the one third of its original value. Therefore, in RC test a shearing rate increases at least by three orders of magnitude going through small strains to intermediate. 

An additional drawback of RC method is the need of damping correction for problems with equipment compliance [18]. This compliance can manage the overestimation of damping, which mainly occurs at small strains. The RC technique requires the measurement of the current flowing through the driving coils. However, very often not the current but the input voltage is measured, which replaces it wrongly. This change does not have much influence on the wave velocities, on the contrary to damping ratios that are significantly overestimated. This happen due to the induction an opposite voltage in the coils by the motion of the magnets [2]. On the other hand, during RC tests small strain stiffness is measured with great accuracy, which is one of the main advantages of this technique.

Determination of soil stiffness at small strains in laboratory is difficult because of insufficient resolution and accuracy of load and displacement of the measuring devices. The usage of local strain transducers in the triaxial apparatus makes it possible to regularly carry out  measurements of small strain stiffness. However, this method can be rather expensive and is usually confined to some research projects. But, it has been proved that the addition of bender elements to a standard triaxial testing system simplifies and reduces costs of the routine tests of Gmax determination.

Bender elements, originally invented for soft soils [5], have opened new possibilities in laboratory testing, since seismic waves measurements are conducted at the same time with standard mechanical static/cyclic tests. These transducers, in addition to the simple installation in any conventional soil experimental device (such as oedometer or triaxial chamber) or more complex systems (resonant column, centrifuges, calibration chambers), enable transmitting and receiving shear as well as compression waves [17].

Detailed descriptions of the bender elements (BE) technique are presented in publications [13, 15]. However, as in the case of the resonant column, it is important to summarize the advantages and drawbacks of this method. The BE technique is attractive because of its simplicity. With the use of a single pulse excitation, one of the transducers, at one end of the sample, is excited. The time, which is required for registration of this excitation by the second transducer (so called receiving bender element), is registered by an oscilloscope. Using the travel length of the signal and the time measured during triaxial tests the seismic wave velocity, VS, is calculated [1, 5] according to the following equation:


VS – small-strain shear wave velocity,
L – the effective travel distance (the tip-to-tip distance),
TS – the travel time of VS

The elastic shear modulus (Gmax) can be subsequently derived from VS using equation (1). In the consequence, piezoelectric transducers present an inexpensive and universal solution for laboratory tests of seismic waves. Furthermore, particularly attractive are their capabilities, for example, in monitoring changes of stiffness with effective stresses (cementation process, load stabilization process etc.) [7].

The main drawback of the BE method is the difficulty in the interpretation of the results due to some degree of uncertainty. A clear identification of the arrival time is not always possible [1], therefore a great number of methods for the interpretation of experimental data from bender elements tests are commonly used, either in time or in frequency domain. Currently, there is not a single and accurate technique that can be adopted as standard [1]. Many foreign [3, 20] and Polish [17] authors described in detail the interpretation techniques of the bender elements data as well as the difficulties they encountered while analyzing the results.

The geometry of the sample, together with BE-related phenomena play also a significant part in the performance of BE tests and their quality. Especially, spatial and boundary conditions (among others: wave reflections, coupling or near-field effects, likewise overshooting at high frequencies) should be mentioned here as some limitations of this technique. Additionally, the lack of knowledge of the actual behavior of the piezoceramic elements inside the soil sample is one of the most important issue in BE testing. While free conditions, the main impact on the response of the transmitter has its own resonant frequency and not the excitation frequency as suspected. Under embedded conditions, the stiffening of the medium causes the increase in the natural frequency as well as in the damping ratio and the decrease in the magnitude of the oscillation [1]. Furthermore, the applicability of BE method is restrained for stiff geomaterials such as compacted soils, cemented soils (naturally or artificially) or weak rocks, because of the greater stiffness resistance between the tested materials and the transducers [7].

Considering the current discussion on the laboratory methods for the determination of soil stiffness at small strains it is interesting to compare results of the application of both tests (RC and BE) on selected cohesive soil from Warsaw area. This seems particularly important since relatively few publications can be found in the literature reporting similar experiments. Before presenting the results, we include a brief description of the test equipment and tested material as well as give some explanation of the testing procedures.


The Resonant Column Apparatus (Fig. 1) used for this work was developed by a British company, GDS Instruments Ltd., as an example of Hardin-Drnevich device, projected in configuration “fixed-free”. A detail description of this device can be found in previous papers of the authors, namely [13, 14]. None the less, the main characteristics of the apparatus are summarized below.

The application of the GDS Resonant Column Apparatus, shortly known as RCA, is the excitation of one end of a solid (or hollow) confined cylindrical soil specimen. The specimen can be isotropically-consolidated and such a situation is presented in this article. A bath of silicon oil is provided all around the specimen to avoid air migration through the membranes.

The sample is vibrated in torsion or flexure mode (flexure means here bending) through an electromagnetic drive system. The electromagnetic drive system incorporates four precision wound coils and consists of sintered neodymium iron boron “rare-earth” four magnets. RCA systems supplied by GDS Instruments are current driven using an transconductance power amplifier. This is due to the fact that the impedance of the RCA system changes with frequency. With higher frequencies, the current will be reduced for the same voltage. The torque will also be lower as it is directly proportional to current. It was considered necessary to apply here a current driven power amplifier.

The accelerometer is used to determine the resonance during RC tests. The internal LVDT is assembled for measurements of sample deformation. Back pressure and cell pressure are controlled by GDS Standard pressure/volume controller (STDDPC) and GDS pneumatic controller respectively. Both controllers and the data acquisition system are computer monitored [8].

Fig. 1. Photography of the GDS Resonant Column Apparatus (authors’ photography)

The Bender Elements employed in this work were manufactured also by GDS Instruments (Fig. 2) and were installed in the GDS Triaxial Automated System (GDSTAS) (Fig. 3). The GDS Bender Element system provides an easy way for measurements of the maximum shear modulus of a soil in the range of small strains in a triaxial cell. Elements are bonded into a standard insert which has two advantages. Firstly, the bender element insert is a modular device and can be simply fitted into a rightly modified top cap or pedestal. Secondly, should an element fail, it is easy and quick to replace the whole insert.

Elements are produced to allow both kinds of wave testing, S- and P-wave, to be performed during measurements (on the contrary to propagation directions). Additionally, the length of the bender element protruding into the soil has been optimised without any negative impact on the power transmitted by or received to the element. The construction of each element assures the maximum flexure at its tip, while only sticking into the sample by a reasonable distance. Thus, the life of the element is prolonged by increased resistance to breakage as well as easier preparation of the sample.

The GDS Bender Element system is connected directly into a master control box, which is combined with a PC [8,15].

Fig. 2. Photography of the top cap and pedestal with – GDS Titanium Bender Elements and Insert (authors’ photography)

Fig. 3. Photography of the GDS Triaxial Automated System (authors’ photography)

Both of the above-presented devices, that is: the Resonant Column Apparatus and triaxial chamber equipped with the Bender Elements, are available in the laboratory of Geotechnical Engineering Department and in Laboratory – Water Centre, Warsaw University of Life Sciences – SGGW.


The samples in this work represent three types of natural cohesive soil: clSa, sasiCl, saCl, very similar to each other in respect of physical properties. The tested soils were collected from the Warsaw area, more specifically, from the test site located near the express route (S2) built recently, between its two nodes “Konotopa-Lotnisko”. The main properties of the specimens are summarized in Table 1. It should be emphasized that each specimen was fairly homogeneous in texture and fabric. All tests were performed on undisturbed samples.

Table 1. Identification and soil index properties of selected cohesive soils from Warsaw area
Soil type
Water content
Bulk density
Liquid limit
Plastic limit
Plasticity index
Liquidity index


A set of CID triaxial tests was performed on 2 samples (01BE, 02BE) whereas a set of CIU triaxial tests was carried out on 3 samples (03BE–05BE). Resonant column tests were conducted on 4 specimens (01RCA–04RCA). The initial test stages for both devices were comparable and are indicated below.

Saturation stage
During the saturation stage a small amounts of cell and back pressure were applied in steps, with a consequent dissolution of the air contained in the intergranular space. A control system generated cell and back – pressure using respectively air or water interfaces. All pressures: cell, back and pore were measured by proper transducers, with the accuracy equal to 0,01 kPa. Volume changes were,  also,  registered by the high sensitivity pressure transducer (accuracy up to 1,0 mm3). When the end of the saturation process was detected, namely after the Skempton’s parameter B reached the value of at least 0,9, first measurements of seismic waves velocity using BEs installed in triaxial chamber were made together with first measurements of resonant frequency in RCA.

Consolidation stage
Each sample was subject to the same back pressure,  achieved in the last step of saturation, while the cell pressure varied, according to the mean effective stress required in the next steps. The consolidation stage was finished after a complete dissipation of pore water pressure and with no volume changes. The process of consolidation itself always consisted of several steps, as outlined in Table 2. During these phases the axial strain was measured using a LVDT transducer.

Shortly after each consolidation step, before the next one, RC readings of resonant frequency were made, followed by the acquisition of waves in the time domain with the BE. An accurate explanation how the authors measured resonant frequency and then how they used it in the further calculations of small-strain shear modulus (Gmax) can be found in [13, 14, 16]. Similarly, the exact procedures for measurements of seismic wave velocity with BE, as well as the chosen method for the interpretation of bender element results are contained in the positions [13, 15].

The last step of each triaxial test was the sample shearing, which was not performed in Resonant Column Apparatus. In order to achieve the strength characteristic (cohesion and friction angle) of the test material, the shearing was performed at the stress level equal in situ stress, as well as at the stress value smaller and bigger than in situ stress.

Table 2. Testing conditions for studied samples
Testing conditions
Isotropic consolidation: 9 stages, p' from 45 kPa up to 450 kPa, shearing at p'=25 kPa
Isotropic consolidation: 9 stages, p' from 90 kPa up to 450 kPa, shearing at p'=180 kPa
Isotropic consolidation: 13 stages, p' from 45 kPa up to 585 kPa, shearing at p'=360 kPa
Isotropic consolidation: 10 stages, p' from 45 kPa up to 450 kPa, shearing at p'=45 kPa
Isotropic consolidation: 12 stages, p' from 45 kPa up to 540 kPa, shearing at p'=90 kPa
Isotropic consolidation: 2 stages, p' from 135 kPa up to 225 kPa
Isotropic consolidation: 6 stages, p' from 45 kPa up to 315 kPa
Isotropic consolidation: 6 stages, p' from 45 kPa up to 315 kPa
Isotropic consolidation: 6 stages, p' from 45 kPa up to 315 kPa


The final results of the variation of shear wave velocity (VS), determined by BE measurements and calculated on the basis of  resonant frequency (f) from RC tests, at various stress conditions for all analyzed samples, are shown in Figure 4. Figure 5 illustrates the variation of small-strain shear modulus (Gmax) at different mean effective stress in RC and BE tests. Both BE and RC measurements present an increase in values of VS and Gmax with changing, i.e. growing, stress conditions. Analysing these two figures one can  notice, that for clayey sand samples (01BE, 05BE, 01RCA, 03RCA, 04RCA) shear wave velocity as well as small-strain shear modulus obtained using resonant column technique reach greater values than the same parameters determined from bender elements method. In Figure 6 the stiffness values are compared for selected two out of five clayey sand specimens (05BE, 03RCA). In this example the rise in the values of small-strain shear modulus with an increase of effective pressure at a power function. The derived equations for the relationship between Gmax and p’, respective to each method, do not show a very good agreement with each other. Only at low pressures (up to around 100 kPa) the stiffness values obtained from RC and BE methods are similar. However, when the stresses increase the determined parameters significantly differ from each other. For example, for p’=315 kPa the value of Gmax from RC tests is around 50% higher (Gmax≈400 MPa) than from BE experiments (Gmax≈200 MPa).  Regarding the sandy clay samples (04BE, 02RCA) it is found that VS and Gmax from BE tests are consistently larger that RC measurements over the range of pressures (Fig. 4, 5). Generally, for a range of effective stresses 45–315 kPa Gmax (RCA) vary between 25 MPa and 400 MPa, while Gmax (BE) reaches the value from 70 MPa to 250MPa.

Fig. 4. Shear wave velocity from BE and RC tests as a function of mean effective stress

Fig. 5. Small-strain shear modulus from BE and RC tests as a function of mean effective stress

Fig. 6. Small-strain shear modulus of two clayey sand samples from BE and RC tests

The RC test is known to be a very accurate dynamic test as it is confirmed in these (and previous) experiments. The large differences in stiffness values from BE and RC measurements obtained by the authors are quite striking, especially since previous literature [6, 21] report that the results from bender elements tests are comparable to the resonant column one. Both methods should provide similar values of VS and Gmax. The reasons for these differences can be twofold. Firstly, the authors used two different apparatus, i.e. the GDS Resonant Column Apparatus and the GDS Triaxial Automated System with Bender Elements. These devices are characterized by slightly different research methods, outlined in the introduction of this paper. Secondly, the tested material was not the same. As shown in table 1., the index properties of the selected cohesive soils are a bit different.

Figure 7 presents the variation in shear modulus (G) with shear strain (g), derived from resonant column tests and triaxial tests with bender elements. The figure shows a further combination of these two techniques. A strong non-linearity and the dependence on stress level is evident. It is confirmed that for small values of mean effective stress, such as p’=45 kPa (Fig. 7), the results from resonant column and triaxial tests (TRX) are in a good agreement. For greater pressures, for example p’=180 kPa (Fig. 7), G values from RCA are higher than from TRX. G value determined from bender elements measurements by assuming that shear strain is equal to γ=10-4% is also indicated on this chart. In this case, with the presumption of the shear strain from BE test on the level of 10-4%, the small-strain shear modulus Gmax (BE) reaches quite big values in comparison with Gmax (RC). This finding suggests some impact of strain level on the results. This significant increase in shear modulus from BE test is probably because of the assumed quite low shear strain level.

Fig. 7. Variation of shear modulus with strain from BE and RC tests


This article addresses the laboratory methods of the determination of small-strain shear modulus (Gmax), recognized as a reference parameter in soil. In the laboratory, at present, the most widely used method of estimating shear wave velocity (VS) and then soil stiffness (Gmax) is the bender elements test. The authors made a comparison of this modern technique (BE test) with resonant column testing, which has been successfully applied for more than 40 years in the field of soil dynamics, providing a reliable method of determining shear modulus (G).

Comparisons were made between the values of shear wave velocity and small-strain shear modulus obtained from  these two techniques. Calculations were based on the studies carried on natural cohesive soils from Warsaw area. Soil index properties and testing procedures were summarized in tables while results were illustrated on graphs.

From the data presented in this paper the following conclusions may be drawn:

It is possible to estimate fair values of shear modulus at small strains using resonant column technique and bender elements method. Both kinds of measurements show the same trend of variations of VS and Gmax with mean effective stress. Stiffness parameters increase with an effective pressure, although it was observed that for clayey sand samples small-strain modulus from RC tests grows faster and its values are consistently higher than those obtained by BE tests. For sandy clay samples the results were opposite; BE method resulted in bigger Gmax values.

Although the grain distribution curves and the main index properties of studied soil samples were similar, different values of shear modulus (G) were obtained. This indicates the need to perform more series of tests on a larger number of specimens taken from one geotechnical profile. Consequently, the planners of the geotechnical constructions should analyze various options for selection of proper geotechnical parameters to construct safely and economically. 

Results demonstrate that these two testing methods compare well, only for relatively small pressures. With increasing consolidation pressure, the results obtained are not comparable. Due to the kind of the resonant column device, the authors of this manuscript employed voltage-based measurements instead of current-based measurements, recommended by some scientists [2] as more precise. Moreover, there is still a great discussion about the proper measurements technique not only in RC testing but also in BE testing, where exist two categories of methods: time domain (TD) and frequency domain (FD) [1]. In the case of bender elements, the authors performed time domain analysis as the one that produces reasonable results [1]. Generally, from both BE and RC tests the stiffness non-linearity and dependence on stress level is evident. From the literature review [11] as well as based on the own laboratory experience, the authors state that a large effect on the value of shear modulus has the excitation frequency (30Hz<f<300Hz for RC test and 1Hz<f<5Hz for BE test) and the method of wave production too, different for described in this article research techniques.

To avoid large discrepancies between the results, the authors suggest a modification of the GDS Resonant Column Apparatus by accommodating bender elements and then performing simultaneous readings of the RC frequency and the BE waves. A necessary future work is the combined measurements, RC and BE, in order to achieve a valid and comparable results.

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Accepted for print: 22.07.2014

Wojciech Sas
Laboratory - Water Centre,
Warsaw University of Life Sciences - SGGW, Poland
Nowoursynowska Str. 159, 02-776 Warsaw, Poland
Phone: + 48 22 59 35401
email: wojciech_sas@sggw.pl

Katarzyna Gabryś
Laboratory - Water Centre,
Warsaw University of Life Sciences - SGGW, Poland
Nowoursynowska Str. 159, 02-776 Warsaw, Poland
Phone: + 48 22 59 35401
email: katarzyna_gabrys@sggw.pl

Alojzy Szymański
Department of Geotechnical Engineering,
Warsaw University of Life Sciences - SGGW, Poland
Nowoursynowska Str. 159, 02-776 Warsaw, Poland
Phone: + 48 22 59 35401
email: alojzy_szymanski@sggw.pl

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