Sources of Uncertainty in Bender Element Testing: Execution and Interpretation Challenges in Reconstituted Sandy Soils

Abstract

This paper discusses the principal sources of uncertainty in the execution and interpretation of Bender Element (BE) tests conducted on reconstituted sand samples. Based on the experience accumulated by the Geotechnical Laboratory of the University of Coimbra, the study addresses three critical stages of the testing process: sample preparation, test execution, and result interpretation. For each stage, the key challenges are identified, and potential solutions are proposed. Particular emphasis is placed on the control of relative density and sample saturation during preparation, as well as on factors affecting signal quality and time lag of the system during test execution. The interpretation of the results is analyzed with respect to the limitations of currently employed methods. The overall reliability of the procedures employed throughout the testing process is also assessed, with the results providing guidance for improving the accuracy and consistency of BE test outcomes.

1. Introduction

In laboratory research, the use of high-quality intact samples is arguably the most important factor in correctly analyzing soil behavior [1]. However, in the case of sandy soils, obtaining intact samples is often unfeasible or expensive. As a result, the study of such materials is typically performed through reconstituted samples [2]. The choice of the reconstitution method, as well as its calibration, is influenced by several factors that, if not properly considered, can compromise the representativeness of the sample [2,3,4,5,6]. However, the challenges involved in testing reconstituted materials extend beyond sample preparation, encompassing the setup and control of the testing equipment itself. A final potential source of uncertainty lies in the interpretation of the obtained results. This aspect is strongly influenced by the aforementioned factors, and is particularly relevant in the case of tests such as the Bender Element (BE) test since its interpretation is strongly dependent on the experience of the user as previous studies have shown [7,8,9].

The use of BE was first introduced by Shirley and Hampton [10], but it was only after the work of Dyvik and Madshus [11] that the method became popular due to its relative simplicity and low cost, but also because it allows for the near-direct determination of the soil small strain shear modulus ($G_{max}$). Another significant advantage of this method is the compact and simple nature of the equipment, which allows for it to be installed in different apparatus, such as oedometers [12], Rowe’s cells [13], cubical cells [14], and most commonly—as in the present study—in standard triaxial cells [15,16,17]. Its use in triaxial cells enables the determination of $G_{max}$ under controlled conditions and facilitates the evaluation of the effects of factors such as the confining pressure. The test principle is based on determining the shear wave velocity ($V_S$) as it travels through a soil sample. To achieve this, a shear wave is generated using a wave generator which is transmitted through the sample by the BE transmitter and received by the BE receiver, typically placed on the opposite end of the sample. Both transmitted and received signals are recorded by an oscilloscope, with $V_S$ calculated using Equation (1) upon the estimation of the wave arrival time ($t_{arr}$),

$$V_S={\displaystyle \frac{L}{t_{arr}}}$$

(1)

where $L$ is the distance between transmitter and receiver. The value of $G_{max}$ can then be obtained using the continuum elasticity expression (Equation (2)), where $\rho$ represents the density of the material.

$$G_{max}=\rho ·V_S^2$$

(2)

However, the main challenge in using BE lies in accurately determining the arrival time of the transmitted wave, as this is often influenced by factors such as the shape and frequency of the transmitted shear wave, near-field effects and sample boundary conditions. Despite the several proposed methods and frameworks proposed to address those factors [18,19,20,21,22,23,24,25,26], there is still no consensus on which interpretation technique yields the most accurate results.

Based on the extensive experience of the Geotechnical Laboratory of the University of Coimbra (LG-UC) [15,27,28,29], this paper analyzes the potential sources of uncertainty and error when performing BE tests on reconstituted sand samples and proposes mitigation measures to enhance result reliability, offering practice-oriented guidance for future experimentalists. The material used in this study was Coimbra sand (Batch L-I), which has been the subject of ongoing research at the LG-UC by several researchers [3,27,30,31,32], as part of a broader research project focused on characterizing materials susceptible to liquefaction. The paper provides a detailed description of the various stages involved in sample preparation, with particular emphasis on the development of a pluviation device to control the target relative density of each sample and on the assessment of the influence of both the setup and the saturation process on the final test results. Subsequently, aspects related to test execution are presented, highlighting the advantages of using multiple frequencies as a transmitted signal and the importance of accurately evaluating the time lag of the system. Finally, the paper addresses the complexities associated with interpreting the BE test results by comparing outcomes obtained through both time and frequency frameworks. As demonstrated by the results, the adoption of the proposed measures at all stages of the testing process ensures the reliability of the BE test results.

2. Materials and Methods

2.1. Equipments

BEs are essentially ceramic piezoelectric transducers composed of two thin ceramic plates, which can be polarized in either parallel or series configurations. Usually, the transmitter is polarized in parallel, as this setup enables the generation of twice the displacement for the same transmitted voltage. Conversely, series configuration is more effective for receiver elements, as it enhances the sensitivity of the output signal [11,17,25,33]. In the present study, the polarization of both the transmitting and receiving BE followed the recommended configurations described above. The BEs were installed in the triaxial apparatus by incorporating the transducers in the pedestal and top cap. Apart from the necessary electrical connections for signal transmission and reception, no additional modifications to the triaxial setup were required.

The triaxial cell used was a hydraulic apparatus of the Bishop and Wesley [34] type, designed to accommodate samples 38 mm in diameter and 76 mm in height, and capable of operating at a maximum pressure of approximately 1000 kPa. The pressures in the cell and in the interior of the sample (base and top) were measured by pressure transducers with a capacity of 1000 kPa, while volume changes were recorded using a volume gauge with a capacity of 50 cm³. Additionally, an external pressure controller connected to the top of the cap was used to apply suction during the sample setup phase.

A TTi TG1010 function generator was used to generate the transmitted signal, while signal acquisition—both transmitted and received—was carried out using a Tektronix TDS220 oscilloscope. The recorded signals were then transferred to a computer, where the data was processed using various interpretation techniques.

2.2. Coimbra Sand

The material used in this study was Batch L-I of Coimbra sand, also commonly referred to as upstream Coimbra sand. The main difference between the batches lies in the location where the sand is collected. In the case of Batch I, the sand originates from the alluvial deposits found along the banks of the Mondego River near Quinta da Portela in Coimbra. The sandy alluvial deposits of the Mondego River, in their natural state, do not provide the desired quality for samples to be used in laboratory testing, primarily due to the well-known almost practical impossibility of collecting intact samples from this type of material. Therefore, in order to mitigate potential issues related to the preparation of specimens with comparable characteristics—crucial for consistent comparative analysis—the sand was subjected to treatment immediately after collection. The initial processing followed the procedure described by Santos [30] and involved three main stages: (i) washing, to remove fines and organic matter; (ii) drying, to eliminate moisture from the sand; and (iii) sieving, in accordance with the ASTM D422 [35] standard, to select the appropriate grain size fraction. To ensure uniformity and avoid segregation, all particles retained on sieve No. 40 (0.425 mm) as well as those passing through sieve No. 100 (0.15 mm) were excluded. Figure 1 presents the particle size distribution curve obtained for Coimbra sand, Batch L-I. As expected, and due to the processing applied, the sand exhibits a nearly uniform gradation. The physical characterization of the sand was carried out according to ASTM standards [35,36,37,38,39], with the most relevant results for the present study summarized in

Table 1. Physical properties of Coimbra sand (Batch L-I).

Particle Size Analysis					Phase Relationships
${\mathit{D}}_{10}$ (mm)	${\mathit{D}}_{50}$ (mm)	${\mathit{D}}_{60}$ (mm)	${\mathit{C}}_{\mathit{U}}$	${\mathit{C}}_{\mathit{C}}$	${\mathit{e}}_{\mathit{m}\mathit{i}\mathit{n}}$	${\mathit{e}}_{\mathit{m}\mathit{a}\mathit{x}}$	${\mathit{\gamma }}_{\mathit{d},\mathit{m}\mathit{i}\mathit{n}}$ (kN/m3)	${\mathit{\gamma }}_{\mathit{d},\mathit{m}\mathit{a}\mathit{x}}$ (kN/m3)	$\mathit{G}$
0.18	0.27	0.34	1.88	1.00	0.48	0.81	14.4	17.6	2.65

.

3. Sample Preparation

3.1. Reconstitution Technique

The characterization of sandy soils presents significant challenges, primarily due to the difficulties associated with collecting intact samples. Consequently, various techniques (Figure 2) have been developed over the years to replicate, as accurately as possible, the natural in situ deposition conditions of such soils, with their effectiveness typically assessed by comparing the structure of the original soil with that of the reconstituted specimen [2,4,5,40]. According to Butterfield and Andrawes [41], sand reconstitution techniques can be broadly classified into two main groups:

• Techniques in which the relative density is adjusted after sand deposition, such as vibration and tamping methods.
• Techniques in which the relative density is controlled during the deposition process, such as gravitational deposition methods and air pluviation.

Additionally, reconstitution can be performed under dry, moist, or wet conditions. Studies have shown that the deposition method directly affects the fabric and, consequently, the mechanical behavior of the sand [4,40,42,43]. As a result, selecting an appropriate reconstitution technique is of paramount importance and should ideally be chosen according with the depositional history of the in situ sand. The moist tamping technique is more appropriate for simulating the in situ compaction of moist sand deposits [44], whereas the air pluviation method is considered by several studies [6,43,44] to most closely replicate the natural deposition process of alluvial sand deposits, making it suitable for studying the behavior of Coimbra sand. Although not perfect—since the air pluviation method tends to induce some anisotropy, with elongated particles aligning horizontally [45]—it offers better control of the target relative density compared to methods that adjust density after sand deposition.

In the air pluviation method, there are four main factors that must be considered: (i) the deposition medium; (ii) the drop height; (iii) the deposition flow; and (iv) the uniformity of the deposition. The most commonly used deposition medium is air dry, although deposition can also be carried out under water. However, unless combined with techniques like vibration or tamping, underwater deposition alone produces low relative density samples, making it difficult to reproduce dense sand conditions [46]. The drop height of particles directly influences their deposition velocity and the resulting relative density. According to Vaid and Negussey [46], kinetic energy increases with drop height only up to about 70 cm, beyond which it remains nearly constant. In underwater deposition, the maximum energy is reached at a drop height of just 0.2 cm, which explains the typically low relative densities obtained in such conditions. Another important aspect to consider is the distance between the pluviation system and the sample, which can be either fixed or variable, with the fixed configuration making it more difficult to ensure uniformity as the drop height varies [2]. The deposition flow is controlled by both the number and diameter of the holes in the pluviation device. Okamoto and Fityus [47] observed a tendency for higher deposition flows when using fewer and larger holes, compared to a larger number of smaller ones. Experimental studies have shown that the relative density increases as the deposition flow decreases [48]. Finally, the process is influenced by the uniformity of sand deposition. To ensure a uniform distribution of sand, several stacked sieves are often employed as sand diffusers [42,49].

Since the effectiveness of the air pluviation technique largely depends on the uniformity of the sand itself—as sands with a wide grain size distribution tend to exhibit particle segregation—a narrow particle size distribution for the Coimbra sand was selected, as described previously, to minimize this problem. After carefully evaluating the different options, air dry pluviation was selected as the deposition medium, with the remaining factors—drop height, number and diameter of the holes and diffuser sieves—being calibrated to achieve the target relative densities to be studied.

3.2. Calibration of the Air Pluviation Device

The evaluation of relative density ($D_r$) is based on the values of the maximum ($e_{max}$), minimum ($e_{min}$) and current ($e$) void ratios of the sand, according to Equation (3).

$$D_r={\displaystyle \frac{e_{max}-e}{e_{max}-e_{min}}}\left[\%\right]$$

(3)

The extreme void ratio values were determined following ASTM Standards D4253 [36] and D4254 [37]. The current void ratio ($e$) was calculated using Equation (4), which relates the density of the soil particles ($G$) with the dry unit weight (${\gamma }_d$). The dry unit weight itself was obtained from Equation (5), based on the dry weight ($P_s$) and volume ($V$) of the sample.

$$e={\displaystyle \frac{G\times 9.81}{{\gamma }_d}}-1$$

(4)

$${\gamma }_d={\displaystyle \frac{P_s}{V}}$$

(5)

The development of a robust and reliable preparation method that allows for the control of relative density is a critical aspect when testing reconstituted sand samples. Since no standardized procedure is available in the literature for preparing samples at specific relative densities, it was necessary to develop a simple yet dependable method capable of reproducing the desired densities. Based on the work of Araújo Santos [3], a funnel fitted with a cap was used as the pluviation device. The outflow was controlled by adjusting the number and diameter of holes made in the cap. Following the characterization work previously carried out on Coimbra sand at the LG-UC [3,27,30,31,32], target relative densities of 40% and 80% were adopted for calibrating the pluviation device. To calibrate the process, several trials were conducted in which sand was poured into a mold with dimensions similar to the intended specimen (76 mm in height and 38 mm in diameter). By weighting the sand and using Equations (3)–(5) and the data presented in Table 1, the achieved relative density was calculated and compared to the target value. The relative density of 40% was achieved by adopting a drop height of 5 cm and 9 holes with a diameter of 3 mm evenly distributed in the cap. To achieve a relative density of 80% it was necessary to place a diffuser sieve, No. 8 (2.36 mm) [35], between the funnel and the mold.

Table 2. Results of the calibration of the pluviation device for Coimbra sand—Batch L-I.

${\mathit{D}}_{\mathit{r}}$ 1 = 40%					${\mathit{D}}_{\mathit{r}}$ 1 = 80%
Trial nº.	${\mathit{P}}_{\mathit{s}}$ (N)	${\mathit{\gamma }}_{\mathit{d}}$ (kN/m3)	$\mathit{e}$ ( )	${\mathit{D}}_{\mathit{r}}$ 2 (%)	Trial nº.	${\mathit{P}}_{\mathit{s}}$ (N)	${\mathit{\gamma }}_{\mathit{d}}$ (kN/m3)	$\mathit{e}$ ( )	${\mathit{D}}_{\mathit{r}}$ 2 (%)
1	1.352	15.68	0.658	44.8	1	1.471	17.07	0.523	84.4
2	1.340	15.55	0.672	40.4	2	1.458	16.91	0.537	80.2
3	1.337	15.51	0.676	39.3	3	1.472	17.08	0.522	84.7
4	1.345	15.60	0.666	42.3	4	1.471	17.07	0.523	84.4
5	1.340	15.55	0.672	40.4	5	1.465	16.99	0.530	82.4

¹ Relative density required in the calibration procedure. ² Relative density measured in the calibration test.

presents the results for five of the final repetitions performed during the calibration trials. In all cases, a consistent margin of error of less than 5% was obtained for both targeted relative densities, demonstrating that the pluviation device developed is feasible and capable of reliably producing samples with consistent characteristics.

3.3. Sample Setup

As there is no standard guideline for setting up reconstituted samples in the triaxial apparatus, the adopted procedure was refined based on experience from pilot tests and insights from previous studies. The cylindrical shape of the sample was achieved using a split mold, with suction applied to ensure that the membrane adhered tightly to the inner surface of the mold, as shown in Figure 3a. The sand was then poured into the mold using the pluviation device to achieve the target relative density. A crucial step in the sample setup, which significantly affects the quality of the results, is the placement of the upper porous stone and the top cap that seals the sample. Levelling the top of the sand is necessary to ensure perfect contact between the BE and the sand (Figure 3b). However, this procedure must be carried out with extreme care, as any unexpected movement may disturb the sample and cause densification—particularly in those prepared at lower relative densities—potentially compromising the test. After sealing the sample, a suction of approximately 7 kPa was applied to preserve the intended shape and conditions of the sample, enabling the removal of the split mold, as shown in Figure 3c. The triaxial cell was then sealed and filled with water. A confining pressure of approximately 15 kPa was applied in the cell, corresponding to an effective stress of around 22 kPa in the sample.

3.4. Sample Saturation

The final step of the sample preparation consists in its saturation. This is arguably the most critical stage, as the introduction of water can cause disturbances within the sample. The initial procedure adopted for this phase (referred to as M1) consisted of the following steps:

• Upward percolation of water from the base to the top of the sample under low pressure in the cell (15 kPa). A pressure differential of about 10 kPa was applied (suction of around 7 kPa at the top and a pressure at the base of about 3 kPa) (Figure 3d). This process was carried out until a significant amount of water (approximately 150 cm³) had percolated through the sample, aiming to remove all entrapped air.
• Subsequently, the suction applied at the top was gradually reduced to zero, while the upward percolation of water through the sample was maintained. This step enabled the controlled removal of the suction system used during sample preparation.
• Afterwards, the pressures in the cell and at the base and top of the sample were gradually increased (at an equal rate) to 215 kPa, 200 kPa, and 197 kPa, respectively, ensuring that the effective stress in the sample remained constant between 15 to 18 kPa. This procedure of water percolation at higher pressures and with a smaller seepage differential (3 kPa) enables a faster and more effective removal of any remaining air bubbles trapped within the sample.
• The degree of saturation was assessed after pressure stabilization inside and outside the sample (i.e., no percolation and an effective stress of 15 kPa) using Skempton’s B parameter [50]. The sample was considered saturated when the B-value reached 0.98 or higher (a value of 1 indicating full saturation).

Despite all efforts to minimize sample disturbance, it was found that the percolation applied during step 1 (i.e., at low pressures) was too intense and caused undesired densification. This effect became evident when comparing the test results obtained using this initial procedure (M1), as the looser samples (relative density of 40%) exhibited stiffness values closer to those of samples prepared at a relative density of 80%, as shown in Figure 4. This behavior was noticeable for all stress levels (p’) applied to the sample. In light of these findings, a new saturation procedure (M2) was adopted. It followed the same steps as M1, except that step 1 was skipped, and the process began directly with the removal of suction (step 2). The results obtained using the M2 procedure are also presented in Figure 4, where a noticeable difference is observed, with the looser samples exhibiting a reduction in stiffness of approximately 30–40%. As expected, the denser samples were much less affected by the change in procedure (on average less than 8%), as they were significantly compacted. The greater sensitivity of the looser samples to the saturation procedures is attributed to their initial density conditions, which proved to be less affected by the percolation effects induced by saturation procedure M2. This procedure employs significantly lower hydraulic gradients and avoids suction effects that could increase the magnitude and non-uniformity of the internal hydraulic gradients, thereby minimizing particle-relative movements and the associated void ratio reduction. Based on these results, the M2 procedure was considered the most appropriate, and all subsequent tests were conducted using this revised saturation methodology.

4. Test Execution

4.1. Test Procedures

Before the test begins, two additional aspects of great significance should be highlighted. The first one concerns the alignment of the BEs, which should be carefully ensured during sample preparation to minimize issues with signal reception. According to Pedro [9] misalignments greater than 30° can lead to a loss of more than 50% in signal strength and a significant increase in distortion compared to cases where the BEs are properly aligned. The second aspect relates to electromagnetic coupling (“crosstalk”) generated by the various devices operating during the test. This noise, which weakens and distorts the received signal, is largely unavoidable, as it originates from the various equipment required for testing. However, some mitigation strategies can be adopted, such as using grounded, short, and shielded cables [33]. Another approach involves filtering the received signal [16,51], though this must be applied with care to avoid inadvertently removing relevant data. In the present study, an analogue filter integrated in the oscilloscope was used to remove high frequencies, above 150 kHz, which were primarily associated with electrical and background noise.

In order to evaluate the stiffness for different stress levels, the cell pressure was gradually increased while maintaining a constant pore water pressure inside the sample, thereby isotropically increasing its effective stress. At predefined stress levels (25, 50, and then every 50 kPa up to 500 kPa), the pressures were allowed to stabilize for about 10 min to ensure the dissipation of any excess pore water pressure in the sample. Only after this stabilization period was the BE test performed at each predefined stress level.

4.2. Transmitted Signal

The selection of an appropriate transmitted signal is critical to facilitate the detection of the shear wave arrival time. Although various studies have examined the type and shape of the signal to be transmitted, there is still no consensus on which option is most suitable [7,25,52]. Nonetheless, previous research [11,53] has shown that there is considerable uncertainty in the response of BE to square wave signals, making it more difficult to compare the received signal with the transmitted one. In contrast, the use of a single centered sinusoidal signal (‘sine wave’) has gained broader acceptance within the scientific community [25], as its waveform allows for a more objective analysis, leading to more reliable results. Previous studies [7,9,25,54,55] have shown that the frequency of the transmitted signal can affect shear wave velocity, with the response being enhanced when the transmitted frequency is close to the resonant frequency of the BE-soil system [21,33]. As demonstrated by Ferreira, et al. [56] and Astuto, et al. [57] the frequency of the transmitted signal significantly influences the predominant type of wave generated. Frequencies above 20 kHz are known to primarily generate compression waves. In contrast, for the generation of shear waves, as intended in this study, lower frequencies, typically below 10 kHz [9,56], are recommended. In this study, and based on the considerations above, single centered sinusoidal waves with frequencies of 1, 2, 3, 4, 5, 6, 8, and 10 kHz and an amplitude of 20 V were transmitted, as illustrated in Figure 5.

4.3. System Delay

As suggested by Yamashita, et al. [25] and ASTM D8295 [58] the entire BE system was checked to ensure proper equipment functioning and to determine the system delay. Theoretically, if the top and bottom BEs are in perfect contact (i.e., zero distance), the received signal should coincide exactly with the transmission of the shear wave. However, this is generally not observed, as a slight delay and distortion in signal reception typically occurs and is inherent to the testing system, making it impossible to eliminate. Wang, et al. [59] observed that the delay induced by the signal generator up to the transmitting BE was less than 0.1 μs, but the delay from the receiving BE to the oscilloscope could be significantly higher and relevant. The overall system delay can be estimated by placing the transmitting and receiving BEs in direct contact (i.e., tip-to-tip), triggering a single sine pulse, and measuring the time difference between the transmitted and received shear wave. In the present study, this test was conducted by triggering a single sine pulse with polarizations of 0 and 180° (inverted) for the different transmitted frequencies. As shown in Figure 6, for a transmitted frequency of 10 kHz, a system delay of 10 μs (0.01 ms) was identified and subsequently subtracted from the arrival times determined during test interpretation in the time domain framework. This delay was observed with insignificant dispersion across all tested transmitted frequencies.

5. Results and Discussion

5.1. Test Interpretation

As mentioned previously, interpreting BE results can be quite challenging, as the received signal is often significantly weakened and heavily affected by noise. One of the commonly debated issues concerns the determination of the distance travelled by the shear wave ($L$). Despite some alternative approaches [26,53,60], it is widely accepted that the tip-to-tip distance provides the best estimative [11,23,25] and, consequently, it was adopted in the present study. Given the small dimensions of the tested samples, it is important that this distance is updated throughout the test, as the increase of the confining stress will cause the sample to shorten. Ideally, this correction should be performed using an internal displacement transducer, although an indirect correction based on the volume variation measurement is also possible. Naturally, increasing the confining stress leads to sample densification. However, in all tests performed, the variation in the relative density between the extreme stress levels considered (25 and 500 kPa) was less than 3%, indicating that the influence of this factor on the results is relatively minor. Nevertheless, the recorded volume change during the test was used to correct the sample height, assuming isotropic compression deformation.

However, the main challenge in BE testing typically lies in determining the arrival time of the shear wave ($t_{arr}$), as its detection is often hindered by multiple factors. One of the most significant is the so-called “near-field effect”, which is associated with the generation of compression waves (P-waves) during the transmission of the shear wave (S-wave). Although these P-waves have lower amplitude, they travel faster through the sample and are detected first at the receiver, interfering with the determination of the arrival time since they produce small deflections in the signal [59,61]. Other factors are related to the reflection of shear waves due to sample size and boundaries [13,33], particle scattering [54] or signal overshooting at high frequencies [21]. All these phenomena have been investigated both experimentally and numerically [13,18,21,62,63,64,65] in an effort to minimize noise and facilitate the detection of the shear wave arrival time. All studies agreed that noise decreases with increasing frequency and suggested that the ratio of the wave path length to wavelength ($L/\lambda$) should be greater than 2 for improved signal clarity [17,18,59]. More recently, Moldovan and Correia [64], through a numerical study, proposed that adjusting the position of the BE receiver could significantly improve the interpretation of the arrival signal, while Roshan, et al. [65] tested, with relative success, the use of new types of lateral boundaries to dampen reflections.

To overcome these ambiguities, several methodologies have been proposed for the interpretation of BE test results, which can generally be categorized into two main approaches: time domain (TD), and frequency domain (FD). TD approaches involve directly comparing the transmitted and received signals to identify a characteristic point common to both signals [7,33]. Alternatively, the arrival time can also be estimated within the TD framework by cross-correlation between transmitted and received signals [23]. FD interpretation, on the other hand, involves transforming the transmitted and received signals from the time domain to the frequency domain using a Fast Fourier Transform (FFT). The resulting cross-spectrum is used to compute a transfer function that characterizes the system, where the phase angle is directly related to the time delay between transmitted and received signals [16,19,23]. Viana da Fonseca, et al. [24] proposed a hybrid method that combines both TD and FD analyses to improve the accuracy in estimating the arrival time of the shear wave. Moldovan and Gomes Correia [20], in an attempt to reduce subjectivity and achieve consistent results, proposed a new method in which $G_{max}$ is determined by maximizing the correlation between the received signals obtained experimentally and those generated through computational simulation. More recently, Flood-Page, et al. [26] proposed the use of a statistical tool, the Akaike Information Criterion, to decompose a BE signal into two stationary processes representing periods before and after the shear wave arrival. They introduced two algorithms to identify the correct segment of the signal for estimating the arrival time, but despite the promising results, the authors acknowledge that the algorithms might still require refinement and validation across different materials.

In the present study, the interpretation of the BE tests was performed using both TD and FD frameworks to assess the influence of each methodology in the determination of the arrival time.

5.2. Time Domain Framework

As referred to previously, the analysis in the time domain attempts to determine the arrival time through an inspection of the received signal. Based on the idealized shape of the signal containing “near-field effects” [33], as shown in Figure 7, four characteristic points are commonly considered for this purpose: the first sharp (point A), first bump (point B), zero crossing (point C) and first peak (point D). The International Parallel Test benchmark [25] revealed that the Start–to–Start (S-S) method, where the arrival time is defined as the time difference between the starting point of the transmitted signal and point C, is the most commonly adopted approach in the literature. In addition, this method, ASTM D8295 [58], also recommends the use of the Peak–to–Peak (P-P) approach, in which the arrival time is defined as the time difference between the first peak of the transmitted signal and point D (first peak) of the received signal. Although theoretically straightforward, identifying the precise arrival time using these characteristic points can be challenging due to the noise and distortions often present in the received signal.

Within the time domain framework, Viggiani and Atkinson [23] proposed the cross-correlation method (CC), in which the transmitted and received signals are cross-correlated to determine the time shift between them. The arrival time is assumed to correspond to the highest peak of the cross-correlation function. While this method has the advantage of being objective, recent studies have questioned the underlying assumption that the transmitted and received signals share identical frequency contents and that the highest peak of the cross-correlation function reliably indicates the actual arrival time [66,67].

Nevertheless, in the present study, all the described methods were employed to evaluate the arrival time. The first approach was conducted using the free tool provided by GDS Instruments [55], which automatically identifies all characteristic points and performs the cross-correlation between the transmitted and received signals. Although the automatic procedure is generally reliable, visual inspection of the received signal is essential to verify the correct identification of the characteristic points, as errors were detected while using the tool.

Figure 8 illustrates the influence of the frequency of the transmitted signal on the received signal. Although most of the received signals were relatively clean, a significant improvement in signal clarity was observed in all cases for transmitted frequencies above 6 kHz. In fact, when a 1 kHz signal was transmitted, the noise level was so high that an accurate BE interpretation was not feasible, regardless of the relative density of the sample or the applied stress level. The analysis of Figure 8a, which displays the received signals for loose sand under a p’ of 200 kPa, confirms that the characteristic points are strongly influenced by the transmitted frequency, particularly for frequencies below 6 kHz. This effect was particularly pronounced for points B, C and D. In contrast, point A remained relatively stable, showing a small variation regardless of the sand relative density or the applied stress level. These results are clearly illustrated in Figure 8b, where the arrival time corresponding to point C is plotted against the transmitted frequencies for three stress levels (p’ = 100, 300, and 500 kPa), considering both loose (closed dots) and dense (open dots) sand samples. The results indicate that the transmitted frequency primarily affects the loose sand, with the differences diminishing as the stress level increases. In contrast, the dense sand appears unaffected by the transmitted frequency, showing consistent values regardless of the stress level applied. The obtained $L/\lambda$ ratios are plotted in Figure 8c and show that values exceeding 2 were only achieved at higher transmitted frequencies and under high stress levels. These findings align with previous studies recommending $L/\lambda >2$ to enhance signal clarity. However, it was observed that reliable arrival time estimations could still be achieved with lower $L/\lambda$ values, provided the sample was dense or the confining stress exceeded 300 kPa, suggesting that this guideline may be influenced by the sand’s relative density and/or the applied stress level.

Given the influence of the transmitted frequency, it was decided to determine the arrival time—regardless of the relative density or stress level—as the average of the estimated values obtained for transmitted frequencies of 6, 8 and 10 kHz. The variations resulting from this approach were found to be lower than 5% in all cases analyzed.

In Figure 9 (check Table S1 for dataset), the received signals corresponding to a transmitted shear wave of 6 kHz are plotted for various stress levels in both loose and dense samples. The estimated arrival time based on point C is highlighted in red to facilitate interpretation. The results indicate that noise and distortions increase significantly with an increasing stress level, making the identification of the arrival time more complex and subjective. This occurs despite the increase of $L/\lambda$ with confinement stress, suggesting that this parameter is stress dependent. Based on the analysis performed, an $L/\lambda$ greater than 2.0 appears to yield the most reliable interpretation at lower stress levels (below 300 kPa), while at higher levels, values between 1.5 and 2.0 provided clearer results. As expected, the arrival time decreases with the increase in the stress level, indicating a corresponding increase in the shear stiffness and $G_{max}$. This trend is evident in both loose and dense samples, with the dense samples consistently exhibiting shorter arrival times at all corresponding stress levels.

5.3. Frequency Domain Framework

Given the uncertainties involved in estimating the arrival time within the TD framework, the use of FD approaches has been suggested by several studies [16,19,23,68,69,70], as a more objective alternative. Among the methods suggested, the cross-spectrum (CS) method [23] is the most commonly employed since it can be applied to single pulse signals, whereas other methods, like the π-point method and continuous-sweep method require continuous harmonic signals to be transmitted [19,24]; consequently, only the CS method was employed in this study. In the CS method, the transmitted and received signals are normalized to ensure they contribute equally to the system’s transfer function, then converted from the time to the frequency domain using an FFT and subsequently correlated to determine the phase angle between them. The arrival time can then be estimated from the slope of the frequency-unwrapped phase plot, as the phase relationship between the transmitted and received signals is expected to be approximately linear. However, the reliability of FD methods has been questioned by several studies [22,25,66], as these approaches assume that transmitted and received signals have the same shape and a constant wavelength throughout the sample, which is typically not true due to noise and distortions. It is also recognized that FD methods tend to estimate higher arrival times, although the underlying reason for this behavior remains unclear [22,24,25].

In the present study, the GDS Instruments tool [55] was employed to calculate the phase relationship between both signals. Although an almost linear relation was observed in the vast majority of the results—regardless of the transmitted frequency—there were cases where more complex relationships were detected. As an example, Figure 10 displays the unwrapped phase relationships obtained for the same stress level (p’ = 500 kPa) in the loose sand sample, for transmitted frequencies of 3 and 10 kHz. While a clear and consistent slope is observed at 3 kHz for most of the system frequencies, the 10 kHz signal results in a broken curve with sudden jumps, making it difficult to determine a representative slope, which is consistent with the results obtained by Rees, et al. [55]. As previously mentioned, this unexpected behavior is most likely due to the significant noise and distortions observed at higher stress levels, which affected the received signal making it difficult to interpret. It should be noted that even when a clear slope appears to be defined, the selection of the interval of points used to fit the trendline can moderately influence the value of the arrival time.

5.4. Comparison of the Two Frameworks

Table 3. Comparison of $G_{max}$ (MPa) obtained using different BE test interpretation methods.

p’ (kPa)	${\mathit{D}}_{\mathit{r}}$ = 40%						${\mathit{D}}_{\mathit{r}}$ = 80%
	A	B	S-S (C)	P-P (D)	CC	CS	A	B	S-S (C)	P-P (D)	CC	CS
25	51.0	45.1	40.2	40.1	28.4	40.5	113.1	86.7	75.3	65.1	65.3	61.3
50	69.0	61.6	54.4	54.8	38.3	55.6	138.6	120.7	106.1	92.0	92.9	91.3
100	98.4	84.1	75.9	60.3	54.8	77.2	205.4	168.4	132.8	130.3	132.8	116.0
150	116.7	102.2	91.4	72.6	67.5	93.1	285.6	216.5	165.7	159.8	160.7	130.0
200	134.5	116.7	102.2	84.9	79.4	108.4	300.8	241.6	176.6	181.3	179.5	137.1
250	151.8	132.5	113.5	93.4	90.2	120.8	335.1	288.5	205.3	207.5	195.0	151.0
300	169.8	147.1	119.9	103.3	99.6	131.5	424.1	282.6	212.6	244.5	212.6	181.0
350	178.5	156.6	134.4	114.8	109.0	134.0	398.8	300.7	241.6	254.0	228.4	201.7
400	187.9	167.0	140.5	123.0	116.6	144.9	516.3	335.0	255.8	280.4	246.2	209.5
450	198.1	178.5	151.7	134.0	126.8	156.5	482.4	358.5	266.0	298.3	260.8	237.0
500	213.0	191.2	156.5	138.0	132.4	164.7	569.9	366.9	300.7	325.0	276.9	268.7

and Figure 11 present the $G_{max}$ values obtained using the previously mentioned BE test interpretation methods for both loose and dense sand samples across all applied stress levels. As expected, all methods estimate higher $G_{max}$ values in the dense sample across all corresponding stress levels. Naturally, given its early position on the received signal, the $G_{max}$ determined using the first sharp (point A) method yields the highest values, followed by the first bump (point B) method. Interestingly, S-S and P-P methods provide similar $G_{max}$ values for the dense sand, whereas for the loose sand, S-S method results in slightly higher values. The CC method yields the lowest $G_{max}$ values for the loose sand (slightly smaller than those obtained with P-P method), whereas for the dense sand, it estimates values identical to the S-S method, being slightly higher at the highest applied stress levels. Finally, the CS method in the FD framework predicts the lowest $G_{max}$ values for the dense sand, while for the loose sand it estimates values similar to those obtained by the S-S method. Overall, if first sharp and first bump methods are excluded due to their early positions in the received signal, the remaining methods yield results within a margin of 14% for the loose sand and 12% for the dense sand samples, respectively. Among these, the S-S method stands out as the one that best represents the average behavior, justifying its generalized recommendation. To some extent, these results also confirm that frequency-based methods consistently yield lower estimates of $G_{max}$ and their use in practical applications should be approached with caution, as recommend by Yamashita, et al. [25].

5.5. Reliability of the Methodology

In order to validate the reliability of the sample preparation method and the interpretation of the results, two tests were conducted for each of the target relative densities. The $G_{max}$ values obtained using the S-S method are illustrated in Figure 12. The results demonstrate a high level of consistency between tests, with differences remaining below 5% for both densities, confirming the robustness of both the sample preparation procedure and the S-S interpretation methodology. Based on the achieved results, it can be concluded that the employed methodology is capable of reproducing tests with high accuracy, even when accounting for the inevitable experimental uncertainties.

6. Conclusions

This study aimed to contribute to a better understanding of the potential sources of uncertainty associated with BE testing on reconstituted sand samples. In addition to identifying the main challenges related to sample preparation, test execution, and result interpretation, it also sought to present practical solutions and establish methodologies to overcome these issues, which can be of extreme value to future experimentalists using BE. The following conclusions can be drawn based on the obtained results:

• Regarding sample preparation, a simple and effective air dry pluviation device was developed, allowing the preparation of samples with the desired relative density in a practical and reproducible manner. Setting up loose samples in the triaxial apparatus is particularly challenging, as any unexpected vibration can lead to densification, irreversibly compromising the test results. It was also observed that water percolation during the sample saturation process at low stress levels can induce significant changes in the density of loose samples, with variations of 30 and 40% being recorded. In contrast, dense samples experience smaller disturbances (less than 8%), although it remains advisable to perform percolation under high stresses but with low pressure differentials at the ends of the sample (preferably below 5 kPa).
• The proper functioning of all the equipment and instruments involved in BE testing is fundamental. Even simple factors, such as the alignment between BE transmitter and receiver, are critical, since misalignments greater than 30° can result in up to 50% signal loss and a significant increase of noise. To minimize electromagnetic noise, the use of short, double-shielded cables is essential. The results also confirmed that transmitting a single sinusoidal pulse at different frequencies is an effective strategy, as the clarity of the received signal depends on frequency. Additionally, it is important to account for the system’s response time, which was determined to be approximately 10 μs for the equipment used in this study.
• Due to the presence of noise and distortions, interpreting BE results becomes complex, especially when the arrival time is assessed in the TD. The results showed that signals transmitted at frequencies smaller than 6 kHz were difficult to interpret, particularly in loose sands and under low stress levels. In contrast, dense sands yielded consistent arrival time values regardless of the applied stress level. Based on these observations, the arrival time was defined as the average value of the values obtained from signals transmitted above 6 kHz, which resulted in a dispersion of less than 5% in all analyzed cases. However, if the lower transmitted frequencies were used in the calculation, significantly higher dispersion would have been obtained, making it essential to critically assess the results and exclude any that appear unreasonable. The findings also confirm that an $L/\lambda >2.0$ is a reliable indicator of signal quality for stress levels below 300 kPa, whereas values between 1.5 and 2.0 yielded the best signal quality for greater stress levels.
• Interpretation in the FD is generally more straightforward, although some phase relationships exhibited poorly defined slopes. Additionally, the selection of the interval of points used to fit the trendline was found to moderately influence the estimated arrival time.
• As expected, first sharp (point A) and first bump (point B) interpretation methods yielded the highest $G_{max}$ values. The S-S, P-P, CC, and CS methods provided results within a margin of 14% for the loose sand and 12% for the dense sand samples. Among these, the S-S method appears to be the most reliable, whereas frequency-based methods consistently yield lower values and should therefore be used with caution.

As demonstrated by the reliability tests, the developed sample preparation methodology and BE interpretation procedures are capable of reproducing results with high accuracy, even when accounting for inevitable experimental uncertainties. These findings confirm that the adopted procedures not only ensure consistency but also offer a robust framework and valuable guidance for future BE testing and interpretation, namely in reconstituted sandy soils. These findings are specific to the material and sample preparation technique employed in this study. Consequently, other challenges may arise when different materials or conditions are used, which were not addressed here and should be carefully considered by future experimentalists. Still, the experimental results strongly suggest that not following consistent sample preparation techniques and testing procedures may result in a significant variation of the measured $G_{max}$ values, which can have a substantial impact on the design of real-world engineering solutions.