Introducing the stiffness as a feature of construction soils allows to adopt computational techniques commonly used for materials of elastic properties. Fundamental mechanical parameter of the soil which describes its stiffness, is soil modulus. Research for the mechanical properties of soils conducted over the last 30 years have led to the conclusion that the basic parameter determining the soil stiffness is shear modulus G . This parameter is the instantaneous value under the conditions specified by strain level and the stress state, which depends on the load history, its velocity and character (static, dynamic, cyclic), drainage conditions as well as the type of structure that transmits the load. The nonlinear nature of the stiffness variability for the shear modulus G can be presented as: (1)
Gmax – initial shear modulus;
F(γ) – stiffness degradation function;
γ – shear strain in soil.
Initial shear modulus Gmax is widely recognized as a parameter of the state of the soil . It can be assumed that the value of the dynamic shear module for very small shear strain is the same as the Gmax modulus for static problems in both “drained” and “undrained” conditions . For this reason, many of the techniques used in dynamic and seismic applications have been adopted for estimating the Gmax module. In dynamic studies based on the measurement of shear wave propagation velocity in a soild medium, the initial shear modulus is derived from a simple relationship: (2)
ρ – volume density of soil;
Vs – shear wave velocity.
This dependence is used in many uncomplicated research methods, where the main goal is to measure the time of the propagation of generated elastic wave on the separated area of natural subsoil or soil specimen [3, 4]. Determined in this way modulus is a value corresponding to very small shear strain level.
Over the past 50 years, a number of correlation formulas have been developed to estimate the initial Gmax strain modulus based on the results of the velocity measurement of the transverse mechanical wave in soil and the study of other geotechnical characteristics. For example, Hardin and Richart  have developed one of the first correlations for uniformly-grained coarse sand (Ottawa sand, No. 2030, d50 = 0.72 mm, CU = 1.2): (3)
e – void ratio,
p’ – average effective stress [kPa],
pref – reference pressure (pref = 100 kPa).
On the basis of further investigations of natural and artificially prepared from sanded soils, Hardin and Black  reported a universal relationship: (4)
Based on research carried out worldwide in the field, it has been confirmed that the value of the initial shear modulus can be functionally linked to the void ratio and the average effective stress : (5)
where A is a constant dependent on soil type, grain properties and distribution, f(e) is a function of void ratio e reflecting the soil density, m is the stress exponent reflecting the confining pressure.
A number of studies conducted in the world on noncohesive soils differing in grading from the d50 diameter and the coefficient of uniformity CU have enabled the development of formulas to determine the value of the Gmax modulus. The formulas shown in Table 1 relate to noncohesive soils with specified grain size distribution characteristics (d50 and CU). Based on the recent research, it has been found that for non-uniformly grained soils, these formulas produce overstated values of Gmax modulus. Research carried out over the last decade, including Wichtmann and Triantafyllidis [8, 9, 10] at the Karlsruhe Institute of Soil Mechanics and Rock Mechanics, have led to develop a universal formula that will take into account the value of the coefficient of uniformity CU by function: (6)
where appropriate constants were proposed as : (7) (8) (9)
In Figure 2 the relationship between Gmax values and coefficient of uniformity Cu is given and in Figure 3 between Gmax and average effective stress p′, for constant value of the void ratio e = 0.37. The value of the Gmax modulus decreases with the increase of the coefficient of graining uniformity Cu and increses with the increase of the average effective stress p’.
This article presents selected issues related to the continuation of non-cohesive soils survey conducted at the Institute of Soil Mechanics and Rock Mechanics in Karlsruhe. The purpose of the present study is to verify the correlation between the characteristics of grain size distribution of non-cohesive soils and the value of the dynamic shear modulus. For this purpose, elastic wave propagation studies were conducted on specimens with artificially generated grain size distribution, using piezoelectric bender elements (BE) and triaxial compression cell. The purpose of the research was also to verify and evaluate the different methods of interpreting BE test results.
2 Bender elements
Currently, the use of bender elements is a commonly used measurement method in geotechnical laboratories, although the idea itself is based on the phenomenon of simple and inverted piezoelectricity discovered in 1880 by Jacques and Pierre Curie. However, first practical attempt was carried out more than 100 years later by Donald Shirley and Loyd Hampton in 1977. Initially it was used to measure shear wave velocity in marine sediments using piezoceramic materials. Since then, numerous upgrades of electronic components such as transducers have been made and in spite of the enlargement of the spectrum of uses, the very idea of research still remains unchanged.
Bender elements are two-layer piezoelectric plates covered with a protective resin (Figure 4). Elements work in pairs. By mounting them to the opposite ends of the cylindrical specimen, the system and components of the triaxial compression apparatus (Figure 5) are most commonly used for this purpose.
Applying electrical voltage of variable amplitude (±10V = 20Vpp) to one of the plates causes alternating and simultaneous shortening of one of the layers of the element and elongation of the second layer. As a result, this leads to a cyclic bending of the entire element and the generation of a elastic mechanical wave inside the tested specimen. The wave propagating along the soil specimen is a shear wave and accompanying compressive waves that propagate perpendicular to the plane of the bender element. The wave reaching the receiver bends the piezoelectric plate generating the voltage that the oscilloscope records (Figure 6).
Measurement of the wave transit time between the transmitter and the receiver is crucial for determining the velocity of propagation of the wave in the tested medium. Then, it leads to the calculation of the practical parameter which is the value of the Gmax module according to the equation (1.2).
The system controlling device allows selection of the waveform generation method in terms of the shape of a single waveform. It is possible to generate rectangular, trapezoidal, triangular waveforms, although sinusoidal waves are most commonly used. It is also possible to generate a continuous or pulse wave according to the selected range of interpretations of the obtained results.
methods of interpreting the results of a single-pulse test:
visual detection method for the first signal level deviation (first arrival FA, visual picking VP) – depends on determining the time that has elapsed since the beginning wave generation until the first recording of the disturbance in the received signal (tFA, tVP); it is most commonly used in practice and also the simplest method of interpretation of results; its major disadvantage is the possibility of interfering with measurement by the disordering influence of compression waves;
peak signal detection (major peak-to-peak, PP) method – depends on the measurement of the time difference (tPP) between the time the peak occurs in the transmitted signal and the time when the first peak in the received signal is received; similar to the first deviation method, it does not require any analysis beyond the visualization of two characteristic points on the oscillogram; its disadvantage is the ambiguity of the result in the case of receiving a number of closely spaced peaks with small amplitude differences resulting from, for example, the heterogeneity of the specimen of the soil being tested;
cross power spectrum analysis (cross-power spectrum, CS) determines wave propagation time tcs: (10)
this method is based on the angle of inclination of a straight line, which is the linear approximation of the phase angle relation φ from the frequency f for the complexity of the cross-power spectrum GXY of the signals X(t) and Y(t), determined according to the following formula: (11)
where LX (f) and LY (f) are linear spectra of the signals X(t) and Y(t), determined by Fourier transformation (FFT): (12)
and L*X(f) is the conjugate coupling of LX(f);
cross correlation method (CC) – is a measure of the degree of matching of a generated and received signal in the frequency domain; determining the time tcc of the best time-domain matching is the result of the analysis of the cross-correlation coefficients CCXY of the signals X(t) and Y(t), recorded at tr time; this coefficient defines the following formula: (13)
where tCC is the time delay between the analyzed signals, indicated unequivocally by the position of the maximum CCXY value; having the cross power spectrum GXY(f) of the signals X(t) and Y(t), CCXY(t) is the inverse Fourier transformation GXY (f): (14)
methods of interpreting the results based on continuous wave generation:
the π-points, phase frequencies method (π-points); this method is based on generating sinusoidal waves of smoothly adjustable frequency (0-30kHz) and recording the selected frequency values for which the transmitted and received signals have this the same or the exact opposite phase (colloquially: the signals are in phase or counter-phase); phase compatibility is adjusted visually by observing the screen of the oscilloscope that should display the signal amplitude values in the mutual function relation (ie the x coordinate is amplitude of the transmitted signal, and y – received; graph in straight line means phase compatibility); chosen frequencies should differ by one step: kπ – phase matching should appear at constant frequency intervals (angular). Due to k value the number of wavelengths N that lie on the Ltt distance is determined: (15)
then the graph of dependency N(f) is performed; the sought value of the shear wave propagation time is the value of the slope of the straight line approximating the points N(f).
continuous sweep input frequency method; it is very similar to cross-power spectrum analysis, except that the wave generation is based on the π-point method, and the received signal is continuously analyzed for power spectrum compatibility and the linearity of the functional relationship between the phase angle and the change in frequency; applying this technique in real time needs to supplement the measurement system with a power spectrum analyzer.
Regardless of the method used, there must be taken into account the fact that the input signal is different from the signal generated. It is related to elastic reaction of the material (mainly the resin surrounding the piezoelectric) and the length of the bender element (working as a cantilever).
The response of the bender receiver is dependent on:
stiffness, damping and dispersion properties of the specimen medium,
distance between transmitter and receiver BE,
the length and frequency of the generated waveform,
interference of component waves,
effects that disorder the measurement of the shear wave component.
It is crucial to highlight that S-wave measurement can be disordered by P-wave interference. Proximity effect of the transmitter is a serious limitation of the method which appears in the case when the transmitter’s distance to the receiver is too small. As the relation λ/A increases (L-distance between receiver and transmitter, λ-generated wavelength), the latency of the arrival of the P-wave to the bender relative to the S-wave is increased – the effect of the proximity phenomenon of the transmitter – the amplitude due to the P-wave decreases, and the final measurement accurately maps the S-wave. According to  it is recommended that L/λS be greater than 4 times the shear wave length.
3 Test material and specimen preparation
Each of the sand specimens P1-P5 was prepared from pre-formed non-cohesive soil fractions, according to the accepted grain size curves (approx. weight of each specimen: 1.5kg). A cylindrical specimen was then formed using a template with rubber installed inside it. Each cylindrical specimen (Figure 12a) was compacted with a hand rammer in five layers of dry sand, great care was taken not to damage the bender element installed on the bottom cap.
The weight and geometrical dimensions of formed triaxial specimens were precisely controlled and meticulously recorded. After acceptance of the quality of prepared specimen, it was placed in the compression chamber and isotropic pressure of 100 kPa was applied (effective stress). If changes in displacements are by LVDT the gauge stopped, bender element tests were performed.
After that, a higher level of isotropic pressure was applied (400 kPa) and procedure was repeated. Then the specimen was unloaded to 100 kPa and it was saturated with CO2 using back-pressure system. Then the specimen was saturated with vacuum de-aired water and whole procedure of testing was repeated (first with 100 kPa and then with 400 kPa, every time after full stabilization of displacements).
4 Results of the research
The results of all performed tests in form of relations between Gmax predicted by Hardin’s formula and Gmax measured using bender elements are collected in Table 3-4 (for all Hardin’s relationships irrespective of their applicable range) and Figure 13. As it can be easily noticed, satisfactory compliance of the results can be observed for specimens in a state of a isotropic pressure of 100 kPa. The points representing the individual comparison of results obtained for specimens tested at isotropic pressure of 400 kPa do not lie on a straight line that is a set of points with identical coordinates (Figure 13). This means that for higher pressures either Hardin’s rule is not correct, or the results of the research are flawed.
Analyzing the results of the research on the basis of the literature, it can be seen that non-cohesive soils with identical grain-size distributions and mineral compositions can significantly differ in mechanical properties due to even slight differences in surface geometry and grain shape. The results of the research indicate the need to include in the formulas on Gmax beyond CU and e yet additional geometrical parameters of grains, eg surface smoothness.
On the other hand, it is very important to mention that interpretation methods and techniques are far from ideal ones. For example, the π-points method used in signal analysis has proved to be extremely problematic due to the interference occurring in certain frequency ranges. Significant fluctuations in the amplitude of the received signal often prevented the observation due to the high level of electrical noise generated by triaxial apparatus and control system. The usefulness of the remaining methods can be assessed in a simple way, using artificial idealized received signal with constant amplitude and constant timedelay. The results of relative error analysis of the timedelay for frequency variation in the range of 10-30 kHz and noise with the amplitude of 0-100% of pure received signal amplitude for two selected interpretation methods are presented in Figure 14.
As it can be noticed, a cross power spectrum analysis is very sensitive on noise level (see Figure 14b). Satisfactory results can be obtained for noise level below 12% of pure signal amplitude. It is very difficult to achieve this, but it is possible thanks to the use of multiple repeated measurements with averaging of the obtained results (the application which controls oscilloscope ADC-212 has built-in such a function). However, in the case of crosscorrelation method (Figure14a) dark fields indicating a low error value are quite chaotic. In spite of good performance with low-level noise (below 3%), quite accurate results can also be obtained for full-level noise (see left upper corner of the graph). On the other hand, a low level noise can completely degrade the results for certain frequencies (see bright spots on the graph). As far as the peak method is concerned, it is not susceptible to interference and frequency changes because its effectiveness depends only on the proper indication of the peak in the received signal. Unfortunately, due to disturbances caused by S- and P-waves reflections and interferences, it is often very difficult to correctly identify the right peak – such identification is very often not as obvious as it might seem. That is why it is so important to use multiple interpretation methods with detailed analysis of the reliability of the obtained results.
5 Concluding remarks
The results of the laboratory tests presented in this paper confirm the correlation between the grain-size characteristics of non-cohesive soils and the value of the dynamic shear modulus for isotropic pressure of 100 kPa. The equation derived by Hardin and extended by Wichtmann & Triantafyllidis has a high applicability in practical estimations of Gmax treated as a nonlinear function of CU and e soil structure parameters. However, it is worth mentioning that the results of the research in the resonant column and the bender elements can be very different. This inconvenience hampers comparative analysis and requires the simultaneous application of many advanced signal processing techniques. The authors used four interpretation techniques and methods to process collected measurement data: peak-to-peak, cross-correlation, cros power spectrum and π-points. Despite the multifaceted processing of measured data, the results obtained for sand specimens at a pressure of 400 kPa have a significant discrepancy. This shows that obtaining unambiguous results from bender element tests is a difficult task in practical applications and direct comparisons with RC results are often not possible. These comparisons are difficult because of the fundamental difference in the generation of small strain: in RC the deformations associated with the torsion develop from the smallest in the center of the specimen to the largest at its circumference; in the BE tests the situation is exactly the opposite. In addition, the RC induces deformation throughout the whole specimen cross section and in BE – it is a matter of the wave propagation phenomenon, which can not be controlled by a researcher. The difficulties with using BE, which were mentioned above, require further research, both in signal processing and in the technical aspects of BE generation of S-waves. The works last.
CCXY cross-correlation coefficient, [-]
CU coefficient of uniformity, [-]
d grain size, [mm]
d50, d60 specific grain size, [mm]
e void ratio, [-]
G shear modulus, [Pa]
Gdyn dynamic shear modulus, [Pa]
Gmax initial shear modulus, [Pa]
Gs specific gravity, [-]
GXY cross-power spectrum of the signals X(t) and Y(t), [-]
LX linear spectrum of the signal X, [-]
conjugate coupling of LX, [-]
Vs shear wave velocity, [m/s]
p′ average effective stress, [Pa]
pref reference pressure (pref = 105 Pa)
tCC time delay, [s]
tCS wave propagation time, [s]
γ shear strain, [-]
ρ volume density of soil, [kg/m3]
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Published Online: 2017-11-23