This study presents a novel self-sensing steel fiber-reinforced polymer composite bar (SFCB). The SFCB combines damage control, self-sensing, and structural reinforcement functions using distributed fiber optic sensing (DFOS) technology. By combining DFOS strains with theoretical and numerical models, a multilevel performance method for damage assessment is proposed from the perspectives of safety, suitability, and durability. Stiffness is a metric used to assess the complete service history of the reinforced concrete (RC) structure, which was used to define the damage variables. Initially, a basic correlation is created between the SFCB strain and several performance characteristics, such as moment, curvature, load, deflection, stiffness, and crack breadth, at characteristic points. The threshold values of damage variables for safety, serviceability, and durability were determined based on loading peak, mid-span deflection limits, and crack width limits corresponding to the damage variables. Then, a modified fiber damage model based on DFOS strain data is proposed to improve identification, quantification, and tracking for fiber damage. Finally, the reliability of the proposed theoretical and numerical models was verified by three-point flexural tests of SFCB-RC beams, and the test beams were analyzed using the proposed method. The results show that increasing the reinforcement ratio can lower the threshold at all levels and improve the ability of the flexural beams to control damage. This study contributes to advancing the intelligence of RC structures and offers valuable insights for the design of intelligent RC structures.
Zenghui Ye, Zhongfeng Zhu, Feng Xing, Yingwu Zhou.
Performance Assessment of Reinforced Concrete Structures Using Self-Sensing Steel Fiber-Reinforced Polymer Composite Bars: Theory and Test Validation.
Engineering, 2026, 57(2): 283-301 DOI:10.1016/j.eng.2024.11.022
Various elements can compromise the structural integrity, resulting in failures and collapses when subjected to abrupt dynamic stresses such as earthquakes, typhoons, or explosions. If damage is not repaired immediately, this can lead to substantial economic losses and put both structural integrity and personnel safety at considerable risk. Therefore, the focus is increasingly on the rapid and effective assessment of damage and the prediction of structural performance under various loads. Structural health monitoring (SHM) is about detecting damage or degradation of the structure by analyzing the structural response using non-destructive sensing techniques [1], [2], [3], [4]. It provides an effective engineering approach to ensuring the safe operation and maintenance of infrastructures. SHM is based on sensing technology, and conventional point sensors have made remarkable progress in health monitoring [5], [6], [7]. However, conventional point sensors pose a major challenge for the overall response monitoring of complex components. This is because of the need for many conventional point sensors to be arranged to complete the monitoring of a single component [5]. Distributed fiber optic sensing (DFOS) technology, characterized by distributed multi-point continuous sensing, high accuracy, and strong anti-interference ability, offers promising applications for monitoring in the overall response of complex structures [4], [5], [8], [9], [10]. Moreover, the integration of DFOS technology with high-performance structural materials can significantly advance the development of structural intelligence [11], [12]. The steel fiber-reinforced polymer composite bar (SFCB) combines the material properties of steel and carbon fiber-reinforced polymer (CFRP) and is characterized by a high elastic modulus, corrosion resistance, and controllable secondary stiffness, making it an excellent material for damage control [13], [14], [15]. Embedding fiber optic sensors in SFCBs can endow them with structural enhancement, self-sensing, and damage control functions and provide a database for structural performance prediction and damage condition assessment.
Numerous studies have investigated the performance and damage-phase assessment of structures using DFOS technology [5], [11], [12]. Tang et al. [11] developed self-sensing basalt fiber-reinforced polymer (FRP) tendons using DFOS technology and installed them as a prestressed near-surface strengthening system on reinforced concrete (RC) beams. Four-point bending tests were performed on RC beams reinforced with near-surface-mounted self-sensing basalt FRP bars. An evaluation method for the displacement, curvature, and flexure capacity was developed based on the combination of self-sensing strain and the fiber model [11]. Tang et al. [12] proposed a method for structural monitoring and damage assessment of RC columns using internally distributed self-sensing basalt FRP bars and the results showed less than 20% error in predicting displacement before failure. Tan et al. [5] achieved the detection, localization, tracking, quantification, and visualization of the crack development process using strains measured by the DFOS technique. However, current strain monitoring methods based on self-sensing bars primarily focus on describing response characteristics such as load, deflection, curvature, and crack width. They are not able to quantitatively characterize the evolution of damage at multiple levels, including safety, durability, and suitability.
The damage variable is commonly used in studies to characterize the damage-phase of a component or structure under load. It can be expressed as an equation incorporating specific response parameters, such as stress, displacement, ductility, stiffness, hysteretic energy, and so forth [16], [17], [18]. Stiffness, a commonly used response parameter in damage degradation modeling, directly describes the mechanical properties of materials, sections, members, and structures [16]. Therefore, a damage model based on stiffness degradation can assess damage at all levels [16]. Moreover, stiffness is a critical parameter for evaluating the safety behavior of a structure. The fiber model solves the problem of damage identification by dividing the section and member into a series of continuous fiber elements. It assumes that the nonlinear behavior of the material is distributed throughout these elements. The fiber model is widely used for performance prediction and damage assessment of structural members owing to its efficient computational speed and highly accurate results [19], [20]. The definition of damage variables at each level, including the fiber, section, member, and structure, is crucial for fiber damage modeling [16]. Guo et al. [16] integrated stiffness degradation and fiber models to create a new damage model for the accurate assessment of damage to frame structures. This model identifies damage values, locations, and phases for each material, section, and member, and enables the prediction of failure paths and modes [16]. Despite its theoretical advantages for the assessment of structural damage, the determination of the strain distribution of fiber elements using the finite element method is complex and computationally intensive. However, coupling DFOS technology with fiber element model-based stiffness degradation offers promising possibilities for real-time monitoring of damage.
As an expression of durability, maximum crack width control is an essential consideration in the design of RC members for the serviceability limit state [21], [22]. Furthermore, cracking affects the behavior of the structure, including stiffness, ductility, and energy dissipation [22]. Therefore, crack width control is of great importance for RC components, and many predictive models for maximum crack width have been developed [22]. In general, the theories for crack width evaluation can be summarized into four categories: bond-slip, no-slip, comprehensive, and statistical theories [22]. For example, the American design guideline ACI-318-19 [23] proposes a crack width prediction model based on a physical model and no-slip theory, which considers the protective concrete cover as the primary factor for crack width. EN-1992-1-1:2004 [24] and the fib model code 2010 [25] specify a crack width predictive model for flexural members based on a comprehensive theory that accounts for the effects of bond slip between concrete and reinforcement as well as concrete strain gradients. The Chinese design code GB/T 50010-2010 [26] derives a predictive formula for crack width based on a statistical theory, combined with a large amount of test data. In addition, DFOS and digital image correlation (DIC) techniques have become suitable tools for investigating the cracking performance of RC components. Galkovski et al. [27] validated a crack width model for tensile and flexural members based on test data of steel bars with fiber optic sensing. The results indicate that Eurocode 2 and the fib model code can predict the crack width well [27]. Typically, structural design codes control the crack development of a structure by setting maximum crack width limits to ensure the durability of the structure [23], [24], [25], [26]. These studies provide a reliable theoretical basis for establishing damage variables for the level of durability in this study.
As a representation of suitability, deflection control is an essential aspect of the design in normal service conditions for bending members [26]. The design code controls the deformation of flexural members by limiting the maximum allowable deflection, thus ensuring their suitability [23], [26]. The service deflection of an RC flexure member is calculated using the elastic deflection equation in combination with a uniform effective moment of inertia (Ie), which is assumed to be constant over the length of the member, or by integrating the curvature over the span of the member [28]. Normally, the use of a single effective moment of inertia for the entire span is based on the conditions of the critical section, and it is assumed that the stiffness variations along the member are accounted for. The American design guidelines (ACI 440. IR-15) [29] and the Canadian design standard (CSA-S806-12) [30] propose a model for calculating deflection based on the equivalent moment of inertia method and provide numerous design recommendations. However, the method of calculating deflection using Ie, has considerable limitations, especially for slabs with low reinforcement ratios, slender walls, and FRP-RC members [28]. Furthermore, the approach using a constant value of Ie is not suitable for some types of loads and boundary conditions, as it does not always correctly account for the stiffness of the uncracked sections of the member [28]. Integrating the curvature along the span using moments of inertia corresponding to each part provides a theoretically correct solution for calculating the deflection but requires more computational effort compared to using the elastic deflection equation with simple effective moments of inertia [28]. The fiber model, combined with DFOS, solves this problem effectively. In general, the above design approach to control deflection provides the theoretical basis for determining the damage variables under the level of suitability in this study.
The objective of this study was to develop a method to quickly predict the residual mechanical performances and evaluate the damage phases of RC beams to provide reference points for the design and maintenance of intelligence structural components. To achieve this objective, a novel SFCB with a self-sensing function was proposed and applied to the critical part of the structure to sense the stress state of the structure (Section 2). A damage definition method considering the stiffness degradation throughout the service life was established. Then a multilevel performance and damage assessment method combining the DFOS strain with theoretical and numerical modeling was proposed (Section 3). Finally, the effectiveness of the developed model was verified and validated using the SFCB-RC three-point flexural beam (4 Experimental testing of SFCB-RC beams, 5 Verification).
2. Self-sensing SFCB
In this study, a novel SFCB combining the functions of self-sensing, structural improvement, and damage control was developed. Considering its superior performance and economic efficiency, it is installed in critical parts of the structure, and a performance and damage assessment method based on monitoring DFOS strain has been developed. The self-sensing SFCB proposed in this study is a composite structure comprising a steel core, a single-mode telecommunication-grade optical fiber, and a CFRP wrapping layer. It uses a DFOS technology with optical frequency domain reflectometry (OFDR). Fig. 1 shows the details of the self-sensing SFCB. When using optical fibers as distributed sensors for monitoring, it is important that they are economical and robust. Therefore, the single-mode fibers, which contain a bend-resistant glass-fiber core (model G657 b3, diameter: 8.2 μm), a coating layer (190 μm), and a cladding layer (diameter: 250 μm), are tightly sheathed with protective jackets (diameter: 900 μm), as shown in Fig. 1. Note that this study focuses on evaluating the performance of the members using fiber-optic data. Therefore, the tests in this study were conducted in rigorous and identical test environments, and environmental factors were not considered to influence the results of the study. In addition, the failure strain of the optical fiber is almost the same as the rupture strain of CFRP (1.375%) and considerably lower than the rupture strain of the steel bar. In this study, CFRP rupture was used to define the structural failure based on the prevailing structural design guidelines. The failure of the optical fiber is solely a self-sensing functional failure and has no effect on the load-bearing capacity of the structure.
Rayleigh scattering is the phenomenon where uneven microstructures in the glass core of optical fibers cause the elastic scattering of light. This anomaly arises in the course of the fiber’s production, and its magnitude is directly linked to the wavelength of the light wave. The principle of OFDR-based measurement and demodulation of strain and temperature is shown in Fig. 2. This process consists of two phases: measurement and reference. The linear scanning light source produces Rayleigh scattering at different locations in the optical fiber, which corresponds to different wavelengths. Perform measurement of the echo signal at each point along the length of the fiber, establish the relationship between amplitude and wavelength, and convert the amplitude-wavelength data into an intensity-frequency relationship by fast Fourier transform (FFT). When a location in the optical fiber is subjected to temperature, pressure, or strain, the optical fiber produces a Rayleigh scattering frequency shift. Cross-correlation operations on the reference and perturbed phases allow the frequency shift along the fiber at each point to be determined. By measuring the Rayleigh scattering frequency shift, both strain and temperature measurements can be performed at a specific location, as well as distributed measurements throughout the entire optical fiber. A data acquisition system (model OSI-S) based on OFDR demodulation technology was used in this study. The measurement accuracy for strain and temperature specified by the manufacturer was ±1 με and ±0.1 °C, respectively, at a maximum measurement distance of 100 m and spatial resolution of 1 mm.
The frequency shift is correlated with the changes in strain ε and temperature T and can be expressed using Eq. (1) [5].
where λ and υ denote the average wavelength and frequency, and KT and Kε denote the calibration constants for temperature and strain, respectively. The spectral shift at constant temperature can be converted to the strain along the fiber by correcting the strain sensitivity coefficient.
3. Methods of damage assessment
Rapid identification and assessment of performance and damage condition of bending beams at multiple levels, including safety, suitability, and durability is required. Therefore, in this study, theoretical and nonlinear numerical methods were combined to establish the relationship between the monitored strain of SFCBs and performance parameters, such as stiffness, moment, deflection, and crack width. Stiffness and moments are the key parameters for assessing the damage condition and safety of flexural members. They can serve as assessment indexes for the entire service life of the members. Deflection and crack width are key indicators for assessing the suitability and durability of the members, and are used as assessment parameters for the normal service life phase. First, the relationship between SFCB strain and key performance parameters was developed based on the flexural beam theory. Then, a modified high-precision fiber damage model was developed using a nonlinear numerical method, and the theoretical results were validated. The fiber damage model takes the theoretically calculated damage threshold based on the stiffness degradation as the criterion for damage assessment. The accuracy of the fiber damage model in detecting damage is then improved by including DFOS strain data for correction. Finally, a multilevel damage and performance assessment method was developed for the rapid analysis of hybrid SFCB-RC flexural beams, as illustrated in Fig. 3.
3.1. Theoretical method
3.1.1. Damage definition and performance parameters
To quantify the performance degradation and damage evolution processes of members under loading, the correspondence between the damage variables and the behavior of component must be established based on the damage concept. The damage variable has the properties of sensitivity, consistency, and cumulative effect. In addition, stiffness is a frequently used metric in damage modeling and can directly describe the mechanical behavior and damage evolution of members. Therefore, a combination of the concept of damage variables and the theory of stiffness degradation can be used to define the damage to a member, as in Eq. (2) [16].
where D denotes damage variable, EI denotes the flexural stiffness of the damaged beam, and EI0 is the original stiffness of the undamaged beam [16].
According to the definition of curvature, the flexural stiffness of a critical section can be expressed as Eq. (3).
$\mathrm{E}\mathrm{I}=\frac{M}{\varphi }$
where M denotes moment, and φ is the curvature.
To ensure that the structure meets the safety level requirements, the flexural members are categorized into four damage phases, according to GB/T 50011-2010 [31], including minor, moderate, serious damages, and failure (Fig. 4(a)). The characteristic points of bending behavior corresponding to the threshold of damage phases are referred to as the yield point, the 0.53 times peak deflection (0.53Δm) point, and the peak point, respectively. Notably, the ratio of peak accelerations for moderate and large earthquakes is approximately 0.53 [32]. This ratio delineates the deflection corresponding to the thresholds of moderate and severe damage as 53% times the deflection associated with those of severe damage and failure. To evaluate the damage of bending members from a safety perspective, a simplified load-deflection curve is constructed for the double-reinforced rectangular beam, which includes characteristic points such as the crack point, yield point, 0.53Δm point, and peak point. Meanwhile, the relationship between the SFCB strain and the two damage variables and characteristic point performance parameters (moment, stiffness, and deflection) was established, as shown in Fig. 4(b). The stress state of the beam is depicted in Fig. 4(c). The basic assumptions and the derivation process are outlined in Section S1 in Appendix A.
The details of the derivation of the crack point are given in Section S1 in Appendix A. In this study, the yield point was defined as the point in which the SFCB reaches the yield strain (εy = 0.002). After yielding, the strain distribution in the beam section is shown in Fig. 4(c), assuming a triangular stress distribution in the concrete at the compression area. The yield moment My is expressed by Eq. (4).
where b is the width of the beam, d is the effective height of the beam, d′ is the distance from the external compression fiber to the compression bars, ρs1, ${\rho }_{\mathrm{s}}^{\text{'}}$, and ρf are the total tensile steel bar, compressive steel bar, and CFRP ratio, respectively. Note that ρs1 includes the steel core area of SFCB. Es, ${E}_{\mathrm{s}}^{\text{'}}$, and Ef are the elastic moduli of tensile steel bars, compressive steel bars, and CFRP, respectively. k is a coefficient related to the height of the compressive concrete.
If My and the yield curvature (φy = εy/(d−kd)) are known, the yield stiffness EIy can be calculated using Eq. (3). According to Eq. (2), the damage variable corresponding to the yield point DI can be expressed using Eq. (5).
where n, n′, and nf are the ratios of the elastic moduli of the tensile steel bars, compressive steel bars, and CFRP (Es, ${E}_{\mathrm{s}}^{\text{'}}$, and Ef) to that of concrete (Ec), respectively. h is the beam height. The yield deflection Δycan be expressed by Eq. (6).
where λ is determined by the loading mode: λ = 1/12 for three-point loading, λ = (3 - 4(Ln/L)2)/24 for four-point loading, Ln is the distance from the loading point [33], [34]. L is the span of the beam.
The peak point is defined as the point at which the maximum strain on the external compression fibers εcu reaches 0.0033. According to ACI 440.1R-15 [29], the moment capacity (Mm) of these beams, featuring compression reinforcement, is determined by considerations of force equilibrium and deformation coordination, and by assuming an equivalent rectangular stress block for concrete in the compression area, as depicted in Fig. 4(c), expressed by Eq. (7).
where εsfm is the peak strain of SFCB, ${\epsilon }_{\mathrm{s}}^{\text{'}}$ is the strain of the compressive reinforcement, and fy is the yield stress of the tensile steel bar. When the compression reinforcement yields (d/c ≤ 1− εy/εcu, c = a/β1, where c is the height of the compressed areas of concrete, and the height equivalency factor β1 = 0.85 according to ACI 440.1R-15 [29]), the height of the rectangular concrete stress block (a) can be expressed by Eq. (8).
where ${f}_{\mathrm{c}}^{\text{'}}$ is the compressive strength of the concrete. When the compression reinforcement is unyielding (d'/c > 1− εy/εcu, c = a/β1), the height of the rectangular concrete stress block can be expressed by Eq. (9).
where εsfu is the SFCB outer layer CFRP rupture strain. The peak curvature φm can be expressed as φm = εsfm/(d − c). By correlating Eq. (3) and Eq. (7), the flexural stiffness at the peak point EIm can be derived. Furthermore, by combining Eq. (2), the damage variable corresponding to the peak point DIII can be calculated as given in Eq. (12).
where Lp= (d + 0.05L) represents the length of the plastic hinge [35]. φs denotes the modified curvature. Based on the experimental data calibrated by Hwang et al. [34], φs ≈ 2.5φm.
When 0.53Δm is reached, the curvature can be expressed as given in Eq. (14).
where c0.53 and εsf0.53m are the compressive concrete height and tensile bar strains at 0.53Δm, respectively.
A comparison of the calculated values of c and c0.53 is presented in Table 1. The analysis indicates that the values of c and c0.53 are very close to each other, with c/c0.53 ranging from 0.95 to 0.96. Thus, εsf0.53m ≈ 0.53εsfm. Similarly, the strain for extremely compressed fibers is approximately equal to 0.53εcu at 0.53Δm. This value is lower than the peak strain (ε0) resulting from the stress-strain relationship of the concrete. Therefore, the stress distribution of the compressed concrete is assumed to be triangular, as shown in Fig. 4(c). The moments M0.53m, which correspond to 0.53Δm can be expressed by Eq. (15).
where k2 is a coefficient related to the height of the compressive concrete at the 0.53Δm point. The curvature corresponding to the 0.53Δm is φ0.53m = 0.53εsfm /(d − k2d). The combination of Eq. (15) and Eq. (3) results in a flexural stiffness at the 0.53Δm point (EI0.53m). According to Eq. (2), that results in a damage variable that corresponds to 0.53Δm, and is expressed by Eq. (16):
The loads at the characteristic points can be determined from the corresponding moments. After calculating the corresponding performance parameters (moment, stiffness, and deflection), damage variables, and SFCB strains for all characteristic points, they can be connected by a line to create a simplified theoretical model curve of SFCB strains versus performance parameters, SFCB strains versus damage variables, and load versus deflection, as shown in Figs. 4(a) and (b).
3.1.2. Method for calculating the crack width
The maximum crack width in the normal service phase is of greater importance for practical engineering applications with regard to the durability of the structure. The prevailing structural design codes (fib model code 2010 [25] and EN-1992-1-1:2004 [24]) focus on the crack width during this phase. In addition, the deformation of the beam after yielding of the reinforcement is unstable, which complicates the calculation of the crack width. Therefore, this study focused on the method to calculate the crack width before yielding.
In this section, a method for identifying multi-crack patterns in flexural beams and calculate crack widths based on DFOS strain is investigated. Studies have shown that there is direct method to recognize the location of cracking by finding local maxima in the strain distribution measured by the DFOS technique. The prevailing structural design codes (fib model code 2010 [25] and EN-1992-1-1:2004 [24]) determine the crack width by assuming that it is twice as large as the slip at the interface between the concrete and the reinforcement at the crack. This slip is calculated as the integral of the strain differential between the reinforcement and the concrete over the crack spacing.
Usually, the cracking behavior of bending beams is modeled using “crack elements” (i.e., RC elements between two cracks) [27], assuming that the concrete cross-section remains plane, as shown in Fig. 5. Studies have shown that the actual stress distribution in concrete is very complex and not uniform, and that usually only the concrete surrounding the reinforcement is effectively tensioned [27]. To account for tension stiffening, it is assumed that the concrete in the neighboring reinforcement has a constant effective cross-sectional area, with the total force of concrete tensile stresses acting on the centroid of the reinforcement, as shown in Fig. 5. The relationship between the crack width of the ith crack, denoted as wcr,i, and the strain of the self-sensing SFCB can be expressed by Eq. (17). Note that this simplified average cracking model calculates an idealized cracking pattern (i.e., cracks that penetrate the entire effective tension area and develop uniformly). Internal invisible and external visible cracks within the cracking element are accounted for.
where hc,eff is the height of the effective concrete area of the beam in tension, as determined by Eq. (19); As1 is the area of the total tensile steel bar; Af is the area of CFRP; ${x}_{0,i}^{-}$ and ${x}_{0,i}^{+}$ denote the 0-slip points on the left and right sides of the ith crack, respectively. The 0-slip points are assumed to be located at the midpoints of the neighboring cracks. εsfcr,i and σsf(εsfcr,i) are the strain and stress of the self-sensing SFCB at ith cracking position, εsf(x) and σsf(εsf(x)) are the strain and stress of the self-sensing SFCB at uncracking position.
where xc is the height of the compressed concrete before the steel bar yield. The derivation process can be found in Section S1 in Appendix A.
3.2. Nonlinear numerical method
Fiber damage models also allow a nonlinear analysis of the cross-section. Their simplified modeling, acceptable accuracy, and fast solution compared to nonlinear finite element methods have greater potential for rapid assessment of structural performance and damage [16]. Therefore, a high-precision nonlinear fiber damage model for hybrid SFCB-RC beams was established in this study based on the strain monitored by distributed fiber technology and the damage variables of stiffness degradation. Fig. 6 shows the strain, stress, and damage distribution in the fiber cross-sections. The model recommended by GB/T 50010-2010 [26] is used to describe the stress-strain relationship and damage-strain relationship of concrete, while a bilinear model characterizes the mechanical properties of steel and SFCB [31], [36], [37]. The model equations and analytical procedures are described in detail in Section S2 in Appendix A.
In this study, the fiber damage model describes the damage evolution of the cross-section using the stiffness degradation-based sectional damage variable proposed by Guo et al. [16]. This sectional damage variable considers the nonlinear modified term of stiffness and the material fiber damage variable. Note that the conventional fiber damage model, which assumes a linear distribution of strain in the cross-section, is consistent with the crack opening process in the critical section and only reflects the damage variables in the cracked cross-section. However, as described in Section 3.1.2, only the effective concrete near the steel bars is stretched in the uncracked section, while almost no tensile damage occurs within the ineffective concrete. Therefore, the conventional fiber damage model overestimates the tensile damage of concrete in an uncracked section. To improve the ability of the fiber damage model to identify damage, it was modified as follows: the crack location is identified by the local peak of the SFCB strain distribution. When calculating the damage variables for the uncracked section, only the tensile damage of the effective concrete is considered. This tensile damage is determined by the mean strain, which is calculated using Eq. (18). The comparison of the strain and damage distribution in uncracked sections between the original and the modified model is shown in Fig. 7.
3.3. Multilevel assessment method
This section deals with the establishment of multilevel performance and damage assessment criteria in terms of safety, suitability, and durability and the determination of their threshold values for damage variables. Safety is a performance parameter throughout the operational phase, while suitability and durability are performance parameters during normal operation. Therefore, the damage variable based on stiffness degradation was consistently used as the damage definition method in this study. The flexural beams designed for ductile failure have sufficient safety redundancy to provide moment-carrying capacity despite the rupture of the outer CFRP layer for hybrid SFCB-RC beams. In this study, DIII was used as the safety performance threshold. ACI 318-19 requires a maximum deflection Δmax of L/180 during the normal service phase [23]. Therefore, when the deflection of the simplified load-deflection model reaches Δmax, it is defined as the suitability damage threshold Ds, which can be obtained by linear interpolation of a simplified theoretical model of the deflection-damage variable.
Crack width is one of the key indexes for evaluating structural durability, and the maximum crack width wmax is limited to 0.5 mm for severe environments and 0.7 mm for other environments according to ACI 440.1R-15 [29]. Therefore, the damage variable, which corresponds to the maximum crack width is used as the damage threshold Dd for durability. The SFCB strain εsf,d corresponding to the critical section at wmax can be derived from the crack width calculation formula in the ACI 440.1R-15 code [29] and can be expressed by Eq. (20).
where β is the ratio of distance between the neutral axis and tension face to distance between neutral axis and centroid of reinforcement, hc is the thickness of cover from tension face to the center of the tension bar, s is the bar spacing, and kb is the bond coefficient.
In Section 3.1, simplified assumptions were used to derive the theoretical values that can be used to calculate the corresponding damage thresholds for each member based on the cross-sectional forms, material properties, and loading methods of the members in practical engineering. After establishing the simplified theoretical model of the SFCB strain-damage variables in Section 3.1, the damage thresholds for durability corresponding to εsf,d can be determined using the linear interpolation method. Finally, based on the damage thresholds and the relationship between the SFCB strains monitoring and the damage variables, the performance and damage phase of the beam are assessed, allowing a quick decision on appropriate maintenance measures.
4. Experimental testing of SFCB-RC beams
4.1. Experimental program
4.1.1. Specimen design
The objective of this study was to investigate whether self-sensing SFCB can effectively monitor the strain of RC beams throughout their service life. In addition, an attempt was made to validate the effectiveness of the proposed performance evaluation method. Based on the ductile design requirements of the prevailing structural design codes GB/T 50010-2010 [26], three RC beams with reinforcement ratios of 0.27%, 0.82%, and 1.67% were designed. The failure mode of these beams is primarily characterized by ductile failure in flexure, with a transition from CFRP rupture to concrete crushing.
Three hybrid SFCB-RC beams with identical dimensions were designed for the tests: a total length of 1900 mm, a span of 1600 mm, a cross-section width of 150 mm, a height of 250 mm, and a concrete cover thickness of 25 mm. To ensure sufficient shear bearing capacity and to induce flexural failure, all beams are equipped with deformed reinforcement with a diameter of 10 mm and a spacing of 100 mm. The calculated theoretical shear bearing capacity is 388 kN, according to GB/T 50010-2010 [26]. The SFCBs, with a length of 2100 mm and a diameter of 10 mm, were positioned at the center of the underside of the beam as tensile longitudinal bars. To protect the fiber optic sensors, the SFCB extends 100 mm from both ends of the beam. The study investigates the effects of different reinforcement ratios on the flexural properties and damage condition of hybrid SFCB-RC beams. Each beam contains two deformed bars with diameters of 8, 14, and 20 mm, designated as B8, B14, and B20, respectively (Table 2). Fig. 8 shows the design details of the test specimens. To prevent slippage between the SFCB and concrete, CFRP-wrapped 50 mm long and 10 mm thick is used as an end anchor 150 mm from the end of the beam (Fig. 8).
4.1.2. Test setup and instrumentation
As illustrated in Fig. 9, all specimens were subjected to a three-point flexural loading test using an actuator with a capacity of 3000 kN and a loading rate of 0.5 mm·min−1. Five linear voltage displacement transducers (LVDTs) were positioned to measure the deformation of the beam at mid-span, quarter-span, and at the supports. In addition, the displacement changes of the beams were recorded using Optotrak Certus system (Northern Digital Inc., Canada) during the loading process to determine the curvature changes of the beams. Crack development on the concrete surface was monitored using the DIC method. The monitoring area, illustrated in Fig. 9, had a height of 250 mm and a length of 1000 mm. A DFOS interrogation device was used to record the strain of fiber optic sensors within the SFCB. The spatial resolution was set to 1 mm, at a sampling frequency of 1 Hz. To ensure stability and accuracy of the measurement, the strain of the SFCB was recorded by applying a load every 5 kN and maintaining it for 3 min.
4.1.3. Material properties
All test specimens were prepared from the same grade of concrete, with a mix ratio design of 1.0 part Portland cement, 0.4 parts water, 1.2 parts sand, and 3.0 parts coarse aggregate. The maximum size of the coarse aggregate was approximately 14 mm. The 28 d average compressive strength of three 100 mm concrete cubes measured 31.5 MPa, according to American Society of Testing Materials (ASTM) C39/39M-21 [38]. Uniaxial tensile tests were performed on three samples each of deformed bars with diameters of 8, 10, 14, and 20 mm [39]. The average yield strengths recorded were 409.4, 392.9, 422.4, and 455.7 MPa, respectively, as listed in Table 2.
As shown in Fig. 10, the stress-strain curve of SFCB is divided into an elastic phase before yielding (Phase A) and a plastic deformation phase of the steel core after yielding (Phase B). The surface strain of SFCB measured by DIC is consistent with the strain of the steel core measured by DFOS, indicating deformation coordination throughout the loading process. In addition, the rupture strain of the CFRP is almost identical to the maximum measurable strain of the optical fiber, which is approximately 1.375% (Fig. 10). Table 3 summarizes the yield strength, initial elastic modulus (EI), post-yield elastic modulus (EⅡ), ultimate strength, and ultimate strain of the smart SFCB, with values of 387 MPa, 207 GPa, 116 GPa, 1748.1 MPa, and 1.375%, respectively.
4.2. Test results and discussion
4.2.1. Failure patterns
The hybrid SFCB-RC beams in this study exhibited uniform ductile failure. This involved initial yielding of the tensile reinforcement, followed by either concrete crushing or rupture of the FRP layer after concrete crushing. As shown in Fig. 11, the failure patterns of specimens B8, B14, and B20 transition from FRP rupture-control mode to concrete crushing with increasing reinforcement ratio. For specimens B8 and B14, the failure pattern consists of yielding of the tensile longitudinal reinforcement, followed by partial crushing of the concrete near the loading point, culminating in FRP rupture. However, the B20 beam, featuring a higher reinforcement ratio, exhibited a failure pattern characterized by concrete crushing, which is a more severe failure mode compared to B8 and B14. In addition, sufficient stirrups were equipped in this study, and the cracks of the beams are mainly dominated by vertical flexural cracks. The flexural cracks increase as the reinforcement ratio of the steel bars increases, and the deformation capacity is improved. This ductile failure mode shows that ductile failure with a balanced reinforcement ratio effectively ensures the safety margin of the structure.
4.2.2. Load-deflection curves and deformation capacity
The load-deflection curves of the hybrid SFCB-RC beams, shown in Fig. 12, are accompanied by the loads and deflections at the yield point, peak point, and ultimate point for each specimen, as summarized in Table 4. The cracking loads for all beams were similar, at approximately 10.6 kN. The load deflection curve has a large slope from the cracking point to the yield point phase, and the slope increases significantly as the steel bar ratio increases. Because of the strengthening effect of the CFRP in the outer layer of SFCB, the beam has a stable post-yield stiffness after yielding, although the slope of the load-deflection curve decreases. After reaching the peak load, B8 and B14 experienced a gradual nonlinear reduction in load due to concrete crushing. When the ultimate load was reached, there was then a significant decrease, which can be attributed to the rupture of the outer CFRP layer of the smart SFCBs. For specimen B20, the corresponding deflection was only 18.5 mm when the peak load was reached, and the load decreased slightly owing to concrete crushing, but was maintained at approximately 95% of the peak load by the compression reinforcement. Specimen B20 was loaded until the load dropped below 85% of the peak load. The CFRP still did not rupture, and as the deformation of the beam increased, the concrete in the compression zone was eventually crushed. Therefore, appropriately increasing the steel bar ratio in hybrid SFCB-RC beams can effectively prevent the rupture of the outer CFRP layer of the SFCB and improve its monitoring ability during service.
4.2.3. Crack progression and SFCB strain distribution
Fig. 13 shows the strain distribution recorded in the SFCB before yielding. The occurrence of cracks in the concrete near the SFCB leads to sudden strain peaks that allow the crack position to be determined. As shown in Fig. 13, Ci (where i = 1-11) denotes cracks, with the numbers indicating the order of their occurrence. In all specimens, cracks originated consistently from mid-span. With increasing load, the peaks became indistinguishable, indicating a lower crack sensing sensitivity. The reason for this phenomenon is the gradual failure of the interface between the SFCB and the concrete as the load increases. As a result, the difference in strain between cracked and uncracked sections is only related to the moment carried. As shown in Fig. 13, there is a greater concentration of stress in the span after cracking, while the strain distribution in the beam becomes more uniform as the steel bar ratio increases. This is because the increase in steel bar ratio reduces the difference in stiffness between before and after cracking. For beams with a small steel bar ratio, the initial cracking occurs in the mid-span section, resulting in a significant stiffness degradation and a rapid increase in strain within the cracked section. This stress concentration phenomenon results in fewer cracks in beams with a smaller steel bar ratio, but these cracks tend to be wider and more widely spaced. As the steel bar ratio increases, the stiffness degradation in the cracked section decreases, allowing for a more uniform transfer of stresses in the reinforcement. This leads to an increase in the number of cracks and a reduction in their width and spacing. These results show that the strain distribution of SFCBs can directly identify the location and spacing of cracks.
5. Verification
5.1. Moment-curvature relationship and load-mid-span deflection
In this section, the proposed simplified theoretical and numerical models for predicting the flexural behavior of hybrid SFCB-RC beams are reviewed and validated. The load-deflection curves and moment-curvature curves of the experimental results are compared with the theoretical and numerical results, respectively, as shown in Fig. 14. The results showed high agreement between them. Although the simplified theoretical results only roughly represent the load-deflection and moment-curvature relationships by the characteristic points, they still represent the flexural behavior of the hybrid SFCB-RC beams with sufficient accuracy. In addition, a total of 20 previous datasets were collected to validate the simplified theoretical and numerical models [40], [41], [42], [43]. The data used to validate the model are summarized in Table 5 [40], [41], [42], [43]. These datasets include SFCB-RC beams and hybrid steel/FRP beams that experienced flexural failure, with beam widths b ranging from 120 to 220 mm, heights h ranging from 250 to 400 mm, and beam spans L ranging from 1200 to 2100 mm. The concrete strengths ${f}_{\mathrm{c}}^{\text{'}}$ varied from 25.7 to 48.7 MPa. The total reinforcement ratios ρs1 ranged from 0.11% to 7.40%, and FRP ratios ρf varied between 0.31% and 9.30%. EI ranged from 72.6 to 132.0 GPa, EII from 17.6 to 55.0 GPa, and Ef from 49.0 to 68.0 GPa.
Table 6 [40], [41], [42], [43] summarizes the loads and deflections at the yield point, 0.53 peak point, and peak point from the test, numerical model, and theory, respectively. Since the reliability of the ACI 318-19 equation for calculating the cracking point has been confirmed by numerous studies and its value has only a minimal influence on the results [23], it is not validated in this study. The ratios of numerical (Num) values to tested values and theoretical (Theor) values to tested values are also compared, respectively. The average (Avg) values of the ratios of numerical values to tested values and of theoretical values to tested values are between 0.96 and 1.03, and the coefficients of variation (COV) are between 0.037 and 0.134. In addition, the numerical model has better consistency in predicting the moment-carrying capacity than the simplified theoretical model because it accounts for the nonlinearity of the material. For example, for yield load (Py), load at 0.53 peak point (P0.53m), and peak load (Pm), the COV of the numerical results are 0.062, 0.069, and 0.037, respectively, while the COV of the simplified theoretical results are 0.128, 0.079, and 0.043, respectively. Thus, the numerical and simplified theoretical models proposed in this study are both capable of accurately predicting the load deflection curves of hybrid steel/FRP-RC beams with different materials, geometric parameters, and reinforcement forms. Their prediction accuracy is within an acceptable range, and their results can be used as a reference for structural performance evaluation.
5.2. Crack width and development
Fig. 15 shows the relationship between load and crack width for the initial cracks (C1) in each specimen and compares the calculated results with the DIC test results. Note that the measured crack width in this study is the average width of the effective tensile concrete around the center axis of the SFCBs. Before yielding, the results of ACI-440.1R-15 [29] and the proposed method show a high degree of agreement with the DIC test results. The empirical equations in ACI 440.1R-15 [29], in which the bond coefficients for B8, B14, and B20 specimens are 1.0, 1.2, and 0.7, respectively, are due to the varying rib heights for different diameters of rebar. The results show that the proposed crack width calculation method can well evaluate the crack development during the normal service phase. At a load of 40 kN, specimens B14 and B20 showed a reduction in crack widths of 62% and 90%, respectively, compared to specimen B8. This indicates that increasing the steel ratio effectively improves control over crack widths.
Fig. 16 presents the modified and original distribution of sectional damage variables for different members at cracking deflection Δcr, yield deflection Δy, and peak deflection Δm. The sectional damage gradually decreases from the mid-span towards the supports. In the modified sectional damage variable distribution, the damage variables of the section at the crack location are significantly larger than those of the uncracked section due to the tensile damage to concrete fibers. The maximum damage variable corresponds to the critical section where the damage is the most severe, which is consistent with the observed test results (Fig. 11). For the B20 specimen, severe crushing failure of the concrete occurs when the peak load is reached approximately 400 mm from the mid-span, which is consistent with the test results. These results show that the modified fiber damage model can adequately identify and reflect damage variables in different sections compared to the original model.
When the strain of the concrete fibers at the bottom of the section reaches the crack strain εc,r, it can be considered that cracks have occurred in the concrete. Consequently, the crack depth can be determined from the strain of the concrete fibers. Fig. 17 shows the crack depths for critical sections and compares the numerical results with the DIC results at the crack point, yield point, and peak point, respectively. The results for the numerical crack depth are largely consistent with the DIC results. In additionally, the numerical results for the section damage variables show that increasing the reinforcement ratio leads to a shift in the predominant damage distribution at critical sections from tensile damage in concrete to compressive damage, effectively preventing the risk of brittle failure in the components. In summary, the modified fiber damage model proposed in this study enables precise localization, tracking, and quantification of structural damage. It systematically quantifies the progression of damage across material, section, and component levels, thus providing valuable guidance for structural damage monitoring and maintenance.
5.3. Damage variables
As shown in Fig. 18, the theoretical and fiber-related numerical values of the damage variables corresponding to characteristic points were respectively compared with the test values. In general, they were largely consistent with the test values, with the mean values of the theoretical values to the test values and the numerical values to test values being 0.990 and 1.000, respectively. The results of the theoretical calculation method are slightly lower than the test results due to the use of the equivalent rectangular stress block method for stress distribution in compressed concrete. However, the error is still within acceptable limits. Table 7 summarizes the damage variables corresponding to the damage phase of the specimen.
Fig. 19 compares the damage variable-mid-span deflection curves obtained from the test, theoretical, and fiber damage models. The results of the fiber damage model are largely consistent with the test results. During the minor damage phase (D < DI), the damage of the member progresses rapidly, mainly due to concrete cracking.
In the moderate damage phase (DI ≤ D < DII), the plastic rotation at the cracks after the yielding of the reinforcement allows the compressive damage of the concrete to develop and dominate, so that the progression of the damage in the members slows down. This phenomenon results from the comparatively slower rate of degradation of the compressive modulus of concrete compared to its tensile modulus. However, with increasing reinforcement ratio in the B8, B14, and B20 specimens, the development rate of damage after yielding accelerates. This acceleration is attributed to the increased area of compressed concrete fibers in critical sections.
During the serious damage phase (DII ≤ D < DIII), there is a slight acceleration in the development rate of damage as the reinforcement ratio of the member increases. This is attributed to the faster rate of degradation of the concrete fibers’ modulus after the compressive peak strength is reached compared to before the compressive peak strength.is reached. The theoretical results, which are associated with the damage variables corresponding to the characteristic points, are able to adequately reflect the process of damage development caused by the nonlinearity of the material. In particular, the damage variables from the cracking to the yield point are underestimated. This phenomenon is attributed to the faster rate of degradation of the tensile modulus of concrete fibers after cracking compared to the rate of degradation of the elastic modulus before the peak compressive peak strength is reached. The above results show that the simplified equivalent stress block method proposed in this study has a smaller error compared with the exact solution of the fiber damage model and can also calculate the degradation of the section stiffness appropriately. Therefore, the theoretically calculated threshold values can be used for the assessment of structural damage and applied to the fiber damage model as an evaluation criterion for the damage phase.
6. Discussion
In this section, specimens B8, B14, and B20 are examined and discussed using the multilevel evaluation method proposed in this study. First, the SFCB strains, material parameters, and geometric parameters of the beam were determined throughout the bending process. The performance parameters and the threshold values of the damage variables for each level were calculated based on the theoretical approach in Section 3 and are summarized in Table 8. The wmax is determined based on the service environment of the member, according to ACI 440.1R-15 [29]. To calculate the durability threshold, this study assumes that all specimens are considered to be in service in an normal environment with a wmax of 0.7 mm [29]. The suitability threshold is calculated by taking the Δmax (= Ln/180) of 8.9 mm [23].
Fig. 20 shows the relationship between the mid-span deflection, damage variables, and crack width. As shown in Fig. 20, the durability threshold and the suitability threshold occur in the moderate damage phase due to the good deformation control capability of the hybrid SFCB-RC beam designed in this study. This shows that the beams designed in this study fulfil the damage and control requirements for durability and suitability in the normal service phase. Table 8 shows that the threshold values of the damage variables for durability, suitability, and safety all decrease with increasing reinforcement ratio. This indicates that increasing the reinforcement ratio can effectively control the stiffness degradation and slow down the development of damage to improve the safety redundancy of the beam. Then, the relationship between the mid-span deflection and the damage variable is obtained using the fiber damage model, and the performance and damage phase of the beam are determined by the damage variable. Finally, appropriate maintenance measures are taken for the beam. For example, in the minor damage stage, the crack locations can be recognized based on the distribution of the sectional damage variables, and the cracks can be repaired. When either the durability threshold or the serviceability threshold is reached, a reduction in the applied load is required. In addition, when safety thresholds are reached, appropriate strengthening measures can be implemented to increase the load-carrying capacity and stiffness of the beam. In summary, the multilevel evaluation method proposed in this study combines theoretical and numerical approaches to establish the relationship between SFCB strains, damage variables, and performance parameters. The safety, suitability, and durability of bending beams can be effectively evaluated using the monitored SFCB strain alone.
7. Conclusions
Based on the results of this study, the main conclusions can be summarized as follows:
(1) The increase of steel reinforcement in SFCB-RC hybrid beams under a balanced ratio prevents cracking in the outer CFRP layer and improves the monitoring capability during service.
(2) The proposed crack width calculation method can effectively estimate the crack width of RC beams before yielding.
(3) The proposed simplified theoretical model can accurately predict the performance parameters and damage variables at the characteristic points of RC beams dominated by flexural failure. These results can serve as a benchmark for the classification of damage phases.
(4) The fiber damage model, modified by integration with monitored DFOS strain data, effectively identifies, and reflects the progression of damage and enables the monitoring of crack localization, tracking, and development processes.
(5) The proposed multilevel damage assessment approach combines numerical and theoretical methods to establish relationships between SFCB strains and damage variables. This enables a rapid assessment of safety, serviceability, and durability using the monitored SFCB strain and relevant material parameters.
This study investigates the application of a self-sensing SFCB-RC structural design and a multilevel damage assessment method based on DFOS technology for advancing structural intelligence and digitalization, though some limitations persist. The simplifications in fiber damage and theoretical models may be able to adequately reflect complex structural deformations or nonlinear responses. Future research can focus on developing new damage assessment methods that combine structural analysis models with machine learning techniques, enhancing both the efficiency and precision of structural monitoring.
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
The authors are grateful for the financial support provided by the National Science Fund for Distinguished Young Scholars of China (52325804), the National Natural Science Foundation of China (U2001226 and 52108230), the Guangdong Basic and Applied Basic Research Foundation (2022B1515120007), and the Shenzhen Basic Research Project (JCYJ20210324095003010).
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