Distributed fiber-optic sensors (DFOSs), which are based on optical frequency-domain reflectometry (OFDR), provide high-resolution strain measurements and have promising application in structural health monitoring. This study introduces a novel steel fibe-reinforced polymer composite bar (SFCB) with self-sensing, structural reinforcement, and damage control features designed to evaluate the response and damage status of concrete members. Investigating the force transfer mechanism between SFCB and concrete is essential for understanding concrete cracking behavior and establishing a reliable damage evaluation approach. Initially, tension tests were conducted on SFCB concrete members to investigate the effects of cover depth and bonding mechanism (concrete type and surface treatment of the SFCB) on the end effects, along with a test procedure designed to effectively eliminate the end effects. The results indicate that the use of members with small cover depths, surface sandblasted SFCB, and geopolymer concrete (GPC) can reduce the impact of the end effects. The tracking and quantification of particular crack progressions were subsequently assessed through the integration of a digital image correlation (DIC) system and a DFOS system. Finally, based on the results from which the influence of end effects has been eliminated, a theoretical model for the response of SFCB concrete tension members was proposed, along with a model for damage variables that is independent of geometry and material behaviors.
The aging of civil infrastructure is a serious threat to economic construction and public safety. According to the 2021 Report Card for America’s Infrastructure [1], an additional 206 billion USD per year will be required for infrastructure maintenance to close the projected 2 trillion USD funding gap over the next 10 years. This large funding gap has driven the priorization of timely maintenance of critical structures to minimize the catastrophic consequences associated with structural failure. Structural health monitoring provides an effective solution for detecting and assessing early damage to infrastructure [2,3]. Cracking is one of the most important characteristics for assessing the overall damage to and durability of reinforced concrete (RC) structures [4]. Cracking development may significantly affect various aspects of structural performance, such as strength, stiffness, and durability [4]. Therefore, timely and reliable assessment of cracks can effectively prevent fatal consequences and enable effective structural repair at an early stage of degradation. However, the complexity and variability of the crack development process limit the performance of health monitoring systems based on traditional point sensors such as strain gauges, accelerometers, and linear variable differential transformers [5].
Recent studies have underscore the significant potential of distributed fiber-optic sensors (DFOSs) technology in applications such as mechanical performance evaluation [6], [7], [8], damage detection [2],[3],[9], [10], [11], and laboratory monitoring of RC structures [12], [13], [14], [15], [16], [17], [18]. DFOS technology is distinguished by its wide-range monitoring capability, high spatial resolution, and robust anti-interference characteristics [6], [7], [8]. Although fiber optic sensors offer superior sensing properties, they are delicate and prone to damage, thus necessitating appropriate encapsulation [19], [20], [21]. Previous research has demonstrated that embedding fiber optic sensors within steel or fiber-reinforced polymer (FRP) bars not only provides protection for the fiber optic sensors but also guarantees high measurement accuracy [22,23], even enabling the detection of strain gradients near the reinforcement ribs [22]. Though steel bar is a preferred construction material due to its high stiffness and ductility, the reduced area of corroded reinforcement weakens the bond between the fiber optic sensor and the reinforcement, affecting measurement reliability and performance evaluation [17], [18],[20],[21],[24]. Furthermore, the unstable local plastic deformation caused by the yielding of steel bars is incompatible with the elastic properties of fiber optic sensors, hindering strain monitoring under extreme loads such as earthquakes and impacts [25,26]. FRP is an ideal material for encapsulating fiber optic sensors due to its high strength, durability, and compatibility with the sensors [6], [7], [8]. However, its low elastic modulus and brittleness limit its widespread use in structural applications. A promising solution is the use of steel-fiber reinforced polymer composite bar (SFCB), which is a novel material that combines a steel inner core with an FRP wrapping layer. This composite material combines the advantages of both reinforcement and FRP, including increased initial stiffness and improved corrosion resistance [27], [28], [29], [30], [31], [32], [33]. Zhou et al. [34] showed that the corrosion resistance of SFCB with carbon-FRP (CFRP) as a wrapping layer is approximately 10 times greater than that of the reinforcement. Furthermore, SFCB has stable and controllable secondary stiffness and excellent recovery capabilities [27], [28], [29], [30], [31], [32], [33]. Therefore, the research group of the first author designed the fiber-optic sensors embedded within a steel core and wrapped CFRP around the surface to manufacture SFCBs with structural enhancement, self-sensing, and damage control features. Moreover, the group plans to install these SFCBs at key parts of RC structures to monitor the responses of the structures in real time and to evaluate the service level of the structures in combination with the developed damage evaluation method, thereby achieving the multifunctional incorporation of structural materials. These high-performance reinforcements are named self-sensing SFCBs, as shown in Fig. 1. Investigating the force transfer mechanism between SFCBs and concrete is essential to obtain an in-depth understanding of the development of concrete cracking and establish a reliable damage evaluation approach.
Tension testing of RC ties is a conventional method used to assess the tension response and cracking behavior of RC structures. Despite the prevalence of direct tension testing of RC prisms as the predominant technique for such assessments [35], the method does not perfectly reflect actual structural behavior [36]. Moreover, standardized test setups have not yet been established. Despite the seemingly simple nature of the tests, the results often contradict the general assumption (Navier-Bernoulli hypothesis) that the average strains are equal in members, reinforcement, and concrete. Furthermore, conventional tests typically yield measurements of average deformation along the reinforcement and concrete surfaces, which oversimplifies the actual distribution of strains within the concrete. This limitation hinders the adequate assessment of deformation and cracking behavior in concrete members under tension [35,36].
In the conventional average deformation model, it is assumed that the tensile forces within the cracked section are carried by the reinforcement. However, this model neglects the softening behavior of the concrete that occurs after cracking; instead, the model uses an idealized cracking pattern. This pattern assumes that cracks penetrate completely through the cross-section and maintain a constant width along the depth [35,36]; this is not consistent with the actual test results, where the crack widths are distributed in a wedge shape. The distribution of crack width, as plotted by Borosnyói and Snóbli [37], is illustrated in Fig. 2(a). Here, the crack width increases almost linearly from the reinforcement (w1 = 0.35 mm) to the concrete surface (w2 = 0.45 mm) [36]. Furthermore, this simple assumption does not allow for differences between internal and external cracked blocks, resulting in an inappropriate evaluation of the end effects [36,37]. The concrete exhibits a complex stress-strain state during cracking, and the cross-section becomes non-planar due to the formation of primary cracks and internal cracks, as depicted in Fig. 2(b) [38]. When concrete is subject to tension, the areas of the concrete are differentiated into effective and ineffective regions. Typically, the boundary of the effective region has a parabolic shape, attributable to the transmission of bond stress between the concrete and the reinforcement, as illustrated in Fig. 2(c) [35,36]. Prevailing design methods typically employ an equivalence approach to determine the effective tensile area in concrete [39], yet this method has notable deficiencies [35]. Specifically, it fails to consider variables such as concrete cover, loading conditions, and stress-strain state [35,37]. Despite numerous studies on the subject, the precise distribution of stresses within concrete remains elusive. Therefore, it is essential to reasonably simplify the concept of the effective area when analyzing cracking and deformation in RC structures to prevent incorrect design recommendations.
As an advanced internal monitoring system, DFOS technology is capable of adequately evaluating the deformation of the reinforcement [2,16]. Concurrently, the digital image correlation (DIC) system proficiently tracks the cracking process of the concrete surface [2]. However, the results obtained from these advanced monitoring techniques often include deformation conditions outside the member, such as end effects [35,36]. Eliminating the impact of end effects is critical for correctly assessing the tensile load response and cracking behavior. Reducing the measurement base is an effective strategy for eliminating end effects [16,36]. Gribniak et al. [36] proposed the concept of representative specimens to separate the studied parameters from the uncontrolled influence characteristics of the sample and developed an experimental procedure for eliminating end effects. The concrete cover and bonding mechanism are important factors in controlling the effective area and end effects [36]. The bonding mechanism between the SFCB and the concrete consists mainly of chemical bonding forces, interlocking forces between the reinforcement and the concrete matrix, and friction forces at the interface [33]. The behavior of the concrete determines the chemical adhesive force. Research has demonstrated the substantial influence of quasibrittle fracture behavior on the cracking process of structural concrete components [4]. Despite its widespread usage, ordinary Portland cement (OPC) concrete is associated with significant CO2 emissions and energy consumption during production [40,41]. As a viable solution, the utilization of limestone calcined clay cement (LC3) concrete and geopolymer concrete (GPC), which are low-carbon cementitious materials, has gained attention. However, their tensile deformation and cracking behavior after being composited with SFCBs have not yet been fully clarified. Furthermore, the interlocking and friction forces, which are dependent mainly on the surface treatment of the SFCB, determine the bond failure pattern. Moreover, a good surface treatment of SFCB is critical for controlling the end effects.
The purpose of this study was to investigate the mechanism of tensile load transfer and cracking behavior in RC ties to establish a load response prediction and damage evaluation approach based on DFOS technology. Additionally, the feasibility of employing the DFOS technique to directly measure the tensile response and deformation of SFCB concrete members was explored. Considering the cover depth and bonding mechanism to be the primary factors controlling the end effects, firstly, tensile tests of SFCB concrete members were carried out. These tests evaluated the influence of cover depth, surface treatment of the SFCB, and concrete type on the end effects in combination with the DIC system. A test procedure was modified to reduce the base, aiming to eliminate the impact of the end effects on the average deformation of the member; this led to obtaining load-averaged strain responses that were consistent with the Navier-Bernoulli hypothesis; this finding was based on strains monitored by the DFOS. A crack width calculation method was subsequently proposed based on the bonding mechanism and the strain monitored by the DFOS. The study further evaluated the cracking behavior of the members by analyzing the surface crack progression captured by the DIC. Finally, damage variables were established for the tensile response of SFCB concrete based on the continuous damage concept.
2. Experimental program
2.1. Specimen design
A total of 20 specimens were designed for tensile testing of SFCB concrete ties. As shown in Fig. 3(a), each sample contained a concrete prism with a length of 600 mm and a square cross-section. A single SFCB, 1100 mm in length, was embedded at the center of the cross-section, extending 250 mm from each end of the concrete. This extension included 200 mm for mounting a strengthened steel pipe and 50 mm for measuring deformation. The concrete prisms had various cross-section sizes: 80 mm × 80 mm, 100 mm × 100 mm, 120 mm × 120 mm, and 160 mm × 160 mm. The primary test variables were concrete type (OPC, LC3, and GPC), cover depth (36, 45, 46, 56, and 72 mm), SFCB ratio (ρs; 0.349%, 0.502%, and 0.785%), and surface treatment of the SFCB (sandblasted or surface-thread), as shown in Fig. 3(b).
As shown in Table 1, the first letter of the specimen designation indicates the type of concrete, the immediately following number indicates the sectional dimension, the next number indicates the diameter of the SFCB, and the last letter indicates the surface treatment of the SFCB, where S indicates sandblasting. For example, O80-8S indicates that the concrete of the tension sample was OPC, the cross-sectional size was 80 mm × 80 mm, the SFCB diameter was 8 mm, and the surface was sandblasted. Notably, the specimens for the tension test had to be strengthened with steel pipes at both ends before the SFCB was placed in the mold.
2.2. Material properties
2.2.1. Self-sensing SFCBs
A self-sensing SFCB with distributed sensing capabilities, which was obtained by embedding fibers optic sensors based on optical frequency-domain reflectometry (OFDR) technology, enabled for real-time strain monitoring. The inner steel core of the SFCB consisted of a bare round steel bar, while the wrapped material was made of CFRP. The mechanical properties of the steel bars and CFRP, which were provided by the manufacturer, are summarized in Table 2.
The steel core external CFRP was prepared by a pultrusion molding process (Fig. 1, Fig. 4). The optical fiber sensor is a material with a small bending stiffness and high brittleness. Directly embedding an optical fiber sensor when the steel core and carbon fiber are infused with epoxy resin increases the probability of the optical fiber sensor rupturing. Previous studies have usually used CFRP tubes or steel tubes to encapsulate fiber optic sensors before they were prepared via the pultrusion molding process [6]. However, in this study, since the SFCB was a composite of steel and CFRP, and the steel had a greater bending stiffness. Therefore, an optical fiber sensor could be encapsulated by embedding it into the steel inner core via a slot; this also ensured the survival ratio of the optical fiber sensor, the stability of the measurement signal, and the production of self-sensing SFCBs in industrial quantities, and allowed for better deformation of the external CFRP, the optical fiber, and the steel inner core.
Embedding a fiber optic sensor involved three important steps: ① A 1 mm deep square groove was created on the edge of the inner core of the steel bar to ensure that the optical fiber remains parallel to the reinforcement (Fig. 4(a)). ② A 0.9 mm polyurethane coated optic fiber sensor was placed at the bottom of the groove and secured with adhesive tape to maintain its position. The optical fiber sensor was then potted with epoxy resin to prevent any relative movement between the optical fiber and the inner core of the steel bar. The optical fiber sensor comprised a 250 μm glass fiber inner core (mode G657b3), an electroplated cladding layer, and a polyurethane coating (Fig. 1). ③ Carbon fiber bundles impregnated with epoxy resin were wrapped around the surface of the round steel core. Owing to the significant potential difference between carbon fibers and steel, an insulating epoxy resin layer was used to avoid direct contact between the carbon fibers and the steel core.
The SFCB uniaxial tension test variables, including FRP thickness and the diameter of the reinforcement inner core, were summarized in Table 3. To balance the constraints on the cost, behavior, and manufacturing process, CFRP thicknesses of 1 and 2 mm were selected in this study to investigate their influence on the elastic modulus and sensing accuracy of self-sensing SFCBs. A total of three samples were designed. In this study, a surface-thread treatment method was selected for the SFCBs in the uniaxial tension test. As shown in Fig. 5, the total length of all the samples is 550 mm, with a test region length of 150 mm. Steel pipes with an outer diameter of 32 mm, an inner diameter of 26 mm, and a length of 200 mm were used at both ends of the samples to reinforce and protect the wrapped CFRP layer (Fig. 5). These steel pipes were filled with expanded cement around the SFCB to ensure satisfactory bonding. Uniaxial tension tests were conducted according to ASTM D7205/D7205M-21 [42].
To evaluate the sensing accuracy of self-sensing SFCBs, the DIC method was also employed to measure the deformation of the length of the test segment. The typical stress-strain relationship for the SFCB test is expressed by Eq. (1) [31,33].
with EI = (EsAs + EfAf)/(As + Af) and EII = EfAf /(As + Af), where EI is the initial elastic modulus of the SFCB, GPa; EII is the elastic modulus of the SFCB after yielding, GPa; Es is the elastic modulus of the inner core of the reinforcement, GPa; Ef is the elastic modulus of the CFRP, GPa; Af is the cross-sectional area of FRP, mm2; and As is the cross-sectional area of the inner core of the steel bar, mm2. σs is the stress of the SFCB, $\overline{{\varepsilon }_{\text{s}}}$ represents the average strain, εsfy is the yield strain, and εsfu is the ultimate strain.
As depicted in Fig. 6, the SFCB stress-strain curve were obtained from the average strains measured by the DFOS and DIC. The stress within the SFCB is determined by the ratio of the applied tensile force P to the cross-sectional area of SFCB A (where A = As + Af, under the the assumption that the reduction in the cross-section due to grooving is eliminated), as illustrated in Fig. 4(a).
Table 4 summarizes the characteristic values of mechanical properties, including the yield strain εsfy, yield strength fsfy, ultimate strain εsfu, and ultimate strength fsfu, for all SFCB uniaxial tension test specimens. The maximum strain εmax,FOS monitored by the SFCB is very close to the rupture strain of the SFCB εsfu monitored by DIC, indicating that the SFCB can be monitored throughout almost its entire service life before failure. Additionally, Table 4 provides the theoretical values of EI and EII calculated via Eq. (1). Generally, the tested values for EI and EII did not exceed these theoretical values, possibly because of misalignment of the CFRP fibers and defects in the epoxy resin during the manufacturing process.
2.2.2. Concrete
The concrete mixture design for this study was developed based on previous studies [40], which included OPC concrete, LC3 concrete, and GPC, as shown in Table 5. The water-to-cement (w/c) ratio was set to 0.5 for both the OPC and LC3. The calcined clay and limestone in the LC3 were powders mixed at a ratio of 2:1, as suggested by Scrivener et al. [41], replacing 35% of the OPC. A mixture of liquid sodium silicate, solid sodium hydroxide, and water was used as the alkaline solution in the GPC, with an exciter alkali equivalent of 6% and a solid-liquid ratio of 0.5 (with fly ash and slag as solids and the exciter solution as the liquid). Three cubic samples 100 mm in size were attached to each set of samples to determine the compressive strength of the concrete. All the samples and subsidiary cubes were placed in a standard curing room at a temperature of (23 ± 3) °C and 95% relative humidity. After 28 d of curing, all the samples were examined, and the cubic mean compressive strengths of the OPC, LC3, and GPC were 39.7, 33.6, and 41.2 MPa, respectively, per ASTM C39/39M-21 [43].
2.3. Testing setup
The load frame is depicted in Fig. 7, and the tension was transferred to the SFCB through a steel frame fixed to the load frame. The test was controlled with a displacement rate of 0.1 mm·min−1. A DFOS interrogation device was used to record the strain on the SFCB, covering the entire measurement length of 1.1 m with a spatial resolution of 1 mm, and data was collected every 1 s. To record the average strain (Base 1), crack width, and loading end deformation (Bases 0 and 2) of the concrete via DIC, scatter spots were placed on the concrete surface and on the bare SFCB at the ends of the sample (Fig. 8). Note that the deformation of the concrete on only one surface was recorded via the DIC system with two cameras in this study. The average strain of the sample can be obtained from Eq. (2) for the test results of the DFOS.
where $\overline{{\varepsilon }_{\text{s}}}$ represents the average strain of the SFCB measured by optical fibers, Δl represents the spatial resolution, n = ln/Δl + 1 represents the total number of measurement points; εs,i represents the SFCB strain at measurement point i, and ln represents the length of the test segment (150 mm for uniaxial tension testing of the bare SFCB and 600 mm for Base 4).
In this study, the test results for the DFOS were corrected through calibration with data from the test results of the DIC system, with the objective of eliminating the influence of end effects on the DFOS results. Deformation monitoring was independently conducted for both the SFCB within the total base (Base 3) and the SFCBs exposed externally (Bases 0 and 2), utilizing the DIC system for comprehensive assessment. Moreover, the deformation of the SFCB embedded within the members was monitored with the DFOS system (Base 4). The average strain of the SFCB monitored by the DIC system, which is equal to the member strain εm with elimination of the end effects, was selected as a reference, as expressed in Eq. (3).
where Δl0, Δl2, Δl3, and Ls represent the deformation of loading end 1, the deformation of loading end 2, the deformation of the SFCB on Base 3 (630 mm), and the bonding length of the SFCB (Base 4: 600 mm), respectively, as shown in Fig. 8.
The strain data monitored by the DFOS for a typical tensile sample and the concrete surface strain clouds monitored by DIC are presented in Fig. 9. To ensure that the strain data monitored by the DFOS and the concrete surface strain data monitored by DIC correspond in coordinates, the coordinate origin is positioned at loading end 1 of the concrete. The coordinate system covers a total length of 600 mm along the concrete and has a spatial resolution of 1 mm, resulting in a total of 601 data measurement points.
3. Test results
3.1. Strain distribution of SFCB and concrete
Fig. 10 shows the strain distribution of the SFCB and the corresponding DIC strain distribution clouds for different cracking phases of sample O100-10S. The cracks are denoted by Cr1, Cr2, and Cr3. Additionally, the bare bar strains and calculated composite section strains are presented in Fig. 10. The bare bar strains represent the average of the strains at the exposed portion of the end. The composite section strains are calculated as σs(εs)A/(EcAc + EIA), where σs(εs) is the SFCB stress obtained by substituting the bare SFCB strain into Eq. (1) and Ec and Ac are the elastic modulus and the cross-sectional area of the concrete, respectively, which are determined based on the elastic moduli of the two materials by converting the strains of the SFCB to the equivalent concrete strains.
As illustrated in Fig. 10, the force was transferred from the bare SFCB at the loading end towards the concrete embedding depth. This transfer resulted in a stress transfer length (lf) of approximately 150 mm, which was related to the maximum bond stress τmax, the cracking strength of the concrete fcr, and the dimensions of the concrete and SFCB [4]. During Phase I (the elastic behavior phase), the SFCB strains in the 150-450 mm length range aligned with the calculated composite section strains, indicating identical deformation between the concrete and the SFCB in the uncracked phase. Moreover, the bonding mechanism at the ends was precisely what led to the inconsistency between the average strains in the concrete and the steel reinforcement. The inconsistency of the test results with the Navier-Bernoulli hypothesis indicated the need to eliminate the influence of end effects on member deformation. When the concrete stress exceeded the cracking strength, Phase II (the crack initiation phase) began. The DIC strain cloud reveals a notable increase in the concrete strain at the cracking location. At this point, the SFCB strain reached a peak close to the strain of the bare bar, demonstrating the ability of the SFCB to identify the crack location precisely. As the load continues to increase, the concrete stress transfer length became insufficient for forming new cracks upon entering Phase III (the stabilized cracking phase). In this phase, the SFCB strain no longer exhibited new peaks; instead, the strain values continued to increase.
The SFCB at the end of the sample serves as the first measurement point. The sum of the tensile forces in the concrete and the SFCB at any measurement point equals the applied external tensile force, denoted as P. Note that the concrete at the ends and cracking locations is not stressed. Therefore, the value of P equals the tension in the SFCB at the first measured point. The total area of concrete is assumed to be effectively tensile. This assumption makes the cracks have an ideal cracking pattern, wherein the cracks fully penetrate the cross-section and develop uniformly. The concrete equivalent method employed in this study, despite not being directly applicable to design, as noted in Refs. [35,36], serves as a practical simplification for examining the internal physical mechanisms. Eq. (4) can then be derived for calculating the predicted value of the equivalent concrete strain εc(x) for each measurement point.
where εs,1 is the SFCB strain at the first measurement point; Ac,eff denotes the effective concrete area; and σs,1(εs,1) is the SFCB stress at the first measurement point, which is obtained by substituting the SFCB strain at the corresponding measurement point into Eq. (1).
Fig. 11 shows a comparison between the concrete surface strains measured via DIC and the equivalent concrete strains as calculated via Eq. (4). Owing to the influence of the cover depth, bonding mechanism, and the equivalence assumptions, both measures of strain (surface and equivalent) within the uncracked concrete section remained relatively small. In fact, these values were less than 0.005%. Importantly, Eq. (4) may lead to an underestimation of the strain in areas of concrete that are considered ineffective, resulting from the approach of assuming that the total area is under effective tension. Nevertheless, the concept of concrete equivalent strain as an idealized form of deformation offers a reasonable approximation for the deformation of concrete. This approximation is influenced significantly by the bonding mechanism. For example, at the end and crack locations, the strain peaks, whereas the neighboring crack midpoint location is the valley. Moreover, the equivalent and surface concrete strains at the center of the neighboring cracks are similar, indicating that for the O100-10S sample, the effective concrete area in this region is consistent with the assumption (Fig. 11). The concrete surface strain reflects an uneven distribution of internal bond stresses, where the regions with an effective tensile area falling below the cover depth correlate with the zones exhibiting zero surface strain. This empirical observation is in agreement with the patterns presented in Fig. 2(c). Note that the peak strain measured by DIC cannot truly reflect the surface strain of the concrete, but it can be used to assess the crack opening tensor [14]. Although neither the equivalent strain nor the surface strain of the concrete is the true strain distribution of the concrete, they can be used to qualitatively assess the difference between the deformation of the concrete and that of the reinforcement. Reasonable separation of these discrepancies can improve the reliability of analyses carried out on cracks.
3.2. Eliminating the end effects
The DFOS system can detect and monitor minute deformation of the SFCB in 1 mm increments. Consequently, it allows for the determination of the average strain on the SFCBs across various measurement bases. This iterative process, which involves comparing results from different bases, is crucial for identifying the minimal distance from the edge of the sample at which strain differentials between adjacent bases become insignificant. The process of locating the end effects consists of five steps depicted in Fig. 12. In Step 1, the samples are prepared for testing, and the test setup is installed according to Section 2.3. The load-averaged strain relationship for the sample is determined in Step 2. Step 3 compares the average SFCB strain (εm,s) monitored by the DFOS system with εm. In this phase, two alternative outputs are possible: ① significantly different average strains and ② insignificant differences. The first case implies that the deformation measurement base of the SFCB must be reduced (Step 4-i; Base 5: Ls,r in Fig. 12). The alternative output indicates that the measurement base is close to the target base length, which is the localization of the end effect (Step 4-ii). Ultimately, the measurement base is reduced, and the member tension response based on the DFOS results is determined.
The load-averaged strain curves for all the samples are presented in Fig. 13, Fig. 14, Fig. 15, including the results from DIC, the DFOS initial bases (Base 4), and the DFOS reduced bases (Base 5). As illustrated in Fig. 13, Fig. 14, Fig. 15, the difference in the average strain between the member, as measured via DIC, and the SFCB for the initial base, as determined by DFOS (Base 5), is a characteristic observed across all the samples. This discrepancy arises from end effects caused by the cover depth and the bonding mechanism and is not consistent with the Navier-Bernoulli hypothesis [36]. Moreover, the differences between Bases 4 and 5 are minor. Furthermore, the results indicate that using the DFOS system to measure the average strain of tensile bars can effectively eliminate the impact of end effects on most samples by applying a modified procedure to reduce the base. Except for members L160-16S and G120-8S, owing to the greater coverage depth, the reduction in the base in the stabilized cracking phase did not fully eliminate the influence of the end effects.
The end effects have a significant effect on the local strain in the elastic phase (Fig. 10). Therefore, this study assessed the end effects by comparing the difference between the average strain of the reinforcement at the initial cracking point between the initial base and the reduced base (Fig. 13, Fig. 14, Fig. 15). The larger difference indicates a more significant impact of the end effects on deformation.
For the samples with an 8 mm diameter SFCB, the average strain difference of the bars |Δεm,s| increased with the cover depth increase, as shown in Fig. 13, Fig. 14, Fig. 15. This is because as the cover depth increases, a greater amount of stress transfer needed, causing the concrete to crack. Additionally, ineffective concrete inhibits crack development as the cover increases, resulting in a significant end effect. For samples (e.g., O80-8S, O100-10S, and O160-16S) with the same reinforcement ratio (0.7850%), a similar result was observed with an increasing cover depth. For specimens of the same size and concrete type, reducing the average strain difference in the reinforcement through SFCB surface sandblasting weakened the impact of end effects on member deformation, as depicted in Fig. 13, Fig. 14, Fig. 15. The reason for this was the sandblasting on the surface. This process enhances the bond stress between the SFCB and the concrete [33]. Interestingly, compared with their sandblasted counterparts, the ends of the surface-thread samples were more prone to slipping and experiencing localized deformation.
To minimize the effects of variables that cannot be controlled, specimens measuring 80 mm by 80 mm were chosen. This size allowed for a more adequate comparison of how different types of concrete influence the end effect. Generally, the end effects observed in the LC3 and GPC members were less pronounced than those in OPC members, as illustrated in Fig. 13, Fig. 14, Fig. 15. The strain difference among the GPC members was the smallest, at 0.0110%. This variance can be attributed to the different bonding mechanisms inherent in the different types of concrete and the SFCB. Research conducted by Zheng and Xiao [44] corroborates these findings, demonstrating that the compressive strength, splitting tensile strength, and bond strength of GPC surpass those of OPC. As a result, the end effects of the GPC tensile members have a lesser impact on deformation than those of their OPC counterparts.
3.3. Crack behavior and width calculation methods
When the strain distribution of the reinforcement is known, the location of cracking (xcr,i) can be determined directly from the strain peaks and analyzed for particular crack widths. The mechanical model proposed in the current prevailing structural design codes (e.g., Eurocode 2 [45] and fib model code for concrete structures 2010 [39]) considers the crack width to be equal to the relative displacement on either side of the crack due to slip between the reinforcement and the concrete. Based on this concept (if the cracks have an ideal cracking pattern), the local slip between the concrete and the SFCB was investigated. Furthermore, a method for calculating the concrete crack width based on the local slip relationship is proposed. For any given position, the slip s(x) between the SFCB and concrete can be expressed by Eq. (5):
where εs(x) and εc(x) are the SFCB strain and the equivalent concrete strain (Eq. (4)), respectively, at the corresponding position.
As shown in Fig. 10, the strain valleys of the SFCB are near the center of each concrete segment. Therefore, it is assumed that the slip at the center of each concrete segment is 0. The total local slip S(xi) for any measured point xi within each segment can be obtained via integration from the 0 slip point x0 to the position of the measured point as follows:
The local slip distribution along the length of the specimen is presented in Fig. 16, where the SFCB exhibits positive slip to the right relative to the concrete and negative slip to the left relative to the concrete. In Phase I, local slip occurred mainly at the two loading ends. When the crack initiation phase began (Phase II), positive and negative peaks appeared on the left and right sides of the cracking location. Consequently, the difference in local slip between the left and right sides of the cracking location represents the total width of the crack (wcr), expressed by Eq. (7).
where x0,i is the 0 slip point on the left side of the cracking point xcr,i, and x0,i+1 is the 0 slip point on the right side of the cracking point xcr,i.
A detailed calculation flowchart is depicted in Fig. 17. A similar method was used by Berrocal et al. [11] to evaluate particular crack widths, and its reliability has been confirmed.
Using the DIC method, the evolution of the three-dimensional displacement field on the studied surface is analyzed. Two points, Aj and Bj, are selected on the concrete surface, with line AjBj perpendicular to the specified crack path. As the crack develops, AjBj moves to Aj′Bj′. The DIC method can continuously and adequately record the motion of these points. Therefore, the crack width wcr_DIC recorded by DIC can be determined via Eq. (8) [46].
As depicted in Fig. 18, considering the discrete character of crack development, two parallel lines (L0 and L1) were used to cut the surface on both sides of the crack. A series of points 2 mm apart were set along each line. Points on two parallel lines (L0 and L1) with the same x-coordinate were considered a set of vectors (e.g., A1B1, A2B2, …, AjBj, where j denotes the number of points on the line), as shown in Fig. 18. The crack width for each set of vectors was calculated via Eq. (8), which represents the crack width at different x-coordinates for each crack. The average crack width obtained via the DIC method was then determined by averaging the results from all the vectors vertically along the cut line (L0 and L1). Note that SFCB concrete members, unlike RC members, were still able to satisfy the Navier-Bernoulli hypothesis after the reinforcement yields because of their stable postyield stiffness, as shown in Fig. 13, Fig. 14, Fig. 15. Considering that the member exhibited a larger average deformation than the RC member did, this study limited its analysis to crack development during phases with a member strain of less than 0.6% (Phase II and part of Phase III).
Fig. 18 shows the initial cracking pattern of a typical sample, with the locations of the cracks labeled. Fig. 19 shows the final cracking pattern, and cracks that formed at more than 0.6% of the average strain were excluded from the study. The average spacing of primary transverse cracks was between 175 and 200 mm.
Fig. 20 presents the relationship between the crack width and member strain for the corresponding cracks, providing a comparison of the results calculated via Eq. (7) with the DIC results. Fig. 20 indicates that the Eq. (7) calculation trends accord with the DIC results. Phase II matches better because there are fewer uncontrollable factors. During Phase III, there was a notable increase in the discrepancy between the results of the two methods when eparticular crack widths were evaluated. Typically, the calculation results from Eq. (7) are greater than those from DIC monitoring. This is primarily because DIC is confined to monitoring the average width of visible cracks on a single surface. In contrast, Eq. (7) calculates idealized average crack widths that include both visible and internal cracks within the spacing of neighboring cracks.
As illustrated in Fig. 19, Fig. 20, several factors contribute to the discrepancy between the two methods: ① The number of vertical split cracks and multiple transverse cracks increases as deformation increases; ② as the depth of the concrete cover increases, some internal cracks may not penetrate through to the surface; and ③ single-surface DIC has various limitations in monitoring overall crack development.
The maximum crack width in practical engineering applications is an important indicator for evaluating the durability of structures [39,45]. The maximum crack width is the maximum value for all measurements of a particular crack. Fig. 20 shows that the maximum crack width was the width of the initial crack Cr1 during loading for samples O80-8, O120-8S, and O100-10S. For O100-8S, the maximum crack width changed from Cr1 to Cr2 because of the formation of vertical cracks and subsequent transverse cracks, leading to sudden stress release (Fig. 19, Fig. 20(b)). Other test samples with final cracking patterns and crack width-member strain relationships are presented in the Appendix A (Figs. S1 and S2).
4. Member tension respond model and validation
4.1. Typical load-member strain relationship
Fig. 21 presents the typical load-member strain response of an SFCB concrete tie. Owing to the stable postyield stiffness of the SFCB, the average cracking model is valid until the rupture of the wrapped CFRP. Therefore, this study focuses on the initial three loading phases at the intersection of the SFCB concrete prism with the bare reinforcement in the load-member strain relationship. During the elastic behavior phase (Phase I), the initial behavior of the member is linear elasticity, with the SFCB transferring stresses to the surrounding concrete. This results in a significant initial stiffness K0 (the ratio of the initial cracking load Pcr to the cracking member strain εcr). When the cracking strength of the concrete increases, the member enters the crack initiation phase (Phase II). The tensile stress of the concrete within the crack region is relieved, leading to the transfer of stresses from the SFCB located at the crack to the surrounding concrete. This process results in noticeable decreases in the slopes of the load-member strain curves. During the stabilized cracking phase (Phase III), transverse cracks fully develop across the member, and as the bond strength gradually decreases, the tension stiffening effect progressively decreases. When the tension response of the member intersects with that of the bare bar, the concrete will no longer be effective, and the tension stiffening effect will be reduced to zero.
4.2. Theoretical model
Fig. 22 shows the tension response of the member concrete materials, and the concrete contribution is generally consistent across all the members. The bond coefficients (β = fc/fcr, fc denotes stress of concrete and fcr denotes the concrete cracking strength) is considered effective index for characterizing the material properties of cracked concrete. This coefficients is independent of the concrete strength, surface treatment of the SFCB, and SFCB ratio. The tension response of concrete after cracking can be adequately predicted via Eq. (9), which was proposed by Fields and Bischoff [47].
where εcr = fcr/Ec is the concrete cracking strain.
This equation has been validated by various scholars for members with different reinforcement ratios [16,48,49]. Furthermore, the response of the entire concrete tensile process can be predicted via Eq. (10), as illustrated in Fig. 22.
To predict the member response, we assumed that the SFCB and concrete share axial tension [16],[47], [48], [49]. The load of the member can be expressed by Eq. (11).
where P is the applied external tensile force, fcr = 0.37 $\sqrt{{f}_{\text{c}}^{\prime }}$ is the cracking strength of the concrete [16,47], and Ec can be expressed by Eq. (12).
where Ls,r is the base length of the SFCB with the end effect length le removed. Xu et al. [4] derived an equation for le, which is presented as Eq. (14).
where d is the diameter of the SFCB. The relationship between le and 3dfcr/(16ρs) is shown in Fig. 23. On the basis of the test results, 1/τmax values of 0.04 and 0.06 can be fitted for the sandblasted and surface-thread samples, respectively.
4.3. Validation
Figs. 24(a) and (b) [50] compare the load versus member strain curves for a typical sample between the test and theoretical results. The theoretical model employs the shared axial tension force model described by Eq. (11). Fig. 24(a) shows that the theoretical model can adequately predict both the cracking load of the sample and the postcracking load response. This finding indicates that the load response and deformation of the member after eliminating the end effects satisfy the Navier-Bernoulli hypothesis. Additionally, the results from three glass-FRP (GFRP) concrete ties were collected and compared with the proposed theoretical model [50]. As depicted in Fig. 24(b) [50], generally, the proposed model captures the behavior of the members well. However, the study by Baena et al. [50] used only GFRP reinforcement (without a steel core), which has a lower elastic modulus than steel bars do. Furthermore, the proposed theoretical model underestimates the test results because it only considers the tensile material response of cracked concrete ties reinforced with steel bars, which have a more insignificant strain localization at the boundary (Phase I) and tension stiffness effect (Phases II and III) than GFRP-RC ties do, as shown in Fig. 24(c). Generally, the model proposed in this paper, which is based on strain data monitored by the DFOS, offers a reasonable and adequate prediction of the tension stiffening performance of SFCB concrete members.
5. Damage development in tension test members
The preceding analysis indicates that damage to tension test members results primarily from concrete cracking, interfacial slip, and SFCB deformation. Approximately 40% of the strain in the SFCB was observed to develop when the member reaches its ultimate strain. Fig. 25 provides a schematic illustration of the average stress development in an SFCB concrete member, considering the area enclosed by the member response curve and the bare SFCB curve as the contribution of the concrete to the tension stiffening effect, as expressed in Eq. (10). The intersection of the two curves represents the ultimate state when the tension stiffening effect of the concrete member decreases to 0. The recommended ultimate member strain (εm,u) for RC members in the Ref. [4] is 0.01. In this study, the average ultimate strains of all the member was assumed to be 0.0064, as indicated in Fig. 21.
Kachanov [51] proposed the concept of a damage variable to describe continuous damage to a material. This variable is defined as the ratio of the tensile strain energy to the total damage strain energy of concrete, as expressed in Eq. (15) [51,52].
$D=\frac{{A}_{\text{d}}}{{A}_{\text{t}}}$
where D is the damage variable; At is the strain energy of the concrete tension, which represents the area enclosed by the member response and the bare SFCB response, and Ad is the damage area, which represents the area enclosed by the member response from point 0 to any damage point (εm,1, P1), the bare SFCB response, and the unloaded stiffness at any damage point (εm,1, P1), where P1 and εm,1 (εm,1 ≤ εm,u) correspond to the load and strain values at any damage location, as shown in Fig. 25.
Therefore, this study introduces a continuous damage method for characterizing the development of damage in tension test members. In this method, member damage is governed primarily by the tensile energy dissipation of concrete, which represents the contribution of the concrete to the tension stiffening effect. The unloaded stiffness of the member is assumed to be identical to the initial stiffness. This approach offers a reasonable description of the process through which tension test member damage accumulates as the member deforms.
Based on the theoretical model proposed in Section 4.2 and the definition of the damage variable in Eq. (15), the damage variable for the tension test of SFCB concrete members can be expressed by Eq. (16):
This was obtained by substituting Eqs. (1), (9), (10), (11), (12), (13), (14) into Eq. (15). According to Eq. (16), the damage variables are influenced primarily by the cracking strain of the concrete, the member ultimate strain, the ratio of the concrete to the SFCB area, and the ratio of the initial elastic modulus of the SFCB to the elastic modulus of the concrete. The results in Section 4.2 reveal that there is no essential difference in tensile behavior among the OPC, LC3, and GPC materials. Variations are observed only in their strengths, which do not significantly impact the cracking strains or ultimate strains of the members. While the variables influencing the damage variables of the members are concentrated in the parameters of Eq. (17), Ae represents the elastic energy recovered from the unloaded stiffness.
However, Ae is very small and can be neglected, as shown in Fig. 25. Therefore, the damage variable can be expressed more simply via Eq. (16).
As illustrated in Fig. 26, the relationship between the damage variable and the member strain of all the samples was established based on the strain monitored by the SFCB. Damage development comprises three phases, with the member damage remaining constant at 0 when the member strain is below the member cracking strain. When the member strain reaches the member cracking strain, damage begins to appear and increases rapidly. Once the member damage variable reaches 1, the member is considered completely damaged, and the contribution of the concrete to the tension stiffening effect is reduced to 0.
Combining Eq. (16) and Fig. 26, the damage variables remain independent of the concrete size, type, strength, SFCB type, and surface treatment of the SFCB. Therefore, this approach for defining damage establishes a close correspondence between the member strains and damage variables, adequately describing the tension damage phase of SFCB concrete members based on strains measured from the SFCB. For example, it is possible to adequately assess when the concrete in the tensile region of a member cracks or ceases working, providing guidelines for the assessment and maintenance of structures.
6. Conclusions
To establish a reasonable and reliable method for assessing the tension response, cracking behavior, and damage evolution of RC tie, a self-sensing SFCB based on DFOS technology was developed in this study. Considering the influence of end effects on the DFOS monitoring results, tension tests were carried out on SFCB concrete members. These tests investigated how cover depth and bonding mechanism (concrete type and reinforcement surface treatment) influence the end effects, and this influence was eliminated. Then, based on the DFOS results after eliminating the influence of the end effects, the development of cracks and the load response of the members were analyzed. Finally, a theoretical model for the response of SFCB concrete tension members is proposed. A damage variable independent of geometry and material properties was also established in relation to the average strain in a member. The main conclusions are as follows:
(1) By reducing the DFOS strain measurement base, the modified test procedure identifies end effects, thereby facilitating an adequate assessment of member deformation and load response.
(2) A shared force model was developed to predict the load-member strain relationship of SFCB concrete tension members via the DFOS strain of removing the end effect length.
(3) A method for calculating crack width based on DFOS strain is proposed for the qualitative assessment of the cracking process. This method describes an idealized crack pattern that includes both internal invisible and external visible cracks.
(4) This study established relationships between the member strain and damage variables that are independent of geometry and material behavior, enabling an adequate assessment of the tension damage phase of SFCB concrete members.
While the presented method offers considerable insights into the damage evolution and structural performance of RC members, it is important to acknowledge the limitations encountered in this study. The limitations include several factors: each case is based on only a single sample, DIC is restricted to monitoring only one surface of the concrete, the predictive models are idealized simplifications, and cracks are inherently discrete and complex. These limitations restrict the current study to a qualitative analysis of the observed cracking patterns. Future research should expand the sample size and explore a broader range of cracking patterns using more advanced monitoring techniques.
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