Distributed fiber-optic sensing of reinforced concrete arch ribs to monitor plastic hinge formation

Abstract

This research aims to evaluate the potential of fiber-optic sensing of reinforced concrete (RC) structures for improved understanding of damage progression in laboratory testing and for rapid postearthquake damage assessment of real structures. A testing campaign was conducted on RC specimens with different detailing options for RC arch bridges, which experience high axial load and bending moment during earthquakes. Several types of fiber-optic cables were embedded within the four RC arch rib specimens, which were subjected to varied axial loads followed by cyclic lateral loading. The fiber-optic cables enable detailed distributed strain measurement within the RC specimen throughout the testing, as well as the residual distributed strain after each load cycle. Results provide new insight regarding the progression of cracking and the measurement of plastic hinge formation; results are compared with previous plastic hinge models. The residual strain measurements led to the development of a novel damage index that enables correlation of residual distributed strain with the maximum drift experienced during cyclic loading; this correlation is essential to evaluate potential damage when continuous fiber-optic sensing is not economically feasible and only postearthquake fiber-optic strain data are available. The findings from the testing will not only have a direct impact on the design of RC arch ribs but also pave the way for distributed fiber-optic sensing-driven postearthquake damage assessment.

Introduction

In recent years, the field of SHM has seen significant advancements with the development of various sensing technologies. Recent innovations in infrastructure monitoring, including terrestrial laser scanning, unmanned aerial vehicle imagery, ¹ and piezoelectric sensors (PZTs), ² offer promising alternatives for structural assessment. Externally surface adhesively bonded PZTs, in particular, have been successfully applied to detect and identify concrete cracking and steel yielding in large-scale reinforced concrete (RC) structural members subjected to shear and flexural monotonic and cyclic loading. These sensors have demonstrated the ability to qualitatively evaluate the propagation and severity level of damage in RC structures.^3,4 However, these technologies are often limited in their spatial coverage, sensitivity to environmental noise, or inability to integrate seamlessly within structural components.^5,6

Fiber-optic sensing technologies,^7–9 on the other hand, represent a powerful tool for SHM due to their unique advantages. These sensors, known for their high sensitivity and accuracy, provide more detailed measurement from which engineers can understand the behavior of RC structures. Unlike traditional monitoring methods, fiber-optic sensors offer the advantages of distributed sensing, allowing for the detection and localization of strain changes over large areas. Their small size, immunity to electromagnetic interference, and ability to withstand harsh environmental conditions make them particularly suitable for embedding within concrete structures. This has opened new possibilities in monitoring critical structural elements, such as arch ribs in bridges, for signs of deformation and damage, including the formation of plastic hinges, which are pivotal in understanding the seismic responses and overall resilience of such structures.

Distributed fiber-optic sensing (DFOS) has been applied successfully in various civil infrastructure projects to measure distributed temperature and deformation and detect damage. ¹⁰ Nevertheless, previous applications using Brillouin optical time domain reflectometry (BOTDR) or quasidistributed fiber-optic sensing like fiber Bragg grating (FBG) were limited by their spatial resolution.^11–16 Advances in DFOS technologies, such as pulse prepump Brillouin optical time domain analysis (PPP-BOTDA) and OFDR,^17,18 have significantly enhanced spatial resolution and measurement accuracy. These improvements enable more precise detection of local damage and support a transition from simple damage identification to detailed quantification, offering greater insights into structural health.

Numerous researchers have recently made contributions to the field of DFOS in RC, with studies ranging from concrete crack monitoring,^19–24 shape sensing, ²⁵ and corrosion monitoring of steel bars,^26–28 to the exploration of SHM challenges and techniques for large-scale structures.^29–31 Zhang et al. ³² conducted an experimental study utilizing OFDR with different fiber-optic cables demonstrates their effectiveness in detecting concrete cracking and reinforcement deformation in RC, leading to practical guidelines for fiber selection based on response to nonlinear behaviors. Zhang et al. ³³ introduced a mechanical spring model to elucidate fiber deformation arising from displacement discontinuities, further corroborating their model with experimental findings from both cable calibration and concrete cracking tests. Another study by Zhang et al., ³⁴ delves into the strain transfer mechanism of various fiber-optic cables embedded in concrete, particularly under conditions of displacement discontinuity, to enhance the interpretation of strain sensing results. It compares the strain transfer characteristics of different cables, examines their mechanical properties, and introduces a parameter for quantifying strain transfer length, aiding in the selection and interpretation of fiber-optic sensors in structural applications. Berrocal et al. ³⁵ studied the effectiveness of robust distributed optical fiber sensors embedded in concrete beams for assessing vertical deflection and crack width, presenting a novel postprocessing technique for enhanced structural condition analysis. Additionally, Zhang et al. ³⁶ explored the use of OFDR for its precision in detecting strain and assessing damage within RC structures, exemplified through laboratory tests on a novel beam-column connection. The study also proposes structural damage indices derived from DFOS results, which show a promising correlation with maximum sustained drift, suggesting the potential of DFOS in assessing RC structural damage. Brault et al. ³⁷ developed a fiber-optic sensing technique for detailed field monitoring of RC beam deflections and crack widths, demonstrating its effectiveness and practicality through tests on varied beam specimens, and showcasing its superiority in data robustness compared with other sensor technologies. In Richter et al., ³⁸ the use of distributed fiber-optic sensing for precise crack monitoring in concrete structures is demonstrated, introducing the “fosanalysis” Python framework for scalable data analysis and demonstrating its effectiveness in accurately measuring crack widths in a RC beam. The study by Hoult et al. ³⁹ focuses on experimental analyses of two RC U-shaped walls, examining their flexural and torsional responses using both traditional and high-definition fiber-optic sensors to understand their behavior until failure.

In this study, a comprehensive testing campaign was carried out, involving a variety of fiber optic cable types embedded in four RC specimens, each subjected to different axial forces and reinforcement configurations. The objectives of this research are to (1) investigate the application of DFOS for dynamic and postearthquake measurements to detect and quantify plastic hinge deformations in critical regions of RC structures; (2) develop an index to quantify plastic damage and establish a correlation between residual strain and maximum strain experienced by RC structures for improved damage assessment; and (3) utilize DFOS data to inform and enhance design of RC structures, specifically aiming to optimize the design of arch ribs in bridge constructions.

Experimental setup

The four specimens examined are part of the second phase of the “Ductile Behavior of Reinforced Concrete Arch Ribs” project. The project aims to assess the performance of RC arch ribs under various axial forces and reinforcement configurations. Fundamentally, it seeks to explore the inelastic deformation capacity of RC members subjected to high levels of axial force combined with lateral deformation reversals.

To fulfill these objectives, the study involved conducting tests on a series of test specimens at one-third of full scale. These specimens were subjected to reversed cyclic lateral displacements with incrementally increasing amplitude while under a constant axial load. The test variables included the volumetric ratio of transverse reinforcement, the configuration of transverse reinforcement, the geometry of the cross section, and the axial force ratio $\rho =P/\left(A_g{f}_c^{\prime }\right)$ , where P is the working axial load, $f_c^{\prime }$ is the compressive strength measured on companion cylinders at the age of testing, and $A_g$ is the gross cross section of the arch rib.

Materials and specimens

The test specimens were modeled after the Shasta Viaduct Replacement Bridge in California. Each specimen comprised two main components: a heavily reinforced foundation block and a segment of the arch rib. The foundation block measures 48 inches by 48 inches by 60 inches (1219 $\times 1219\times$ 1524 mm). The arch rib segment, 168 inches (4267 mm) in length, features a square cross-section with 28-inches (711 mm) sides.

The reinforcement in the arch rib includes 28 No. 5 longitudinal bars arranged in a circular pattern, encased within closely spaced No. 3 circular hoops, all conforming to ASTM A706 Grade 60 specifications. A detailed comparison of the reinforcement configurations for all four specimens in the test series is presented in Figures 1 and 2. Specimen 2 is distinguished by a hollow section containing a 16-inch (406 mm) diameter steel pipe at its center, designed to reduce the mass of arch rib by creating an air void and providing internal reinforcement to counteract the dilation of compressed concrete into the void. This pipe was not intended to enhance the moment capacity at the critical section and thus did not extend into the foundation block. All specimens underwent testing about 30 days postcasting, targeting a concrete compressive strength of 4 ksi (27.6 MPa).

Figure 1.

Cross-sectional views of the specimens: (a) Specimen #1 and (b) Specimen #2.

Figure 2.

Cross-sectional views of the second batch specimens: Specimen #3 and #4. (Note: The figure represents both specimens with the primary difference being the stirrup spacing and welded headed bars spacing — 2” for Specimen #3 and 3” for Specimen #4.).

Load test setup and procedure

To measure the strains in the longitudinal and transverse reinforcement, strain gauges were employed. Local shear and flexural deformations along the arch rib’s length were measured using an array of linear variable differential transformers (LVDTs), while global displacement at the arch rib tip was recorded via wire potentiometers.

The axial force, applied at the outset of the test, remained constant throughout. Upon reaching the designed axial load, a cyclic lateral force was introduced in incremental steps, each featuring two displacement cycles. The target displacement for each successive step was 1.4 times that of its predecessor. The axial forces were calculated based on the designed compressive strength of each arch rib to simulate the supported gravity. Specifically, specimen 1 was subjected to an axial force of 850 kips (3781 kN), specimen 2 to 550 kips (2447 kN), and specimen 3 and 4 each to 680 kips (3024 kN).

Sensing fibers

Figures 1 to 3 depict the layout of optical cables within the specimens. The BRUsens DSS V9 fiber-optic cable by Solifos was selected for its armored central metal tube and structured polyamide outer sheath. Previous uniaxial tension tests on six different fiber-optic cable types, as documented in Liu et al., ⁴⁰ demonstrated that this cable offers optimal sensitivity and durability in concrete.

Figure 3.

Plan (a) and elevation (b) views of specimen #1 (note that specimens 2, 3, and 4 were similarly instrumented.

Strain measurements were measured using an ODiSI 6100 sensor interrogator, ⁴¹ capable of measuring strains up to 1% (10,000 micro $\mu$ strain). Each specimen incorporated four optical cables, designated as V1, V2, H1, and H2, with “V” representing “Vertical” and “H”“Horizontal.” These cables were measured on distinct channels and looped inside the concrete arch rib, entering and exiting through the foundation block. To maximize data collection, especially in the event of damage-induced cable breakage, cables V1 and V2 were configured to carry optical signals in opposite directions. This arrangement ensures that even if significant damage at the top of the specimen results in breakage, complete strain profiles can still be retrieved, with data loss occurring only beyond the breakage point. Given the vertical direction of the cyclic loading, severe damage was anticipated primarily at the top and bottom, likely causing earlier breakage in V1 and V2 compared with H1 and H2. The gauge length (spatial resolution) for each channel was set at 2.6 mm, with a sampling rate of 6.25 Hz.

As depicted in Figure 4(b), the optical cables were embedded directly within the concrete and secured to the longitudinal bars at 1 m intervals within the steel cage using zip ties. The zip ties were intentionally applied loosely to minimize any interference with the strain transfer between the cables and the surrounding concrete. Throughout the attachment process, meticulous attention was paid to avoid sharp bends that might damage the cables. For comprehensive details on fiber-optic cable installation in RC structures, refer to the works of Zhang et al. ³² and Liu et al. ⁴⁰

Figure 4.

Instrumentation setup details: (a) RC specimen positioned in the loading machine and (b) attachment of fiber-optic cables to the steel cage.

Results and discussion

Strain and curvature distribution

The tests were primarily aimed to evaluate structural performance, with a secondary goal of exploring the use of robust DFOS for monitoring plastic hinge formation in RC structures and analyzing residual strain postloading cycles. This section presents and discusses DFOS results at drift ratios of 0.18%, 0.34%, 1.57%, and 2.28% in specimen 2. The drift ratio is defined as the displacement of the arch rib section at the point of lateral force application, measured relative to a tangent to the foundation block, and then divided by the rib segment length. Figure 5 displays the strain and curvature distribution at an early loading stage (0.18% drift), a phase with no visible damage or significant cracks. Figure 5(a) illustrates the drift ratio evolution from LVDTs, marking the loading stage under investigation. An axial compressive force of 550 kips (2446.5 kN) was applied to simulate the gravity load in the prototype structure. The curvature distribution within the arch rib section is derived from analyzing the top and bottom segments of cables V1 and V2, effectively measuring the curvature from these vertically oriented sensors. Regarding the horizontally looped cables H1 and H2, their contribution to curvature analysis is slightly different; curvature is calculated by combining data from one segment of H1 and one segment of H2, both on the same side of the arch rib. Additionally, the gauge length over which curvature is measured is the same as the fiber optic strain gauge length, which is 2.6 mm.

Figure 5.

Specimen 2 loading protocol (a), strain distributions at 0.18% drift in cables V1 (b), V2 (c), H1 (d), and H2 (e), and beam curvature at 0.18% drift; results, illustrate initial loading response with minimal damage.

Figure 5(b) to (e) shows the raw DFOS measurements and the net strain for cables V1, V2, H1, and H2, after accounting for initial compressive strain. The data reveal minor cracks formation near the joint in the cross-sectional areas of V1 and V2. In contrast, cables H1 and H2, being closer to the neutral axis, do not exhibit significant strain peaks indicative of cracking. The symmetrical placement of these four optical cables enables curvature distribution analysis within the arch rib. At 0.18% drift, Figure 5(f) indicates a nearly linear curvature distribution along the length of the arch rib segment. At this early stage, the assumption that longitudinal strain varies linearly through the depth, a principle where plain sections remain plain, holds true according to the FO measurements. However, this assumption might be challenged as the degree of damage to the arch rib increases, a phenomenon that will be further explored at higher drift ratio levels.

At a higher drift ratio of 0.34% (Figure 6), the results indicate increased crack formation and damage propagation beyond the arch rib-foundation joint. Notably, Figure 6(e) shows about five cracks on the left side of the rib, compared with only two on the right, suggesting asymmetric behavior near the neutral axis. Despite this, the curvature distribution remains approximately linear, implying a nearly linear, constant section property along the rib segment. This pattern indicates that there are no plastic regions, as a highly nonlinear distribution would suggest otherwise, providing evidence that plastic damage has not occurred at this loading level.

Figure 6.

Specimen 2 loading protocol (a), strain distributions at 0.34% drift in cables V1 (b), V2 (c), H1 (d), and H2 (e), and beam curvature at 0.34% drift; results illustrate cracks beginning to form.

The DFOS results at a drift ratio of 1.57% are illustrated in Figure 7. At this stage, a plastic damaged region formed near the arch rib-foundation joint, specifically between 2.2–2.8 m and 10.5–11.0 m along cable V1, as depicted in Figure 7(b) and (c). Each concrete crack produces a distinct peak in the strain profile, with numerous cracks near the joint causing a wider, more noticeable peak. This peak is indicative of the plastic damage region. Figure 7(b) further shows that at a drift level of 1.57%, the arch rib can be segmented into three sections: plastic, cracked, and uncracked. This segmentation results from the increasing moment from the arch rib end to the joint, leading to a corresponding increase in strain in the optical fibers and curvature within the arch rib. As shown in Figure 7(f), within 0.5 m range from the joint, the curvatures recorded by cable V1/V2 exhibit a significant variance when compared with those of H1/H2, indicating a highly nonlinear distribution. This discrepancy suggests the onset of plastic damage and the breakdown of constant section properties, implying a plastic hinge length (PHL) of approximately 0.5 m.

Figure 7.

Specimen 2 loading protocol (a), strain distributions at 1.57% drift in cables V1 (b), V2 (c), H1 (d), and H2 (e), and beam curvature at 1.57% drift; results illustrate significant structural damage.

When the drift ratio increases to 2.28%, as shown in Figure 8, the plastic damage extends toward the neutral axis of the arch rib near the joint. At this drift level, the strain and curvature distribution in cables H1/H2 also presents the aforementioned three distinct sections. It is important to note that the ODiSI 6100 sensing platform, with a strain measurement range of $\pm 12,000$ microstrain, may experience data quality degradation near the joint, as observed in Figure 8(b) and (c). Despite this, the curvature distribution patterns from the horizontal (H1/H2) and vertical (V1/V2) configurations generally align well in general, even in the plastic region. The estimated PHL remains approximately 0.5 m at this elevated drift level (see also Figure 8f).

Figure 8.

Specimen 2 loading protocol (a), strain distributions at 2.28% drift in cables V1 (b), V2 (c), H1 (d), and H2 (e), and beam curvature at 2.28% drift; results illustrate extensive plastic deformation.

The observations noted above are substantiated by the recorded photographs of the tested specimen. Figure 9 displays the damage states of the arch rib-foundation joint damage states at both the top and bottom sections under varying drift levels. At drift levels of −0.18% and −0.34%, external inspection does not reveal any visible cracks, yet the DFOS strain profile indicates that minor internal cracks have already formed within the arch rib. When the drift level reaches −1.57%, signs of longitudinal reinforcement yielding and the onset of concrete spalling are evident. With a further increase in drift to −2.28%, more pronounced symptoms such as reinforcement buckling and extensive concrete spalling become apparent. An important observation is the less-pronounced concrete spalling at the top of the arch rib, primarily due to gravitational forces. Specifically, gravity tends to pull the spalled concrete bits downwards at the bottom face, leading them to separate from the core concrete and fall to the laboratory floor. In contrast, at the top face, gravity pulls the spalled bits against the core, hindering their detachment and fall, thereby affecting the apparent severity of spalling observed between the top and bottom faces of the arch rib.

Figure 9.

Depiction of arch rib-foundation joint damage states at both the bottom (left column) top (right column) of specimen 2 for various drift levels.

Damage quantification

Evaluating postearthquake structural damage with DFOS necessitates interpreting residual strain measurements, as it is typically not economically feasible for the sensing platform to be actively recording during seismic events. Key indicators such as maximum drift and PHL are crucial in assessing the load-carrying and deformation capacities of RC structures. Consequently, the primary goal of this section is to establish correlations between residual strain, maximum drift, and PHL.

Peak strain

Figure 10 illustrates the history of the drift ratio and its relationship with the vertical force. Three characteristic loading instants merit attention: (1) positive peaks: marked by down triangles, corresponding to the maximum positive drift in each loading cycle; (2) negative peaks: indicated by up triangles, corresponding to the maximum negative drift; (3) residual points: represented by circles, denoting the moments when the applied vertical force returns to zero. Challenges arise in correlating strain peaks with partially closed cracks during optical cable compression, due to slip between the fiber layers, deformation of the fiber coating, and debonding from the concrete. ⁴⁰ Consequently, this study focuses solely on tensile strain sensing data. For cable V1, this pertains to the bottom part during positive (upwards) drift, and the top part during negative (downwards) drift. Figure 11 represents the evolution of strain distribution within the arch rib at both positive and negative peaks, demonstrating a marked change at a drift level of $\pm 1.60\mathrm{\%}$ . At this critical drift ratio, a distinct nonlinear strain distribution begins to emerge near the rib-foundation joint, indicating a departure from the previously observed linear strain trend that is characterized by minor peaks but maintains a linear progression for drift levels below 1.6%. This nonlinear pattern suggests the initiation of plastic hinge formation, providing a compelling visual cue of the structural behavior transitioning due to increasing deformation. Here, “raw strain” refers to the initial strain data measured directly from the FO sensors, prior to any processing.

Figure 10.

(a) Historical progression of the drift ratio and (b) relationship between applied force and drift ratio.

Figure 11.

Distribution of raw strain at: (a) positive peak drifts and (b) negative peak drifts.

To enable the application of our findings across a broader spectrum of RC structures, characterized by diverse geometries and mechanical properties, we normalize the strain data measured from the FO sensors, referred to as raw strain ( ${ϵ}_{\mathrm{raw}}$ ) by the theoretical concrete strain ( ${ϵ}_{\mathrm{c},\mathrm{theo}}$ ). This normalization process aims to mitigate the effects of variations in specimen size and geometry on the strain measurements, thereby facilitating a more universal analysis. The theoretical concrete strain within the concrete, as defined in this study, originates from the moment-curvature relationship generated by XTRACT, a computer software for the sectional analysis of RC. ⁴² The calculation of theoretical strain involves a multistep process. At a specific drift ratio level, the moment distribution along the arch rib’s length is determined based on the recorded force-tip displacement history. Subsequent linear interpolation within the XTRACT-derived moment-curvature relationship yields the theoretical curvature distribution. Assuming that the neutral axis remains fixed at the geometric center, and by referencing the exact locations of the cables, we can then calculate the theoretical strain in the concrete. It is important to clarify that in XTRACT’s framework, the principle that plain sections remain plain is adopted, thus implying that the strain discussed is sectional strain. The exact position within the cross section where this theoretical strain is calculated align with the location of the fiber to ensure the relevance and accuracy of the comparison between theoretical and observed strains. Figure 12 demonstrates an instance of raw strain at a 2.27% drift and its corresponding theoretical concrete strain. These values are closely aligned in the uncracked linear section (around 3–4 m) but begin to diverge with the formation of more cracks. The net strain is then calculated as follows:

$${ϵ}_{\mathrm{net}}={ϵ}_{\mathrm{raw}}-{ϵ}_{\mathrm{c},\mathrm{theo}}$$

(1)

Figure 12.

Comparison of raw strain vs theoretical strain for a specific loading instance, highlighting the calculation of net strain.

Echoing the format of Figure 11, Figure 13 shows the evolution of net strain distribution at both positive and negative peaks. The theoretical concrete strain is often overestimated because it fails to consider strain reduction in areas adjacent to cracks. This discrepancy arises because FO sensors measure a strain that exceeds the theoretical value, attributed to the slippage occurring at the crack locations, which are not accounted for in the theoretical model. Consequently, the area under each net strain curve represents a lower bound for total crack opening; thus, indicating the damage level within the arch rib. To further extend these findings to a broader range of RC structures, both net and raw strains are normalized using the theoretical concrete strain corresponding to each drift ratio. Figure 14 depicts the relationship between normalized peak strain integration (NPSI) and maximum drifts. It is important to note that two distinct criteria, as defined in Equations (2) and (3), were employed:

$$\mathrm{NPSI}-1=\frac{{\int }_0^L\mathrm{abs}\left({ϵ}_{\mathrm{net}}\right)}{{\int }_0^L{ϵ}_{\mathrm{c},\mathrm{theo}}}$$

(2)

$$\mathrm{NPSI}-2=\frac{{\int }_0^L\mathrm{abs}\left({ϵ}_{\mathrm{raw}}\right)}{{\int }_0^L{ϵ}_{\mathrm{c},\mathrm{yield}}}$$

(3)

Figure 13.

Net peak strain profiles corresponding to: (a) positive peak drifts and (b) negative peak drifts.

Figure 14.

Depiction of the relationship between normalized residual strain integration and maximum drifts, shown for:(a) positive drifts and (b) negative drifts.

where $L$ denotes the arch rib’s length, and ${ϵ}_{\mathrm{c},\mathrm{yield}}$ represents the theoretical concrete strain at the onset of yielding in the first longitudinal reinforcing bar at the arch rib-foundation joint. The analysis reveals a linear correlation between the raw strain integration at peak drifts and maximum drift. Additionally, a least-squares fitting indicates a similar linear relationship in both positive and negative directions. However, the NPSI-1 results exhibit a nonlinear relationship: as maximum drift increases, the normalized peak integration rises more slowly in the positive direction than in the negative. Notably, for maximum drifts below 1%, the normalized peak strain in the positive drift direction is negligible. A lower NPSI-1 ratio suggests closer alignment between raw strain and theoretical concrete strain, indicating reduced plastic damage. Plastic damage accumulation accelerates beyond a 1.5% drift level in both directions, with the arch rib’s bottom appearing stronger and stiffer compared to the top. This asymmetry could stem from coarse aggregate settlement during the compaction process, potentially impacting the homogeneity and long-term durability of the hardened concrete. ⁴³

Residual strain

The definition of residual strain in our study, observed as the specimen is unloaded after maximum deformation, is specifically tied to the loading methodology, where the specimen is deformed to a peak and subsequently relaxed to zero force. This process contrasts with the behavior of structures during an earthquake, which typically do not reach a single peak and then return to zero forces; instead, they continue to oscillate under ongoing loading or free vibration. It is crucial to recognize that the residual condition measured in the test specimens may not accurately reflect the residual state of real structures postearthquake, due to the difference in dynamic response and loading conditions. As in the previous section, the residual strain profile can be normalized using the theoretical concrete strain. However, since the residual strain corresponds to instances of zero forces, the associated ${ϵ}_{\mathrm{c},\mathrm{theo}}$ is uniformly zero throughout the arch rib. Consequently, the normalized residual strain integration (NRSI) is defined as follows:

$$\mathrm{NRSI}=\frac{{\int }_0^L\mathrm{abs}\left({ϵ}_{\mathrm{raw},\mathrm{res}}\right)}{{\int }_0^L{ϵ}_{\mathrm{c},\mathrm{yield}}}$$

(4)

Here, ${ϵ}_{\mathrm{raw},\mathrm{res}}$ represents the raw strain measurements from DFOS at zero-force. Note that the residual strain is defined as the strain observed upon unloading the specimen after reaching its maximum deformation, recognizing that, in an earthquake scenario, multiple additional deformation cycles could follow the peak deformation. This definition equates the residual condition with the state before any loading is applied, which might seem unconventional. However, this approach enables the examination of unloading effects on fiber-optic measurements, acknowledging that the actual residual condition postearthquake could vary and be random. As depicted in Figure 15, the distribution of residual strain at the arch rib’s bottom under positive drift markedly differs from that at the top under negative drift. The negative strain observed in the first 0.5 m near the joint, indicative of concrete crushing, is corroborated by Figure 9. Additionally, buckling in the longitudinal reinforcement suggests a shortening effect on the arch rib, contributing to compressive strain in the optical fibers. Table A1 in Appendices presents a comprehensive summary of the arch rib shortening observed at the peak of each load cycle for all tested specimens. However, this interpretation warrants caution; prior tensile elongation of the reinforcement can lead to buckling upon deformation reversal, even though the net strain within the section at the location of the reinforcement might still be tensile.

Figure 15.

Residual strain profiles under zero-force conditions: (a) upon unloading from positive drifts and (b) upon unloading from negative drifts.

Figure 16 illustrates the relationship between the NRSI and maximum drifts. The figure indicates that for maximum drifts below approximately 1.6%, a linear relationship exists between NRSI and maximum drift, consistent for both the top and bottom portions of the arch rib. This relationship is quantified through a least-square fitting as follows:

$$\mathrm{NRSI}=0.12·\mathrm{Mmaximum}\mathrm{Ddrift}$$

(5)

Figure 16.

Relationship between normalized residual strain integration and maximum drifts at different levels.

However, beyond a maximum drift of 1.6%, the occurrence of concrete spalling and crushing leads to significant plastic damage and arch rib shortening. This results in less consistency in the NRSI criterion, reflecting the increased severity of structural damage.

Plastic hinge length

In Figures 7 and 8, the curvature and strain distribution from DFOS are categorized into three sections: plastic, cracked, and uncracked sections. This section introduces a practical approach to quantitatively identify the boundaries between these sections.

In the uncracked section, the strain in the optical fiber closely aligns with the theoretical concrete strain, both exhibiting a smooth profile, resulting in a net strain ${ϵ}_{\mathrm{net}}$ near zero. In contrast, the cracked section is characterized by minor peaks in the strain profile, indicative of displacement discontinuities near the cracks, thereby increasing strain variance, as illustrated in Figure 17. This discrepancy between the raw and theoretical strains suggests underlying mechanical phenomena, which can be attributed to several factors as captured by FO strain measurements. These factors include (1) debonding of the optical fiber from the concrete adjacent to cracks, (2) deformation of the fiber coating, and (3) slippage between the fiber layers. ⁴⁰ Such dynamics contribute to the observed increased strain variance in the FO measurement. In the plastic section, the strain magnitude is notably higher due to (1) an increased number of cracks and the amalgamation of multiple strain peaks from closely spaced cracks and (2) wider crack widths. The rolling standard deviation along the arch rib at a drift level of approximately 1.6% is presented in Figure 17. Here, “rolling std” is defined as the moving standard deviation calculated over a 50 mm window, effectively measuring the strain variance attributable to cracks. By setting two distinct thresholds for the rolling standard deviation, we can delineate the boundary between the plastic and cracked sections (“bound-1”), and between the cracked and uncracked sections (“bound-2”).

Figure 17.

Delineation of plastic, cracked, and uncracked sections in the specimens at approximately 1.6% drift: (a) specimen 1, (b) specimen 2, (c) specimen 3, (d) specimen 4. Strains are plotted for fiber location V1-Top for all specimens.

Figure 18 presents analogous findings for identifying the plastic, cracked, and uncracked regions, but at a higher drift ratio of 2.3%. This further demonstrates the efficiency of the rolling standard deviation method in tracing the evolution of damage across different stages of structural deformation.

Figure 18.

Identification of plastic, cracked, and uncracked sections in the specimens at approximately 2.3% drift: (a) specimen 1, (b) specimen 2, (c) specimen 3, (d) specimen 4. Strains are plotted for fiber location V1-Top for all specimens.

The location of “bound-1” correlates with the PHL near the arch rib-foundation joint. For comparative analysis, the results from several empirical PHL models,^44–46 and a data-driven machine learning model by Feng et al. ⁴⁷ are included in Figures 17 and 18. It is crucial to clarify, however, that the PHL defined may not directly correspond to the measurements we are presenting. Our approach to identifying PHL, particularly in the context of DFOS, might encompass nuances and specifics that differ from those traditionally outlined in empirical or machine learning-based models. What we measure using DFOS can more accurately be described as a “plastic length” over which strains and curvatures deviate from a linear cracked section property, rather than the traditional RC definition of PHL. This conventional term refers to a nominal length that, when multiplied by the maximum curvature, yields a displacement aligning with the displacement beyond the onset of yielding. For members that cantilever from foundation blocks, such as those in our study, this includes reinforcement slip effects from the foundation block and may account for tension shift due to the interplay between moment and shear effects.

The PHL derived “bound-1” from the rolling standard deviation method generally aligns well with these existing PHL models at 1.6% drift; at 2.3% drift, the DFOS data are more noisy and variable, and the automated method of detecting the PHL struggles. Regardless, plastic hinge formation is clearly evident in the data, and other methods of automatically detecting the PHL from FO data could be derived.

Comparative strain analysis: specimens and cable types

The preceding sections primarily focused on specimen 2, examining curvature analysis and damage quantification. These methodologies are applicable to other test specimens as well. However, owing to space limitations and the need for conciseness, a comprehensive presentation of the results is omitted here. Instead, in this section, we will concentrate on the variations in strain distribution observed in specimen 1, 2, and 4. Additionally, specimen 4 incorporates a new type of cable, the EpsilonSensor by Nerve-Sensors, positioned adjacent to the top segment of V2. This pre-manufactured cable, available in fixed lengths, offers several main advantages:

• (i) Its monolithic section, devoid of intermediate layers, ensures accurate strain readouts.

• (ii) Its design facilitates a sharp and precise detection of local phenomena, particularly concrete cracks, due to improved bonding with the concrete. The specially selected composite material of the cable enhances its precision and sensitivity, making it particularly suitable for laboratory settings.

• (iii) The lightweight and ease of installation of the sensor—ready for use straight from the drum—make it ideal for embedding into new structural elements (whether concrete or soil) or integrating into existing structures. ⁴⁸

Figure 19 illustrates the strain distribution comparison in three specimens at four different drift ratio levels: approximately 0.35%, 0.72%, 1.40%, and 2.45%. At these levels, specimen 1 consistently exhibits higher strain magnitudes in the crack regions, particularly near the arch rib-foundation joint, indicating wider cracks and more extensive damage. For instance, at a drift ratio of 0.35%, the steeper strain slope in the uncracked linear segment of specimen 1 suggests lower stiffness, while specimen 2 and 4 display similar stiffness characteristics, evidenced by their alignment in the uncracked section and strain patterns in cracked area.

Figure 19.

Comparative analysis of strain distribution in Nerve and Solifos fiber-optic cables at location V2 at approximately the following drift ratios: (a) 0.34%, (b) 0.72%, (c) 1.4%, (d) 2.4%.

As the drift ratio increases to 0.72%, plastic damage begins to accumulate in specimen 1, yet specimens 2 and 4 maintain a relatively linear profile. However, at this stage, specimen 4 starts showing a divergence from specimen 2, edging closer to the behavior observed in specimen 1. This trend continues at a higher drift ratio of 1.40%, with ongoing plastic damage in specimen 1.

At the highest examined drift ratio of 2.45%, all three specimens exhibit signs of plastic hinge formation near the arch rib-foundation joint, around 0.5 m. Notably, specimen 2 displays almost zero strain outside the plastic damage zone, indicating a more localized and concentrated damage pattern. Across all drift levels, the strain distributions in the Nerve cable (specimen 4) closely match those in the Solifos V9 (specimen 4), though with greater fluctuations and more pronounced peaks. This gives confidence in both results and also suggests a heightened sensitivity of the Nerve cable, enabling better detection of minor concrete cracks. This observation correlates the stated advantage of the Nerve cable: its sharp and precise detection of concrete cracks, a result of the superior bonding between the cable’s coating and the concrete, compared to the Solifos cable.

Conclusion

This article has considered the application of DFOS for RC structures, specifically concerning post-earthquake damage assessment, plastic hinge formation, and the use of DFOS in the design of RC components like bridge arch ribs. The use of different types fiber-optic cables into several RC specimens under varied axial loads and reinforcement configurations, enabled new insights regarding both structural performance and sensing capabilities and limitations. The research culminated in the creation of a new index for quantifying plastic damage that enables correlation between residual and peak strain measurements due to seismic loading. This demonstrates the potential for the use of embedded fiber-optic cables to measure structural integrity post-earthquake, particular when the extent of minor-to-moderate damage is uncertain and cannot be readily observed externally. Additionally, the study provides additional data regarding the progression and extent plastic hinge formation. The integration of DFOS data to enable deeper understanding of competing design options (e.g. reinforcement detailing) could also help to enhance the resilience and safety of RC structures, particularly in seismically active regions.