1. Introduction

2. Acoustic Emission and Attenuation of Elastic Waves in Concrete Constructions

3. Materials and Methods

3.3. Data Processing

The analysis of the measurement data was conducted using two distinct processes, each comprised of multiple steps. First, the time-domain features peak-amplitude $A_{\mathrm{peak}}$ and energy E were derived. To achieve this, signals were standardized at a preamplification level of zero dB, and a consistent threshold of 90 dB was established. Following this, both features were calculated directly from the raw data of the time-amplitude signal. The peak-amplitude is typically measured in decibels ($d{B}_{AE}$), whereas energy is calculated over the signal’s duration and is expressed in $V^2·s$. For enhanced comparability, the energy was also converted to decibels in this study using the formula

$$E_{\left(\mathrm{dB}\right)}=10·{log}_{10}\left(E_{(V^2·s)}\right).$$

(4)

The features were then represented in decibels against the distance between the source and sensor, as illustrated in Figure 7, Figure 9 and Figure 12. To account for the variance of individual measurements, a 90 % confidence interval was depicted as an envelope curve for each feature. In processing the data, the two types of attenuation —geometric and dispersive— could not be distinguished in detail due to insufficient knowledge about their specific contributions to the attenuation observed. Nevertheless, converting the data to a logarithmic scale linearized the exponential function’s appearance, facilitating the determination of the attenuation coefficient through linear regression across the entire range of values, i.e., all distances on the x-axis [21]. Additionally, to enhance the empirical distribution statistically, a bootstrapping method with 1000 iterations per attenuation profile was employed, justified by the adequate size of the original sample set. The regression’s calculated coefficient and the standard deviation of its residuals are documented in Table 2, Table 3 and Table 4. Utilizing these two statistical parameters, the characteristics of the phenomena under investigation were assessed.

Figure 5. Frequency spectra of the wire-break and rebound-hammer signals, i.e., longitudinal signal emission into the tendons, for girder B of the Stolpe (BT500) and Witzenhausen girders. The signals were measured at a distance of 0.3 m from the source location. The individual spectra are shown superimposed, with the mean value highlighted in black.

Figure 5. Frequency spectra of the wire-break and rebound-hammer signals, i.e., longitudinal signal emission into the tendons, for girder B of the Stolpe (BT500) and Witzenhausen girders. The signals were measured at a distance of 0.3 m from the source location. The individual spectra are shown superimposed, with the mean value highlighted in black.

In the second process, the focus was on the frequency dependence of attenuation. To this end, signals were normalized to a preamplification level of zero dB. A threshold is not required for frequency analysis. The signals were trimmed to a unified length with pre- and post-triggers of 200 and 1000 µs, respectively. This segmented signal was then processed, resampling at 10 MHz and applying a Hanning window to mitigate leakage effects before being transformed into the frequency domain with Fourier transformation. To investigate the frequency dependence of attenuation, the resulting frequency spectrum was segmented into partial frequency bands, each 10 kHz wide, spanning from 0 to 100 kHz, as illustrated in Figure 5a. The investigation’s upper frequency limit was set at 100 kHz. This was based on the observation that the frequency spectra of the measured signals from both signal sources show a significant decrease within the 70 to 100 kHz range, already evident near the source location (approximately 0.3 m), as depicted in Figure 5. The subsequent increase between 115 and 120 kHz remains below the amplitudes noted in the range of 10 to 60 kHz and is therefore excluded from the evaluation, presumably resulting from a harmonic of the sensor’s resonance frequency, around 60 kHz. For each of the delineated partial frequency bands, the mean spectral amplitude was calculated. These calculations served as the basis for creating the attenuation profiles shown in Figure 6, Figure 8, Figure 10 and Figure 11. In these figures, a representative curve for each structure is displayed, regardless of the number of sensors analysed. The mean amplitudes were plotted against the distance between the signal source and sensor, similarly to the previous time-domain features. For the frequency bands extending up to 50 kHz, the attenuation coefficients and the standard deviations of the residuals were determined using linear regression and subsequently documented in Table 2, Table 3 and Table 4. The analysis again utilized bootstrapping to expand the sample size.

The resulting statistical parameters from the regressions of both processing approaches were finally transformed into box plots to illustrate the statistical trends and variations in Figure 13.

4. Results

4.1. Wire-Breaks on Griders

The wire-breaks were generated on the girders from Stolpe, Roding, and Witzenhausen. Figure 5 shows the amplitudes for the frequency bands depending on the distance between the signal source and the sensor. Each girder (A and B) is depicted in a separate subplot. Figure 6 shows the time-domain features. In general, the attenuation profiles presented demonstrate the expected trend where amplitudes diminish as the distance from the signal source increases. This attenuation is least noticeable at lower frequencies, where the relationship between distance and amplitude reduction can sometimes appear nearly linear. However, with an increase in frequency the amplitudes decline more sharply, leading to nonlinear profiles characterized by a steep decrease in amplitude over distances up to 3 to 4 m. Beyond this point, the slope of the decrease lessens, suggesting that the frequency components have been almost entirely filtered out. This effect varies in magnitude across different girders; for instance, it is clearly visible in the case of girder A from Stolpe, whereas it is less pronounced in girder A from Witzenhausen.

The observation is similarly reflected in the time-domain features. In Figure 7, the plots represent data each from one measurement campaign conducted on one tendon per girder. The nonlinear relationship between the variables is also evident to varying degrees. Despite this nonlinearity, employing linear regression across the full range of the abscissa for calculating the attenuation coefficients is deemed sufficiently precise for each scenario, aligning with standard practices. It is important to acknowledge, however, that this methodology tends in absolute terms to overestimate the attenuation coefficient at greater distances.

Figure 6. Frequency dependent attenuation profiles of wire-break signals for the girders from Stolpe (BT500), Roding and Witzenhausen.

Figure 6. Frequency dependent attenuation profiles of wire-break signals for the girders from Stolpe (BT500), Roding and Witzenhausen.

Figure 7. Attenuation profiles of the time-domain features peak-amplitude in (a) and energy in (b) for wire-break signals for the girders from Stolpe (BT500), Roding and Witzenhausen.

Figure 7. Attenuation profiles of the time-domain features peak-amplitude in (a) and energy in (b) for wire-break signals for the girders from Stolpe (BT500), Roding and Witzenhausen.

In Table 2, the attenuation coefficients documented reveal that the girders from Roding exhibit the highest values, indicating a significant increase in attenuation as the frequency rises as well, with girder A being particularly notable. This is partly attributed to the shorter length of the Roding girders, which limited the range of distances between the source and sensor that could be examined. For girder A, it was only possible to measure distances up to approximately 3 m, leading to an overestimation of the attenuation coefficients in this case. Due to this lack of comparability, these values were excluded from the statistical analysis presented in Section 5. Moreover, there is a considerable variation in the cross-sectional dimensions of the girders. The girders from Witzenhausen are characterized by their slender profile in the section of the wires, while the Stolpe girders feature hollow bodies that locally reduce the cross-section, as detailed in [13]. In contrast, the Roding girders are significantly more massive and contain a large number of tendons, which act as major scatterers and contribute to increased attenuation. Nevertheless, the cross-sectional dimensions are likely the critical factor influencing these observations. The geometric dimension of the cross-section becomes apparent once the spherical wavefront originating from the source encounters the component’s surface. In slender components this occurs after shorter travel distances when compared to more massive components. Subsequently, the wavefront ceases to expand, apart from reflections, and when the curvature of the wavefront decreases, it effectively transforms itself into a planar wave. This phenomenon mitigates the impact of geometric attenuation in slender components. This hypothesis is supported in the analysis of time-domain features, where, with the exception of girder A from Stolpe, the Roding girder demonstrated higher attenuation levels.

Table 2. Attenuation coefficients and residual standard deviations for wire-break signals for the girders from Stolpe (BT500), Roding and Witzenhausen in their respective frequency bands and for the time-domain features.

Table 2. Attenuation coefficients and residual standard deviations for wire-break signals for the girders from Stolpe (BT500), Roding and Witzenhausen in their respective frequency bands and for the time-domain features.

Partial Spectra (kHz) and Feature (dB)	Attenuation Coefficient (dB/m)						Residual Standard Deviation (dB/m)
	Stolpe (BT500)		Roding		Witzenhausen		Stolpe (BT500)		Roding		Witzenhausen
	A	B	A	B	A	B	A	B	A	B	A	B
0-10	-4.7	-4.5	-5.8	-6.9	-4.2	-5.3	7.3	5.2	7.3	6.2	7.4	5.5
10-20	-5.2	-7	-10.2	-7.6	-4.4	-5.8	9.6	6.1	7.3	5	8.3	6.3
20-30	-4.4	-8.8	-11.7	-8.7	-5	-7.7	10.6	7.8	7.7	5.6	7.5	6.7
30-40	-5.7	-10.1	-12.5	-10.6	-6.2	-8.8	8.8	7.7	8.1	4.9	8	6.6
40-50	-8.2	-12.3	-15.6	-11.8	-6.8	-9.6	7.6	6.8	8.1	4.7	8.3	6.9
${\mathrm{A}}_{\mathrm{peak}}$	-5.5	-2.8	-7.1	-4.9	-3.7	-4.6	4.7	6.3	2.5	4.8	2.9	5
E	-4.9	-2.2	-5.4	-3.8	-2.6	-3.3	4.5	6.5	2.5	4.9	2.3	4.2

4.2. Rebound-Hammer Impacts on Girders

In Figure 8 and Figure 9 the experiments utilizing the rebound-hammer on the girders are illustrated. These tests involved generating signals on the concrete surface orthogonal to the tendon axis and at the prestressing steel at locations where the cross-sections of the tendons were accessible. The figures provide a side-by-side comparison of these two signal input methods for the girders B of each structure. Additionally, Table 3 aggregates the statistical parameters derived from these measurements.

Table 3. Attenuation coefficients and residual standard deviations for rebound-hammer impulses on the tendon (longitudinal) and the concrete surface (orthogonal) for the girders B from Stolpe (BT500), Roding, and Witzenhausen in their respective frequency bands and for the time-domain features.

Table 3. Attenuation coefficients and residual standard deviations for rebound-hammer impulses on the tendon (longitudinal) and the concrete surface (orthogonal) for the girders B from Stolpe (BT500), Roding, and Witzenhausen in their respective frequency bands and for the time-domain features.

Partial Spectra (kHz) and Feature (dB)	Attenuation Coefficient (dB/m)						Residual Standard Deviation (dB/m)
	Stolpe (BT500)		Roding		Witzenhausen		Stolpe (BT500)		Roding		Witzenhausen
	Tendon	Concrete	Tendon	Concrete	Tendon	Concrete	Tendon	Concrete	Tendon	Concrete	Tendon	Concrete
0-10	-4.8	-4.6	-5.5	-7.4	-5.2	-5.5	6.4	4.5	3.5	6.1	4.1	3.7
10-20	-5.2	-5.6	-4.7	-9	-6	-6.1	6.1	4.8	4	4.3	5.3	5
20-30	-7.7	-8.7	-5.8	-11	-8.1	-8.4	7.2	5.5	3.3	3.8	6.1	3.7
30-40	-9.7	-8.8	-7.4	-12.4	-8.8	-9.1	8.3	5.8	3.6	4.5	6	5.4
40-50	-11.5	-10.5	-8.4	-14	-10.4	-10.3	6.7	3.3	4.1	4.8	6.4	5.5
${\mathrm{A}}_{\mathrm{peak}}$	-2.7	-2.2	-3.4	-5.3	-3.8	-3.6	2.7	2.5	2.5	7.9	4.5	2.1
E	-2.3	-1.4	-2.4	-4.4	-3	-2.8	3.5	2.1	16.2	7.5	3.3	2.3

Figure 8. Frequency-dependent attenuation profiles of rebound-hammer impulses on the tendon (longitudinal) in the subfigures (a), (c), and (e), as well as on the concrete surface (orthogonal) in the subfigures (b), (e), and (f) for the girders B from Stolpe (BT500), Roding and Witzenhausen.

Figure 8. Frequency-dependent attenuation profiles of rebound-hammer impulses on the tendon (longitudinal) in the subfigures (a), (c), and (e), as well as on the concrete surface (orthogonal) in the subfigures (b), (e), and (f) for the girders B from Stolpe (BT500), Roding and Witzenhausen.

Figure 9. Attenuation profiles of the time-domain features peak-amplitude in (a) and energy in (b) for rebound-hammer impulses on the tendon (longitudinal) and the concrete surface (orthogonal) for the girders from Stolpe (BT500), Roding and Witzenhausen.

Figure 9. Attenuation profiles of the time-domain features peak-amplitude in (a) and energy in (b) for rebound-hammer impulses on the tendon (longitudinal) and the concrete surface (orthogonal) for the girders from Stolpe (BT500), Roding and Witzenhausen.

The signal profiles resulting from the rebound-hammer impulses share general characteristics with those of wire-break events, demonstrating similar behaviour in terms of attenuation and signal propagation. Despite these similarities, closer examination also reveals differences in the attenuation coefficients between the two types of sources. Notably for the frequency bands, in some cases signals from rebound-hammer impulses tend to exhibit slightly higher attenuation compared to those from wire-breaks, e.g., for the Witzenhausen girder. This observation, however, is not confirmed by the analysis of time-domain features, which suggests the opposite for Stolpe and Witzenhausen. This discrepancy might originate from the methodological differences in data processing between the two analyses. Specifically, for the frequency band analysis only the initial part of the signal was considered, focusing on the signal onset for extraction and evaluation. In contrast, the time-domain feature analysis took the entire signal into account. Moreover, the comparison between the two methodologies is inherently constrained due to the composite nature of time-domain features, which superimpose all frequency components without delineating their distinct contributions, notably in the context of peak-amplitude. Given the practical importance of time-domain features in assessing attenuation, this feature should be considered as decisive in the analysis.

In the comparison of excitation types with the rebound-hammer, the time-domain features for Stolpe and Witzenhausen exhibit higher attenuation coefficients when the excitation is applied to the tendon, thus emphasizing the analogy to wire breaks. Yet the time-domain features reveal the reverse pattern, except for the Roding girder, where the opposite can be observed, and the inversion represents a "safer" scenario for extrapolating results. In this case, the attenuation coefficients obtained with the rebound-hammer are higher than those for wire-breaks. However, the quality of correlations for this case are the weakest, as illustrated in Figure 9.

These findings highlight that both wire-break and rebound-hammer excitations exhibit very similar attenuation behaviour, independent of the method of excitation. Nonetheless, it also underscores the complexity in interpreting individual measurements. Consequently, a statistical examination of the data will be undertaken in Section 5 to facilitate a better understanding of how results can be transferred between the two sources based on a broader perspective. This analysis will prioritize time-domain features, reflecting their widespread adoption in both academic and practical applications.

4.3. Rebound-Hammer Impacts on Bridges

The girders discussed in Section 4.1 and Section 4.2 which, due to their smaller dimensions compared to actual bridges, may yield signal propagation influenced by effects such as reflections, potentially skewing the assessed attenuation coefficients. To evaluate the findings from these girders further, measurements from six bridges were included. The attenuation was evaluated in both the longitudinal and transverse directions for each of these structures. Figure 10 depicts the frequency-dependent attenuation profiles for the structures in the longitudinal direction, while Figure 11 shows them in the transverse direction. Each structure is represented by a sensor’s characteristic profile, even when multiple datasets were available; see Table 1. Figure 12 presents the time-domain features for both directions of analysis, with coefficients and standard deviations detailed in Table 4.

Compared to the girders’ profiles, it is notable that in the bridge measurements the initial amplitudes across different frequency bands resemble each other more than those observed in the girder measurements; see Figure 8, concrete surface. This distribution pattern is somewhat mirrored in the case of girder B from the Roding bridge; see Figure 8d. Given that the Roding bridge’s girders were more heavily reinforced than the others, it suggests that the increased reinforcement and the proximity of tendons to the surface contribute to the reduced attenuation of high-frequency components. Along the bridges’ longitudinal measurements, such as on the web, reinforcement such as tendons was also present. Similarly, the transverse profiles of the BT500 Bridge and Ruhrwald Bridge, measured in the load-bearing areas, which are characterized by significant reinforcement due to shear loads and the anchorage of the tendons, exhibited comparable patterns, as depicted in Figure 10.

Figure 10. Frequency-dependent attenuation profiles of rebound-hammer impulses in longitudinal direction of the structure for the Stennert Bridge (a), High Bridge (b), Werra Bridge (c) and South Ring Bridge (c) for one exemplary sensor position.

Figure 10. Frequency-dependent attenuation profiles of rebound-hammer impulses in longitudinal direction of the structure for the Stennert Bridge (a), High Bridge (b), Werra Bridge (c) and South Ring Bridge (c) for one exemplary sensor position.

Figure 11. Frequency-dependent attenuation profiles of rebound-hammer impulses in transverse direction of the structure for the High Bridge (a), South Ring Bridge (b), BT500 Bridge (c) and Ruhrwald Bidge (d) for one exemplary sensor position.

Figure 11. Frequency-dependent attenuation profiles of rebound-hammer impulses in transverse direction of the structure for the High Bridge (a), South Ring Bridge (b), BT500 Bridge (c) and Ruhrwald Bidge (d) for one exemplary sensor position.

Figure 12. Attenuation profiles of the time-domain features peak-amplitude in (a) and energy in (b) for rebound-hammer impulses on the bridges in longitudinal and transverse direction of the structure for one exemplary sensor position.

Figure 12. Attenuation profiles of the time-domain features peak-amplitude in (a) and energy in (b) for rebound-hammer impulses on the bridges in longitudinal and transverse direction of the structure for one exemplary sensor position.

Moreover, the attenuation profiles in Figure 10 align with prior observations, showing significant attenuation of high-frequency components up to 50 kHz within a spread of approximately 3 m from the source to the sensor, in contrast to a much lesser decrease observed for low frequencies. Notably, at distances of around 5 to 6 m from the source, the amplitudes for frequencies below 30 to 40 kHz begin to increase again in the Werra Bridge and in the High Bridge. This phenomenon may be attributed to the structural design and variations in reinforcement configuration along the structure’s length, both of which influence signal pathways. Additionally, it is remarkable that the pre-existing damage to the Werra Bridge’s concrete due to ASR manifests only minor differences in the frequency range analysis when compared to the Stennert Bridge; yet a more pronounced correlation is observed in the time-domain features. This highlights the challenges associated with individual assessments and underscores the necessity of performing detailed statistical analysis with particular emphasis on time-domain features, as outlined in Section 5.

The investigation of attenuation anisotropy is examined by means of three examples, each containing results in both the longitudinal and transverse directions. For the South Ring Bridge, the directional difference in attenuation is surprisingly small, with the transverse attenuation being at most 30 % greater than the longitudinal, as depicted in the time-domain features of Table 4. This may be attributed to the sensor’s placement on the side surface of the main girder, unlike the positioning on the underside with other bridges. At the High Bridge, transverse attenuation is roughly twice that of the longitudinal direction. The presence of hollow bodies within the slab, contributing to enhanced wave scattering, is likely a significant factor in such increased attenuation. Furthermore, combining the results from the Stolpe´s BT500 girder and the BT500 Bridge —owing to their use of identical prefabricated elements— reveals a pronounced increase in transverse attenuation over longitudinal. Time-domain features indicate that the transverse attenuation is about four times greater than the longitudinal. This difference is attributed to the construction. The BT500 Bridge’s slab comprises individual prefabricated elements joined by an in-situ concrete layer, elevating the number of construction-related transition zones and challenging the assumption of seamless acoustic coupling between elements. Consequently, for monitoring purposes each prefabricated element might require individual surveillance if minor transmission is possible. The Ruhrwald Bridge, constructed similarly to the BT500 Bridge, also demonstrates evident transverse propagation, supporting the viability of a two-dimensional sensor concept for both structures. Remarkably, both the Ruhrwald and BT500 Bridges exhibit lower attenuation than the High Bridge, underscoring the predominant influence of hollow bodies on attenuation behaviour.

Table 4. Attenuation coefficients and residual standard deviations for rebound-hammer impulses on the bridges in the longitudinal and the transverse directions of the structure in their respective frequency bands and for the time-domain features.

Table 4. Attenuation coefficients and residual standard deviations for rebound-hammer impulses on the bridges in the longitudinal and the transverse directions of the structure in their respective frequency bands and for the time-domain features.

Partial Spectra (kHz) and Feature (dB)	Attenuation Coefficient (dB/m)								Residual Standard Deviation (dB/m)
	Longitudinal				Transversal				Longitudinal				Transversal
	Stennert Br.	Werra br.	South Ring Br.	High Br.	South Ring Br.	High Br.	Ruhrwald Br.	BT500 Br.	Stennert Br.	Werra br.	South Ring Br.	High Br.	South Ring Br.	High Br.	Ruhrwald Br.	BT500 Br.
0-10	-4.5	-3.8	-3.5	-1.9	-3.5	-8.9	-4.5	-7.5	6.1	8.9	6.1	4.4	4.7	2.6	3	5.2
10-20	-4.2	-4.9	-3.7	-4.3	-4.2	-9.8	-7.2	-8.4	3.7	8.3	4.4	3	2.5	4.3	3.7	5.1
20-30	-5.2	-8.5	-4.3	-4.7	-5.8	-11.8	-8.8	-11.7	5.3	5.7	3.8	3.2	2.8	1.9	5.1	7.6
30-40	-9.7	-10.9	-5.3	-7	-6.3	-11.2	-11.3	-12.5	5.5	8.3	4.1	5.3	4.2	5.1	6	8.1
40-50	-10.8	-11.1	-6.6	-7.1	-6.7	-14.1	-10.7	-11.1	7	11.8	4.7	5.5	3.3	4.1	8.2	10.6
${\mathrm{A}}_{\mathrm{peak}}$	-1.7	-5.7	-2.7	-4.5	-4.3	-10.5	-6.8	-7.4	2.5	3.9	2.4	4	3.9	1.8	2.6	2.3
E	-1.9	-6.1	-2.7	-4.4	-3.9	-10.8	-6.7	-7.4	1.8	3.6	2.1	3.6	4.4	2.7	2.8	3.4

5. Aggregation of Results and Discussion

Figure 13. Statistical distributions of the attenuation coefficients and residual standard deviation of the regressions in their respective frequency bands and for the time-domain features for wire-break and rebound-hammer signals at the girders in (a) and (c) and for rebound-hammer signals at bridges in longitudinal and transverse direction in (b) and (d).

Figure 13. Statistical distributions of the attenuation coefficients and residual standard deviation of the regressions in their respective frequency bands and for the time-domain features for wire-break and rebound-hammer signals at the girders in (a) and (c) and for rebound-hammer signals at bridges in longitudinal and transverse direction in (b) and (d).

6. Conclusions

Abbreviations

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