Probability-Based Performance Degradation Model and Constitutive Model for the Buckling Behavior of Corroded Steel Bars

Abstract

The corrosion of steel bars significantly influences the safety running of concrete bridges. The current bridge design code always ignores the changes in mechanical properties of the materials during their service life which results in an overestimation of the seismic capacity of concrete bridges Therefore, a hysteretic constitutive model that considers the influence of reinforcement corrosion is necessary. This study established a probability-based model for the mechanical property degradation of corroded steel bars using statistical analysis to estimate the mechanical properties under specific corrosion rates. The validity of the proposed degradation model was verified by a series of monotonic tensile tests on corroded steel bars. Moreover, a constitutive model of corroded steel bars based on the Reinforcing Steel Model in OpenSees was obtained. This model considers the influence of corrosion on the slenderness ratio, yield strength, ultimate strength, and elongation. A comparison with the experimental data demonstrated the accuracy of the proposed model. The proposed model is straightforward because its parameters can be predicted using existing research data. The proposed model can provide a basis for the seismic performance evaluation of corroded reinforced concrete (RC) members.

1. Introduction

Concrete bridges worldwide often fail to reach their projected service life just because of the corrosion of steel bars under the influence of various environmental factors [1,2,3]. Corrosion reduces the effective cross−sectional area of steel bars, degrading their mechanical properties. When corrosion expands, the concrete cover would crack or even spall, reducing the bearing capacity of the reinforced concrete (RC) member. The current bridge design code typically overestimates the seismic capacity of concrete bridges by assuming the mechanical properties of the materials are unchanged during the service life of the structure [4,5]. Using such assumptions to predict the seismic performance of piers will significantly overestimate their seismic performance which will have a negative impact on the accuracy of sustainable structural safety monitoring. Additionally, from the perspective of sustainability, corrosion behavior will greatly affect the sustainability of reinforced concrete structures which cannot be ignored. Therefore, in order to accurately evaluate the seismic performance of concrete bridges, the hysteretic constitutive model that considers the influence of reinforcement corrosion is necessary. The goal of this study is to obtain a convenient and accurate hysteretic constitutive model of reinforcement to help engineers better predict the seismic performance of piers and improve the reliability of sustainable safety monitoring.

Scholars have conducted relevant research on the effect of corrosion on the mechanical properties of reinforcements [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]. These studies include a large amount of actual test data. However, these data lack systematic collation. In this paper, a degradation calculation formula based on these data has been proposed which considers the yield strength, elastic modulus, ultimate strength, elongation, and other parameters affected by corrosion.

Buckling is a common failure mode of steel bars under earthquake, and corrosion also significantly influences the buckling behavior of steel bars [22,23]. Many researchers have studied the buckling behavior of uncorroded steel bars [24,25,26]. However, few studies have focused on the constitutive model for the buckling behavior of corroded steel bars. Kashini et al. [15,27] determined the material parameters of corroded steel bars through a series of monotonic tension and compression tests. They proposed a calculation method for the buckling strength of corroded steel bars through cyclic loading tests. This method can predict the post−buckling behavior of corroded steel bars. However, it requires more monotonic compression tests of corroded bars to determine material parameters which are difficult to apply in engineering practice.

In general, at the present stage, there are sufficient experimental data on the monotonic tensile properties of corroded steel bars, and the research on the buckling constitutive model of corroded steel bar is not sufficient. The existing models often need to carry out new corroded steel bar material tests which is not convenient to use in practical engineering. The aim of this study is to obtain an accurate and easy−to−use buckling constitutive model of corroded steel bars. In order to overcome the shortcoming of the previously existing constitutive model, a prediction method for mechanical property degradation based on probability distribution was established by collecting a large amount of previous monotonic tensile test data of corroded steel bars. Subsequently, a batch of steel bar samples with different corrosion degrees was obtained through an accelerated corrosion test in a salt spray environment to verify the prediction method. Eventually, a constitutive model for the buckling behavior of corroded steel bars is proposed based on the Reinforcing Steel Model in OpenSees [28]. The model uses the degradation law proposed in this study to consider the influence of corrosion on the slenderness ratio, yield strength, and ultimate strength of steel bars. The research flow is shown in Figure 1.

2. Degradation Model for the Mechanical Properties of Corroded Steel Bars

The mechanical properties of corroded RC members are primarily affected by the degree of steel corrosion. The corrosion process, distribution of corrosion pots, and inhomogeneity of the surface geometry are random processes [20,29]. The corrosion of reinforcements becomes increasingly uneven as corrosion progresses. In general reinforcement failure often occurs at the minimum cross section. However, determining the minimum cross section is challenging in engineering practice. Therefore, the average cross−sectional area is typically used to measure the changes in the mechanical properties of the steel bars. The calculation method for the average cross−sectional area is given by Equation (1).

$$A_{av}=A·(1-{\eta }_s),$$

(1)

where η_sc is the corrosion rate, A the initial cross-sectional area, and A_av the average cross−sectional area of the corrosion rate.

This study collected a large amount of corroded monotonic tensile test data as shown in

Table 1. Statistical sample for the mechanical properties of corroded steel bars.

Source of Specimen	Number	Test Method
Almusallam [6]	48	Current accelerated corrosion
Apostolopoulos (2008) [7]	23	Salt spray accelerates corrosion
Apostolopoulos (2012) [8]	27	Salt spray accelerates corrosion
Apostolopoulos (2016) [9]	18	Salt spray accelerates corrosion
Chen Hui [10]	25	Current accelerated corrosion
Diamantogiannis [11]	38	Salt spray accelerates corrosion
Drakakaki [12]	24	Salt spray accelerates corrosion
Fernandez [13]	40	Current accelerated corrosion
Guo Chao [14]	96	Current accelerated corrosion &Simulating natural corrosion
Kashini [15]	42	Current accelerated corrosion
Li Fenglan [16]	46	Current accelerated corrosion
Luo Xiaoyong [17]	12	Current accelerated corrosion
Wu Xun [18]	46	Current accelerated corrosion
Xu Gang [19]	18	Current accelerated corrosion
Zhang Weiping [20]	67	Current accelerated corrosion
Zhang Yanfang [21]	135	Current accelerated corrosion
summary	705

. The probabilistic distribution model for the steel bar performance parameter under a specific corrosion rate and the degradation law of the distribution parameters with the corrosion rate were determined through statistical methods using the collected data.

2.1. Probability Distribution Model for the Mechanical Properties of Corroded Steel Bars

Steel bars with similar corrosion rates in the collected data were classified into the same group. The elastic modulus E₀, yield strength loss rate $\mathsf{\Delta }{\sigma }_y$, ultimate strength loss rate $\mathsf{\Delta }{\sigma }_u$, and elongation distribution $\mathsf{\Delta }{\epsilon }_u$ of steel bars in the same group were statistically analyzed. The Kolmogorov–Smirnov test [30] and Shapiro–Wilk test [31] were used to verify the normality of the data.

Table 2. Normal distribution verification results.

Corrosion Rate	Kolmogorov–Smirnov Test			Shapiro–Wilk Test
	Yield Strength	Ultimate Strength	Elongation	Yield Strength	Ultimate Strength	Elongation
1~2	0.200	0.200	0.200	0.656	0.267	0.110
2~3	0.039	0.001	0.097	0.001	0.030	0.010
3~4	0.093	0.055	0.200	0.137	0.223	0.421
4~5	0.199	0.200	0.200	0.160	0.749	0.375
5~6	0.200	0.087	0.103	0.100	0.090	0.026
6~7	0.172	0.054	0.200	0.180	0.115	0.179
7~8	0.200	0.200	0.021	0.855	0.169	0.011
8~9	0.191	0.016	0.004	0.394	0.024	0.002
9~10	0.009	0.200	0.123	0.003	0.320	0.003
10~11	0.019	0.005	0.135	0.078	0.000	0.087
11~12	0.200	0.011	0.072	0.149	0.000	0.048
12~13	0.200	0.200	0.024	0.852	0.554	0.050
13~14	0.102	0.067	0.200	0.111	0.054	0.691
14~15	0.200	0.200	0.200	0.926	0.417	0.515
15~16	0.200	0.200	0.025	0.434	0.666	0.018
16~17	0.200	0.043	0.200	0.697	0.011	0.193
17~18	0.200	0.200	0.200	0.467	0.976	0.308
18~19	0.200	0.200	0.200	0.805	0.162	0.653
19~20	0.200	0.200	0.200	0.850	0.551	0.401
20~21	0.200	0.000	0.200	0.674	0.000	0.431
21~22	0.200	0.200	0.200	0.855	0.300	0.422
22~23	0.200	0.180	0.200	0.701	0.240	0.249
23~28	0.770	0.200	0.200	0.148	0.442	0.509
28~33	0.970	0.200	0.200	0.081	0.938	0.137
33~45	0.128	0.030	0.200	0.108	0.862	0.183

Kolmogorov–Smirnov test or Shapiro–Wilk test result greater than 0.05 indicates that the data obey normal distribution.

lists the verification results. Except for a few sets of data, the data followed a normal distribution. The corresponding mean value $\mu$, variance ${\sigma }^2$, and variation coefficient $\delta$ can be calculated. Accordingly, the probability distribution model for the mechanical properties of steel bars under the same corrosion rate can be obtained as shown in Equation (2).

$$f\left(x\right)=\frac{1}{\sqrt{2\pi }\delta \mu }{e}^{\left[-\frac{{\left(x-\mu \right)}^2}{2{\left(\delta \mu \right)}^2}\right]},$$

(2)

where x represents the required performance parameters, such as the elastic modulus, yield strength, ultimate strength, and elongation, $\mu$ is the mean value, and $\delta$ is the variation coefficient.

2.2. Random Process Model for the Material Parameter Degradation of Corroded Steel Bar

2.2.1. Elastic Modulus Loss

Since the elastic modulus of different batches of steel bar specimens cannot be compared directly, this study used the elastic modulus of the uncorroded steel bar as the reference value and counted the elastic modulus loss rates with different corrosion rates in the same batch of collected data. Figure 2 shows the results. After excluding extreme data points, the elastic modulus of the steel bars did not change significantly with an increase in the corrosion rate but fluctuated within a certain range. Therefore, the corroded steel bars can directly use the elastic modulus data obtained from the monotonic tensile test of the uncorroded steel bars.

2.2.2. Strength and Elongation Loss

The strength loss rate is divided into the yield and ultimate strength loss rates. These are the differences between the yield and ultimate strengths of the same batch of steel bars before and after corrosion divided by the uncorroded yield strength or ultimate strength. The elongation loss rate is the difference in elongation before and after corrosion divided by the uncorroded elongation. Figure 3 summarizes the results.

The strength and elongation losses follow a normal distribution under the same corrosion rate. The mean value and coefficient of variation in Equation (2) are expressed as functions of the corrosion rate ($\mu \left({\eta }_s\right)$,$\delta \left({\eta }_s\right)$). Accordingly, Equation (2) can be rewritten as Equation (3):

$$f\left(x\left({\eta }_s\right)\right)=\frac{1}{\sqrt{2\pi }\delta \left({\eta }_s\right)\mu \left({\eta }_s\right)}{e}^{\left[-\frac{{\left(x\left({\eta }_s\right)-\mu \left({\eta }_s\right)\right)}^2}{2{\left(\delta \left({\eta }_s\right)\mu \left({\eta }_s\right)\right)}^2}\right]}.$$

(3)

The mean value and variation coefficient of the yield strength, ultimate strength, and elongation loss rates were obtained by fitting the data in Figure 3.

Table 3. Relationship between mean value, coefficient of variation, and corrosion degree.

Stochastic Process	$\mathit{\mu }\left({\mathit{\eta }}_{\mathit{s}}\right)$	$\mathit{\delta }\left({\mathit{\eta }}_{\mathit{s}}\right)$
$\mathsf{\Delta }{\sigma }_{\mathrm{y}}$	1.421x − 1.855	0.353x + 0.026
$\mathsf{\Delta }{\sigma }_u$	1.228x + 1.240	0.387x + 0.546
$\mathsf{\Delta }{\epsilon }_u$	2.424x + 13.776	0.101x + 1.125

shows the fitting results.

2.3. Verification Test

Ribbed HRB400 steel bar specimens with different degrees of corrosion were prepared to verify the validity of the mechanical property degradation law. A series of monotonic tensile tests were performed to collect the material property parameters after corrosion. The results were then compared with those of the previously fitted formula.

2.3.1. Test Introduction

The salt spray test is typically used to corrode steel bar specimens. The test is primarily divided into the neutral salt spray (NSS) test, acetate spray (ASS) test, and copper−accelerated acetic acid salt spray (CASS) test according to different corrosive media. The ASS and CASS tests add acidic reagents to the salt spray, increasing the corrosion rate; however, waste liquid treatment can easily cause environmental pollution. Therefore, this study used the NSS test to accelerate steel bar corrosion. The relevant provisions of the NSS test and the parameter setting range of the salt spray corrosion chamber were based on the GJB548B-2005 (test methods and procedures for microelectronic devices).

Table 4. Environmental control parameter of corrosion test.

Temperature/°C	NaCl Solution Concentration/%	PH	Sedimentation Rate/(mL/80 cm2/h)	Setting Angle/%
35	5.0	7.0	1~2	45

shows the environmental control parameters used in this study.

Figure 4a shows the internal structure of the test chamber. Figure 4b shows the placement of the reinforced bar samples. The test chamber was sprayed automatically according to environmental parameters.

2.3.2. Test Specimen and Test Procedure

HRB400 ribbed steel bars were used in the test. Figure 5 shows the dimensions of the test pieces. The accelerated corrosion time was divided into three batches: 30 days, 60 days, and 90 days. Each batch had 10 steel bars, totaling 30 specimens. Twelve specimens were stretched to fracture, and the remaining 18 were used to calibrate the Giuffre–Menegotto–Pinto (G−M−P) [32] model parameters.

A rust remover was used to remove surface rust and passive film from 30 rusted steel test pieces. The steel was then labeled after rinsing, drying, and weighing. Epoxy resin paint was applied to the clamping sections at both ends to prevent rusting of the clamping section in the tensile test. Figure 6 shows the conditions of the specimens after a period of corrosion.

The reinforced bar samples were taken out and rinsed with water after reaching the predetermined corrosion time. A layer of red and black rust can be observed on the surface of the samples. The rust was removed with a diluted hydrochloric acid solution. Excessive reaction speed will cause residual hydrogen in the gap of corrosion pits, affecting the mechanical properties of the steel bar; therefore, adding a corrosion inhibitor to the acid solution is necessary. The specific ratio was 10% by volume of hydrochloric acid to 0.25% of corrosion inhibitor. The rusted steel bar was immersed in the descaling solution for a few hours to remove the surface rust completely. The residual acid solution was removed with lime water. The anti−rust coating on both ends of the steel bar was removed before weighing for washing and drying.

2.3.3. Monotonic Tensile Test of Corroded Steel Bar

The steel bar test was performed on an INSTRON8802−100KN electro−hydraulic servo fatigue tester as shown in Figure 7. The test was performed using strain control, and an axial extensometer was used to accurately determine the average strain over the effective length of the test piece. The extensometer had a 12.5 mm gauge length and a 50% pull−down range. To protect the extensometer from excessive displacement, the loading was controlled by the displacement when the strain was large. The test strain was controlled between −0.5% and 2% because steel bars are prone to compressive buckling.

Figure 8 shows the stress–strain curves obtained from the monotonic tensile fracture tests of the intact and corroded steel bars. The figure shows that the influence of corrosion on the yield and ultimate strengths of the steel bars was primarily reflected in the cross−section reduction. The effect of corrosion on the ductility of steel bars was more significant. Only a small amount of corrosion caused a significant decline in the elongation. When the corrosion rate reached 10%, the elongation was only half that of intact steel bars. The yield platform disappeared completely after the corrosion rate reached 10%. Therefore, for ordinary steel bars, the critical corrosion rate η_sc for the disappearance of the yield platform was set to 10%.

Table 5. Material parameters of HRB400 ribbed steel.

Corrosion Time (Day)	E0/MPa	σy/MPa	σu/MPa	εu/%
0	2.06 × 105	464.8	643.6	26.6
30	2.02 × 105	452.7	628.6	21.7
60	1.91 × 105	428.8	609.6	19.8
90	1.93 × 105	408.8	583.1	15.2

lists the average steel material parameters obtained from the monotonic tensile tests: σ_y is the yield strength, E₀ is the elastic modulus, σ_u is the tensile strength, and ε_u is the fracture elongation. The intact HRB400 ribbed steel test results showed that the ratio of tensile strength to yield strength exceeded 1.38, and the fracture elongation exceeded 25%. These results indicated good ductility and that brittle damage did not occur easily, satisfying the criterion requirements. The ductility of steel decreased with an increase in the degree of corrosion.

In Figure 9, the elastic modulus, yield strength, ultimate strength, and elongation change rates of the specimen are represented by the longitudinal axis, the corrosion rate is represented by the horizontal axis, and the straight line is the fitting result of the material performance degradation obtained by Equation (2). The fitting result is also applicable to the actual data obtained in this experiment. This proves that the performance degradation law of corroded steel bars obtained previously can also be referenced for other corroded steel bars.

3. Hysteretic Constitutive Model for the Buckling Behavior of Corroded Steel Bars

Post−earthquake damage statistics show that longitudinal steel bar buckling is a common failure mode of RC structures, particularly compression members, such as piers and columns. Under the action of an earthquake, the concrete cover of an RC pier or column gradually cracks until spalling, and the longitudinal bar buckles rapidly and bulges after losing the restraint of the concrete cover. Consequently, the bearing capacity decreases rapidly under compression, significantly weakening the seismic performance of the structure. Therefore, the effect of buckling is an important factor that should be considered in the hysteretic constitutive model for steel bars. Corrosion leads to the degradation of the mechanical properties of steel bars, and the corrosion pits on the surface become defects in the steel bars. This significantly affects the buckling behavior. The corrosion of steel bars is an inevitable trend in RC structures with long construction times or in corrosive environments. Therefore, establishing a hysteretic constitutive model for the buckling behavior of corroded steel bars is necessary.

Many studies on the hysteretic constitutive model for uncorroded steel bars have been conducted [24,25,26]. This study used the Reinforcing Steel Model in OpenSees and added the effect of corrosion. This model was proposed by Kunnath et al. [28] based on the skeleton curve of the Dhakal model [24], using the hysteresis equation of the G−M−P [32] and Gomes–Appleton (G−A) [25] models.

3.1. Monotonic Tensile Curve of Corroded Steel Bar

The Dhakal model can be used to describe the monotonic tension stress–strain relationship of the steel bars straightforwardly and accurately as shown in Figure 10a. The degradation equation in Equation (2) can be used to determine the strength and elongation loss rates for a steel bar with a specific corrosion rate.

It should be noted that the Dhakal model contains a yield platform; however, according to previous tests [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21] and the verification test in this study, the yield platform of corroded steel bars will shorten rapidly until it disappears with the deepening of the corrosion degree. The corrosion rate at which the yield platform disappeared is defined as the critical corrosion rate ${\eta }_{sc}$ (10% for regular steel bars). The yield platform changed linearly within the critical corrosion rate interval. Therefore, the Dhakal model should be modified. When the corrosion rate exceeded the critical corrosion rate, the model degenerated into only two stages as expressed in Equations (3) and (4). Figure 10b shows a schematic of this process.

$${\sigma }_s=\{\begin{array}{cc}{E}_0{\epsilon }_{\mathrm{s}},\hfill & \hfill {\epsilon }_{\mathrm{s}}\le {\sigma }_y\left({\eta }_{\mathrm{s}}\right)/E_0 \\ {\sigma }_{\mathrm{y}}\left({\eta }_{\mathrm{s}}\right),\hfill & \hfill {\sigma }_y\left({\eta }_{\mathrm{s}}\right)/E_0<{\epsilon }_{\mathrm{s}}\le {\epsilon }_{\mathrm{sh}}\left({\eta }_{\mathrm{s}}\right)\\ {\sigma }_{\mathrm{y}}\left({\eta }_{\mathrm{s}}\right)+\left[{\sigma }_{\mathrm{u}}\left({\eta }_{\mathrm{s}}\right)-{\sigma }_{\mathrm{y}}\left({\eta }_{\mathrm{s}}\right)\right]{\left[\frac{{\epsilon }_{\mathrm{s}}-{\epsilon }_{\mathrm{sh}}\left({\eta }_{\mathrm{s}}\right)}{{\epsilon }_{\mathrm{u}}\left({\eta }_{\mathrm{s}}\right)-{\epsilon }_{\mathrm{sh}}\left({\eta }_{\mathrm{s}}\right)}\right]}^P,\hfill & \hfill {\epsilon }_{\mathrm{sh}}\left({\eta }_{\mathrm{s}}\right)<{\epsilon }_{\mathrm{s}}\end{array},$$

(4)

where

$$\{\begin{array}{c}{\sigma }_{\mathrm{y}}\left({\eta }_{\mathrm{s}}\right)={\sigma }_{\mathrm{y}}\left(1-\mathsf{\Delta }{\sigma }_{\mathrm{y}}\right)\\ {\sigma }_{\mathrm{u}}\left({\eta }_{\mathrm{s}}\right)={\sigma }_{\mathrm{u}}\left(1-\mathsf{\Delta }{\sigma }_{\mathrm{u}}\right)\\ {\epsilon }_{\mathrm{sh}}\left({\eta }_{\mathrm{s}}\right)={\sigma }_{\mathrm{y}}\left({\eta }_{\mathrm{s}}\right)/E_0+{\epsilon }_{\mathrm{sh}}\left(1-\frac{{\eta }_{\mathrm{s}}}{{\eta }_{\mathrm{sc}}}\right)\hfill & \hfill ({\eta }_{\mathrm{s}}\le {\eta }_{\mathrm{sc}})\\ \begin{array}{l}{\epsilon }_{\mathrm{u}}\left({\eta }_{\mathrm{s}}\right)={\epsilon }_{\mathrm{u}}\left(1-\mathsf{\Delta }{\epsilon }_{\mathrm{u}}\right) \\ P=\frac{\log \left[\frac{{\sigma }_{\mathrm{u}}\left({\eta }_{\mathrm{s}}\right)-{\sigma }_{sh1}\left({\eta }_{\mathrm{s}}\right)}{{\sigma }_{\mathrm{u}}\left({\eta }_{\mathrm{s}}\right)-{\sigma }_{\mathrm{y}}\left({\eta }_{\mathrm{s}}\right)}\right]}{\log \left[\frac{{\epsilon }_{\mathrm{u}}\left({\eta }_{\mathrm{s}}\right)-{\epsilon }_{sh1}\left({\eta }_{\mathrm{s}}\right)}{{\epsilon }_{\mathrm{u}}\left({\eta }_{\mathrm{s}}\right)-{\epsilon }_{\mathrm{y}}\left({\eta }_{\mathrm{s}}\right)}\right]}\end{array}\end{array},$$

(5)

σ_s is the stress, ε_s is the strain, σ_y is the yield strength, E₀ is the elastic modulus, σ_u is the tensile strength, ε_u is the ultimate strain, ε_sh is the strain of the end of the yield platform, Δσ_y is the yield strength loss rate, Δσ_u is the ultimate strength loss rate, Δε_u is the elongation loss rate, η_s is the corrosion rate, η_sc is the critical corrosion rate, P is the shape parameters of the strengthening stage, and ε_sh1 is the intermediate point strain which can be valued by ${\epsilon }_{\mathrm{sh}1}=0.5\left({\epsilon }_{\mathrm{sh}}+{\epsilon }_{\mathrm{u}}\right)$, and σ_sh1 is the intermediate point stress which can be valued by ${\sigma }_{\mathrm{sh}1}={\sigma }_{\mathrm{y}}+0.75\left({\sigma }_{\mathrm{u}}+{\sigma }_{\mathrm{y}}\right)$.

3.2. Hysteretic Curve of Corroded Steel Bar Buckling

The hysteresis curve in the reinforcing steel model comprised non−buckling and modified paths after compression buckling. The non−buckling path follows the G−M−P model, and the post-buckling path is modified by the G−A model. Figure 11 shows the hysteresis curve of the model.

The monotonic stage, OA, used in Equation (3) to determine the stress–strain relationship of the steel bar when sections AB and BD do not buckle can be expressed by Equations (5) and (6).

$${\sigma }^{*}=b{\epsilon }^{*}+\frac{\left(1+b\right){\epsilon }^{*}}{{\left(1+{\epsilon }^{*R}\right)}^{1/R}},$$

(6)

where

$$\{\begin{array}{l}{\epsilon }^{*}=\left({\epsilon }_{\mathrm{s}}-{\epsilon }_{\mathrm{r}}\right)/\left({\epsilon }_0-{\epsilon }_{\mathrm{r}}\right)\\ {\sigma }^{*}=\left({\sigma }_{\mathrm{s}}-{\sigma }_{\mathrm{r}}\right)/\left({\sigma }_0-{\sigma }_{\mathrm{r}}\right)\\ b=E_1/E_0\\ R\left(\xi \right)=R_0\left[1-a_1\cdot \xi /\left(a_2+\xi \right)\right]\\ \xi =\left|\left({\epsilon }_{\mathrm{m}}-{\epsilon }_0\right)/{\epsilon }_{\mathrm{y}}\right|\end{array}.$$

(7)

In Equations (5) and (6), σ_s and ε_s are the current stress and strain, respectively, b is the strain-hardening coefficient defined as the ratio of the elastic modulus E₀ to the hardening modulus E₁, R is a curvature parameter used to adjust the shape of the transition curve, (σ_r,ε_r) is the reversal point of unloading or reloading (point B in Figure 11), and (σ₀,ε₀) is the intersection point of two asymptotic lines (point O in Figure 11). Equation (5) describes the transition equation from an asymptotic line with slope E₀ to an asymptotic line with slope E₁.

The FE section is the post−buckling softening section of compression. The stress–strain curve in this section can be expressed by Equations (7) and (8).

$${\sigma }_s=\gamma f_{\mathrm{u}}-\frac{{\mathsf{\Omega }}_{\mathrm{b}-2}+\gamma }{1+\gamma }\left(\gamma f_{\mathrm{u}}-{\sigma }_{ub}\right),$$

(8)

where

$${\mathsf{\Omega }}_{\mathrm{b}-2}=\beta \frac{\sqrt{32}}{3\pi l_{\mathrm{sr}}}\frac{\left(1-r\right)}{\sqrt{{\epsilon }_{\mathrm{s}}-{\epsilon }_0}}+r.$$

(9)

In Equations (7) and (8), σ_ub is the stress of the corresponding strain with the unbuckling pathway, f_u is the monotonic tensile strength of the reinforcement, and β, r, and γ are the corrective parameters. β is used to adjust the location of the onset point of buckling at the intersection of the buckling and unbuckling pathways, r is a factor used to adjust the softening degree of the compressive curve and can only be a real number between 0 and 1 (r = 0 and r = 1 represent the maximum buckling degree and no buckling, respectively), and γ is used to adjust the pinching degree of the re−tensioning pathway which can only be a real number between 0 and 1 (γ = 0 represents the highest degree of reduction). Figure 12 shows the effect of these parameters.

Parameters, such as the yield strength, ultimate strength, slenderness ratio, and elongation of steel bars, changed owing to the influence of corrosion. The mean values of these parameters under specific corrosion rates were calculated using the previous mechanical performance degradation model. The parameter degradation expression is given by Equations (9)–(12).

$$l_{\mathrm{cor}}=l_{\mathrm{sr}}=\frac{D_0\sqrt{\left(1-{\eta }_s\right)}}{L},$$

(10)

$${\sigma }_{\mathrm{ycor}}={\sigma }_{\mathrm{y}}\left(1-\mathsf{\Delta }{\sigma }_{\mathrm{y}}\right),$$

(11)

$${\sigma }_{\mathrm{ucor}}={\sigma }_u\left(1-\mathsf{\Delta }{\sigma }_u\right),$$

(12)

$${\epsilon }_{\mathrm{ucor}}={\epsilon }_u\left(1-\mathsf{\Delta }{\epsilon }_{\mathrm{u}}\right).$$

(13)

The stress–strain curve of the corroded steel bar at a specific corrosion rate η_s can be obtained by substituting the modified parameters into Equations (4)–(8). This study selected the cyclic loading test of corroded steel bars conducted by Kashini [15,27] and used the proposed hysteretic constitutive model to simulate the test data. Figure 13 shows the results.

The results in Figure 13 shows that when the corrosion rate and slenderness ratio were small, the results of the proposed hysteretic constitutive model for corroded steel bars were consistent with the experimental results. The equivalent slenderness ratio l_cor increased with an increase in the corrosion rate. After l_cor reached 12, the prediction results of the model were consistent with the experimental data in the buckling softening path; however, the bearing capacity of the steel bar in the reloading path was underestimated, resulting in a conservative prediction result. However, the proposed hysteretic constitutive model does not require additional tests and exhibits accuracy. Old RC structures typically only have the uncorroded steel material data and estimated corrosion rate of the current steel bar. Moreover, finding the same batch of steel bars to perform the corrosion test is challenging. Therefore, the proposed model can be used as a basis for evaluating the seismic performance of a structure.

The error between the prediction model and the actual situation under the large equivalent slenderness ratio may be due to the failure to consider the geometric deformations after the compression buckling of the reinforcement. These deformations will be partially restored after the reinforcement is re−stretched. Ignoring this recovery will lead to the underestimating of the bearing capacity of the reinforcement. Moreover, the proposed model does not take into account the effect of fatigue of the reinforcement. In future works, the concept of the boundary surface in the steel hyperbolic model [33,34] can be introduced into the proposed model to improve the calculation accuracy. The boundary surface can be modified in each cycle to consider the effects of geometric deformation, strength degradation, fatigue, and other factors.

In general, the proposed model can be used for the safety monitoring of reinforced concrete structures to replace the traditional hysteretic constitutive model. Especially for those structures that have been completed and cannot find the same batch of materials to carry out new mechanical performance tests of corroded materials, the proposed model can still predict its performance and has a certain accuracy. From the perspective of feasibility, all the parameters of the proposed model are derived from the open test data, and it is easy to collect and establish the performance database of corroded steel bar materials.

4. Conclusions

Corrosion of reinforcement will greatly affect its mechanical properties and hysteresis history after buckling which cannot be easily ignored when assessing the seismic performance or monitoring the safety of reinforced concrete structures. Many studies on the degradation of mechanical properties of corroded reinforcement have been completed by previous scholars, but relatively few studies have been conducted to consider the effect of corrosion on the buckling behavior of reinforcement. Only a few scholars, such as Kashini [15,27], have proposed a buckling hysteretic constitutive model for corroded reinforcement, but most of these models rely on experiments to determine the material parameters and are difficult to be applied to the general evaluation of structural seismic performance. The aim of this study is to establish a probability−based hysteretic constitutive model for steel bar buckling. This model needs to be both accurate and easy to use, which means that the required parameters cannot be obtained by additional tests. In order to achieve this goal, a database of mechanical properties of corroded steel bars was first established by collecting a large amount of test data, and then a probability−based degradation law of steel bar mechanical properties was obtained by numerical analysis of these data. As a result, the mechanical properties of steel bars under a given corrosion rate can be predicted. Then, by introducing the degradation law into the Reinforcing Steel Model in OpenSees, a buckling hysteretic model of steel bars considering corrosion is finally obtained. The proposed model can accurately predict the stress–strain curve of steel bars under a given corrosion rate and provide a reference for seismic performance evaluation of existing RC structures after corrosion. The primary observations and results of this study are as follows:

• The 705 steel bar specimens were grouped according to their corrosion rates. Based on the mechanical property data of the uncorroded specimens, the elastic modulus, strength, and elongation loss rates of the steel bar specimens under different corrosion rates were determined, and the normality was verified. Except for a few errors in each group, the loss of mechanical properties of the steel bars under each corrosion rate followed a normal distribution.
• The statistics and regression analysis of the mechanical performance parameters of steel bars under different corrosion rates in the collected data showed that the elastic modulus of the corroded steel bar remained unchanged under different corrosion rates. Moreover, parameters, such as the yield strength, ultimate strength, and elongation at different corrosion rates, can be obtained by linear fitting. A series of monotonic tensile tests of corroded steel bars performed in this study proved that the performance degradation equation fitted by previous data is applicable to the data obtained by the test.
• This study obtained a hysteretic constitutive model for corroded steel bars based on the reinforcing steel model in OpenSees. The model considers the influence of corrosion on the slenderness ratio, yield strength, ultimate strength, and elongation. The proposed model was used to simulate the cyclic loading test of corroded steel bars conducted by Kashini [15,27]. The results demonstrated that the proposed model simulated the test results accurately. The proposed model underestimated the bearing capacity of the re−tension path when the equivalent slenderness ratio reached 12; however, the parameters of the proposed model can be predicted using existing research data, and tests on corroded steel bars need not be performed. Therefore, the proposed model can provide a reference for the seismic evaluation of RC structures.