A Novel SMFL-Based Assessment Method for Corrosion Nonuniformity of Rebar and Its Application in Reliability Analysis of Corroded RC Beam

Qiu, Junli; Jing, Yue; Zhang, Weiping; Jiang, Chao

doi:https://doi.org/10.1155/2024/9988034

Structural Control and Health Monitoring

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Abstract Introduction Results and Discussion Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2024 | Article ID 9988034 | https://doi.org/10.1155/2024/9988034

A Novel SMFL-Based Assessment Method for Corrosion Nonuniformity of Rebar and Its Application in Reliability Analysis of Corroded RC Beam

Junli Qiu,^1,2Yue Jing,^1,2Weiping Zhang ,^1,2and Chao Jiang^1,2

Academic Editor: Francesc Pozo

Received06 Nov 2023

Revised07 Feb 2024

Accepted24 Feb 2024

Published13 Mar 2024

Abstract

The reliability of corroded reinforced concrete (RC) structures relies on the accurate minimum cross-sectional area of corroded rebar. In this study, the accurate morphologies and self-magnetic flux leakage (SMFL) field strengths of twenty-eight non-uniformly corroded rebars were obtained using 3D structural light scanning and micromagnetic detection technologies, based on which three indices of the SMFL field variation ratio dH, the corrosion non-uniformity degree , and the cross-sectional area ratio K_0.25 are proposed. The statistical results show that the probability densities of and K_0.25 obey the Weibull distribution and Gamma distribution at the 95% confidence level, respectively, and their distribution parameters are linearly or inversely proportional to dH. The probability density distribution of the minimum cross-sectional area of corroded rebar can be determined using indices and K_0.25, based on which a feasible SMFL-based reliability assessment method of corroded RC structures is proposed. The case study of a real specific corroded RC beam shows that the reliability assessment error of the SMFL-based method is only 1.2%, which is much lower than the 20.7% error of the existing method. This SMFL-based method provides a novel idea that can automatically and accurately assess the effect of rebars’ corrosion non-uniformity on the reliability of specific in-serviceRC structures.

1. Introduction

Reinforced concrete (RC) structures commonly used in civil infrastructure face long-term adverse effects from the complex environment, such as chloride ion attack, sulfate ion attack, and carbonation, resulting in the developing corrosion of the embedded rebars [1–5]. Corrosion of rebars brings many negative consequences, one of which is the substantially reduced reliability of RC structures [6, 7].

Many reliability assessment methods for RC structures have been developed, such as the out-crossing rate method [8], the first-order and second-order methods [9, 10], the probability density evolution method [11, 12], and the novel adaptive method for the small-failure probability analysis [13]. However, these well-developed methods confronted difficulties in establishing and solving complex mathematical models, for which the Monte Carlo simulation (MCS) to approximate failure probability by failure frequency of a large number of simulations is an effective alternative solution [14–16]. When the rebar’s average cross-sectional area is known, the accuracy of MCS depends largely on the assessment accuracy of the minimum cross-sectional area of corroded rebars and the negligence of longitudinal corrosion non-uniformity leads to an unacceptable misestimation of structural reliability [14, 17]. To solve this problem, our group have proposed a cross-sectional area spatial heterogeneity factor R (ratio of the average and minimum cross-sectional areas), based on which the effect of corrosion non-uniformity on the reliability of RC beams is successfully considered [17–19].

However, the existing probability density distribution (PDD) model of factor R is a predictive model based on the statistical analysis of the cross-sectional area data of artificially corroded rebar samples [17, 20, 21], which is not comprehensive enough and may not be suitable to describe the random corrosion non-uniformity of rebars in a specific in-service RC structure. Generally, the corrosion of rebars has a strong randomness closely related to environmental factors, material properties, and structural stress. Chloride ion attack and concrete carbonization are two main reasons that lead to the corrosion of rebars, directly influencing the initiation time of reinforcement corrosion. The transport of chloride ions in concrete is a stochastic process that substantially depends on the mix composition and binder addition of concrete, surface chloride concentration, temperature, and relative humidity in the surrounding environment [3, 22–25]. Concrete carbonation is also a stochastic process affected by the concrete itself and environmental conditions [4, 26–28]. In addition, both of them will be accelerated by stress-induced cracks [26, 29–31], resulting in more significant non-uniform corrosion morphology [32].

With the help of detection or monitoring information, the actual reliability and service status of the RC structures can be assessed more accurately [33, 34]. The non-uniform corrosion defect of the ferromagnetic rebar causes a self-magnetic flux leakage (SMFL) field variation, which can be accurately detected or monitored with high-precision micromagnetic sensors [35–43]. Our group and some other researchers have confirmed the good linear correlation between corrosion degree and the corrosion-induced SMFL field variation amplitude [38–41]. Experimental studies and engineering applications have shown that the SMFL field variation can characterize the exact corrosion defects of rebars non-destructively and conveniently, and have nothing to do with the random development process of non-uniform corrosion [37, 42, 43]. For the issue of magnetization state difference of rebars, some solutions have been proposed, and applicable SMFL-based nondestructive quantitative methods for the corrosion degree of rebars have been established by our group [43–45]. These results show that the SMFL-based characterization of the non-uniform corrosion of rebars is a good choice to avoid the trouble of the stochastic corrosion process and the extensive statistical work of establishing the PDD model for factor R.

In this study, the accurate morphologies and SMFL field strength data of twenty-eight non-uniformly corroded rebars were acquired using 3D structural light scanning and micro-magnetic detection technologies. The SMFL-based assessment of the minimum cross-sectional area of corroded rebar is realized, based on which the effect of corrosion non-uniformity on the reliability of the corroded RC beam is explored.

2. Concepts

As shown in Figure 1, the reliability index β of a corroded RC beam can be calculated using the MCS method with the following four steps [17, 18, 46]. First, the corroded RC beam is divided into a series system consisting of s beam segments with the same length (for example, 100 mm). Second, the PDDs of the load, material strength, cross-sectional sizes, and rebar’s minimum cross-sectional area of the corroded RC beam are determined, based on which a set of values for them are randomly sampled. Third, the ultimate flexural bearing capacity M_u and applied bending moment M_S of all beam segments are calculated according to this set of values. In the i-th simulation, the corroded RC beam is regarded to fail once the of any segment exceeds its , and the failure number . Fourth, the probability of an event is estimated with the frequency of the incident in N times of simulation, for example, N = 1000000. The estimation value failure probability is

Then, the reliability index β is

The is a key part of the MCS-based reliability assessment procedure of the corroded RC beam. Generally, only the average corrosion degree of rebars can be obtained using theoretical prediction, non-destructive monitoring, and destructive detection [47–49], based on which the rebar’s average cross-sectional area can be obtained. However, as shown in Figure 1, the minimum cross-sectional area of a certain segment with a width of 100 mm of an actual corroded rebar is significantly smaller than . Using instead of in the reliability assessment of corroded RC beams avoids errors of up to 200% [14], and thus how to accurately assess when is known is a problem.

As shown in Figure 1(a), method 1 to solve this problem is to use the Gumbel PDD of the cross-sectional area spatial heterogeneity factor R [17, 20, 21]. According to this Gumbel distribution, s values of R used to calculate the for the s beam segments can be randomly sampled in each simulation of MCS. However, as mentioned before, these R values are predictive results based on experimental data [17, 20, 21], which may not be suitable for a specific in-service RC structure.

Therefore, SMFL-based method 2 for assessing the is proposed. The SMFL non-destructive detection has a good capability to assess the actual random corrosion degree of a specific in-service RC beam. As shown in Figure 1(b), the commonly used rebars are usually longitudinally (x-axis) magnetized under the actions of hot rolling and tension, that is, the minor components and of the magnetization vector V can be ignored compared to the major component [44, 50]. For rebar that has a local V-shape corrosion defect with a width of and a depth of d, the magnetization component leads to a steady-state magnetic charge distribution on the circumferential ridge lines at both ends of the corrosion defect with a density of [44]. According to Coulomb’s law of magnetism, the components and of the SMFL field strength vector H generated by the magnetic charges ± ρ at a spatial point P (x, y, z) are as follows [44]:where r is the radius of the rebar, is the vacuum magnetic permeability, and and are functions of , and .

Based on equation (3), set , y = 0, z = 30 mm, = 25 mm, r = 5 mm, and then the numerical solutions (see Figure 2(d) for calculating flow) of and of five local V-shape corrosion defects with the maximum relative corrosion depths d/r of 0.5, 0.4, 0.3, 0.15, and 0 are shown in Figure 1(b). The corrosion defects cause local variations in the and curves, and the variation amplitudes are proved to be positively linearly correlated with d/r [38–41]. Since is positively correlated with d/r and is negatively correlated with when is known. Therefore, the corrosion-induced SMFL field variation has great potential to assess the of corroded rebar.

Different from the R-based prediction of , the SMFL-based non-destructive assessment obtains the real of corroded rebars, which is practical and widely applicable to the reliability assessment of specific in-service RC structures.

3. Experiment and Finite Element Modeling Details

The experiment and finite element modeling processes are shown in Figure 2, which is described in detail in our previous work [51]. As shown in Figure 2(a), the twenty-two naturally corroded rebar specimens S-NC-1 − S-NC-22 were taken out from the longitudinal girder of a wharf bridge that naturally corroded in the marine environment for twenty-six years, and six artificially corroded specimens S-AC-1 - S-AC-6 were obtained by corroding uncorroded rebars taken out of this girder using the impressed current method. The impressed current corrosion of specimens S-AC-1 - S-AC-6 aimed to create random corrosion morphology with different corrosion degrees similar to naturally corroded rebars. Therefore, we did not set a target corrosion degree and these six rebars were corroded with a constant corrosion current density but random corrosion duration.

As shown in Figure 2(b), the 3D structural light scanning with an accuracy of up to 0.02 mm was employed to capture precise corrosion morphologies of all the rebar specimens. On this basis, the accurate actual corrosion degrees for every cross-section of each rebar specimen can be obtained.

The TSC-7M-16 device was then used to collect the SMFL field variation data of the rebar specimens, as shown in Figure 2(c). Five longitudinal SMFL scanning paths directly above the corroded rebar specimen were conducted at different lift-off heights (LFH) of 5, 10, 30, 50, and 100 mm. Along these paths, four micromagnetic sensors (1#–4#) and a displacement pedometer mounted on the small trolley were utilized to measure the SMFL strength and corresponding measurement location with an accuracy of 0.1 A/m and 1 mm, respectively. The measured data was recorded by the host and transmitted to a computer for further analysis. For brevity, only SMFL strength data measured by micromagnetic sensor 2# was employed in this study.

Since the complex corrosion morphologies of corroded rebar specimens, the analytical solution of its SMFL field according to equation (3) is not feasible. Therefore, the finite element simulation was carried out using the COMSOL^R software (COMSOL Inc, Stockholm, Sweden), whose simulation flow is shown in Figure 2(d). In the simulation, the real entity model of the rebar specimen obtained by structured light scanning was inputted. A cuboid air domain was created with the corroded rebar serving as the center, and the boundary condition of the air domain was then set to the external magnetic flux density. The relative magnetic permeability and of the rebar and air domain were set to 200 and 1.0 [51], respectively, where is closely related to the magnetization induced by geomagnetic field :where is the magnetization vector induced by .

In Figure 2(d), the components of were measured values. The rebar’s magnetization vector , where was the residual longitudinal linear magnetization vector of the rebar generated by manufacturing and tension. According to the ideal longitudinal magnetization hypothesis, the components of was set as , , and , where a and b were adjusted to the suitable values according to the measured SMFL strength. Consequently, the finite element model was ultra-finely meshed using free tetrahedron and then solved. When using the default ultra-refinement meshing setting, the solution converged rapidly, typically within 50 iterations or fewer. Thereby, the finite element simulation of the in-service condition of the SMFL field of bare corroded rebar in Figure 2(c) was realized.

4. Results and Discussion

4.1. Cross-Sectional Area and SMFL Field Variation of Corroded Rebars

For brevity, the naturally and artificially corroded specimens S-NC-8 and S-AC-1 are taken as examples to demonstrate features of the corrosion-induced cross-sectional area S and SMFL field components and , as shown in Figures 3(a) and 3(b) [51]. The S curves are actual ones measured from the rebars’ entity models, whose fluctuations and spikes are induced by the corrosion non-uniformity and cathodic protection of stirrups [52], respectively. The features of both the experimental and simulated and induced by natural and artificial corrosion are almost the same, whose fluctuations represent the SMFL field variation caused by the non-uniform corrosion morphology of the corroded rebar. This result shows that the theoretical analysis in Figure 1(b) is established in actual situations. However, the and curves only reveal limited information on non-uniform corrosion and cannot be used to characterize .

(a)

(b)

(c)

(d)

In this study, both the measured and simulated and curves were stepped. However, to facilitate further analysis, the and curves must be smoothed. To this end, we utilized the Savitzky-Golay (SG) smoothing [53] and Adjacent-Average (AAv) smoothing [54]. The SG smoothing is also called polynomial smoothing, with two control parameters of n and m. The parameter n determines the degree of polynomial fitting and the parameter m determines the amount of data involved in the SG smoothing. Given that secondary fitting sufficed, the parameter n was set as 2. It is important to note that increasing m would result in a larger difference between the SG smoothed value and the original value, leading to the loss of SMFL field variation information [51]. Consequently, we chose a relatively small value of 5 for the parameter m.

For theoretical description, the finite element modeling of rebar has five local V-shape corrosion defects with maximum relative corrosion depth d/r of 0.2, 0.1, 0.07, 0.05, and 0.03 as shown in Figure 3(c) was carried out according to Figure 2(d). Based on the simulated data of , the real and simulated SMFL field variations and are respectively defined as follows:where and are the SG smoothing results (n = 2 and m = 5) of the simulated of the V-shape corroded rebar in Figure 3(c) and its corresponding smooth uncorroded rebar, respectively, and is the AAv smoothing results of the simulated of the V-shape corroded rebar.

When there is only the SMFL detection data of corroded rebars, only can be obtained while cannot. The and curves are shown in Figure 3(c), where the numbers on the curves are the value of AAv smoothing parameter m. It can be seen that the curve with m = 50 agrees with the curve better, especially when corrosion degree d/r ≤ 0.1. The curve with smaller d/r deviates significantly from the curve when m > 50. In addition, the curve with a larger m has a larger distortion part that means more data loss. Therefore, the curve with m = 50 is the best choice to characterize the real SMFL field variation when there is only the SMFL detection data of corroded rebars. Accordingly, based on the principle of eliminating the difference in residual magnetization of rebars [44], the quantitative SMFL variation degree N- is defined as follows:where H[i] is the total SMFL field strength and L (units: mm) is the rebar’s length.

The curve of specimen S-NC-8 is shown in Figure 3(d), which quantitatively and intuitively reflects the SMFL field variation induced by the changing cross-sectional area S compared with the original results in Figures 3(a) and 3(b). According to Figure 3(d), three indices SMFL field variation ratio dH, the corrosion non-uniformity degree [51], and the cross-sectional area ratio K are defined as follows:where subscripts and are the abscissas of any two adjacent extreme values on the curve, and are the maximum, average, and minimum cross-sectional areas within the interval .

Using the and S curves of each specimen, data of dH and the corresponding and K can be obtained, which satisfies the

Therefore, the SMFL-based idea of obtaining when is known in Figure 1(b) can be realized based on indices K and that are characterized according to the SMFL field variation ratio dH.

4.2. SMFL-Based PDDs of and K

4.2.1. Results and PDD Estimation Method of and K

According to the definitions in equation (7) and the N-ΔH_SA curves of all the specimens, dH and its corresponding dS_n [51] and K are obtained as shown in Figure 4. As shown in Figure 4(a), when LFH ≤30 mm, the experimental is well linearly correlated with dH with a fitted slope of about 1.0. When LFH >30 mm, this linear correlation is no longer good due to the decay and mutual interference of the SMFL field originating from different adjacent corrosion defects. The simulated and dH have linear relationships similar to the experiment results, which adequately reveals that the corrosion-induced cross-sectional area loss causes an equal-proportional magnetic flux leakage of the rebar. Both experimental and simulated data points of show large discreteness with an increasing trend as the increase of LFH. As shown in Figure 4(b), both the experimental and simulated K are distributed in the interval [0.25, 4.0], which does not apparently change with the increasing LFH. To describe the distribution characteristics of and K more accurately, statistical analysis is further implemented. The reliability of the measured SMFL data was verified through the consistency observed between simulation results and experimental results. Building upon this foundation, further analysis was performed using the experimental data depicted in Figure 4.

(a)

(b)

The statistical analysis of and K are based on the nonparametric kernel density estimation (KDE) method [55, 56], which fits the PDD exactly based on the samples instead of certain precedent probability hypotheses. Suppose a set of samples obeys an unknown continuous distribution p(x), then the kernel density estimation at any point x is defined as follows:where is the kernel function, and h is the bandwidth.

satisfies the definition of the probability density function and has many forms, where the Gaussian kernel function (standard normal distribution function) is employed:

The bandwidth h affects the relative weight of each sample by changing the independent variable value of :

Substituting equations (11) into (9) to get

The bandwidth h can be optimized to minimize the estimation error (generally the mean integral square error (MISE)) of the KDE method. The MISE is defined as follows:

With the weak assumptions:

There is an optimal bandwidth that minimizes MISE(h):

In this study, this optimal bandwidth is used.

4.2.2. SMFL-Based PDD of Index

Based on the KDE method and the samples in Figure 4(a), the true PDD of can be obtained. According to Figure 4(a), the distribution of samples is affected by the increasing LFH. Considering that the concrete cover thickness of the common RC structure is generally no more than 30 mm, and thus the sample (2923 data points) with LFH = 30 mm is estimated first. As shown in Figure 5, when LFH = 30 mm, 97.4% of data points are in the interval dH = [0, 0.03], while only 2.6% are in the interval dH = [0.03, 0.15]. To ensure the data volume for statistical analysis, the dH interval [0, 0.03] is equally divided into ten intervals while the interval [0.03, 0.15] is not divided, and the mean value of dH in each interval is represented by .

The KDE results of for the eleven dH intervals are shown in Figures 6(a)–6(k), which exactly reflects the true PDDs of represented by the histograms. The two-dimensional PDD of shown in Figure 6(m) is obtained by merging these KDE results. To explore the distribution nature of , the PDD type of is determined based on these KDE results. The Kolmogorov-Smirnov (K-S) test [57] is employed to test the PDD type of of each dH interval. The test results in Table 1 indicate that the hypothesis that the of each dH interval obeys the Weibull distribution is accepted (, H = 0) at the 95% confidence level. The Weibull distribution is defined as follows:where λ and k are the scale and shape parameters of the Weibull distribution.

The KDE results of are fitted using equation (16), as shown in Figures 6(a)–6(k). The fitting coefficient ranges from 0.9315 to 0.9905, indicating that the PDDs of can be accurately expressed by the Weibull distribution. As shown in Figure 6(l), as dH increases, both the average value AV and standard deviation STD of increase approximately linearly, which shows that the greater the SMFL field variation ratio, the greater the non-uniform corrosion degree. In addition, the distribution parameters λ and k of are approximately linearly and inverse proportionally increased with the increase of . Thus, they are fitted using linear and inverse proportional functions, respectively, as shown in Figures 6(n) and 6(o). Substituting the fitting results of λ and k into (16), the fitted PDD of shown in Figure 6(p) is obtained, which is similar to the KDE result shown in Figure 6(m) and shows the good adaptation of Weibull distribution in the determination of the PDD of .

Likewise, the PDDs of of other LFHs are obtained, as shown in Figure 7. The K-S test indicates that the PDDs of of different LFHs still obey the Weibull distribution. As shown in Figures 7(a) and 7(b), as increases, the parameter k shows an inversely proportional increase while the parameter λ evolves from a linear increase to an inversely proportional increase. The fitting result of the parameter λ has high accuracy, while that of k is not very good, which leads to some differences between the fitting results shown in Figures 7(c)–7(j) and the KDE results. However, this difference becomes smaller with the decrease of and can be ignored in the small region that contains most of the data points.

4.2.3. SMFL-Based PDD of Index K

The PDD of K is also assessed using the KDE method. As shown in Figure 4(b), the sample points of K are distributed in the interval [0.25, 4.0], and the adjusted cross-sectional area ratio K_0.25 = K−0.25 is more suitable for the PDD estimation according to trial calculation. The K-S test results show that the hypothesis that the PDD of K_0.25 obeys the Gamma distribution is accepted at the 95% confidence level. The Gamma distribution is defined as follows:where a and b are the shape and inverse scale parameters of the Gamma distribution.

As shown in Figures 8(a) and 8(b), the change of LFH has little effect on the shape and magnitude of the PDD of K_0.25. However, different from LFH, the KDE results in Figures 8(c) and 8(d) reveal the PDD of K_0.25 in the interval has a local mutation that contains 90.1% of the data volume, and thus the influence of dH need to be considered. As shown in Figures 8(g) and 8(h), the parameters a and b of K_0.25 are approximately inversely proportional to and can be well fitted by inverse proportional functions. Substituting the fitted results of a and b into equation (17), the fitted results of K_0.25 in Figures 8(e) and 8(f) are obtained and similar to the KDE results in Figures 8(c) and 8(d), which shows that it is appropriate to employ the Gamma distribution to determine the PDD of K_0.25.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

4.3. The Application of SMFL-Based Assessment of in Structural Reliability

4.3.1. The SMFL-Based PDD of

Generally, RC structures have a 30 mm thick concrete cover, which leads to an LFH of no less than 30 mm. Taking LFH = 30 mm, then N values of and K_0.25 are randomly sampled according to equations (16) and (17), Figures 7 and 8, and the value of . Accordingly, N values of are calculated according to the N values of and K_0.25 and equation (8), based on which the PDD of is obtained, as shown in Figure 9. As increases, the PDD curves of become flatter and the probability density peak and its corresponding value decrease gradually, which indicates that a greater SMFL field variation degree (increasing ) means more non-uniform corrosion morphology and a smaller . When is known, the PDD of can be easily acquired, that is, the SMFL-based PDD assessment of is implemented.

Since is obtained in the SMFL detection, the SMFL-based PDD of is real, which has the same function as factor R but is free from adverse effects from rebar element length, corrosion current density, environment, and the prediction’s cumulative error.

4.3.2. Feasibility of SMFL-Based Reliability Assessment of Non-Uniformly Corroded RC Beam

Since the SMFL-based PDD of is obtained, the reliability assessment of RC beams with non-uniformity corrosion can be realized. However, there is a corrosion product layer and a concrete cover between the embedded rebars and the micro-magnetic sensors when detecting the SMFL field. Fortunately, the relative magnetic permeabilities of corrosion products and concrete are almost the same as that of air [41, 42, 51, 58]. Therefore, the PDD of shown in Figure 9 is directly used in the reliability assessment of RC beams.

The MCS method detailed in Figure 1 is used for calculating β. As shown in Figure 10(a), a simply supported beam with a size of 5000 mm × 200 mm × 500 mm is reinforced by four hot-rolled rebars with a diameter of 14 mm, and the concrete cover is 30 mm thick. As shown in Figure 10(b), this corroded RC beam is assumed to bear uniformly distributed dead load G and sustained live load Q, and the rebars are non-uniformly corroded with an average cross-sectional corrosion degree ( is the cross-sectional area of the uncorroded rebar). The random sample of load, geometry, and material strength is according to Table 2. The clear span of the beam is divided into forty-eight 100 mm long beam segments, and the calculation of β is based on the bending resistance limit state with the assumption that the ends bent rebars are well-anchored.

(a)

(b)

(c)

(d)

(e)

(f)

Then, β of corroded RC beams with different and values are calculated, as shown in Figure 10(c). For a corroded RC beam with a certain , its β decreases with the increase of , which demonstrates the potential harm of non-uniform corrosion to the safety of RC beams. In addition, the fitting results of the distribution parameters λ, k, a, and b of and K_0.25 deviate from the actual values, whose effects on β need to be clarified. Therefore, the envelops and actual value curves of β shown in Figure 10(c) are obtained based on the fitting’s 95% confidence levels and the actual values of parameters λ, k, a, and b listed in Table 3, respectively.

In Figure 10(c), the actual values of β with different all fall within these β envelopes. To quantify the exact effect of the fitting errors of λ, k, a, and b on β, the reliability index error is defined as follows:where and are the upper and lower limits of the β envelop, respectively, and is the corresponding actual value of β.

with different and are given in Figure 10(d). When of different do not exceed 3.2%, and when has an increasing trend and reaches a maximum value of 11%. Generally, the of corroded RC beams is not very large, and thus these results indicate that the fitting errors of the parameters λ, k, a, and b will not lead to unacceptable errors in β.

In addition, the factor R-based [17, 18] and SMFL-based β curves of this corroded RC beam are compared in Figure 10(e). With the increase of , the value corresponding to the intersection point of the R-based and the SMFL-based β curves gradually increases, which is consistent with the viewpoint of the factor R that the corrosion non-uniformity degree increases with the increase of η_av [17]. However, for a specific in-service RC structure, the relationship between corrosion non-uniformity degree and is uncertain, which is a deficiency of R-based reliability assessment. For example, when , the R-based β corresponds to , which in practice hardly occurs (see Figures 6–8). Of course, due to the limited original sample volume, the β with obtained based on the fitting results may not be exact. Overall, the R-based and SMFL-based β assessment methods have good consistency especially when is small.

To reveal the effect of non-uniform corrosion on the reliability of RC beams intuitively, the failure probability growth ratio of a certain is defined as follows:where and are the failure probabilities of and , respectively.

As shown in Figure 10(f), the of different increase exponentially with the increase of corrosion non-uniformity degree (increasing ). When monotonically increases with the increase of , and the maximum of different range from 1.8 to 7.6 when . These results show the exact adverse effect of non-uniform corrosion on the reliability of the corroded RC beam and are also generally consistent with the existing research results of the R-based or other methods [17, 18], indicating the feasibility of the SMFL-based reliability assessment of non-uniformly corroded RC beams.

4.3.3. Case Study of the Automatic Assessment of the Real Reliability of Corroded RC Beams Based on Detected SMFL Data

The true reliability of a specific corroded RC beam can be automatically assessed using its SMFL detection data and the MCS method. Taking the beam with a clear span of 5000 mm, a width of 300 mm, and a height of 1000 mm shown in Figure 2(a) as the case, it is assumed to bear dead load G = 150 kN/m and live load Q = 50 kN/m, and its clear span is divided into fifty 100 mm long beam segments, as shown in Figure 11(a). In Figure 11(b), S is the true cross-sectional area of the rebar obtained by the structural light scanning in Figure 2(b), and are the average and minimum cross-sectional areas of each beam segment calculated based on S. Since both ends of the rebars are well anchored, β of this beam is calculated based on the bending resistance limit state. The coefficients of variation and material strength distributions in Table 2 are used in the MCS calculation of β of this beam.

On the basis of the known , β of this beam can be calculated by SMFL-based or R-based method. As shown in Figure 11(c), the real dH of this beam is automatically calculated from the detected SMFL data, based on which the maximum value of each beam segment is accordingly obtained. Then, the PDDs of S_min of all beam segments are assessed by and refer to Figure 8. Accordingly, the SMFL-based β of 4.0 and failure probability p_f of 3.08 × 10⁻⁵ are acquired, as shown in Figure 11(d). According to the PDD of factor R [17], the PDDs of of all beam segments are obtained, based on which the R-based β of 3.21 and p_f of 6.52 × 10⁻⁴ are acquired, as shown in Figure 11(d).

Based on the real value in Figure 11(b), the true β of 4.05 and p_f of 2.5 × 10⁻⁵ of this beam are acquired, as shown in Figure 11(d). In contrast, the SMFL-based β and p_f are close to the truth, with a reliability error of only 1.2% and a failure probability error of 23.2%. However, there is a big difference between the R-based β and p_f and the truth, with a reliability error of up to 20.7% and a failure probability error of an astonishing 64200%. To avoid this error, one effective approach is to establish a PDD model of factor R-based on data from those naturally corroded girders shown in Figure 2(a). However, the process of establishing such PDD model of factor is arduous and labor-intensive. This is precisely why we have developed this non-destructive SMFL-based method as an alternative to the R-based method for assessing the corrosion non-uniformity and reliability of corroded RC structures.

This case study demonstrates the high accuracy of SMFL-based reliability assessment of corroded RC structures, in which the reliability misestimation due to non-uniform corrosion is greatly reduced. In addition, the SMFL-based reliability assessment of a specific RC structure can be automatically realized using the MCS method once its SMFL data is detected. Therefore, the SMFL-based reliability assessment method has great potential in practical applications.

5. Conclusions

In this study, the SMFL-based assessment of the minimum cross-sectional area of corroded rebar is realized, based on which the reliability assessment case study of a non-uniformly corroded RC beam is carried out. The following conclusions can be drawn:(1)The value of the SMFL field variation ratio dH is an effective index to characterize the corrosion non-uniformity degree of rebar quantitatively.(2)At the 95% confidence level, the corrosion non-uniformity degree and the cross-sectional area ratio K_0.25 obey the Weibull distribution and Gamma distribution with distribution parameters linearly or inversely proportionally correlated to dH, respectively.(3)For non-uniformly corroded RC beam with a certain average corrosion degree and SMFL field variation ratio dH, the rebar’s minimum cross-sectional area can be determined based on and the PDDs of and K_0.25 that depend on dH. The reliability assessment example of a non-uniformly corroded RC beam indicates that the increase of dH means a substantial increase (up to 180%–760%) in the failure probability.(4)A novel SMFL-based method with better adaptability to the automatic reliability assessment of specific in-service corroded RC beams is developed. A real case shows that this SMFL-based method can reduce the reliability assessment error from 20.7% of the existing R-based method to a very low level of 1.2%.

Some aspects, including the weakened strength of corroded rebar and the effects of stress and stirrups on the SMFL field, have not been considered in this study temporarily. In addition, the effects of concrete cover and corrosion products on this proposed SMFL-based assessment method of corrosion non-uniformity of rebars and reliability of corroded RC structures also need to be deeply explored and solved. Related issues will be discussed in detail in our future work.

Nomenclature

a:	Parameter of gamma distribution
AAv:	Adjacent-average
b:	Parameter of gamma distribution
d:	Depth of V- shape corrosion
dH:	SMFL field variation ratio
:	Average value of dH
:	Maximum value of dH
:	Corrosion non-uniformity degree
:	Reliability index error
G:	Dead load
:	Failure probability growth ratio
h:	Bandwidth of kernel density estimation
:	Optimal h
:	Geomagnetic field
:	x component of
:	y component of
:	z component of
H:	SMFL field strength
:	x component of H
:	y component of H
k:	Shape parameter of Weibull distribution
K:	Cross-sectional area ratio
K_0.25:	Adjusted cross-sectional area ratio
K(·):	Kernel function
KDE:	Kernel density estimation
LFH:	Lift-off height of SMFL scanning
m:	Parameter of SG or AAv smoothing
:	Applied bending moment
MCS:	Monte Carlo simulation
MISE:	Mean integral square error
:	Ultimate flexural bearing capacity
n:	Parameter of SG smoothing
N:	Times of Monte Carlo simulation
:	Quantitative SMFL variation degree
:	Failure probability
PDD:	Probability density distribution
Q:	Live load
r:	Radius of the rebar
R:	Cross-sectional area spatial heterogeneity factor
RC:	Reinforced concrete
S:	Cross-sectional area of corroded rebar
:	Average cross-sectional area
:	Rebar’s maximum cross-sectional area
:	Rebar’s minimum cross-sectional area
:	SMFL variation amplitude
:	Cross-sectional area of uncorroded rebar
SG:	Savitzky-golay
SMFL:	Self-magnetic flux leakage
V:	Magnetization vector
:	x component of V
:	y component of V
:	z component of V
:	Geomagnetic field-induced magnetization vector
:	Residual magnetization vector
x:	x-axis coordinate
y:	y-axis coordinate
z:	z-axis coordinate
:	Half-width of V- shape corrosion
β:	Reliability index
ρ:	Symbol of magnetic charge
λ:	Scale parameter of Weibull distribution
:	Average cross-sectional corrosion degree of rebar
:	Vacuum magnetic permeability
:	Relative magnetic permeability of air
:	Relative magnetic permeability of rebar
:	Real SMFL field variation
:	Simulated SMFL field variation.

Data Availability

The data can be obtained from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors sincerely thank Ruilin Wang, the doctoral candidate of the Department of structural engineering of Tongji University, for his help in writing the calculation codes. This research was funded by the National Natural Science Foundation of China (51878486) and the Program of Shanghai Science and Technology Committee (22dz1203603).

References

C. Jiang, H. Ding, X. L. Gu, and W. P. Zhang, “Failure mode-based calculation method for bending bearing capacities of normal cross-sections of corroded reinforced concrete beams,” Engineering Structures, vol. 258, Article ID 114113, 2022.
View at: Publisher Site | Google Scholar
C. Jiang, H. Ding, X. L. Gu, and W. P. Zhang, “Calibration analysis of calculation formulas for shear capacities of corroded RC beams,” Engineering Structures, vol. 286, Article ID 116090, 2023.
View at: Publisher Site | Google Scholar
U. M. Angst, “Predicting the time to corrosion initiation in reinforced concrete structures exposed to chlorides,” Cement and Concrete Research, vol. 115, pp. 559–567, 2019.
View at: Publisher Site | Google Scholar
Y. Zhou, B. Gencturk, K. Willam, and A. Attar, “Carbonation-induced and chloride-induced corrosion in reinforced concrete structures,” Journal of Materials in Civil Engineering, vol. 27, no. 9, Article ID 04014245, 2015.
View at: Publisher Site | Google Scholar
K. X. Liao, Y. P. Zhang, W. P. Zhang, Y. Wang, and R. L. Zhang, “Modeling constitutive relationship of sulfate-attacked concrete,” Construction and Building Materials, vol. 260, Article ID 119902, 2020.
View at: Publisher Site | Google Scholar
K. J. Zhang, J. Z. Xiao, and Q. T. Zhang, “Time-dependent reliability analysis of recycled aggregate concrete cover cracking induced by reinforcement corrosion,” Journal of Building Engineering, vol. 39, Article ID 102320, 2021.
View at: Publisher Site | Google Scholar
J. Bojórquez, S. Ponce, S. E. Ruiz et al., “Structural reliability of reinforced concrete buildings under earthquakes and corrosion effects,” Engineering Structures, vol. 237, Article ID 112161, 2021.
View at: Publisher Site | Google Scholar
D. Veneziano, C. A. Cornell, and M. Grigoriu, “Vector-process models for system reliability,” Journal of the Engineering Mechanics Division, vol. 103, no. 3, pp. 441–460, 1977.
View at: Publisher Site | Google Scholar
D. M. Frangopol, K. Y. Lin, and A. C. Estes, “Reliability of reinforced concrete girders under corrosion attack,” Journal of Structural Engineering, vol. 123, no. 3, pp. 286–297, 1997.
View at: Publisher Site | Google Scholar
A. Der Kiureghian, H. Z. Lin, and S. J. Hwang, “Second-order reliability approximations,” Journal of Engineering Mechanics, vol. 113, no. 8, pp. 1208–1225, 1987.
View at: Publisher Site | Google Scholar
M. Akiyama, D. M. Frangopol, and H. Ishibashi, “Toward life-cycle reliability-risk-and resilience-based design and assessment of bridges and bridge networks under independent and interacting hazards: emphasis on earthquake, tsunami and corrosion,” Structure and Infrastructure Engineering, vol. 16, no. 1, pp. 26–50, 2020.
View at: Publisher Site | Google Scholar
J. Li and J. B. Chen, Stochastic Dynamics of Structures, John Wiley & Sons, New York, NY, USA, 2009.
H. Y. Guo, Y. Dong, and P. Gardoni, “Efficient subset simulation for rare-event integrating point-evolution kernel density and adaptive polynomial chaos kriging,” Mechanical Systems and Signal Processing, vol. 169, Article ID 108762, 2022.
View at: Publisher Site | Google Scholar
M. G. Stewart and A. Al-Harthy, “Pitting corrosion and structural reliability of corroding RC structures: experimental data and probabilistic analysis,” Reliability Engineering and System Safety, vol. 93, no. 3, pp. 373–382, 2008.
View at: Publisher Site | Google Scholar
M. Y. Zhang, H. J. Song, S. Lim, M. Akiyama, and D. M. Frangopol, “Reliability estimation of corroded RC structures based on spatial variability using experimental evidence, probabilistic analysis and finite element method,” Engineering Structures, vol. 192, pp. 30–52, 2019.
View at: Publisher Site | Google Scholar
H. Hamidane, A. Chateauneuf, A. Messabhia, and A. Ababneh, “Reliability analysis of corrosion initiation in reinforced concrete structures subjected to chlorides in presence of epistemic uncertainties,” Structural Safety, vol. 86, Article ID 101976, 2020.
View at: Publisher Site | Google Scholar
W. P. Zhang, B. B. Zhou, X. L. Gu, and H. Dai, “Probability distribution model for cross-sectional area of corroded reinforcing steel bars,” Journal of Materials in Civil Engineering, vol. 26, no. 5, pp. 822–832, 2014.
View at: Publisher Site | Google Scholar
X. L. Gu, H. Y. Guo, B. B. Zhou, W. P. Zhang, and C. Jiang, “Corrosion non-uniformity of steel bars and reliability of corroded RC beams,” Engineering Structures, vol. 167, pp. 188–202, 2018.
View at: Publisher Site | Google Scholar
X. G. Liu, W. P. Zhang, X. L. Gu, and Z. W. Ye, “Probability distribution model of stress impact factor for corrosion pits of high-strength prestressing wires,” Engineering Structures, vol. 230, Article ID 111686, 2021.
View at: Publisher Site | Google Scholar
M. Otieno, H. Beushausen, and M. Alexander, “Prediction of corrosion rate in reinforced concrete structures–a critical review and preliminary results,” Materials and Corrosion, vol. 63, no. 9, pp. 777–790, 2012.
View at: Publisher Site | Google Scholar
Y. C. Ou, Y. T. T. Susanto, and H. Roh, “Tensile behavior of naturally and artificially corroded steel bars,” Construction and Building Materials, vol. 103, pp. 93–104, 2016.
View at: Publisher Site | Google Scholar
C. Chalhoub, R. François, and M. Carcasses, “Critical chloride threshold values as a function of cement type and steel surface condition,” Cement and Concrete Research, vol. 134, Article ID 106086, 2020.
View at: Publisher Site | Google Scholar
C. Song, C. Jiang, X. L. Gu, Q. Zhang, and W. P. Zhang, “Calibration analysis of chloride binding capacity for cement-based materials under various exposure conditions,” Construction and Building Materials, vol. 314, Article ID 125588, 2022.
View at: Publisher Site | Google Scholar
C. Jiang, C. Song, X. L. Gu, Q. Zhang, and W. P. Zhang, “Modeling electrochemical chloride extraction process in cement-based materials considering coupled multi-ion transports and thermodynamic equilibriums,” Journal of Cleaner Production, vol. 401, Article ID 136778, 2023.
View at: Publisher Site | Google Scholar
C. B. Cheah, L. E. Tan, and M. Ramli, “The engineering properties and microstructure of sodium carbonate activated fly ash/slag blended mortars with silica fume,” Composites Part B: Engineering, vol. 160, pp. 558–572, 2019.
View at: Publisher Site | Google Scholar
Q. W. Qiu, “A state-of-the-art review on the carbonation process in cementitious materials: fundamentals and characterization techniques,” Construction and Building Materials, vol. 247, Article ID 118503, 2020.
View at: Publisher Site | Google Scholar
C. Jiang, X. L. Gu, Q. H. Huang, and W. P. Zhang, “Carbonation depth predictions in concrete bridges under changing climate conditions and increasing traffic loads,” Cement and Concrete Composites, vol. 93, pp. 140–154, 2018.
View at: Publisher Site | Google Scholar
C. Jiang, Q. H. Huang, X. L. Gu, and W. P. Zhang, “Modeling the effects of fatigue damage on concrete carbonation,” Construction and Building Materials, vol. 191, pp. 942–962, 2018.
View at: Publisher Site | Google Scholar
V. Carević and I. Ignjatović, “Influence of loading cracks on the carbonation resistance of RC elements,” Construction and Building Materials, vol. 227, Article ID 116583, 2019.
View at: Publisher Site | Google Scholar
C. Jiang, Q. H. Huang, X. L. Gu, and W. P. Zhang, “Experimental investigation on carbonation in fatigue-damaged concrete,” Cement and Concrete Research, vol. 99, pp. 38–52, 2017.
View at: Publisher Site | Google Scholar
H. G. Min and W. P. Zhang, “Effects of coupled heat and moisture and load damage on chloride transport in concrete,” Magazine of Concrete Research, vol. 74, no. 13, pp. 659–671, 2022.
View at: Publisher Site | Google Scholar
J. Y. Chen, W. P. Zhang, Z. J. Tang, and Q. H. Huang, “Experimental and numerical investigation of chloride-induced reinforcement corrosion and mortar cover cracking,” Cement and Concrete Composites, vol. 111, Article ID 103620, 2020.
View at: Publisher Site | Google Scholar
J. Alam, L. A. C. Neves, H. Zhang, and D. Dias-da-Costa, “Assessment of remaining service life of deteriorated concrete bridges under imprecise probabilistic information,” Mechanical Systems and Signal Processing, vol. 167, Article ID 108565, 2022.
View at: Publisher Site | Google Scholar
A. Kamariotis, E. Chatzi, and D. Straub, “Value of information from vibration-based structural health monitoring extracted via Bayesian model updating,” Mechanical Systems and Signal Processing, vol. 166, Article ID 108465, 2022.
View at: Publisher Site | Google Scholar
H. Zhang, H. X. Li, J. T. Zhou, K. Tong, and R. C. Xia, “A multi-dimensional evaluation of wire breakage in bridge cable based on self-magnetic flux leakage signals,” Journal of Magnetism and Magnetic Materials, vol. 566, Article ID 170321, 2023.
View at: Publisher Site | Google Scholar
K. Tong, J. T. Zhou, R. Q. Zhao, W. X. Hu, Y. H. Qu, and C. S. Cheng, “Experimental study on rebar stress measurement based on force-magnetic coupling under excited magnetic field,” Measurement, vol. 189, Article ID 110620, 2022.
View at: Publisher Site | Google Scholar
H. Zhang, X. T. Ma, H. J. Jiang, K. Tong, Y. Zheng, and J. T. Zhou, “Grading evaluation of overall corrosion degree of corroded RC beams via SMFL technique,” Structural Control and Health Monitoring, vol. 2023, Article ID 6672823, 23 pages, 2023.
View at: Publisher Site | Google Scholar
D. Yang, J. Qiu, H. Di, S. Zhao, J. Zhou, and F. Yang, “Quantitative evaluation of corrosion degrees of steel bars based on self-magnetic flux leakage,” Metals, vol. 9, p. 952, 2019.
View at: Publisher Site | Google Scholar
H. Zhang, J. B. Qi, Y. Zheng, J. T. Zhou, and J. L. Qiu, “Characterization and grading assessment of rebar corrosion in loaded RC beams via SMFL technology,” Construction and Building Materials, vol. 411, Article ID 134484, 2024.
View at: Publisher Site | Google Scholar
J. H. Huang, Z. Dong, C. Q. Fu et al., “Evaluation of nonuniform corrosion of steel in concrete based on two-yoke magnetic sensor,” Journal of Materials in Civil Engineering, vol. 34, no. 9, Article ID 04022204, 2022.
View at: Publisher Site | Google Scholar
C. Q. Fu, J. H. Huang, Z. Dong, W. J. Yan, and X. L. Gu, “Experimental and numerical study of an electromagnetic sensor for non-destructive evaluation of steel corrosion in concrete,” Sensors and Actuators A: Physical, vol. 315, Article ID 112371, 2020.
View at: Publisher Site | Google Scholar
J. R. Zhang, C. Liu, M. Sun, and Z. J. Li, “An innovative corrosion evaluation technique for reinforced concrete structures using magnetic sensors,” Construction and Building Materials, vol. 135, pp. 68–75, 2017.
View at: Publisher Site | Google Scholar
J. L. Qiu, J. T. Zhou, S. Y. Zhao, H. Zhang, and L. Liao, “Statistical quantitative evaluation of bending strength of corroded RC beams via SMFL technique,” Engineering Structures, vol. 209, Article ID 110168, 2020.
View at: Publisher Site | Google Scholar
J. L. Qiu, H. Zhang, J. T. Zhou, and W. P. Zhang, “An SMFL-based non-destructive quantification method for the localized corrosion cross-sectional area of rebar,” Corrosion Science, vol. 192, Article ID 109793, 2021.
View at: Publisher Site | Google Scholar
R. C. Xia, H. Zhang, J. T. Zhou, L. Liao, Z. Y. Zhang, and F. X. Yang, “Probability evaluation method of cable corrosion degree based on self-magnetic flux leakage,” Journal of Magnetism and Magnetic Materials, vol. 522, Article ID 167544, 2021.
View at: Publisher Site | Google Scholar
S. Srivaranun, M. Akiyama, P. Bocchini et al., “Effect of the interaction of corrosion pits among multiple tensile rebars on the reliability of RC structures: experimental and numerical investigation,” Structural Safety, vol. 93, Article ID 102115, 2021.
View at: Publisher Site | Google Scholar
Y. Wang, C. Liu, Y. Wang, Q. Li, and B. Yan, “Semi-empirical prediction model of chloride-induced corrosion rate in uncracked reinforced concrete exposed to a marine environment,” Electrochimica Acta, vol. 331, Article ID 135376, 2020.
View at: Publisher Site | Google Scholar
J. E. Ramón, J. M. Gandía-Romero, R. Bataller, M. Alcañiz, M. Valcuende, and J. Soto, “Potential step voltammetry: an approach to corrosion rate measurement of reinforcements in concrete,” Cement and Concrete Composites, vol. 110, Article ID 103590, 2020.
View at: Publisher Site | Google Scholar
V. Talakokula, S. Bhalla, R. J. Ball et al., “Diagnosis of carbonation induced corrosion initiation and progression in reinforced concrete structures using piezo-impedance transducers,” Sensors and Actuators A: Physical, vol. 242, pp. 79–91, 2016.
View at: Publisher Site | Google Scholar
J. L. Qiu, H. Zhang, J. T. Zhou, H. Ma, and L. Liao, “Experimental analysis of the correlation between bending strength and SMFL of corroded RC beams,” Construction and Building Materials, vol. 214, pp. 594–605, 2019.
View at: Publisher Site | Google Scholar
J. Qiu, W. Zhang, and Y. Jing, “Quantitative linear correlation between self-magnetic flux leakage field variation and corrosion unevenness of corroded rebars,” Measurement, vol. 218, Article ID 113173, 2023.
View at: Publisher Site | Google Scholar
P. Zhao, G. Xu, Q. Wang, and G. J. Tang, “Influence of sustained load on corrosion characteristics of reinforced concrete beams under galvanostatic accelerated corrosion,” Construction and Building Materials, vol. 215, pp. 30–42, 2019.
View at: Publisher Site | Google Scholar
A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Analytical Chemistry, vol. 36, no. 8, pp. 1627–1639, 1964.
View at: Publisher Site | Google Scholar
J. S. Simonoff, Smoothing Methods in Statistics, Springer Science & Business Media, Berlin, Germany, 2012.
E. Parzen, “On estimation of a probability density function and mode,” The Annals of Mathematical Statistics, vol. 33, no. 3, pp. 1065–1076, 1962.
View at: Publisher Site | Google Scholar
M. Rosenblatt, “Remarks on some nonparametric estimates of a density function,” The Annals of Mathematical Statistics, vol. 27, no. 3, pp. 832–837, 1956.
View at: Publisher Site | Google Scholar
W. J. Conover, Practical Nonparametric Statistics, John Wiley & Sons, New York, NY, USA, 1999.
Z. Li, Z. Jin, T. Zhao et al., “Use of a novel electro-magnetic apparatus to monitor corrosion of reinforced bar in concrete,” Sensors and Actuators A: Physical, vol. 286, pp. 14–27, 2019.
View at: Publisher Site | Google Scholar

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Copyright © 2024 Junli Qiu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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