Principal Component Analysis (PCA)

The structure of correlations in Table 2 proved that these six variables are not independent and therefore they could not be treated as such when modelling their effect on creep. Principal Component Analysis (PCA) was used to condense the information they describe into a reduced set of variables.

After these variables were centered and scaled to unit variance, principal components were extracted by singular value decomposition of the correlation matrix [11], and a Varimax rotation was applied [12]. The first three principal components were retained as sufficiently informative, explaining 97.11 % of the total variance in the original variables.

Each one of the principal components is a linear combination of the original variables, and therefore they define new rotated axes PC1, PC2 and PC3. By plotting the weights of the original variables in these linear combinations against the new coordinate system defined by PC1, PC2, and PC3, the two biplots in Fig. 1 were obtained.

Table 2 Correlation matrix

 fc fL fR1 fR2 fR3 fR4 fc (1.000) 0.310 0.055 0.090 0.066 0.076 fL (1.000) 0.654 0.633 0.598 0.572 fR1 (1.000) 0.954 0.939 0.908 fR2 (1.000) 0.949 0.902 fR3 (1.000) 0.978 fR4 (1.000)

Fig. 1 Biplots for interpretation of the principal components after the PCA

The four residual flexural strength parameters formed a very clear cluster. This cluster defined the first component PC1, and therefore PC1 was interpreted as representative of the residual load-bearing capacity. Furthermore, as the distances between fR1, fR2, fR3 and fR4 were negligible, it followed that PC1 values were proportional to /rj + fR2 + f R3 + fR4,

Concrete compressive strength, on the other hand, was clearly apart from this cluster, consistently with what the bivariate correlations indicated. Therefore, the second component PC2 was defined by compressive strength alone.

These two components PC1 and PC2 described 89.70 % of the total variance. The third component (PC3), accounted for only 7.41 %, and had no clear interpretation, as neither the limit of proportionality nor any of the other variables were found to have a determining weight.

Redefinition of the Variables Characterising FRC Mixes

The results of the PCA clearly indicate that the mechanical performance of FRC mixes as described by the six initial variables can be effectively reduced to only two variables while retaining nearly 90 % of the total variance:

• - Concrete compressive strength, fc, in MPa.
• - Toughness, defined as T = fR1 + fR2 + fR3 + fR4, in MPa.