3.1. Responses of the Compression Behavior of the SFRC: Analysis with Physical Magnitudes
From the information of the database, the responses that describe the compressive strength of the SFRC, which are , , , , and , have been studied. Each one of them has been analyzed by means of the following factors: the compressive strength of the base concrete or matrix without fibers, , the maximum size of the aggregate, , the parameters that define the geometry of the fiber , , and , the fiber volume ratio, , and a parameter that takes into account the relationship between the sizes of fiber and the aggregate, .
Figure 3 shows a scheme of the process followed to apply the RSM to the results from the database. All the responses were initially adjusted to two models of complete first and second degree polynomic regression. However, it was verified that the second degree models did not correctly predict the responses in the definition domain of the levels of the factors, for which reason they were disregarded for their use in the analysis. Furthermore, these models had excessively high VIF values in the majority of their terms, which indicated the presence of a strong multicollinearity. As stated above, this phenomenon caused the loss of precision in the estimation of the coefficients of the factors, as well as alterations in the detection of their statistical significance. It bears emphasizing that the fiber factor,
, was not found to be significant in the two degree polynomial models, neither in the quadratic nor in the combined adjustments, as shown in the Pareto diagrams in
Figure 4. The corresponding VIFs were 3.85 and 2.85, respectively. Thus, it can be assumed that the level of multicollinearity for
in both cases was low, and consequently, the low significances given by ANOVA were reliable.
Therefore, the analysis of the responses has been performed exclusively with first degree models (Equation (
9)). After verifying that the linear models correctly predicted the responses in the domain of the definition of factors, the study of the VIF revealed that they were greater than 10 in the majority of factors, except for
and
, for which VIF ≃ 1. This fact demonstrated that multicollinearity still existed, for which reason it would be appropriate to make a selection and reduce the number of factors with which the responses are analyzed.
The factors that were statistically significant were determined by making use of the Pareto diagrams.
Table 2 contains coefficients of the
adjustment of the
factors in the linear models for each one of the responses, as well as the terms that were statistically significant. Each response can be calculated as:
When making an adjustment among the variables that have different units, as is the case, some of the coefficients of the factors also require units to conserve the dimensional stability. Thus, the coefficients will have the corresponding response units (for example, for the response , is expressed in MPa, and for the response, does not have any units). For the remaining coefficients of the factors, the definition of their units is as follows: the numerator corresponds to the response unit, and the denominator corresponds to the unit of the factor to which it is associated (for example, for the response, the coefficient, associated with the factor, has GPa/mm as a unit). Subsequently, a non-dimensional analysis of the variables will be performed so that the coefficients will be expressed without units, which facilitates the manipulation of expressions.
The diagrams of the residues obtained in the analysis indicated that they were distributed randomly, thereby adhering to a function of normal probability (
Figure 5 shows two of the diagrams, for the compressive strength and for the volumetric deformation work in the post-peak). This confirms the premise that the origin of the experimental data must demonstrate normality in its distribution. The value of the
goodness adjustment coefficient was not so important insofar as the interest of this first analysis concerns the identification of the factors that really affect the variation of the response. These factors were different for each of the responses studied, although it was certain that a series of factors appeared that had an effect on the majority of them:
was statistically significant in all the responses, except in
;
and
were statistically significant in all except for
;
was not significant only in
. Nevertheless, it must be highlighted what happens with the
and
factors, the latter of which had a statistical significance in all the responses (note that
was not affected by the strength of the matrix, which seemed to make sense since this work corresponded to the post-crack branch of the matrix). In addition, both
and
were the only factors with VIF
, which was symptomatic of the true nature of the problem, as may be verified when studying the variables of a non-dimensional manner.
3.2. Responses of the Compression Behavior of the SFRC: Non-Dimensional Analysis
Presented below are the results uncovered when applying the RSM in a non-dimensional manner. Each one of the responses and factors was divided by a parameter that had the same physical dimension: the responses were divided by the value of the same variable corresponding to the concrete matrix without fibers from which it proceeded (for example, the response resulted from the division of by ). Some factors with dimensions of length were divided by an auxiliary parameter, , which was the average value of the fiber lengths included in the database, which was roughly equal to 30 mm. The analysis of the factor was eliminated from the analysis since it was actually present in the non-dimensional stresses. Besides, we sought to only study the influence of those factors that were related to the materials (coarse aggregate and fiber) and not with the strength of the matrix (in the previous section, the statistical significance of was verified in each one of the responses except in ).
The objective that was pursued with this analysis was two-fold: on the one hand, to observe how each one of the factors affected the increase or the decrease of the responses and, on the other hand, to identify those factors that had a statistical significance that may be used in the proposed model, which describes the compression strength of the SFRC [
42]. The responses were adjusted by means of the linear model.
Table 3 presents the linear coefficients of the
adjustment of the
factors for each one of the responses (all are expressed in a non-dimensional form).
From this analysis, it follows that did not have a significant relationship with any factor. This result was in line with that demonstrated by the scientific literature regarding the disparity of conclusions concerning the effect that the addition of fiber has on the variability of the elastic modulus.
Once again, as a consequence of the number of factors that were introduced in the analysis, the VIFs obtained were higher in some terms (only
maintained a VIF ≃ 5, as occurred in variables with dimensions). This fact may conceal factors that are significant and that do not appear as such and vice versa. For this reason, there will be a reduction in the number of factors to study in the compression behavior model of SFRC proposed by Ruiz et al. [
42]. Three parameters:
,
, and
, characteristic of fiber reinforcement and expressed in a non-dimensional manner, were selected. The reason for this choice was that the first two defined the geometry of the fiber and the last made reference to its content. In addition,
was the only factor that had a statistical significance in each of the responses, and
was the next factor with greater significance for all of them. On the other hand, these three parameters were those that intervened in the majority of the models existing in the literature that describes the
-
compression response of SFRC.
3.3. Stress–Strain Model of the Compression Response for SFRC
This section will study the influence that the factors
,
, and
had on the SFRC compression model proposed by Ruiz et al. [
42]. The most important aspect that must be highlighted is the gain in the capacity of energy absorption that the addition of steel-fibers in the concrete provides. It may increase up to 45% in the segment prior to the peak load and reach almost three times more in the segment subsequent to the peak load in average values with respect to the concrete without fiber reinforcement. In addition, once the maximum load has been surpassed,
may reach a maximum value five times greater than the matrix without fiber. This fact demonstrates the increase in ductility that a structural element of SFRC, requested at uniaxial compression, may exhibit.
Table 4 shows the statistical values of each one of the analyzed responses of the compression strength of SFRC obtained from the tests included in the database.
From the linear adjustment and the statistical analysis of the relationships between the defined variables, the
coefficients of the model were calculated. They are included in
Table 5.
Figure 6,
Figure 7,
Figure 8 and
Figure 9 represent the Pareto diagrams for the
,
,
, and
responses, as well as the adjustment diagrams of their response surfaces with respect to the statistically significant factors.
Once again, the absence of the statistically significant relationship of with any factor was verified, as occurred in the prior analysis, which included a majority of factors in the model, thereby reinforcing the idea of the absence of the relationship between the variability of the elasticity modulus and the inclusion of the fiber in the concrete. The physical reason behind this fact maybe that, as the fiber content increases, the SFRC porosity also increases, which in turn leads to a decrease in the modulus of the base concrete that compensates the stiffening effect due to the presence of fibers.
Insofar as
, the SFRC compression strength, it was only significantly affected by
, thereby demonstrating a slight increase as fiber was added. Despite the low value of
(5.3%), the statistical significance, demonstrated by
, and the positive value of its adjustment coefficient in the model indicated the tendency of the increase of
, in mean values. This is how this effect must be interpreted, as observed in
Figure 10. The value of
may be easily measured through the uniaxial strength test [
67] or calculated by means of Equation (
10), which was obtained from a new linear correlation between
and
alone:
The RSM can also provide the characteristic value, understood as the value corresponding to a cumulative probability of 5% in the distribution curves, as shown in
Figure 10. The result is:
Furthermore,
may be experimentally measured with the uniaxial compression test, although it also may be estimated with the coefficients of
Table 5.
The residual compressive strength,
, defined in the model of Ruiz et al. [
42] (equal to the non-dimensional compressive stress associated with a deformation
= 3) is calculated in Equation (
12):
In the analysis performed,
and
were statistically significant, for which reason a new analysis was executed again with the result that both of these factors were once again significant.
and
were then related to
by means of Equation (
13):
Figure 11 represents the Pareto diagram and the response surface of the statistically significant factors of
. Its corresponding characteristic value is:
It should be noted that Equations (
13) and (
14) are defined only for concretes with a minimum fiber content and thus, they do not provide negative values (indeed, all the SFRCs in the database yielded positive residual strengths since the minimum values in the database for
and
were 0.0024 and 20, respectively). It should be reminded that
can also be obtained from experimental
-
records using Equation (
8). It bears emphasizing that both
and
were calculated in mean values since the database contained exclusively the tests’ results. Nevertheless, in light of the concept of characteristic resistance (expressed as that corresponding to the quantile of 5% in the distribution of resistances obtained in a group of tests on similar specimens), Equations (
11) and (
14) were determined. They permit calculating the characteristic values of
and
, if one wishes to work with them.
The residual compressive strength can also be related to the residual flexural strengths
and
, which are the flexural strengths for crack mouth opening displacements of 0.5 mm and 2.5 mm. Ruiz et al. [
71] showed that the dimensional and non-dimensional expressions relating these parameters, on average and in characteristic fashion, are:
The residual flexural strengths
and
in Equations (
15)–(
18) were introduced in MPa; the residual compression strengths
and
were obtained in MPa, whereas
and
were non-dimensional. Equations (
15)–(
18) were derived from an RSM study on a flexural database with experiments reported by Tiberti et al. [
72]. Note that non-dimensional values of the residual compressive strength depended only on
because the ratio
was not found to be significant for them [
71]. It bears emphasizing that fibers greatly enhanced both flexural and compressive SFRC behavior as compared with the same values of the corresponding base concrete. However, so far, it was only the flexural behavior of SFRC that was accounted for by the standards in [
68,
69], to the extent that the characteristic residual flexural strengths
and
were taken as indices to perform an SFRC classification [
69]. Consequently, as stated in [
71], it is appropriate to relate the residual compressive strength
to its flexural counterparts, not only to the fiber parameters.
In the responses
,
,
, and
, the
factor was statistically significant, and its effect was positive on their variation.
was also significant on
,
, and
with a positive effect on variability. This demonstrated that the ductility and capacity of energy absorption from the maximum strength in the phase subsequent to the cracking of the SFRC were governed by a parameter related to the geometry of the fiber, such as
, and for the fiber content,
. This result was in line with that set forth by Shah et al. [
31,
32] for SFRC and by Fanella et al. [
31,
33] for steel-fiber reinforced mortars. As previously explained, the importance of the statistical analysis lies with the determination of the significance of the factors and not with the
goodness adjustment coefficient since this permits determining if a factor has a real influence on the explanation of the response variability. The coefficients
only indicated the dispersion of the data around the mean adjustment surface of the selected linear model. Finally, it must be emphasized that the VIF statistic had a value ≤ 5 in almost all of the analyzed responses. Actually, the VIF reached a maximum value of 5.3, which may be accepted in terms of the moderately low multicollinearity in the statistical analysis of the model that was performed.