2.4.1. One-Way ANOVA

The most commonly used statistical method to analyze the contribution of each level in one factor of experimental results is One-way ANOVA and *F*-tests method, which can take advantage of sums of squares to separate the overall variance in the response into variances caused by measurement error and processing parameters [36].

In One-way ANOVA, the sum of square *SST* can be calculated by

$$SS\_T = \sum\_{i=1}^{m} \sum\_{j=1}^{n} y\_{ij}^{\prime} - \frac{1}{N} (\sum\_{i=1}^{m} \sum\_{j=1}^{n} y\_{ji})^2 \tag{7}$$

where *m*, *n* are the number of levels in each factor and the number of experimental results in each level, respectively. *yij* is the *j*th experimental result of level *i*. *N* is the number of all experimental results, which is equal to *m* × *n.*

The level square of deviance *SSi* can be calculated by

$$SS\_i = \frac{1}{n} \sum\_{i=1}^{m} \left(\sum\_{j=1}^{n} y\_{ij}\right)^2 - \frac{1}{N} (\sum\_{i=1}^{m} \sum\_{j=1}^{n} y\_{ji})^2 \tag{8}$$

The error square of deviance *SSe* can be obtained by

$$SS\_{\mathfrak{e}} = SS\_T - SS\_{\mathfrak{i}} \tag{9}$$

The estimate of variance is given by

$$MS\_i = SS\_i / d\_{f\_i} \tag{10}$$

$$MS\_c = SS\_c / d\_{f\_c} \tag{11}$$

where *MSi* and *MSe* are the estimate of variance for level *i* and error, respectively. *dfi* and *dfe* are corresponding degrees of freedom, which can be obtained by

$$d\_{f\_i} = m - 1\tag{12}$$

$$d\_{f\_e} = N - 1\tag{13}$$

*F*-value is given by

$$F = MS\_i / MS\_\varepsilon \tag{14}$$

In this method, whether the influence factor is significant or not depends on *F*-value. The larger *F*-value is, the higher the influence factor is.

#### 2.4.2. Turkey's HSD Test

Meanwhile, Turkey's HSD is a statistical test procedure. It is a post-hoc and single-step multiple comparison procedure, which can be performed combining with analysis of variance method to determine means that are significantly different from each other or not [37,38].

In Turkey's HSD test, HSD value is the critical value to judge whether the influence of factor is significant or not. HSD critical value can be calculated by

$$HSD\_a = q\_a(m, d\_{f\_\varepsilon}) \times \sqrt{\frac{MS\_\varepsilon}{n}} \tag{15}$$

where *qα*(*<sup>m</sup>*, *df*e ) is studentized range; *α* is significant level; *m* is number of levels in each factor; *df*e , *MSe* and *n* are number of degrees of freedom, estimate of variance of error, number of test results in each level, respectively. In this study, the range values of means between each two levels in specific factor will be calculated and compared with *HSD<sup>α</sup>*. If the range value of means is larger than *HSD<sup>α</sup>*, the two levels are said to have significant effects on rebar corrosion.

#### **3. Results and Discussion**

#### *3.1. Recommendation of Reasonable Electrochemical Test Method*

As an example, *icorr* values of sample B2 calculated from 3 measurements are analyzed to illustrate the reasonable electrochemical test method for rebar in the RC corrosion system.

In TPP measurements, anodic and cathodic potentiodynamic polarization curves for rebar in sample B2 are shown in Figure 8. The curve tested before the chloride penetration is defined as initial curve, while the curve tested after the chloride penetration is defined as final curve.

**Figure 8.** TPP curves of rebar in sample B2.

The anodic and cathodic curves are extrapolated up to their intersection at a point where corrosion current density and corrosion potential can be obtained, as shown in Figure 5. The anodic Tafel slopes (*ba*), cathodic Tafel slopes (*bc*), Stern-Geary coefficient (*B*), corrosion current density (*icorr*) and corrosion potential (*Ecorr* vs. SCE) obtained from TPP curves are listed in Table 3.

It can be concluded *icorr* increases and *Ecorr* (vs. SCE) decreases under the action of chloride penetration. Meanwhile, both *ba* and *bc* increase, which illustrates the anodic and cathodic reactions are all accelerated and affected by the environmental loading.

**Table 3.** Electrochemical parameters from TPP curves of rebar in sample B2.


In LP measurements, the current density can be also calculated by Equation (1). In Equation (1), *B* is obtained from TPP curves and *Rp* is calculated from line polarization curves by Equation (4). The calculated *Rp* and *icorr* from LP measurements are listed in Table 4.


**Table 4.** Electrochemical parameters from liner polarization (LP) curves of rebar in sample B2.

It can be seen that *Rp* is reduced while *icorr* increases under the action of chloride penetration, which infers that rebar is continuously corroded.

In addition, the corrosion current density can be also obtained by EIS measurements. Figure 9 depicts the impedance spectra of rebar in sample B2. The curves before and after the chloride penetration experiment are defined as one initial curve and three final curves, respectively. From Figure 9, it can be seen that the radius of the capacitive loop shrinks after the chloride penetration experiment decreases, which indicates the corrosion resistance of rebar is declined. The equivalent circuit is fitted with impedance spectra by Equation (5), and the results are shown in Table 5.

In Table 5, *icorr* is calculated by Equation (1), where *B* is obtained from TPP curves, and *Rp* is equal to *Rp*3. From Table 5, it can be concluded that the impedance spectra of rebar in sample B2 obtained from three continuous tests is reproducible. Besides, both *Rp*2 and *Rp*3 decrease after the chloride penetration, which indicates that material properties of concrete were degraded and the rebar was corroded gradually under the process of chloride penetration.

**Figure 9.** Impedance spectra of rebar in sample B2.

**Table 5.** Fitting parameters from electrochemical impedance spectroscopy (EIS) of rebar in sample B2.


The corrosion current densities obtained from 3 measurements are compared and shown in Figure 10. It can be seen the corrosion current densities from EIS measurements are larger than those of TPP and LP measurements. The reason is that *Rp* in TPP and LP measurements contain not only the resistance of rebar but also the resistance of concrete in the test conductive circuit, while *Rp* from EIS measurements is exactly the resistance of rebar. Thus, *Rp* from EIS measurements is smaller than those from TPP and LP measurements. According to Equation (1), *icorr* calculated from EIS measurements is greater than the others. Inductively, the reasonable way to calculate *icorr* of rebar is EIS measurement combined with TPP measurement, namely, *Rp* and *B* should be calculated by EIS measurement and TPP measurement. After that, *icorr* can be calculated by Stern-Geary equation. In this way, the corrosion behavior of rebar can be judged more accurately compared with only a single measurement.

**Figure 10.** Comparison of corrosion current densities obtained from 3 measurements.

#### *3.2. The Effect of Crack on Corrosion Behavior of Rebar*

*icorr* of rebar in each specimen could be obtained by the same test method of B2 discussed above. The variation of corrosion current densities is calculated by

$$i\_{corr}^n = i\_F{}^n - i\_I{}^0(n=1,2,3) \tag{16}$$

where *incorr*, *iFn* and *iI*0 are the *n*th variation of *icorr*, the *n*th corrosion current density and initial corrosion current density, respectively. The results are listed in Table 6. The average value for 3 variations of *icorr* and corresponding standard deviations are also calculated. The results can be used to analyze effects of crack width, number and spacing on corrosion behavior of rebar under the aggressive environment. Comparative evaluations were conducted based on One-way ANOVA and Turkey's HSD test.

**Symbol Crack Width (mm) Crack Number Crack Spacing (mm)** *<sup>i</sup>***1***corr* **(**μ**A/cm2)** *<sup>i</sup>***2***corr* **(**μ**A/cm2)** *<sup>i</sup>***3***corr* **(**μ**A/cm2) Average Value (**μ**A/cm2) Standard Deviation (**μ**A/cm2)** N 0 0 - 4.00 3.88 3.63 3.84 0.19 A1 0.05 1 - 7.81 7.76 7.68 7.75 0.07 A2 0.1 1 - 8.06 8.12 8.04 8.07 0.04 A3 0.2 1 - 8.92 8.86 8.88 8.89 0.03 B1 0.1 2 15 9.48 9.36 9.44 9.43 0.06 B2 0.1 2 25 9.24 9.32 9.34 9.30 0.05 C1 0.1 3 15 12.86 12.75 12.78 12.80 0.06 C2 0.1 3 25 12.44 12.54 12.45 12.48 0.06

**Table 6.** Variation of corrosion current density in each sample.

In One-way ANOVA, corrosion current densities of rebar in sample N, A1, A2 and A3 are used to analyze the effect of crack width on rebar corrosion. One-way ANOVA results for crack width are listed in Table 7.

**Table 7.** One-way ANOVA for crack width.


\*\* Correlation is significant at the 0.01 level.

Besides, corrosion current densities of rebar in sample N, A2, B1 and C1 are used to analyze the effect of crack number on rebar corrosion. Results of One-way ANOVA for crack number are listed in Table 8.

**Table 8.** One-way ANOVA for crack number.


\*\* Correlation is significant at the 0.01 level.

In addition, corrosion current densities of rebar in sample B1 and B2 are used to analyze the effect of crack spacing on rebar corrosion. Results of One-way ANOVA for crack spacing are listed in Table 9.


**Table 9.** One-way ANOVA for crack spacing.

\* Correlation is significant at the 0.1 level.

From Tables 7–9, it can be concluded that *F*-values of crack width and crack number are larger than *F*0.01 (3,8), which means the probability is 99% that crack width and crack number present a significant influence of rebar corrosion. Meanwhile, *F*-value of crack number is larger than that of crack width, which means the influence of crack number is larger than crack width. Besides, *F*-value of crack spacing is between *F*0.1 (1,4) and *F*0.05 (1,4), which means that the probability is 90% that crack spacing is the significant effect on rebar corrosion.

According to the conclusions discussed above, it is more likely that crack width and crack number are the statistically significant effects of rebar corrosion. However, it doesn't mean that there is a larger magnitude between these two factors [39]. Therefore, Turkey's HSD test as a post-hoc multiple comparisons method should be conducted to determine the significant effects of three influence factors on rebar corrosion.

Similar to One-way ANOVA, means of corrosion current densities of rebar in sample N, A1, A2 and A3 are performed Turkey's HSD test to determine if crack width is the significant effect on rebar corrosion. Results of Turkey's HSD test are listed in Table 10.

**Table 10.** Turkey's honest significant difference (HSD) test results for crack width.


\*\* Correlation is significant at 0.01 level. - Correlation is not significant.

In Table 10, most range values between each two levels ( *K*0.05–*K*0 etc.) in crack width factor is larger than *HSD*0.01, which means the influence between each two levels is significant at 0.01 level. It is also proved that crack width has a statistically significant influence on rebar corrosion.

Besides, means of corrosion current densities of rebar in sample N, A2, B1 and C1 are adopted to perform Turkey's HSD test to determine if crack number is the significant effect on rebar corrosion. Results are listed in Table 11.

**Table 11.** Turkey's HSD test results for crack number.


\*\* Correlation is significant at 0.01 level.

In Table 11, all range values between each two levels (*K*1–*K*0 etc.) in crack number factor is larger than *HSD*0.01, which means the influence between each two levels is significant at 0.01 level. It is also proved that crack number has a statistically significant influence on rebar corrosion.

In addition, means of corrosion current densities of rebar in sample B1 and B2 are adopted to perform Turkey's HSD test to determine if crack spacing is the significant effect on rebar corrosion. Results are listed in Table 12.


**Table 12.** Turkey's HSD test results for crack spacing.


In Table 12, the range value of means between 15 mm spacing level and 25 mm spacing level in crack spacing factor is not significantly different, which can be concluded that crack spacing is not a significant effect on rebar corrosion. Moreover, the results of Turkey's HSD test reveal that crack width and crack number possess a greater effect on rebar corrosion. In summary, the influence degrees of three factors including crack width, number and spacing on rebar corrosion can be ranked as crack number, crack width and crack spacing from the greatest to the least.

Crack leads to the greater capillarity and osmotic pressure, resulting in a serious corrosion of rebar in the vicinity of cracks. Meanwhile, the larger crack width causes a more oxygen and adverse ion aqueous solution to diffuse into concrete. It is said that rebar corrosion rate maybe depend on total crack width. To some extent, the total crack width would increase with increasing of crack number. Therefore, crack number presents the most significant effect on corrosion of rebar.
