*2.3. Data Analysis and Indicator System Establishment*

There are many independent indicators affecting PE and IC debonding of the strengthened beams and it is unclear whether there is a relationship between each independent indicator and the debonding failure, so correlation and grey correlation analysis were used to identify the redundant indicators.

#### 2.3.1. Correlation Analysis

The correlation between the indicators is expressed by Pearson correlation coefficient, with a correlation coefficient close to 1 for a positive correlation and close to −1 for a negative correlation [79]. The formula for its calculation is given in Equation (1).

$$r\_{xy} = \frac{Cov(X, Y)}{S\_x S\_y} \tag{1}$$

where *Cov*(*x*, *y*) = *n* ∑ 1 (*Xi*−*X*).(*Yi*−*Y*) *<sup>n</sup>*−<sup>1</sup> , *Sx* <sup>=</sup> *<sup>n</sup>* ∑ 1 (*Xi*−*X*) 2 *<sup>n</sup>*−<sup>1</sup> and *Cov*(*x*,*y*) denotes the covariance of *x* and *y*, *Sx* denotes the standard deviation of variable *x,* and *Sy* denotes the standard deviation of variable *y*

The correlation coefficients (expressed as *P*) between independent indicators were calculated as shown in Figure 3. In it, the absolute value of the correlation coefficient of the indicator is compared with 0.5, and the correlation between indicators is strong if it is greater than 0.5 and the correlation is weak if it is less than 0.5.


**Figure 3.** *p*-value of independent indicators of PE and IC failure.

In Figure 3, A1 to A11 are *L*, *b*, *h*, *Lf*, *bf, tf*, *f'c*, *ρsv*, *fy*, *Ef*, *fpy*; B1 to B11 are *L*, *b*, *h*, *Lf*, *bf, tf*, *f'c*, *ρsv*, *fy*, *Ef*, *ρs*. It can be seen from Figure 3 that there is a significant correlation between A1 and A2, A3, A4, a significant correlation between A2 and A3, A5, a significant correlation between A3 and A2, A4, A9, a significant correlation between A4 and A1, A3, A9, a significant correlation between A5 and A2, a significant correlation between A6 and A10, A11, a significant correlation between A8 and A11, a significant correlation between A9 and A3, A4, a significant correlation between A10 and A6, and a significant correlation between A11 and A6. Additionally, there is a significant correlation between B1 and B2, B3, and B4, a significant correlation between B2 and B1 and B4, a significant correlation between B3 and B1 and B4, a significant correlation between B4 and B1, B2, B3, and B6, and a significant correlation between B6 and B7. In summary, the correlations among the independent indicators of PE failure are complicated, and most have significant correlations; the relationships among some indicators of IC failure are complicated and cannot be eliminated; therefore, mixed indicators are considered in establishing the indicator system. The correlation analysis results of the mixed indicators are shown in Figure 4.

In Figure 4, a1 to a7 are *λ*, *Lu*/a, *bf*/*b*, *f'*c, *fy*, *ρsv*, *Eftf*; b1 to b8 are *λ*, *Lu*/a, *bf*/*b*, *f'*c, *ρs*, *ρsv*, *fy*, *Eftf*. It can be seen from Figure 4 that the mixed indicators of PE failure indicators are weakly or lowly correlated (because the *P*-value between these indicators are less than 0.5), and no indicators need to be removed; the mixed indicators of IC failure are all weakly or lowly correlated with each other also, and as with PE failure, there is no need to eliminate any indicator.

**Figure 4.** *p*-value of mixed indicators of PE and IC failure.

#### 2.3.2. Grey Correlation Analysis

Grey correlation analysis is used to determine the degree of influence of each factor on the system [80]. The basic steps are as follows.


$$\left(\xi(\mathbf{x}\_{0}(k),\mathbf{x}\_{i}(k)) = \frac{a+\rho b}{|\mathbf{x}\_{0}(k) - \mathbf{x}\_{i}(k)| + \rho b}, \forall i,k \tag{2}$$

where *a* denotes the minimum difference of the data in the subsequence and *b* denotes the maximum difference of the data in the subsequence; *ρ* is the resolution factor and generally taken as 0.5.

(d) Search for correlation. Because the correlation coefficient is the value of the degree of correlation between the comparison series and the reference series at each point, it has more than one number, and the information is too scattered to facilitate a holistic comparison. Therefore, it is necessary to pool the correlation coefficients at each point into one value; that is, to find its average value, as a quantitative representation of the degree of correlation between the comparison series and the reference series, the correlation degree is noted as *ri*, and its formula is as follows in Equation (3).

$$r\_i = \frac{1}{N} \sum\_{k=1}^{N} \xi\_i(k) \tag{3}$$

The grey correlation of each indicator of PE and IC with failure is shown in Figure 5.

From Figure 5, it can be seen that the degree of the influence of the parameters on PE debonding are FRP stiffness (Eftf), concrete strength (f'c), the ratio of sheet width to beam width (bf/b), stirrup reinforcement ratio (ρsv), tensile strength of tensile reinforcement (fy), shear span ratio (λ), and location of FRP cut-off point (Lu/a); the output parameter is the shear strength (*Vdb,end*). As for IC debonding, the degree of influence of the parameters on it are concrete strength (f'c), FRP stiffness (Eftf), shear span ratio (λ), stirrup reinforcement ratio (ρsv), the ratio of sheet width to beam width (bf/b), tensile strength of tensile reinforcement (fy), tensile reinforcement ratio (ρs), and location of FRP cut-off point (Lu/a); the output parameter is the debonding strain of FRP (*εfd*). It can be obtained that FRP stiffness (Eftf) and concrete strength (f'c) have a large effect on both PE and IC debonding and this conclusion is roughly the same as the models suggested by codes [15,18–20,22–24].

**Figure 5.** Indicator systems of the debonding failure.
