Regression Analysis of Bond Parameters between Corroded Rebar and Concrete Based on Reported Test Data

Reinforcement corrosion is a major cause of degradation in reinforced concrete structures. The fragile rust layer and cracking and spalling of the cover caused by splitting stress due to rust expansion can alter bond behaviors significantly. Despite extensive experimental tests, no stochastic model has yet incorporated randomness into the bond parameters model. This paper gathered published experimental data on the bond-slip parameters of pull-out specimens and beam-end specimens. Regression analysis was carried out to identify the best fit of bond strength and the corresponding slip value in the context of different corrosion levels from the recollected test results. An F-test confirmed the regression effect to be significant. Residual data were also analyzed and found to be well described by a normal distribution. Crack width data of the tested specimens were also collected. A regression analysis of the bond strength and maximum crack width was carried out given the comparative simplicity of measuring crack width versus rebar area loss. Results indicate that maximum crack width can also be used to predict bond strength degradation with similar variation magnitude.


Introduction
The civil engineering field generally accepts that reinforced concrete structures possess durability problems.One of the harshest environments for concrete structures is the marine environment, as it contains corrosive chloride ions [1].Chloride ions can diffuse into concrete, accumulate at the reinforcement surface, and depassivate the protective layer.Corrosion of reinforcement occurs when chlorides surpass a threshold value; it is the main cause of performance degradation in aged concrete structures.Corrosion products of reinforcement expand in volume with a strength much lower than that of steel, reducing the effective reinforcement area.Expansion of reinforcement rust can also lead to cracking and spalling of the concrete cover when expansion stress surpasses the tensile strength of concrete [2].
The effects of reinforcing-bar corrosion on bond behaviors have been widely studied by many scholars [3][4][5][6][7] for over 30 years using different test setups.Abdullah et al. [8] studied the bond behavior of reinforced concrete members including ultimate bond strength, free-end slip, and failure modes in the precracking, cracking, and postfracture stages.Fang et al. [9] conducted a pull-out test to evaluate the effects of corrosion on bond and bond-slip behavior in specimens with and without stirrups that provided confinement.Wei et al. [10] designed beam-end specimens to study the bond between corroded steel and concrete.However, bond strength reduction due to the fragile corrosion products between concrete and steel has been often neglected in the field.
The research disparity persists between industry and academia in the absence of a unified and feasible model that considers the degradation of bond behaviors.Several factors influence bond performance and the complex nature of interface behaviors between concrete and reinforcement, especially when accounting for the effects of reinforcement corrosion [11].The roughness of the reinforcement surface and confinement of concrete have been shown to exert significant effects on bond performance.Rebar corrosion can certainly change the surface between rebar and concrete; the expansion of corrosion products can also lead to cracking and spalling of the concrete cover, thus degrading confinement.Recent studies [12,13] have revealed that stirrup corrosion International Journal of Corrosion can also degrade the confinement of rebar and concrete, potentially altering the bond failure type, strength, and ductility of confined concrete.In some research, these factors are considered collectively; as experiments often implement different test setups [9,14], the results of each test are often unique.Additional studies on the effects of corrosion on bond parameters are necessary to further develop a widely applicable bond-slip model for assessment of corroded reinforcement concrete structures.
This preliminary study recollected the test data on bond strength and corresponding slip value and maximum crack width from published literatures.Regression analysis was applied to obtain the best fit for the above three parameters.An -test [15] was carried out to verify the regression analysis results.Residuals were also analyzed and modeled as a normal distribution to consider variation in bond behaviors.

Pull-out Specimens
Two different rebars were used for pull-out specimen tests: plain rebar (Table 1) and deformed rebar (Table 2).These two types of rebar exhibit different bond behaviors; the rib of deformed rebar can hook to concrete and change the stress field around the rebar and concrete interface, whereas plain rebar bonds have no similar working mechanism.Each type will be discussed separately in the following sections.1 lists the collected test parameters, where  is the rebar diameter,  is the concrete cover depth, and   is the bond length."-" indicates the parameter was not reported in the literature.As setups involved test specimens with different levels of concrete strength, dimensionless bond strength was used in this paper.The dimensionless bond strength of a corroded rebar was defined as    max / 0 max , where    max and  0 max denote the tested bond strength of corroded steel and noncorroded steel, respectively.In tests with more than one noncorroded specimen,  0 max represents the mean value of the bond strength of the tested noncorroded specimens.

Dimensionless Bond Strength. Table
Figure 1(a) shows the recollected test results of dimensionless bond strength and corresponding corrosion level (the mean rebar mass loss is identical to area loss).Nonlinear regression analysis of the test data included the following equation: where   is the proportion (i.e., percentage) of the extent of steel corrosion. 2 is 0.445, and  2 ⊂ [0, 1] is the coefficient of determination, interpreted as the proportionate reduction in total variation associated with the predictor variable; the closer it is to 1, the greater the degree of association between the predictor and response variables.As shown in Figure 1(a), the dimensionless bond strength first increased and then decreased with an increase in corrosion level.
An -test was further applied to test the significance of the regression curve: where   = ∑(  − ) 2 is the explained sum of squares,   = ∑(  −   ) 2 is the residual sum of squares,   is the predicted value corresponding to the abscissa,  is the average of all predicted values,   is the value of dimensionless bond strength, and  is the total number of data.When  is greater than  1− = (1,  − 2), the regression effect is considered significant at .
The  value of pull-out specimens with plain rebar was 21.0, greater than the corresponding  1− = (1,  − 2) = 7.04 at a significance level of  = 0.01; hence the regression effect was significant.The bond between corroded steel bars and concrete is also affected by other factors, such as concrete strength, protective layer thickness, the existence of stirrups, the diameter of corroded steel bars, and test setup.The central limit theorem states that, under some conditions (including finite variance), the averages of sample observations of random variables drawn from independent distributions converge in a normal distribution; that is, they become normally distributed when the number of observations is sufficiently large.In the present study, dimensionless values have also been used to try to eliminate the effects of these factors; residuals were further assessed to verify the above assumption.Figure 1(b) plots the residuals after nonlinear regression analysis; the residuals are distributed randomly along the horizontal axis.Figure 1(c) presents a histogram of the distribution of residuals, which appear to fit a normal distribution with a mean value () of -0.00738 and standard deviation () of 0.355.The regression formula and normal distribution of residuals will be applied to model the bonds of corroded plain rebar and concrete later in this paper.

Dimensionless Slip Value Corresponding to Bond
Strength.Given different setups of test specimens with different levels of concrete strength, the dimensionless slip value corresponding to bond strength was studied as follows.The dimensionless slip value corresponding to bond strength of corroded rebar was defined as    / 0 , where    and  0 are the slip values corresponding to the bond strength of corroded steel and noncorroded steel, respectively.In cases with more than one noncorroded specimen,  0 denotes the mean value of the noncorroded specimens as that of bond strength.
Figure 2(a) shows the variation in dimensionless slip value corresponding to bond strength at certain corrosion levels.Nonlinear regression analysis of the test data employed the following equation: 2 is 0.540, indicating the existence of unexplained factors.Figure 2(b) shows that the residuals are randomly distributed along the horizontal axis on the negative and positive sides.Figure 2(c) indicates the residual data is well fitted by a  normal distribution with an expectation () of 0.0133, which is close to zero, and a standard deviation () of 0.263. 2 shows collected references of pull-out tests with deformed rebars.Deformed rebar specimens can be divided into two subgroups: with and without stirrups.As confinement can change bond behaviors significantly, these subgroups will be discussed separately in subsequent sections.Stirrup density can also affect the bond properties of corroded steel bars.However, because stirrup density was randomly distributed in these tests, it was not considered in this preliminary study; a dimensionless value was used in relevant analyses instead.(  1.8 + 13.9)

Dimensionless Bond Strength without Stirrups
2 is 0.384.The  value of pull-out specimens with deformed rebar (i.e., no stirrups) was 187, while the corresponding critical value was 6.72 at a significance level of  = 0.01.Thus, the regression effect was significant, and the residuals could be analyzed.Figure 3(b) illustrates that the residuals are distributed randomly around the horizontal axis.Figure 3(c) displays a histogram of residuals on a curve fitted with a normal distribution with a mean value () of -0.000425 and standard  deviation () of 0.253, suggesting the residual data fit the proposed normal distribution well.

Dimensionless Slip Value Corresponding to Bond
Strength without Stirrups.Figure 4(a) shows the dimensionless slip value corresponding to bond strength compared to the corrosion level.The nonlinear regression analysis of the test data used the following equation: The corresponding  2 is 0.377.Figure 4(b) shows the residuals are nearly symmetrically distributed along the horizontal axis.Figure 4(c) presents the histogram of the residuals, which fit a normal distribution well with expectations () of -0.00110, which is close to zero, and a standard deviation () of 0.234.

Dimensionless Bond Strength with Stirrups. Figure 5(a)
shows the variation in dimensionless bond strength with the corrosion level.The nonlinear regression analysis of the test data employed the following equation: 2 is 0.505.The  value of pull-out specimens with a deformed rebar (stirrups) was 211, larger than the corresponding critical value of 6.76 at a significance level of  = 0.01.Thus, the regression effect was significant, and the residuals could be analyzed.Residuals are shown in Figure 5(b), randomly distributed well along the horizontal axis.Figure 5(c) plots the histogram of the residuals and indicates they fit with a normal distribution.The expected value () is nearly zero (0.0000770), and the standard deviation () is 0.190.

Dimensionless Slip Value Corresponding to Bond
Strength with Stirrups.Figure 6(a) illustrates the variation in dimensionless slip value corresponding to bond strength with the corrosion level.Nonlinear regression analysis of the test data involved the following equation: 2 is 0.224, relatively smaller than the previously mentioned cases, presumably due to highly scattered data.Figure 6(b) shows that the residuals are also nearly symmetrically distributed along the horizontal axis.Figure 6(c) presents the histogram of the residuals fitting to a normal distribution with a mean value () of -0.0242 and standard deviation () of 0.274.

Beam-End Specimens
3.1.Dimensionless Bond Strength.Table 3 lists the collected beam-end tests.Figure 7(a) shows the variation in the dimensionless bond strength of beam-end specimens by     corrosion level.The nonlinear regression analysis of the test data employed the following equation: 2 is 0.422.The  values of beam-end specimens confirmed the regression effect was significant, and the residuals could be analyzed.Figure 7(b) plots the residuals after nonlinear regression analysis, randomly distributed along the horizontal axis.Figure 7(c) is the histogram of the residuals, which fit a normal distribution well.The expected value () is 0.000970, and the standard deviation () is 0.170.

Dimensionless Slip Value Corresponding to Bond Strength.
Figure 8(a) shows the variation in dimensionless slip value corresponding to bond strength with the corrosion level of the beam-end specimens.The nonlinear regression analysis of the test data shows the following equation: The corresponding  2 is 0.402.Residuals analysis suggested they are randomly distributed well along the horizontal axis as shown in Figure 8(b).Figure 8(c) provides the histogram of the residuals, which were well fitted to a normal distribution with a mean value () of -0.00488, close to zero, and standard deviation () of 0.215.

Discussions
The above regression analysis outlines the degradation of bond parameters based on the measured rebar corrosion level.In engineering practice, it is nearly impossible to measure rebar area loss without structural damage to the site; however, crack width can be easily measured without resultant damage.Fortuitously, many recollected test results also recorded the maximum crack width involved in rebar corrosion.Figures 9(a) and 10(a) show dimensionless bond strength degradation with an increase in maximum crack width.The two different specimens, pull-out and beamend, were discussed individually.The dimensionless bond strength clearly decreased as maximum crack width increased.The regression analysis involved the following formula for degradation: where  is the maximum crack width.The corresponding  2 values are 0.231 and 0.397 for pull-out specimens and beam-end specimens, respectively.The mean value () and  standard deviation () of the residuals are 0.00304 and 0.297 (see Figure 9(c)) for pull-out specimens.The mean value () and standard deviation () of the residuals are -0.0142 and 0.193 (see Figure 10(c)) for beam-end specimens.Results confirmed that the crack opening could serve as an index to stochastically evaluate bond strength degradation.
From the above studies, the degradation of dimensionless bond parameters can be modeled by the following stochastic process: where parameters , , , , , , , , and  are the variables determined from regression analysis and (,

Conclusions
This paper carried out nonlinear regression analysis of recollected test data on bond strength, corresponding slip value, and crack width following rebar corrosion.Four different test groups were discussed: pull-out tests of plain rebars, pullout tests of deformed rebars with stirrup confinement, pullout tests of deformed rebars without stirrups, and beam-end specimens.An -test indicated the regression formulas were significant.The residuals were further analyzed and fitted by normal distributions.Unified formulas for a stochastic process describing dimensionless bond strength, corresponding slip value as corrosion level, and maximum crack width increases were proposed.Our findings suggest the following: (      showing a standard deviation () of 0.355.The dimensionless slip value decreased as the corrosion level increased.
(2) For deformed rebars of pull-out specimens and beam-end specimens, the regression analysis revealed the dimensionless bond strength initially degraded slightly as rebar corroded and then degraded significantly as the corrosion level continued to rise.Clear bond strength variations appeared between specimens with and without stirrups.As the corrosion level increased, the dimensionless slip value declined significantly.
(3) The dimensionless bond strength could also be predicted from the maximum crack width.Regression analysis showed the crack opening could serve as an index to stochastically evaluate bond strength degradation.
This paper represents a preliminary study based on recollected test specimens with a corrosion level below 20% (i.e., the proposed formula applies only to specimens with a corrosion level lower than 20%).The standard deviations reported in this paper were large in some cases, requiring additional experimental tests to further calibrate and improve the proposed formulas.

Data Availability
The collected data that support the findings of this study are available from the corresponding author, L. X. Li

Table 2 :Figure 1 :
Figure 1: Dimensionless bond strength of pull-out specimens with plain rebar: (a) dimensionless bond strength versus corrosion level; (b) residuals versus corrosion level; (c) histogram of residuals of dimensionless bond strength.
Figure 3(a) shows the variation in dimensionless bond strength at different corrosion levels.Nonlinear regression analysis of the test data applied the following equation: 0133 = 0.263 (c) Histogram of residuals of dimensionless slip value

Figure 2 :
Figure 2: Dimensionless slip value corresponding to bond strength of pull-out specimens with plain rebar: (a) dimensionless slip value versus corrosion level; (b) residuals versus corrosion level; (c) histogram of residuals of dimensionless slip value.

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Histogram of residuals of dimensionless bond strength

Figure 3 :
Figure 3: Dimensionless bond strength of pull-out specimens with deformed rebar (no stirrups): (a) dimensionless bond strength versus corrosion level; (b) residuals versus corrosion level; (c) histogram of residuals of dimensionless bond strength.

Figure 4 :
Figure 4: Dimensionless slip value corresponding to bond strength of pull-out specimens with deformed rebar (no stirrups): (a) dimensionless slip value versus corrosion level; (b) residuals versus corrosion level; (c) histogram of residuals of dimensionless slip value.

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Histogram of residuals of dimensionless bond strength

Figure 5 :
Figure 5: Dimensionless bond strength of pull-out specimens with deformed rebar (stirrups): (a) dimensionless bond strength versus corrosion level; (b) residuals versus corrosion level; (c) histogram of residuals of dimensionless bond strength.

Figure 6 :
Figure 6: Dimensionless slip value corresponding to bond strength of pull-out specimens with deformed rebar (stirrups): (a) dimensionless slip value versus corrosion level; (b) residuals versus corrosion level; (c) histogram of residuals of dimensionless slip value.
Histogram of residuals of dimensionless bond strength

Figure 7 :
Figure 7: Dimensionless bond strength of beam-end specimens: (a) dimensionless bond strength versus corrosion level; (b) residuals versus corrosion level; (c) histogram of residuals of dimensionless bond strength.

Figure 8 :
Figure 8: Dimensionless slip value corresponding to bond strength of beam-end specimens: (a) dimensionless slip value versus corrosion level; (b) residuals versus corrosion level; (c) histogram of residuals of dimensionless slip value.
00304 = 0.297 (c) Histogram of residuals of dimensionless bond strength

Figure 9 :
Figure 9: Dimensionless bond strength versus maximum crack width of pull-out specimens with deformed rebar: (a) dimensionless bond strength versus maximum crack width; (b) residuals versus maximum crack width; (c) histogram of residuals of dimensionless bond strength.
0142 = 0.193 (c) Histogram of residuals of dimensionless bond strength

Figure 10 :
Figure 10: Dimensionless bond strength versus maximum crack width of beam-end specimens: (a) dimensionless bond strength versus maximum crack width; (b) residuals versus maximum crack width; (c) histogram of residuals of dimensionless bond strength.

Table 1 :
Parameters of pull-out specimens with plain rebar.

Table 3 :
Parameters of beam-end specimens.
2) is the normal distribution of residuals.Variables and standard deviations are listed in Table4.The above analysis was based on recollected data indicating a general area loss of less than 20%; as such, these results are only applicable to corrosion levels below 20%.Additional data are still needed to refine the proposed formulas.

Table 4 :
Variables and standard deviations of the proposed stochastic model.
, upon reasonable request.