Small-strain stiffness of selected anthropogenic aggregates from bender element tests

BEs comprise two thin piezoceramic plates that are rigidly attached to a central metal plate. Due to the properties of the piezoceramic material, when excited by an input voltage, one piezoceramic plate stretches and the other shortens. This acts like a transmitter, producing a mechanical stimulus. When a bending element is mechanically excited, it produces an electrical output and acts as a signal receiver. From the theory of wave propagation, and assuming that the mass of soil is semi-infinite, homogeneous, and elastic, the modulus of elasticity in small strains can be calculated from the following equation (1): (1) Gmax=ρ⋅Vs2 {G_{max }} = \rho \cdot V_s^2 where r is the bulk density of soil and V_S is the shear wave velocity (Viggiani, 1995, Brignoli et al., 1996, Kawaguchi et al., 2001).

To determine V_S using the BE technique, two measurements are required: i) the travel distance (L_tt) and ii) the travel time (t). The shear wave velocity can be calculated by equation (2) (Brignoli, et al., 1996): (2) VS=Ltt/t {V_S} = {L_{tt}}/t

The L_tt parameter is easy to measure as it is defined as the distance between the ends of BEs. A more challenging aspect is to identify the travel time. So far, various methods have been tested to determine the wave propagation time. The methodologies for estimating the travel time of the shear wave are classified broadly into two groups: TD- and FD-based approaches. At present, however, neither is universally accepted as a perfect technique for estimating the appropriate travel time for any soil or test condition.

The TD-based approach involves the problem of the “near-field effect,” which often results in significant errors in travel time measurement because of the bias in the early part of the output signal. Since there is still some uncertainty about how much the near-field effect will distort the output signal in a given state, the experimenter is forced to rely on trial and error to arrive at an appropriate delay. Meanwhile, the FD-based approach has the advantage of minimizing human error when measuring. However, it has been found that the resulting travel times are, in general, highly scattered and are smaller than those obtained by the TD approach. Thus, the difficulty in estimating the appropriate travel time remains. The meaning of the resulting output signal is not understood well enough theoretically to solve the problem (Ogino, 2019).

The laboratory tests presented in this paper were carried out in the GDS Instruments (a division of Global Digital Systems Ltd) GDS RC apparatus on the compounds of crushed concrete waste originating from a demolition site in Warsaw, Poland. The test material was provided by the capital’s supplier, directly in a shredded form. The recycled concrete delivered consisted of well-graded aggregate, which included gravel, sand, and silt-sized grains. Therefore, further crushing of the coarser gravel fraction as well as sieving were conducted before testing, and sand-sized or sand with soil-sized aggregates was the focus of this study. Six different compositions with the specimen codes M1, M2, M3, M4, M5, and M6 were studied and the details of these compositions are given below: –

M1, M2, M3, M4 – a group of mixtures in air-dry conditions, with a grain size of 0/2 and an FF content of RCA of 0%, 10%, 20%, and 30%, respectively;

The grading curves for all the materials are given in Fig. 1, together with an image of the parent concrete aggregate. Table 1 gives a summary of the physical properties of the tested RCA. Based on the physical characteristics and the particle size distribution curves, the blends were classified as poorly graded fine SANDS (M1, M2, and M5 blends) (FF ≤ 10%), poorly graded SANDS with silt (M6 blend) (FF = 15%), and well-graded SANDS with silt (FF > 15%) according to the unified soil classification system specified in, for example, PN-EN ISO 14688-2:2018-05. The samples were produced by evenly mixing two materials, that is, pure fRCA and the calculated amount of FF for this fRCA. A detailed description of the mixes’ preparation and a report from other tests carried out on this material can be found in Gabryś et al. (2023b).

Figure 1:

The GDS RC apparatus, an example of the fixed-free type of Stokoe RC system with BE inserts implemented, was used in the present study. A schematic of the testing system is available in many publications like He et al. (2018) and Gabryś et al. (2023b). The RC system adopted by the authors houses a specimen with dimensions of 14 cm in height and 7 cm in diameter. A range of isotropic pressures from 90 to 270 kPa was applied to each specimen. Both the top cap and the base were fitted with piezo-inserts. They protrude 3 mm out of the platen. The width and thickness of BEs are 11 and 1 mm, respectively. The BE test setup is shown in Fig. 2. When configured to the BE mode, one of the piezo-inserts triggers shear waves (S-waves) and the other acts as a receiver (Fig. 3), while in the extender element (EE) mode, the roles of sender and receiver are exchanged to generate and receive primary waves (P-waves) (Lings and Greening, 2001).

Figure 2:

Figure 3:

Compacted specimens were prepared in a two-piece metal split mold of four layers of similar heights from the oven-dried RCA. For the last two mixtures, that is, M5 and M6, RCA was moistened to a value close to the optimum moisture content (OMC). OMCs were set at 13.5% and 16% for the M5 and M6 mixes, respectively (Gabryś et al., 2023b). To compact the specimens, a lightweight rod was used, targeting a relative density (D_r) of more than 65%. Before removal of the mold, a small vacuum (~5–10 kPa) was applied to hold the specimen. Its dimensions were then measured and recorded to calculate the initial void ratio (e₀) (Table 1). All the mixtures were consolidated isotropically in progressive stages. The BE tests were conducted at three different pressures, that is, 90, 180, and 270 kPa, after each stage of consolidation had been completed. Such values cover typical stress levels encountered in practice in many geotechnical applications of anthropogenic materials (Tasalloti et al., 2020). The input signal, a sine pulse of 14 V amplitude, was generated by a function generator. Sine-wave pulses have become more popular within different input wave shapes, as these have been shown to give more reliable time measurements primarily (Blewett et al., 2000). The magnitude of the input signal was selected at 14 V as the typical one preferred when using the GDS BE system. Several frequencies were applied, ranging from 3.3 to 25.0 kHz, depending on the mixture and the effective stress. The frequency range is limited by a value of 25.0 kHz due to the characteristics of the transducers, filters, and amplifiers and the resolution of the acquisition system.

The appeal of the BE technique lies in its apparent simplicity, both in performing the test and in interpreting the results. However, over the years, several researchers have demonstrated several potential (inherent and induced) sources of error involved in both BE testing and interpretation. Among these, the most common are near-field effects, wave interferences at the rigid boundaries, specimen geometry, transducer resonance, overshooting, electric noise, and grounding/shielding issues (Viana da Foncesa et al., 2009).

To avoid some of these errors, it is first recommended to comply with several technical requirements and boundary conditions (Lee and Santamarina, 2005). These include properly shielded and grounded, properly connected and encapsulated sensors, leakproof connections, and an environment free of noise. There are also other aspects involved, particularly spatial factors, including the orientation of BE, the wave being reflected off the edges and sides of the sample, and the relative distance between the transmitter and the receiver. Poor contact between BE and the soil, leading to poor coupling, especially at low confining pressures, and overshoot, as BE changes its dominant mode shape at high frequencies and the response becomes complex, are further issues that require consideration.

The input excitation frequency may greatly affect the results of BE tests because of the near-field effects and noise. The near-field component of the wave causes the transmitted wave to be distorted at its origin point, and the wave starts with a downward or upward deflection as described and presented in Ingale et al. (2017). As the shear wave propagates, two different wave signals are produced. One component of this signal wave propagates in the longitudinal direction and the other component propagates as a shear wave, which makes it difficult to distinguish the S-wave propagation time due to the early arrival of the P-wave component. This effect is more prominent when the transmitter and the receiver come closer. To minimize the near-field effect, some authors have suggested using higher input frequencies and wavelength ratios (Wang et al., 2007). The wavelength ratio (Rd) is defined according to equation (3) (Scarrow and Gosling, 1986) as follows: (3) Rd=Ltt/λ=Ltt/VS⋅fin {R_d} = {L_{tt}}/\lambda = \left( {{L_{tt}}/{V_S}} \right) \cdot {f_{in}} where f_in is the excitation frequency and l is the wavelength of input signals. As widely reported in the literature (e.g., Arroyo et al., 2003; Arulnathan et al., 1998), the wavelength ratio values R_d <2 or >9 are associated with prevailing near-field effect, which masks the actual arrival of the received signal that is consequently difficult to be read. According to Sanchez-Salinero et al. (1986), to completely separate the P- and S-wave signals, the R_D values must be greater than 4.

The graphical representation of the test results for the M1 and M4 blend and all applied input signal frequencies for this mix is presented in Fig. 4a and b. Similar plots can be obtained for all other mixtures and pressures. The wavelength ratio (Rd) for the M1 mixture varies between 2.6 and 11.6 for frequencies between 3.8 and 16.7 kHz, whereas for the M4 mixture, Rd falls between 1.7 and 11.0 for f_in from 3.8 to 16.7 kHz. From the data shown in Fig. 4, it is observed that irrespective of the specimen tested, at low frequencies (i.e., 3.3–5 kHz), the wavelength ratio for the studied RCA is lower than 4, which is required to avoid near-field effects in the case of sand or sand with silt (e.g., Zhou et al., 2010; Juneja and Endait, 2013). It should be recalled here that the mixtures tested were classified as sands and sands with silt in terms of grain size distribution. It is also worth noticing that the received signal, where the R_D ratio is less than 2 (Fig. 4b), has no obvious peak due to the near-field effect. There are other factors as well that affect the correct interpretation of the signal at lower frequencies, like wave attenuation. At a higher R_D ratio, however, the noise level appears to increase causing some problems in S-wave travel time determination. This is primarily because a high-frequency signal is more prone to attenuation or interference during propagation, suggesting sensitivity of the signal quality toward L_tt/λ.

Figure 4:

The latest research on natural soils and their compounds indicates that an excitation frequency of the input signal (f_in) close to the resonant frequency (f_r) can produce a very clear received waveform (Viana da Fonseca et al., 2009). As reported in the resonance tests of RCA tested in the current study and described as well in a previous study by Gabryś et al. (2023b), the estimated f_r values were lower than 1 kHz for all the RCA mixtures and all the effective stresses. These resonant frequencies were obtained for the full system under continuous signals (steady state). Unlike the natural soils, the criterion for the selection of the best excitation frequency in BE tests near the resonant frequency is not satisfied regarding the RCA used. Hence, the reading of the actual arrival time of the received signal is very difficult here for f_in = f_r.

Fig. 4a indicates that at p′ = 90 kPa, determination of the travel time is quite straightforward; however, Fig. 4b shows that at higher stresses (p′ = 270 kPa), interpretation is ambiguous. Similar observations were recorded for natural soils, residual soil from Porto granite, and Toyoura sand in studies by Viana da Fonseca et al. (2009).

In Fig. 5, the small-strain modulus G_max at different wavelength ratios for the M4 mixture, as an example, is illustrated. The G_max values were calculated through the peak-to-peak method. In this graph, the previously mentioned limits for the R_d values are marked. In this way, a certain narrow range of frequencies (between 5 and 14 kHz for p′ = 90 kPa, and up to 20 kHz for p′ = 180 and 270 kPa) was created. It may be a great suggestion if the BE tests are performed on the RCA mixtures to obtain clearer waveforms. It is shown that in this defined frequency range, the G_max modulus has a slight upward trend. The average value of 11G_max was a maximum of 4%. Compared to the tests performed on natural soils (Wang et al., 2021), there are no stable values of the small-strain shear modulus as the Rd or excitation frequency increases. Most likely, the reason for this is the self-compaction with simultaneous crushing of the concrete aggregates with increasing frequency excitation. Due to the very irregular shape of crushed concrete grains, the crushing effect is observed even in the case of dynamic tests. The phenomenon of breakage of the RCA grains was described in detail by He and Senetakis (2016), who revealed that breakage of the RCA grains for samples subjected to pressures even smaller than 1 MPa was greater than the corresponding breakage in sands.

Figure 5:

Various methods have been proposed over the years for interpreting the received signals, ranging from the simplest, based on direct observation of the waveforms and measurement of the time interval between the starting points, to more sophisticated techniques supported by signal processing and spectrum analysis tools. Both TD and FD methods were initially taken into account for signal interpretation in BE testing of RCA. A short description of the main principles and applications of each method is given.

The first group of methods is based on the observation of the transmitted and received wave signals and the calculation of their difference as the travel time in the soil sample. With this technique, the arrival time of the shear wave is affected by the near field, the influence of compression wave signals, and other electrical noise and reflections. As a result, it is often difficult to read the arrival time (Sas et al., 2016). Viggiani and Atkinson (1995) compared the initial arrival of the received signal at different possible locations. Fig. 6 gives an example of different potential points of the resulting output signal. Point A is the first deflection; point B is the first inflection (first peak; point maximum); point C is the first zero after the inflection (zero crossing); and point D is the second inflection. Characteristic points of the input and output waves, such as peaks, troughs, and zero intercepts, are easy to identify, and the intervals between the corresponding points can be considered to represent the travel time of the S-wave, again under the assumption of plane wave propagation and absence of reflections or refractions (Viggiani and Atkinson, 1995). 15

Figure 6:

Based on the same assumptions as above, Viggiani and Atkinson (1995) suggested using a cross-correlation (CC) function, which measures how two signals are correlated. CC of a low-frequency impulse and the response produce a time displacement, which can be used as the time of flight of the wave between the two points (Mohsin and Airey, 2003). Such a technique is strictly applicable to signals of the same amplitude, requiring frequencies of both waves of the same magnitude (Santamarina and Fam, 1997). However, it is not clear what, if any, benefit is there in using CC to match a single-pulse input signal and a more complex output signal.

The second group of methods calculates the cross spectrum of the transmitted and the received waves, producing the relations of the amplitude and phase angle with the frequency axis. The arrival time is then computed from the phase spectrum slope. As the frequency characteristics of the input and output waveforms are used, this is sometimes called the FD method (Yamashita et al., 2007).

Both groups of procedures require an experienced researcher with a proper knowledge of how to interpret the correlated result. There is also a subjective aspect to this testing technique.

Moreover, the same method used by the same researcher on the same test apparatus can provide different levels of interpretation reliability, simply because of the different properties of the soil. In the subsequent section of the article, only travel time interpretation issues for the RCA specimens will be examined.

In Fig. 7, the results of the small-strain shear modulus (G_max) from the BE test using different analysis methods versus the mean effective stress (p′) from one representative specimen, the pure fRCA (the M1 blend) with FF = 0%, are illustrated. The arrival points A, B, C, D, and peak to peak 2 method were read automatically in the GDS BEAT program (Multi-Method Automated Tool for Travel Time Analyses). The summary of this program is given in the other work of the authors (Gabryś et al., 2021), whereas the details can be found in Rees et al. (2013). These five techniques were subsequently compared to two nonautomated subjective analyses. The first is the peak-to-peak technique that is defined by the researcher in the process of executing the BE test, hereinafter referred to as peak to peak 1. The results of the G_max modulus from the RC test were also integrated into the comparison. It is noteworthy that the test procedures for these two types of tests, BE and RC, are somewhat different. The main differences include different levels of strains and different ranges of frequencies (smaller values of both strains and resonant frequencies occur for RC tests). Despite these varieties, many researchers do not avoid comparing the small-strain stiffness results of natural soils using both techniques (Molina-Gómez et al., 2023).

Figure 7:

As presented in Fig. 7, regardless of the excitation frequency, all procedures gave satisfactorily close values for all the tests. In this regard, it should be noted that the input signals f_in = 10.0 and 12.5 kHz were the only ones repeated for each mixture and, most importantly, met the wavelength criterion described earlier. For the input signal frequency of 10.0 kHz, the average difference between the G_max results from various travel time interpretation methods was approx. 30.0 MPa, whereas for f_in = 12.5 kHz, DG_max, avg amounted to 27.3 MPa. For both frequencies offered, the lowest initial stiffness values for the RCA studied in the present research were obtained with the method D-major first peak. The maximum values were obtained during the RC tests (for p′ = 90 and 180 kPa) and with the method A- first deflection (for p′ = 270 kPa). Despite the previously mentioned differences in the BE and RC tests, the divergence between the G_max results on fRCA relative to the average G_max is of the order of 13%. As shown in Fig. 7, the smallest differences in the G_max values (~2.0 MPa) were obtained in the case of two methods, peak to peak 2 and C-zero crossing, independently from the mean effective stress level. These are the results that were considered average. Then, the focus was kept only on these two signal interpretation methods. However, in assessing the travel time by the C-zero crossing method, difficulties were encountered in following the standard criterion because, in some of the transmissions, background noise, dispersion, and near-field effects made picking of the start of the received signal rather difficult. Given the above reasons, the peak to peak 2 technique was selected to identify the travel time to monitor the stiffness variations and, thus, the development of reaction in the fRCA blends. It was adopted for further analysis, as it provides the most consistent results for the studied anthropogenic aggregates.

By contrast, the BE tests carried out by the authors, but on natural Warsaw glacial quartz sands (Gabryś et al., 2021), have eliminated this technique, giving preference to the C-zero crossing and the CC procedures.

All tested fRCA compounds were analyzed in a similar way to Fig. 7, and the conclusions also apply to other mixtures.

In the next order, the G_max values as a function of the p′ stress for all blends studied, provided by the recommended earlier method (peak to peak 2), were plotted in Fig. 8. As observed in Fig. 8, there is a clear variability and divergence in the initial stiffness of the individual mixtures, regardless of the f_in values. The variation in the results amounted from 20% up to almost 40%. These apparent differences are stress and FF content dependent. It seems that the preparation of the specimen (air-dry/moist conditions) is also an important factor. It can be found that for the lowest stress, the mixes of pure anthropogenic aggregates (the M1 blend, FF = 0%) were characterized by the lowest initial stiffness. This trend is confirmed revered with increasing p′, where for p′ =270 kPa, the M1 specimen had the highest stiffness. Considering only the mixtures prepared by the dry tamping method (M1 FF = 0%, M2 FF = 10%, M3 FF = 20%, M4 FF = 30%), the higher the FF content, the initially high stiffness decreased with pressure. To see if this trend of change continues, the research presented here would need to be further developed to include tests at pressures above 270 kPa and a greater number of pressures. In the case of moisturized mixtures (M5 FF = 5% and M6 FF = 15%), irrespective of p′, the lower the fines, the greater is the material shear modulus. It can be assumed that if the M1 mixture had been prepared in moist conditions, its shear modulus would have been higher than that of the M5 and M6 compounds.

Figure 8:

The obtained experimental results are in agreement with those of other researchers, for example, He et al. (2018), dealing with the stiffness of recycled composite aggregate. They also concluded that the material with finer fractions has lower stiffness in comparison to the coarser fractions. When the soil contains a certain amount of smaller and softer (or weaker) material, the contact response is more likely to be dominated to some degree by the softest component. The surface contact behavior, in selected anthropogenic aggregates, might be changed (dropped) drastically compared to the pure fRCA, significantly reducing the wave propagation velocity and thus the mixture stiffness characteristics. In the present study, this drop in the particle contact response and, therefore, lower shear moduli only happened at higher pressures. At low pressures, the effect of the fines was not obvious.