# Bayesian Fatigue Life Prediction of Corroded Steel Reinforcing Bars

Authors: Wei Wang; Jie Chen; Bo Diao; Xuefei Guan; Jingjing He; Min Huang
Journal: Advances in Civil Engineering
 (2021)

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## Abstract

This paper presents a general method for fatigue life prediction of corroded steel reinforcing bars. A fatigue testing on standard specimens with pitting corrosion is carried out to obtain corrosion fatigue data. The maximum corrosion degree (MCD), characterizing the most severe site of the corrosion pit, is identified to have a log-linear relationship with the fatigue life. A fatigue life model incorporating the MCD and the stress range for corroded steel reinforcing bars is proposed. The model parameters are identified using the testing data, and the model is considered as the baseline model. To utilize the proposed model for life prediction of corroded steel reinforcing bars with different geometries and working conditions, the Bayesian method is employed to update the baseline model. The effectiveness of the overall method is demonstrated using independent datasets of realistic steel reinforcing bars.


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## Body

## 1. Introduction

Fatigue phenomenon in civil engineering is an increasingly prominent concern and cannot be ignored [1–7]. It has been widely reported that the fatigue failure of reinforced concrete (RC) bridge structures is controlled by the failure of steel reinforcing bars [8, 9]; therefore, the fatigue performance of reinforcing bars plays a significant role in the safety of RC structures subjected to cyclic fatigue loading. To make things even worse, in the harsh environment of chloride concentration or relatively severe concrete carbonization, the passive film on reinforcing steel bars is disrupted and the bars are likely to get corroded prematurely over the service life of the RC structures [10–12]. Corrosion can have serious impacts on steel reinforcing bars, including the reduction of the cross-sectional area, passive changes of the mechanical properties (especially for fatigue), and stress concentration near the corrosion pit [11, 13, 14]. Brittle fracture may occur more often and faster for the corroded steel reinforcing bars subjected to fatigue loading compared to the uncorroded bars.Studies of fatigue behavior of corroded reinforcing bars have been carried out over the past few decades. Fernandez et al. [13] performed a series of fatigue tests consisting of 140 corroded reinforcing bars under 3 stress ranges and reported a negative exponential model correlating the fatigue life with the corrosion degree, which is defined as the ratio between the mass loss and the initial mass. Tang et al. [15] conducted an experimental study on 22 deformed reinforcing bars with corrosion degrees (mass loss ratio) of 0%, 5%, 10%, and 20%. In the study, corrosion damages were made using the impressed current technique, and fatigue life data with different corrosion degrees were analyzed. Li et al. [16] studied the performance of tensile fatigue using 15 plane steel bars with naturally carbonation-induced corrosions. Corrosion degrees ranging from 15% to 30% under 3 stress ranges were used. Results showed that fatigue life of steel reinforcement with natural corrosions has a power law dependence with the corrosion degree, and the log-scale slope parameter increases with the increasing of the stress range. Ma et al. [17] created semielliptical and triangular corrosion pits on reinforcing bars by removing materials directly. Fatigue life data of the steel reinforcing bars with different notch shapes and sizes were presented. Zhang et al. [18] obtained the fatigue lives of both artificially and naturally corroded reinforcing bars and found that the artificially corroded reinforcing bars are more susceptible to corrosion damage. An empirical fatigue life prediction model was proposed to incorporate the combined influence of the corrosion degree and the stress range.Although a great deal of efforts has been made in fatigue life prediction of corroded steel reinforcing bars, there still remain challenges for practical engineering applications. In particular, most of the reported fatigue life prediction models are specific to certain circumstances and applications. For instance, model parameters need calibrations when the geometry of the bars or the working condition changes significantly. This is due to the uncertain nature of the corrosion process and the variation of the environment. In addition, the mass loss ratio, which is widely used in those models, characterizes an average influence of the corrosion pit to the fatigue life. It rarely accounts for the most severe location of a corrosion pit. It has been shown that the minimum cross section of a local corrosion pit is the most critical site for fatigue crack initiation, propagation, and final fracture due to stress concentration and cyclic fatigue loads [19–21]. As a result, the average treatment can introduce additional uncertainty into fatigue life prediction. Therefore, a more general fatigue life model taking the local geometry of the corrosion pit into account is highly needed for a reliable life prediction of corroded reinforcing bars.This study presents a general method for fatigue life prediction of corroded reinforcing bars. To overcome the potential limit of the mass loss ratio parameter, a new parameter defined based on the cross-sectional area reduction is used. To resolve the issue that a model tunned with one set of testing data can yield inaccurate results when it is directly applied to applications with different geometry and working conditions, Bayesian method is used to update the initial baseline model with the observed data specific to the application. The rest of the study is structured as follows. First, experimental work on standard reinforcing steel specimens with pitting corrosion is introduced, and fatigue life testing data are obtained. Next, a regression model is proposed to associate the stress range and the maximum corrosion degree (MCD) with the fatigue life. Following that, Bayesian updating is employed to update the initial baseline model for more general applications, and the effectiveness of the proposed method is demonstrated using data of realistic corroded reinforcing bars. Finally, conclusions are drawn based on the current study.


## 2. Fatigue Testing of Corroded Specimens

The overall method development process is illustrated in Figure1. Experimental testing is designed and performed on 14 precorroded standard specimens of steel reinforcements to investigate the corrosion fatigue life under different stress ranges. Two variables, the stress range and the MCD defined as the maximum reduction of the cross-sectional area, are identified as the characterizing factors of the fatigue life. A baseline fatigue life prediction model is proposed by incorporating the two variables. To account for uncertainties from different usage conditions, the Bayesian method is employed to update the baseline model allowing for more general applications. An independent dataset is used for Bayesian updating and method verification.Figure 1
Overall method development process.


### 2.1. Materials and Specimens

A total number of 14 corroded standard specimens made from steel reinforcing bars are prepared, and the fatigue testing data are acquired to build the baseline model. Artificial corrosion pits are developed using the impressed current technique in the NaCl solution. The maximum corrosion degree is measured with the three-dimensional (3D) scan technique. Following this, the axial fatigue testing is carried out under three different stress ranges, and the results of fatigue life are acquired. The degree of corrosions in this study is tentatively controlled to be in the range between 10% and 40% in terms of the maximum cross-sectional area reduction to represent the actual degree of corrosions in practice. Three levels of applied loads are considered to simulate the nominal and overload conditions.The dimension and an actual specimen used in this study are shown in Figure2. The specimen is prepared by removing the surface layer of hot-rolled ribbed steel bars (HRB400) with an initial diameter of 20 mm. The chemical composition of the material is shown in Table 1. The geometry and dimension of the final specimens follow the code requirement for both statistic [22] and fatigue testing [23]. The elastic modulus and yield strength of the four uncorroded specimens are shown in Table 2. The mean values of the mechanical properties are calculated and are used as references for fatigue testing.Figure 2
An actual specimen and its dimension (unit: mm).Table 1
Chemical composition of HRB400 bars.
Chemical compositionCSiMnPSVPercentage0.220.451.340.0280.0310.033Table 2
Mechanical properties of HRB400 bars.
Specimen no.M1M2M3M4Mean valueElastic modulusE (MPa)210330212698201833205854207679Yielding strengthσy (MPa)460.29463.88446.09450.23455.12


### 2.2. Precorrosion and 3D Measurement

The corrosion pit is developed on each of the 14 specimens using an accelerated corrosion process in the solution of 5% NaCl with a current density of 0.6 mA/cm2. Before corrosion, the specimen is wrapped using watertight insulation tapes to protect the surface from the solution. An elliptical shape of the tape is removed at the location for corrosion damage to expose the metal surface to the solution, as shown in Figure 3(a). Both ends of the specimen are sealed by hot-melt adhesive with one end connected with a wire. A direct-current power supply is used for the corrosion process during which the specimen serves as anode and a separate stainless-steel bar as the cathode, as shown in Figure 3(b). After the corrosion process, the specimen is cleaned and stored in a dry condition for fatigue testing.Figure 3
Treatment of specimen and precorrosion setup. (a) Treatment of specimen before corrosion process. (b) Corrosion environment setup.
(a)(b)Due to the nonuniform corrosion process, the width and depth of the corrosion pit vary along the axial direction. Consequently, the cross-sectional area also varies along the axial direction. The concept of maximum corrosion degree (MCD),ηmax, is proposed to characterize the most severe condition of the corrosion pit and is defined as follows:(1)ηmax=1−AminA0,where Amin is the minimum cross-sectional area of the corrosion pit and A0 is the initial uncorroded cross-sectional area.A high-resolution 3D laser scanner is used to acquire the surface data of the specimens [24]. The laser scanner can produce a point cloud of the surface. The raw data are processed using the Geomagic wrap software to reconstruct a 3D surface for digital measurement [25]. The reconstructed 3D surface model is imported into the Pro/Engineer software environment to form a solid volume for cross-sectional measurement [26]. The cross-sectional slices are sampled from the solid volume at an interval of 0.1 mm along the axial direction in the corroded region. The areas of the cross-sectional slices are automatically computed in the Pro/Engineering software. The minimum cross-sectional area is identified as Amin. One of the specimens (FC3) is used for illustration. The actual corrosion pit of the specimen is shown in Figure 4(a), and the reconstructed 3D surface model is shown in Figure 4(b). The model shown in Figure 4(b) is imported into Pro/Engineer and the minimal cross section in terms of MCD is identified, as shown in Figure 5. The whole digital processing and measurement process is performed for each of the specimens, and the results of MCD are listed in Table 3.Figure 4
Specimen FC3 and the reconstructed 3D surface model. (a) The actual specimen and the corrosion pit and (b) the reconstructed 3D surface model of the specimen using laser scanning data.
(a)(b)Figure 5
3D surface profile measurements and pit geometry of the specimen FC3.Table 3
Fatigue testing results of corroded specimens.
SpecimenNominal stress rangeS (MPa)MCDηmax (%)Actual stress rangeσ (MPa)Fatigue lifeN (cycle)FC1204.817.1247.04060752FC2204.821.1259.61415913FC3204.823.9269.1670622FC4204.830.1293.0338315FC5204.835.1315.6274103FC6286.719.6356.699580FC7286.721.3364.3153070FC8286.721.6365.7201968FC9286.722.8371.493330FC10286.729.8408.435814FC11327.713.8380.1247064FC12327.714.6383.7348902FC13327.716.9394.3125903FC14327.723.8430.027611


### 2.3. Fatigue Testing Results and Discussion

The uniaxial fatigue testing is carried out in air condition at room temperature using a sinusoidal load with a stress ratio of 0.1. An electrohydraulic fatigue testing machine (Instron 8801, 100 kN) is used to apply cyclic loads as shown in Figure6. The loading frequency of the fatigue testing can be configured using the control software. The maximum loading frequency of the electrohydraulic machine is 20 Hz. The maximum frequency is used to reduce the testing time. Based on the testing results of the mechanical properties, the material has a mean yield strength of 455.12 MPa. Three maximum nominal stress levels are chosen as 0.5σy, 0.7σy, and 0.8σy. The stress levels are chosen to represent the normal service load and two overload conditions [27]. Given the stress ratio of 0.1, the corresponding nominal stress ranges are 0.45σy, 0.63σy, and 0.72σy, respectively. The fatigue testing can adopt two loading schemes, namely, the force-controlled (ASTM E466) and the displacement-controlled (ASTM E606). Depending on the context, they are also called stress-controlled or strain-controlled, respectively. For low-cycle fatigue problems where significant plastic strains are involved and the resulting fatigue life is not larger than 105 [5], the displacement (strain)-controlled testing scheme is usually used. For high-cycle fatigue where the total strain is elastic strain dominant, the force (stress)-controlled testing scheme is suitable. In this study, the maximum loading force is 0.8σy; therefore, the force-controlled testing scheme is employed. The stress ratio and the minimum stress remain constant during the testing process. The nominal stress range is defined as the (Fmax − Fmin)/A0. Terms Fmax and Fmin are the maximum and the minimum forces, respectively, and A0 is the uncorroded cross-sectional area. The fatigue testing stops when the specimen is completely fractured and the number of applied load cycles at rupture is recorded as the fatigue life. The applied loads in terms of the normal stress ranges and the actual stress ranges at the MCD cross section for each of the specimens and the associated fatigue life results are shown in Table 3. All the specimens ruptured at their MCD cross sections due to the maximum local stress at the cross section.Figure 6
Fatigue testing setup.Based on the testing results, it can be seen that the fatigue life in general decreases as the actual stress range increases. The observation is consistent with the usual fatigue testing results. The minimal and maximum values of actual stress range in this case are 247.0 MPa and 430 MPa, respectively, which correspond to the largest and smallest fatigue lives of 4060752 and 27611 cycles, respectively. Due to the uncertain corroded surface conditions and the shapes of the MCD, the fatigue life results exhibit the inherent interspecimen uncertainty. For example, there is a noticeable difference in fatigue life between FC7 and FC8 although the actual stress ranges of the two specimens are almost identical.


## 2.1. Materials and Specimens

A total number of 14 corroded standard specimens made from steel reinforcing bars are prepared, and the fatigue testing data are acquired to build the baseline model. Artificial corrosion pits are developed using the impressed current technique in the NaCl solution. The maximum corrosion degree is measured with the three-dimensional (3D) scan technique. Following this, the axial fatigue testing is carried out under three different stress ranges, and the results of fatigue life are acquired. The degree of corrosions in this study is tentatively controlled to be in the range between 10% and 40% in terms of the maximum cross-sectional area reduction to represent the actual degree of corrosions in practice. Three levels of applied loads are considered to simulate the nominal and overload conditions.The dimension and an actual specimen used in this study are shown in Figure2. The specimen is prepared by removing the surface layer of hot-rolled ribbed steel bars (HRB400) with an initial diameter of 20 mm. The chemical composition of the material is shown in Table 1. The geometry and dimension of the final specimens follow the code requirement for both statistic [22] and fatigue testing [23]. The elastic modulus and yield strength of the four uncorroded specimens are shown in Table 2. The mean values of the mechanical properties are calculated and are used as references for fatigue testing.Figure 2
An actual specimen and its dimension (unit: mm).Table 1
Chemical composition of HRB400 bars.
Chemical compositionCSiMnPSVPercentage0.220.451.340.0280.0310.033Table 2
Mechanical properties of HRB400 bars.
Specimen no.M1M2M3M4Mean valueElastic modulusE (MPa)210330212698201833205854207679Yielding strengthσy (MPa)460.29463.88446.09450.23455.12


## 2.2. Precorrosion and 3D Measurement

The corrosion pit is developed on each of the 14 specimens using an accelerated corrosion process in the solution of 5% NaCl with a current density of 0.6 mA/cm2. Before corrosion, the specimen is wrapped using watertight insulation tapes to protect the surface from the solution. An elliptical shape of the tape is removed at the location for corrosion damage to expose the metal surface to the solution, as shown in Figure 3(a). Both ends of the specimen are sealed by hot-melt adhesive with one end connected with a wire. A direct-current power supply is used for the corrosion process during which the specimen serves as anode and a separate stainless-steel bar as the cathode, as shown in Figure 3(b). After the corrosion process, the specimen is cleaned and stored in a dry condition for fatigue testing.Figure 3
Treatment of specimen and precorrosion setup. (a) Treatment of specimen before corrosion process. (b) Corrosion environment setup.
(a)(b)Due to the nonuniform corrosion process, the width and depth of the corrosion pit vary along the axial direction. Consequently, the cross-sectional area also varies along the axial direction. The concept of maximum corrosion degree (MCD),ηmax, is proposed to characterize the most severe condition of the corrosion pit and is defined as follows:(1)ηmax=1−AminA0,where Amin is the minimum cross-sectional area of the corrosion pit and A0 is the initial uncorroded cross-sectional area.A high-resolution 3D laser scanner is used to acquire the surface data of the specimens [24]. The laser scanner can produce a point cloud of the surface. The raw data are processed using the Geomagic wrap software to reconstruct a 3D surface for digital measurement [25]. The reconstructed 3D surface model is imported into the Pro/Engineer software environment to form a solid volume for cross-sectional measurement [26]. The cross-sectional slices are sampled from the solid volume at an interval of 0.1 mm along the axial direction in the corroded region. The areas of the cross-sectional slices are automatically computed in the Pro/Engineering software. The minimum cross-sectional area is identified as Amin. One of the specimens (FC3) is used for illustration. The actual corrosion pit of the specimen is shown in Figure 4(a), and the reconstructed 3D surface model is shown in Figure 4(b). The model shown in Figure 4(b) is imported into Pro/Engineer and the minimal cross section in terms of MCD is identified, as shown in Figure 5. The whole digital processing and measurement process is performed for each of the specimens, and the results of MCD are listed in Table 3.Figure 4
Specimen FC3 and the reconstructed 3D surface model. (a) The actual specimen and the corrosion pit and (b) the reconstructed 3D surface model of the specimen using laser scanning data.
(a)(b)Figure 5
3D surface profile measurements and pit geometry of the specimen FC3.Table 3
Fatigue testing results of corroded specimens.
SpecimenNominal stress rangeS (MPa)MCDηmax (%)Actual stress rangeσ (MPa)Fatigue lifeN (cycle)FC1204.817.1247.04060752FC2204.821.1259.61415913FC3204.823.9269.1670622FC4204.830.1293.0338315FC5204.835.1315.6274103FC6286.719.6356.699580FC7286.721.3364.3153070FC8286.721.6365.7201968FC9286.722.8371.493330FC10286.729.8408.435814FC11327.713.8380.1247064FC12327.714.6383.7348902FC13327.716.9394.3125903FC14327.723.8430.027611


## 2.3. Fatigue Testing Results and Discussion

The uniaxial fatigue testing is carried out in air condition at room temperature using a sinusoidal load with a stress ratio of 0.1. An electrohydraulic fatigue testing machine (Instron 8801, 100 kN) is used to apply cyclic loads as shown in Figure6. The loading frequency of the fatigue testing can be configured using the control software. The maximum loading frequency of the electrohydraulic machine is 20 Hz. The maximum frequency is used to reduce the testing time. Based on the testing results of the mechanical properties, the material has a mean yield strength of 455.12 MPa. Three maximum nominal stress levels are chosen as 0.5σy, 0.7σy, and 0.8σy. The stress levels are chosen to represent the normal service load and two overload conditions [27]. Given the stress ratio of 0.1, the corresponding nominal stress ranges are 0.45σy, 0.63σy, and 0.72σy, respectively. The fatigue testing can adopt two loading schemes, namely, the force-controlled (ASTM E466) and the displacement-controlled (ASTM E606). Depending on the context, they are also called stress-controlled or strain-controlled, respectively. For low-cycle fatigue problems where significant plastic strains are involved and the resulting fatigue life is not larger than 105 [5], the displacement (strain)-controlled testing scheme is usually used. For high-cycle fatigue where the total strain is elastic strain dominant, the force (stress)-controlled testing scheme is suitable. In this study, the maximum loading force is 0.8σy; therefore, the force-controlled testing scheme is employed. The stress ratio and the minimum stress remain constant during the testing process. The nominal stress range is defined as the (Fmax − Fmin)/A0. Terms Fmax and Fmin are the maximum and the minimum forces, respectively, and A0 is the uncorroded cross-sectional area. The fatigue testing stops when the specimen is completely fractured and the number of applied load cycles at rupture is recorded as the fatigue life. The applied loads in terms of the normal stress ranges and the actual stress ranges at the MCD cross section for each of the specimens and the associated fatigue life results are shown in Table 3. All the specimens ruptured at their MCD cross sections due to the maximum local stress at the cross section.Figure 6
Fatigue testing setup.Based on the testing results, it can be seen that the fatigue life in general decreases as the actual stress range increases. The observation is consistent with the usual fatigue testing results. The minimal and maximum values of actual stress range in this case are 247.0 MPa and 430 MPa, respectively, which correspond to the largest and smallest fatigue lives of 4060752 and 27611 cycles, respectively. Due to the uncertain corroded surface conditions and the shapes of the MCD, the fatigue life results exhibit the inherent interspecimen uncertainty. For example, there is a noticeable difference in fatigue life between FC7 and FC8 although the actual stress ranges of the two specimens are almost identical.


## 3. Model Development

### 3.1. Fatigue Life Prediction Model

The fatigue testing results show that both the stress range and MCD have significant influences on the fatigue life, and they are considered as model independent variables. It is known that the same stress range has different effects on different materials; therefore, the normalized stress range,S/σy, is used for model development. The normalized stress range is defined as the ratio between nominal stress range and the yielding stress of the material. Unless stated otherwise, the stress range in the text hereafter refers to the normalized nominal stress range. To identify the relationship between the fatigue life and MCD at different stress ranges, fatigue life versus MCD are presented in Figure 7. A log-linear relationship between the fatigue life and the MCD can be observed under different stress ranges. At the same level of MCD, the fatigue life decreases as the stress range increases.Figure 7
Relationship between fatigue life and MCD at three stress levels.In view of aforementioned features, a fatigue life model incorporating the combined influence of stress range and MCD is proposed as follows:(2)lnN=A1+A2lnSσy+A3lnηmax,where N is fatigue life and A1, A2, and A3 are model parameters. It is noteworthy that other forms of regression models can also be adopted. For example, an interaction term ln(S/σy)lnηmax can be appended to the right-hand side of equation (2) if it can be justified. The formal and rigorous method for model selection is beyond the scope of this study and can be referred to in other studies [28].


### 3.2. Probabilistic Parameter Identification

The parametersA1, A2, and A3 of the model equation (2) can be obtained using experimental data. For the deterministic estimation, the parameters can trivially be obtained using the standard least square method; however, the deterministic fatigue life prediction makes little sense for practical applications due to the stochastic nature of fatigue process. To incorporate uncertainties, the probabilistic approach is usually preferred. In this study, the model parameters are identified statistically using the Bayesian estimator [29]. The core of the Bayesian method is to make inference based on prior information and data. The Bayesian posterior reads(3)pθ|D∝pθ⋅pD|θ,where p(θ) is the prior probability density function (PDF) of a parameter (vector) θ, D is the data, p(D | θ) is the likelihood, and p(θ | D) is the posterior distribution. The posterior distribution of parameters θ can be asymptotically obtained using sampling methods, such as Markov chain Monte Carlo (MCMC) simulation and its variants [30, 31]. Direct applications of the Bayesian method include parameter estimation and model updating. Without loss of generality, the model prediction error can be modeled as a zero-mean Gaussian variable. Based on this general assumption, the Bayesian estimator for the model parameters can be expressed as follows [32]:(4)pA1,A2,A3,σe∝1σe2πσenexp−12∑i=1nlnN−A1−A2lnS/σy−A3lnηmaxσe2,where σe is the standard deviation of the error variable, n is the number of available measurements, and p(A1, A2, A3, σe) is the posterior joint PDF. Using the experimental data of the specimens given in Table 3, 200,000 samples are drawn using MCMC simulations. The resulting distributions of model parameters A1, A2, A3 and the standard deviation σe of the error variable are shown in Figure 8.Figure 8
Distributions ofA1, A2, A3, and σe obtained using experimental data of standard specimens.The model parameter vector (A1, A2, A3) follows a multivariate normal distribution. The mean vector μ and the variance-covariance matrix Σ of the parameter vector (A1, A2, A3) can be estimated from the MCMC samples as follows:(5)μ=2.4301−7.0829−3.9423,(6)Σ=0.66010.31980.31080.31980.28570.10710.31080.10710.1643,respectively. The standard deviation of the error variable σe follows a Gamma distribution with a shape parameter of 17.82 and a scale parameter of 0.0195. Using the mean of parameter vector (A1, A2, A3), the mean prediction of the model is obtained as follows:(7)lnN=2.4301−7.0829lnSσy−3.9423lnηmax.To verify the effectiveness of the proposed model, the results of the mean prediction and the actual fatigue life are visually compared in Figure9. A good agreement between model predictions (Npre) and experimental data (Nexp) can be observed. The model with the prior distribution of (A1, A2, A3) is used as the baseline model for Bayesian updating.Figure 9
The experimental fatigue lives and model predictions.


## 3.1. Fatigue Life Prediction Model

The fatigue testing results show that both the stress range and MCD have significant influences on the fatigue life, and they are considered as model independent variables. It is known that the same stress range has different effects on different materials; therefore, the normalized stress range,S/σy, is used for model development. The normalized stress range is defined as the ratio between nominal stress range and the yielding stress of the material. Unless stated otherwise, the stress range in the text hereafter refers to the normalized nominal stress range. To identify the relationship between the fatigue life and MCD at different stress ranges, fatigue life versus MCD are presented in Figure 7. A log-linear relationship between the fatigue life and the MCD can be observed under different stress ranges. At the same level of MCD, the fatigue life decreases as the stress range increases.Figure 7
Relationship between fatigue life and MCD at three stress levels.In view of aforementioned features, a fatigue life model incorporating the combined influence of stress range and MCD is proposed as follows:(2)lnN=A1+A2lnSσy+A3lnηmax,where N is fatigue life and A1, A2, and A3 are model parameters. It is noteworthy that other forms of regression models can also be adopted. For example, an interaction term ln(S/σy)lnηmax can be appended to the right-hand side of equation (2) if it can be justified. The formal and rigorous method for model selection is beyond the scope of this study and can be referred to in other studies [28].


## 3.2. Probabilistic Parameter Identification

The parametersA1, A2, and A3 of the model equation (2) can be obtained using experimental data. For the deterministic estimation, the parameters can trivially be obtained using the standard least square method; however, the deterministic fatigue life prediction makes little sense for practical applications due to the stochastic nature of fatigue process. To incorporate uncertainties, the probabilistic approach is usually preferred. In this study, the model parameters are identified statistically using the Bayesian estimator [29]. The core of the Bayesian method is to make inference based on prior information and data. The Bayesian posterior reads(3)pθ|D∝pθ⋅pD|θ,where p(θ) is the prior probability density function (PDF) of a parameter (vector) θ, D is the data, p(D | θ) is the likelihood, and p(θ | D) is the posterior distribution. The posterior distribution of parameters θ can be asymptotically obtained using sampling methods, such as Markov chain Monte Carlo (MCMC) simulation and its variants [30, 31]. Direct applications of the Bayesian method include parameter estimation and model updating. Without loss of generality, the model prediction error can be modeled as a zero-mean Gaussian variable. Based on this general assumption, the Bayesian estimator for the model parameters can be expressed as follows [32]:(4)pA1,A2,A3,σe∝1σe2πσenexp−12∑i=1nlnN−A1−A2lnS/σy−A3lnηmaxσe2,where σe is the standard deviation of the error variable, n is the number of available measurements, and p(A1, A2, A3, σe) is the posterior joint PDF. Using the experimental data of the specimens given in Table 3, 200,000 samples are drawn using MCMC simulations. The resulting distributions of model parameters A1, A2, A3 and the standard deviation σe of the error variable are shown in Figure 8.Figure 8
Distributions ofA1, A2, A3, and σe obtained using experimental data of standard specimens.The model parameter vector (A1, A2, A3) follows a multivariate normal distribution. The mean vector μ and the variance-covariance matrix Σ of the parameter vector (A1, A2, A3) can be estimated from the MCMC samples as follows:(5)μ=2.4301−7.0829−3.9423,(6)Σ=0.66010.31980.31080.31980.28570.10710.31080.10710.1643,respectively. The standard deviation of the error variable σe follows a Gamma distribution with a shape parameter of 17.82 and a scale parameter of 0.0195. Using the mean of parameter vector (A1, A2, A3), the mean prediction of the model is obtained as follows:(7)lnN=2.4301−7.0829lnSσy−3.9423lnηmax.To verify the effectiveness of the proposed model, the results of the mean prediction and the actual fatigue life are visually compared in Figure9. A good agreement between model predictions (Npre) and experimental data (Nexp) can be observed. The model with the prior distribution of (A1, A2, A3) is used as the baseline model for Bayesian updating.Figure 9
The experimental fatigue lives and model predictions.


## 4. Bayesian Updating for General Applications

The baseline model above is constructed using data of the standard specimens. The dimension and size of the specimen can be quite different from those of a realistic corroded reinforcing bar in service [33]. Uncertain factors, such as residual stress introduced by the machining process of a component and the inherent dimensional deviations due to imperfect manufacturing, are difficult to quantify accurately without enough information [34–36]. To make the model more reliable, a scientific calibration of the model must be made. Bayesian updating is a rational way to incorporate new data into a baseline model. By performing the updating, the model is gradually calibrated to favor data of the actual component more than the prior state of knowledge [37].To demonstrate the above idea and validate the effectiveness of method, Bayesian updating is performed using data reported in [13]. In that study, corroded steel reinforcing bars with two different diameters (10 mm and 12 mm) are extracted from reinforced concrete beams exposed to different corrosion degrees. The corrosion degrees are measured using the mass loss ratio ranging from 8% to 28%. Sinusoidal fatigue loads with nominal stresses of 150 MPa, 200 Mpa, and 300 MPa are applied at the frequency of 15 Hz. The details of the testing can be referred to in [13]. It should be noted that the corrosion damage in [13] is measured using the mass loss ratio. To utilize the proposed model, the formulation provided in [35] is used to transfer the mass loss ratio to MCD and is given as(8)AminAave=1−0.83γ,where γ is the mass loss ratio, Amin is the minimum cross-sectional area, and Aave is the averaged reduced cross-sectional area expressed as(9)Aave=A01−γ.The termA0 is the uncorroded section area defined as before. Combining equations (1), (8), and (9), the MCD is established as follows:(10)ηmax=1−1−γ1−0.83γ.Figure10 presents the fatigue life prediction Npre by the prior baseline model and the actual fatigue life data Nexp of the corroded steel reinforcing bars. The vertical bar of each prediction point represents 95% confidence intervals. The dashed lines and dot-dashed lines are the boundaries deviating from the actual fatigue life by the multiplication factors of 2 and 5, respectively. For example, given an x-coordinate value of Nexp, the values for the corresponding life factor 2 and life factor 5 are 2Nexp and 5Nexp in the y-coordinate. The overall trends of the model prediction agree with the experimental data, but the deviation of the prediction can be larger than a factor of 5. This means that the proposed model can capture the influence of stress range and MCD, but the direct application of the baseline model of equation (7) is not reliable due to the uncertainties between the specimens and the real components.Figure 10
Fatigue life prediction results of the prior mean model in equation (7). (a) Steel bars with diameter of 12 mm and (b) steel bars with diameter of 10 mm.
(a)(b)The actual data of the bars are used for Bayesian updating. The posterior PDF of the model parameters writes(11)pA1,A2,A3|D∝12πΣexp−12A1,A2,A3−μΣ−1A1,A2,A3−μT×12πσenexp−12∑i=1nlnN−A1−A2lnS/fy−A3lnηmaxσe,where the mean vector μ and the covariance matrix Σ are given by equations (5) and (6), respectively. The mean value of σe is estimated as 0.34 from the prior distribution and is used in equation (11).


### 4.1. Case 1—Steel Bars with a Diameter of 12 mm

A total number of 16 specimens are arbitrarily chosen from [13] to represent the observation data for Bayesian updating, and they are denoted as updating points. The total 16 sets of (S, ηmax, N) are listed in Table 4 and are used in the posterior of equation (11). The posterior results updated with 1, 4, 8, and 16 data points in Table 4 are evaluated using MCMC simulations. For each of the four updates, 200,000 samples are drawn, and the samples are used to estimate the PDFs of the model parameter vector (A1, A2, A3). The prior and posterior distributions of the model parameters A1, A2, and A3 are presented in Figures 11(a)–11(c), respectively. As more data are used for updating, the distributions are gradually tuned to be more specific to the actual components.Table 4
Testing results (12 mm-diameter bars) arbitrarily chosen from [13], representing observed data for Bayesian updating.
No.Nominal stress rangeS (MPa)MCDηmax (%)Fatigue lifeN (cycle)120034.25909230029.227798330023.463818420024.999873530030.712353630026.848224720030.979579820022.4343394920035.574851020038.0250641120030.1328521230029.5548701330023.0621381430025.9920511520019.52640101615015.3943043Figure 11
The prior and posterior distributions of model parameters (diameter: 12 mm). (a) ParameterA1, (b) parameter A2, and (c) parameter A3.
(a)(b)(c)The fatigue life prediction results of the specimens in [13] other than those used for updating are obtained using the resulting MCMC samples and equation (7). The calculation procedure is as follows. For each of the specimens, that is, the discrete hollow points in Figure 12, the stress (S) and MCD (ηmax) associated with the specimen, and a random instance of the resulting MCMC samples is used in equation (7) to compute one sample of the fatigue life Npre. By using a total number of 200,000 random instances, a total number of 200,000 random samples of the fatigue life of the specimen are evaluated, and the mean and 95% confidence bounds are estimated from the fatigue life samples. Figure 12 presents the prediction results of the updated model with 1, 4, 8, and 16 updating points. The red solid points are the data used for updating and the blue hollow points are prediction results of the rest of the specimens in the dataset. The vertical bar of each prediction point represents 95% confidence intervals. It can be observed from Figure 12 that the fatigue life predictions gradually converge to the experimental results as more and more data are used for updating. It indicates that as more and more relevant information is integrated into the model through Bayesian updating, the features of the realistic reinforcing bars tend to dominate the posterior, yielding more accurate prediction results.Figure 12
Fatigue life prediction results for 12 mm-diameter bars using the model updated with (a) 1 point, (b) 4 points, (c) 8 points, and (d) 16 points.
(a)(b)(c)(d)


### 4.2. Case 2—Steel Bars with a Diameter of 10 mm

The same Bayesian updating procedure is performed for the steel bars with a diameter of 10 mm. A total number of 16 sets of testing results on the 10 mm-diameter bar are arbitrarily chosen from [13] to represent the actually observed data presented in Table 5, which are used for Bayesian updating. Figures 13(a)–13(c) present the posterior distributions of model parameters A1, A2, and A3, respectively, updated with 1, 4, 8, and 16 points in Table 5 using the posterior of equation (11). The prediction results of the updated model are shown in Figure 14. It can be seen that the updated model yields more and more accurate results as more data are incorporated into the baseline model.Table 5
Testing results (10 mm-diameter bars) arbitrarily chosen from [13], representing observed data for Bayesian updating.
No.Nominal stress rangeS (MPa)MCDηmax (%)Fatigue lifeN (cycle)130040.81785230030.18603320037.435988430018.682951520033.75706620035.916346730021.355970830019.6122934930029.057151030034.031571130035.9129591220029.0319591330021.4448831420038.0688121520017.51020301620018.8192984Figure 13
The prior and posterior distributions of model parameters (diameter: 10 mm). (a) ParameterA1, (b) parameter A2, and (c) parameter A3.
(a)(b)(c)Figure 14
Fatigue life prediction results for 10 mm-diameter bars using the model updated with (a) 1 point, (b) 4 points, (c) 8 points, and (d) 16 points.
(a)(b)(c)(d)The above two cases demonstrate that the initial baseline model cannot yield accurate results due to the deviation between the two sets of specimens under fatigue testing. However, as more observation data are incorporated into the baseline model via Bayesian updating, the predictions results can be largely improved. In particular, when the baseline model is updated with 16 data points, most of the life prediction results shown in Figures12(d) and 14(d) are within the bounds deviated from the actual life by a factor of 5. The updated model with 16 data points has the ability to estimate the fatigue life in different corrosion specimens.


## 4.1. Case 1—Steel Bars with a Diameter of 12 mm

A total number of 16 specimens are arbitrarily chosen from [13] to represent the observation data for Bayesian updating, and they are denoted as updating points. The total 16 sets of (S, ηmax, N) are listed in Table 4 and are used in the posterior of equation (11). The posterior results updated with 1, 4, 8, and 16 data points in Table 4 are evaluated using MCMC simulations. For each of the four updates, 200,000 samples are drawn, and the samples are used to estimate the PDFs of the model parameter vector (A1, A2, A3). The prior and posterior distributions of the model parameters A1, A2, and A3 are presented in Figures 11(a)–11(c), respectively. As more data are used for updating, the distributions are gradually tuned to be more specific to the actual components.Table 4
Testing results (12 mm-diameter bars) arbitrarily chosen from [13], representing observed data for Bayesian updating.
No.Nominal stress rangeS (MPa)MCDηmax (%)Fatigue lifeN (cycle)120034.25909230029.227798330023.463818420024.999873530030.712353630026.848224720030.979579820022.4343394920035.574851020038.0250641120030.1328521230029.5548701330023.0621381430025.9920511520019.52640101615015.3943043Figure 11
The prior and posterior distributions of model parameters (diameter: 12 mm). (a) ParameterA1, (b) parameter A2, and (c) parameter A3.
(a)(b)(c)The fatigue life prediction results of the specimens in [13] other than those used for updating are obtained using the resulting MCMC samples and equation (7). The calculation procedure is as follows. For each of the specimens, that is, the discrete hollow points in Figure 12, the stress (S) and MCD (ηmax) associated with the specimen, and a random instance of the resulting MCMC samples is used in equation (7) to compute one sample of the fatigue life Npre. By using a total number of 200,000 random instances, a total number of 200,000 random samples of the fatigue life of the specimen are evaluated, and the mean and 95% confidence bounds are estimated from the fatigue life samples. Figure 12 presents the prediction results of the updated model with 1, 4, 8, and 16 updating points. The red solid points are the data used for updating and the blue hollow points are prediction results of the rest of the specimens in the dataset. The vertical bar of each prediction point represents 95% confidence intervals. It can be observed from Figure 12 that the fatigue life predictions gradually converge to the experimental results as more and more data are used for updating. It indicates that as more and more relevant information is integrated into the model through Bayesian updating, the features of the realistic reinforcing bars tend to dominate the posterior, yielding more accurate prediction results.Figure 12
Fatigue life prediction results for 12 mm-diameter bars using the model updated with (a) 1 point, (b) 4 points, (c) 8 points, and (d) 16 points.
(a)(b)(c)(d)


## 4.2. Case 2—Steel Bars with a Diameter of 10 mm

The same Bayesian updating procedure is performed for the steel bars with a diameter of 10 mm. A total number of 16 sets of testing results on the 10 mm-diameter bar are arbitrarily chosen from [13] to represent the actually observed data presented in Table 5, which are used for Bayesian updating. Figures 13(a)–13(c) present the posterior distributions of model parameters A1, A2, and A3, respectively, updated with 1, 4, 8, and 16 points in Table 5 using the posterior of equation (11). The prediction results of the updated model are shown in Figure 14. It can be seen that the updated model yields more and more accurate results as more data are incorporated into the baseline model.Table 5
Testing results (10 mm-diameter bars) arbitrarily chosen from [13], representing observed data for Bayesian updating.
No.Nominal stress rangeS (MPa)MCDηmax (%)Fatigue lifeN (cycle)130040.81785230030.18603320037.435988430018.682951520033.75706620035.916346730021.355970830019.6122934930029.057151030034.031571130035.9129591220029.0319591330021.4448831420038.0688121520017.51020301620018.8192984Figure 13
The prior and posterior distributions of model parameters (diameter: 10 mm). (a) ParameterA1, (b) parameter A2, and (c) parameter A3.
(a)(b)(c)Figure 14
Fatigue life prediction results for 10 mm-diameter bars using the model updated with (a) 1 point, (b) 4 points, (c) 8 points, and (d) 16 points.
(a)(b)(c)(d)The above two cases demonstrate that the initial baseline model cannot yield accurate results due to the deviation between the two sets of specimens under fatigue testing. However, as more observation data are incorporated into the baseline model via Bayesian updating, the predictions results can be largely improved. In particular, when the baseline model is updated with 16 data points, most of the life prediction results shown in Figures12(d) and 14(d) are within the bounds deviated from the actual life by a factor of 5. The updated model with 16 data points has the ability to estimate the fatigue life in different corrosion specimens.


## 5. Conclusion

This paper proposes a general method for fatigue life prediction of corroded reinforcing bars using testing data of standard specimens and Bayesian updating. To investigate the influencing factors of the fatigue life, a systematical fatigue testing is performed. The local geometry details of the corrosion pits are acquired using the advanced 3D laser scanning technique. Based on 3D measurements of the corrosion pits and fatigue life data, the maximum corrosion degree and stress range are found to be effective in characterizing the fatigue life. A fatigue life model is proposed incorporating the two factors. The parameters of the model are statistically obtained using the testing data, and the resulting model is used as a baseline model for general applications. Due to the variations of the dimension and working condition of an actual reinforcing bar, the Bayesian method is used to update the baseline model using data from the actual reinforcing bars. The proposed method is demonstrated using independent fatigue testing data of steel reinforcing bars. Based on the current results, the following conclusions are drawn:(1)
The normalized nominal stress range and maximum corrosion degree are effective factors to quantify the fatigue life of corroded steel reinforcing bars. Experimental data show that the fatigue life has a log-linear relationship with the maximum corrosion degree under different stress ranges. A regression model incorporating the two factors can be used to characterize the fatigue life of corroded reinforcing bars.(2)
The proposed baseline model can be applied to more general applications where the dimension and working condition of a reinforcing bar differ from that of the standard specimen. This is achieved by updating the initial model using the Bayesian method. The effectiveness of the method is demonstrated using reinforcing bars with different diameters under different fatigue loads.(3)
The model obtained from Bayesian updating with 16 data points is observed to have the ability to predict the fatigue life of different specimens accurately. It is shown that most of the prediction results are within the bounds deviated from the actual life by a factor of 5.It is worth mentioning that the corrosion-free fatigue testing results are indispensable as a reference for comparison purposes. Due to the time and resource limitations on the current study, the corrosion-free fatigue testing will be performed in the future study.



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*Source: 4632152-2021-12-28.xml*