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Principal Component and Multiple Regression Analysis for Steel Fiber Reinforced Concrete (SFRC) Beams
International Journal of Concrete Structures and Materials volume 7, pages 303–317 (2013)
Abstract
This study evaluates the shear strength of steel fiber reinforced concrete (SFRC) beams from a database, which consists of extensive experimental results of 222 SFRC beams having no stirrups. In order to predict the analytical shear strength of the SFRC beams more precisely, the selected beams were sorted into six different groups based on their ultimate concrete strength (low strength with {f}_{c}^{\prime}<50\phantom{\rule{0.277778em}{0ex}}\text{MPa} and high strength with {f}_{c}^{\prime}<50\phantom{\rule{0.277778em}{0ex}}\text{MPa}), spandepth ratio (shallow beam with a/d ≥ 2.5 and deep beam with a/d < 2.5) and steel fiber shape (plain, crimped and hooked). Principal component and multiple regression analyses were performed to determine the most feasible model in predicting the shear strength of SFRC beams. A variety of statistical analyses were conducted, and compared with those of the existing equations in estimating the shear strength of SFRC beams. The results showed that the recommended empirical equations were best suited to assess the shear strength of SFRC beams more accurately as compared to those obtained by the previously developed models.
1 Introduction
The concept of fiber reinforcing in concrete was introduced more than a century ago, and since then, various types of fibers have been utilized in concrete. Steel fibers were first utilized as effective concrete reinforcements in 1960, and they have been used extensively in many applications of large industrial floors, bridge deck overlays, airport runways, pavements, spillways, dams, slope stabilizations, and many precast products (Dinh 2007, 2010). The steel fibers form bridges through developing cracks in the concrete that provide more resistance against crack growth (Narayanan and Darwish 1987; Li et al. 1992; Lim and Oh 1999). Thus, the inclusion of steel fibers to an ordinary reinforced concrete beam suppresses shear failure in favour of more ductile behaviour (Mansur et al. 1986; Narayanan and Darwish 1987; Ramakrishna and Sundararajan 2005). The steel fibers can prevent excessive diagonal tensile cracking and localization of the tensile crack damage (Choi et al. 2007). A number of existing studies have clearly established the potential use of fiber reinforcement for enhancing the shear capacity of reinforced concrete beams (Batson et al. 1972; Swamy and Bahia 1985; Li et al. 1992; Khuntia et al. 1999; Dupont and Vandewalle 2003; Kang et al. 2011). The increase in shear strength contributed by the steel fibers widely varied from 13 to 170 % (Narayanan and Darwish 1988), 58–125 % (Greenough and Nehdi 2008) and 22–89 % (Swamy et al. 1993) due to the variations of fiber volume, its aspect ratio and anchorage condition, tensile reinforcement ratio, and compressive strength of concrete beams, respectively.
Tables 1 and 2 shows the empirical models and constrains in calculating the shear strength of SFRC beams. As can be noted, the shear behavior of SFRC beams mainly depends on concrete compressive strength ({f}_{c}^{\prime}), tensile reinforcement ratio (ρ), spandepth ratio (a/d), fiber aspect ratio (l_{ f }/l_{ d }) and the amount of fiber in concrete (v_{ f }). The ultimate shear strength of SFRC beams decreases with an increase in the spandepth ratio of beam (Narayanan and Darwish 1987; Mansur et al. 1986; Ashour et al. 1992; Li et al. 1992; Imam et al. 1994; Noghabai 2000; Dinh 2007) and increases with increasing flexural reinforcement ratio (Narayanan and Darwish 1987; Imam et al. 1994; Dinh 2007) and ultimate compressive strength (Narayanan and Darwish 1987; Kwak el al. 2002; Dinh 2007). The shear strength also depends on the amount/volume of steel fibers in the concrete mixture (Khaloo and Kim 1997; Dinh 2007; Madan et al. 2007; Yakoub 2011). Additionally, the aspect ratio and anchorage conditions of the steel fibers greatly influence the shear strength of the SFRC beams (Narayanan and Darwish 1987; Li et al. 1992; Khaloo and Kim 1997).
The shear strength prediction models suggested by the previous research studies were mostly complex and confined to nonlinear regression equations. The past investigations were lacked of simple models, whose preferences are not just for philosophical but also for practical aspects. Multiple linear regression model is a typically form of simple models where more than one independent variables are present. The principal component regression (PCR), special types of regression, can also be utilized on the dataset of multiple linear regression. The PCR uses a set of values of linearly uncorrelated variables called principal components (PCs). The principal component analysis (PCA) compresses and classifies data by evaluating a new set of variables, smaller than the original set of variables, which holds most information of the dataset. PCA determines a set of orthogonal vectors, called loading vectors, which can be ordered by the amount of variance explained in the loading vector directions.
A multiple linear regression model is shown in Eq. (1), and its leastsquares solution is given by Eq. (2), where X^{T}X is singular because of the variables in X exceeds the number of observations or the collinearities. In order to elude the singularity of X^{T}X, the principal component regression (PCR) decomposes X into orthogonal scores T and loadings P, as shown in Eq. (3). As such, regressing Y does not only depend on X itself but also the first a columns of scores T. In the principal component regression, the scores are present by the left singular vector of X multiplied with the corresponding singular values, while the loadings are shown the right singular vectors of X. In PCR, the X matrix consists of the first a principal components (PCs), usually obtained from the singular value decomposition (SVD). Equation (4) presents the X in terms of scores T and loadings P. Finally, and Eq. (5) shows the regression coefficients of the scores.
where the subscript a has been dropped for clarity.
A general agreement existed among the previous research studies that a substantial gain in shear strength of SFRC beams is experienced due to the increase in compressive strength, tensile reinforcement ratio, and fiber volume, and due to the decrease in the spandepth ratio of beam. The existing models in predicting shear capacity of SFRC beams often generated results those are typically in excess of or smaller than the experimental values. As such, an accurate model is extensively needed for the shear strength prediction of SFRC beams. The purpose of this study is to develop empirical models for predicting the shear capacity of SFRC beams using the existing experimental results.
2 Research Significance
The previous investigations in assessing the shear strength of SFRC beams were mostly confined to a limited number of beams having a narrow range of ultimate compressive strength, spandepth ratio of beam, tensile reinforcement ratio, fiber aspect ratio and the volume of steel fibers. Additionally, the existing models were mostly unable to accurately predict the shear strength of SFRC beams. Therefore, the development of an accurate model/equation for predicting the shear strength of SFRC beam is needed for the development of valid design codes, which will enable SFRC to be used as a common building material. This study enhances the existing research studies in introducing the multiple regression models and the principal component regression models in predicting the shear capacity of SFRC beams.
3 Existing Experimental Results
A database containing 222 SFRC beams without stirrups was compiled from 23 existing experimental studies conducted by Batson et al. (1972), Swamy and Bahia (1985), Mansur et al. (1986), Sharma (1986), Murty and Venkatacharyulu (1987), Narayanan and Darwish (1987), Ashour et al. (1992), Li et al. (1992), Swamy et al. (1993), Tan et al. (1993), Imam et al. (1994), Shin et al. (1994), Khaloo and Kim (1997), Lim and Oh (1999), Noghabai (2000), Kwak et al. (2002), Cho and Kim 2003, Rosenbusch and Teutsch (2003), Cucchiara et al. (2004), ACI 318R05 (2005), Adhikary and Mutsuyoshi (2006), Minelli and Vecchio (2006), Greenough and Nehdi (2008), and Dinh et al. (2010). The experimental data of the selected SFRC beams were divided into two groups of shallow and deep beams based on the beam spandepth ratio of greater than or equal to 2.5 and that of less than 2.5, respectively (Choi and Park 2007). The shallow and deep beam groups were also sorted based on their ultimate compressive strength of lower than 50 MPa (low strength concrete) and that of greater than or equal to 50 MPa (high strength concrete) (Esfahani and Rangan 1998; ACI318R05 2005), and the anchorage of steel fibers (hooked or plain and crimped or the combination of plain, crimped and hooked). As a result, the selected SFRC beams were grouped into six categories according to their ultimate compressive strength, spandepth ratio and the shape of steel fibers. They are: (a) low strength beams with a/d < 2.5 and hooked fibers (LSDBH), (b) low strength beams with a/d < 2.5 and plain and crimped fibers (LSDBPC), (c) High strength beams with a/d < 2.5 and plain, crimped and hooked fibers (HSDBPCH), (d) High strength SFRC beams with a/d ≥ 2.5 and plain, crimped and hooked fibers (HSSBPCH), (e) Low strength SFRC beams with a/d ≥ 2.5 and hooked fibers (LSSBH), and (f) Low strength beams with a/d ≥ 2.5 and plain/crimped fibers (LSSBPC). The number of SFRC beams in various configurations incorporated in this study is 41, 58, 31, 49, 33 and 10 for the LSDBH, LSDBPC, HSDBPCH, HSSBPCH, LSSBH and LSSBPC, respectively. The detailed parameters of the SFRC beams are shown in Table 3.
4 Results and Discussions
Figure 1 shows the relationship among the independent variables of compressive strength (CS), reinforcement ratio (RR), spandepth ratio (SDR), aspect ratio (AR), fiber volume (FV) and fiber type (FT) and the response variable of shear strength (SS). As can be shown in Fig. 1, the ultimate shear strength of SFRC beams increased with an increase in the ultimate compressive strength, decreased with an increase in spandepth ratio, and was inconsistent with the reinforcement ratio and fiber aspect ratio. Furthermore, no particular trend was observed between the strength of SFRC beams and the aspect ratio of steel fibers used in concrete.
4.1 Proposed Shear Strength Predict Model
4.1.1 Multiple Regression Analysis
In order to predict the analytical shear strength of SFRC beams more accurately, the experimental shear strength of the selected six types of SFRC beams was analyzed with the five parameters of ultimate compressive strength ({f}_{c}^{\prime}), tensile reinforcement ratio (ρ), spandepth ratio (a/d), shape factor of steel fiber (l_{ f }/d_{ f }) and the volume of steel fibers (v_{ f }). A general pattern of multiple regression analysis, as shown in Eq. 6, was conducted in optimizing the regression parameters of a, b, c, d, e and f. The interactions among the parameters had been neglected as suggested by other researchers (Sharma 1986; Narayanan and Darwish 1988; Li et al. 1992; Khuntia et al. 1999; Kwak et al. 2002; Greenough and Nehdi 2008).
Here, Y is the shear strength of SFRC beams, X_{1} is the ultimate compressive strength, X_{2} is the tensile reinforcement ratio, X_{3} is the spandepth ratio, X_{4} is the shape factor of steel fiber, X_{5} is the volume of steel fibers expressed as a percentage of the total volume of concrete, and a, b, c, d, e and f are the respective coefficients.
Several statistical analyses, such as the performance factor (PF = V_{ EXP }/V_{ ANA }), coefficient of variation (COV), coefficient of determination (R^{2}), average absolute error (AAE), and Chi squared value (λ^{2}), were conducted for the proposed simplified and nonlinear models. Additionally, the mean, standard deviation (SD), coefficient of variation (COE) and confidence at the significance level (α) of 0.05 for the performance factor were evaluated. The above mentioned statistical parameters were also compared with those generated by the previously developed equations in predicting the shear strength of SFRC beams as presented in Tables 1 and 2 except for the models that consisted of the shear and moment of plain concrete beams (Mansur et al. 1986), and the beams that were too specific and designed only for two fiber aspect ratios of 29 and 58 (Khaloo and Kim 1997).
The proposed models in predicting the shear strength of the LSDBH, LSSBH, LSSBPC, HSSBPCH, LSSBH and LSSBPC beams are shown in Eqs. 7–12, respectively. The analytical shear strength of the above mentioned SFRC beams was also determined and compared with that obtained by the existing models. The statistical analyses for the existing and proposed models for shear capacity of LSDBH, LSSBH, LSSBPC, HSSBPCH, LSSBH and LSSBPC beams are presented in Tables 4–9, respectively. The experimental and analytical shear strength values for all above mentioned SFRC beams were correlated, and the results are shown in Fig. 2.
For the beams containing hooked fibers and having low compressive strength with a/d < 2.5, Table 4 illustrates that the previously developed models generated a wide range of low R^{2} values (0.00–0.10) and high Chi squared values (39.16–168.45). Additionally, the mean of the performance factor differed from 0.55 to 1.82, the standard deviation of PFs fluctuated from 0.35 to 1.31, and COV of PFs varied from 50.07 to 72.04. The existing models generated a large amount of scattered data, which predicted an under or overestimation of shear capacity. On the other hand, the proposed model outperformed all previous models with a higher R^{2} value of 0.72, a lower λ^{2} factor of 12.10, and the mean of PFs of 1.04, the standard deviation of PFs of 0.33, and the COV of 31.68. The model also produced the least amount of scattered data (Fig. 2a), which indicated less error in the analytical results as compared to those obtained by the existing models for generating the shear strength of LSDBH beams.
For the low strength SFRC beams with a/d < 2.5 and plain/crimped fibers (LSDBPC), Table 5 demonstrates that the model suggested by Khuntia et al. (1999) produced the highest R^{2} value of 0.96, and the equation proposed by Li et al. (1992) yielded AAE of 33.35, λ^{2} of 4.13, and mean of V_{ EXP }/V_{ ANA } of 0.85. The proposed model generated R^{2} values of 0.99, the AAE values of 5.09, and λ^{2} values of 0.15, respectively. Additionally, the suggested equation produced the mean of PFs of 1.00 with the standard deviations of 0.07, and COVs of 7.02. Moreover, the proposed models produced the narrowest dispersion (Fig. 2b), which indicated more precision in determining the shear strength of the selected SFRC beams. Thus, the proposed model was best suited to determine the analytical shear strength of SFRC beams as compared to that obtained by the previously developed models.
Table 6 shows that the existing models in shear strength prediction for the high strength beams with a/d < 2.5 and plain, crimped and hooked fibers (HSDBPCH) generated a wide range of R^{2} (0.29–0.69), AAE (30.38–122.16), λ^{2} (29.50–328.36) and the standard deviation of PFs (0.63–2.40). The proposed equation yielded R^{2} values of 0.77, λ^{2} value 15.67, and AAE value of 18.12. Additionally, the suggested model showed the mean of PFs of 1.02 with the lowest standard deviations of the PFs of 0.29, and COVs of the PFs of 28.79, and the confidences of 0.08 at the 95 % significance level. The results of the study indicated that the existing equations were not able to better predict the shear strength of HSDBPCH beams, whereas the proposed empirical model superseded all the existing models in evaluating the shear strength of SFRC beams.
Among all the models in predicting the shear capacity of high strength SFRC beams with a/d ≥ 2.5 and plain, crimped and hooked fibers (HSSBPCH), as shown in Table 7, the equations proposed by Li et al. (1992) was most inaccurate, and those recommended by Ashour et al. (1992) underestimated the shear capacity of HSSBPCH beams. Of the existing models, the procedure suggested by Kwak et al. (2002) showed the best results with an R^{2} value of 0.64, λ^{2} of 17.43 and AAE of 15.57. Based on the findings from Table 7 and Fig. 2g and h, the proposed model generated the most precise results compared to all existing models.
Based on the statistical significance of the shear strength prediction models for the low compressive strength SFRC beams with a/d ≥ 2.5 and hooked fibers (LSSBH), as shown in Table 8, the most overestimated mean of performance factor (1.68) and the highest standard deviation (1.03) were generated by Khuntia et al. (1999). The highest R^{2} value from the existing models was shown to be 0.42 (Kwak et al. 2002), whereas the proposed model generated an R^{2} value of 0.82. Figure 2i and j demonstrate that the proposed model outperformed the existing models in estimating the shear strength of LSSBH beams.
As can be seen in Table 9, the existing models in predicting the shear strength of SFRC beams with a/d ≥ 2.5 and plain/crimped fibers (LSSBPC) covered a relatively wider range of R^{2} (0.12–0.43), AAE (17.62–174.25), Chi squared (3.96–51.04) and the mean of PF (0.84–1.29) values. However, the recommended model produced R^{2} value of 0.60, λ^{2} of 1.36, AAE of 9.80, and mean of PF of 1.00, which indicated that the suggested model showed the highest precision in determining the shear strength of LSSBPC beams.
The coefficients of ultimate compressive strength (a), tensile reinforcement ratio (b), spandepth ratio (c), fiber aspect ratio (d) and fiber volume (e) for the proposed model for shear strength prediction of the six groups of SFRC beams, as shown in Eqs. 7–12, are presented in Fig. 3. As can be seen, the ultimate compressive strength had more influence on the shear strength of the LSDBPC beams, whereas a negative impact was observed for the low compressive strength SFRC beams containing hooked fibers (Fig. 3a). The shear capacity of the SFRC beams increased with an increase in the tensile reinforcement ratio, and it varied mainly on the beam configurations (Fig. 3b). Additionally, the shear strength of the SFRC beams varied depending on the beam geometry (spandepth ratio). As can be seen from Fig. 3c, the SFRC beams having a/d ratios of less than 2.5 had no significant effect on its shear capacity.
The study also revealed the identical conclusion conducted by Madan et al. (2007), the average shear stress of the SFRC beams decreases with increasing beam depth. The fiberaspect ratio of the SFRC beams was also an influencing parameter on the shear capacity (Fig. 3d). Regardless of the shape and the aspect ratio of the steel fibers, the shear capacity of the SFRC beams was significantly affected by the volume of steel fibers (Fig. 3e). It was shown that the beams having a/d ratios of less than 2.5 contributed more resistant to shear capacity than those having a/d ratios of greater than or equal to 2.5. The results also indicated that the beams prepared with steel hooked fibers had more resistance to shear forces than those made with the plain and crimped steel fibers.
4.2 Principal Component Regression (PCR) Analyses
4.2.1 PCR Model on Entire Dataset
The principal component analysis (PCA) was conducted with all independent variables of compressive strength (CS), reinforcement ratio (RR), spandepth ratio (SDR), aspect ratio (AR), fiber volume (FV), and fiber type (FT). The result showed that the first four principal components (PCs) explained a total of 94 % variability of dataset. A biplot of the first principal component (PC1) versus the second principal component (PC2) is shown in Fig. 4, where the samples were displayed as points, and the variables were displayed as a vector. Moreover, the variables contributed similar information was grouped together. As can be shown from Fig. 4, the compressive strength (CS) and fiber type (FT) showed similar information and they were positively correlated. The two variables of the reinforcement ratio (RR) and fiber volume (FV) also expressed identical information. The distance to the vector from the origin also conveyed important evidence; farther away from the origin showed the strongest influence on the model. As such, the aspect ratio (AR) had less impact on the model as compared to the influence on the model by the remaining five variables (CS, FT, RR, FV and SDR).
The principal component regression analysis was performed on the shear strength and the scores of the first four principal components. The model is presented in Eq. (13). The predicted shear strength of all SFRC beams, evaluated by the PCR model of Eq. (13), was correlated with the observed shear capacity of the respective beams, and the relationship is presented in Fig. 5. As can be shown, a very good correlation with an R^{2} value of 82.2 % was observed.
where SS is the shear strength of SFRC beams, and s1, s2, s3 and s4 are the scores of the first, second, third and fourth principal components, respectively
4.2.2 PCR Model on Various Groups of SFRC Beams
In this analysis, the entire dataset was divided into six groups based on the ultimate compressive strength, spandepth ratio and the shape of steel fibers used in the beams, as shown in Table 3. For each group of SFRC beams, the principal component analysis was conducted on the five variables of compressive strength (CS), reinforcement ratio (RR), spandepth ratio (SDR), aspect ratio (AR) and fiber volume (FV). It was shown that the first four principal components explained 90.77, 94.68, 92.90, 92.50, 92.14 and 97.15 % variations in the dataset of LSDBH, LSDBPC, LSDBPC, HSSBPCH, LSDBPC and LSSBPCH beams, respectively.
The biplot of the first two principal components for each SFRC beam group is shown in Fig. 6. As can be shown, a wide range of correlations was existed among the five variables. The relationships among the variables depended mostly on the configurations of SFRC beams. For instance, for the LSDBH beams, the RR, FV and AR variables expressed similar information and they were closely correlated, while the relationships among these variables were wide for the LSDBPC and LSSBPC beams. Additionally, fiber volume (FV) contributed positive impact on the LSDBPC beams, whereas it’s influence was negative on the HSDBPCH, LSSBH and LSSBPC beams. The FV had most influence on the HSSBPCH beam, and on the other hand, it’s contribution was minor on the HSDBPCH beams. The FV and AR variables were negatively correlated on the LSDBPC beams, and they were positively associated on the LSDBH beams. It can be demonstrated that each variable contributed differently on the SFRC beams depending on the ultimate compressive strength, spandepth ratio and shape of the fiber used in the beams.
For each type of SFRC beams, the principal component regression analysis was performed on the shear strength and the scores of the first two principal components. Equations (14)–(19) show the model in predicting the shear capacity of the LSDBH, LSDBPC, LSDBPC, HSSBPCH, LSDBPC and LSSBPCH beams, respectively. The predicted shear strengths of various types of SFRC beams was evaluated using the PCR models of Eqs. 14–19, and they was compared with the observed shear capacity of the corresponding beams. The results are shown in Fig. 7. As can be shown, a good correlation existed with R^{2} values of 0.834, 0.987, 0.985, 0.945, 0.928 and 0.991 between the predicted shear strength and that obtained by the experimental procedures for the LSDBH, LSDBPC, HSDBPCH, HSSBPCH, LSSBH, and LSSBPC beams, respectively.
where SS is the shear strength of SFRC beams, and s1, s2, s3 and s4 are the scores of the first, second, third and fourth principal components, respectively
5 Conclusions
The outcomes of this research study can be summarized as follows:

1.
The shear capacity of the SFRC beams increased with an increase in the tensile reinforcement ratio, the shear spandepth ratio of the beam and the ultimate compressive strength.

2.
Regardless of the shape and the aspect ratio of the steel fibers, the shear capacity of SFRC beams increased with an increase in the fiber volume with an exception of high strength beams made with plain, crimped and hooked fibers and having spandepth ratios of more than or equal to 2.5.

3.
The shear strength of the SFRC beams predicted by the previously suggested models was typically in excess of or smaller than the experimental results. Of all the suggested existing models, the model suggested by Narayanan and Darwish (1988) was the most accurate for the shear strength prediction of the LSDBH and LSSBH beams, and that recommended by Li et al. (1992) for the HSDBPCH and LSSBPC beams, respectively. The empirical equation proposed by Kwak et al. (2002) for the LSDBPC beams, and the model recommended by Khuntia et al. (1999) for the HSSBPCH beams, respectively. The study also showed that the model suggested by Sharma (1986), which is currently being used by the ACI, was the most inaccurate in generating the shear strength of SFRC beams. The study showed that the predictions made by all existing models varied quite significantly and indicated inaccurate when they were compared with a large experimental database.

4.
The study indicated that the theoretical shear strength of SFRC beams computed by the suggested model were more precise and more accurate compared to those evaluated by the existing shear strength prediction models.

5.
Principal component regression models outperformed not only the existing equations but also the multivariate regression model presented herein in evaluating the shear strength of the SFRC beams.
Abbreviations
 a :

Shear span, mm
 a/d :

The ratio of shear span and effective depth
 A _{ sw } :

Crosssectional area of shear reinforcement, mm^{2}
 b :

Beam width, mm
 d :

Effective beam depth, mm
 d _{ a } :

Maximum aggregate size, mm
 d _{ f } :

Fiber diameter, mm
 F :

Fiber factor
 F _{ y } :

Yield strength of shear reinforcement, MPa
 {f}_{c}^{\prime} :

Concrete compressive strength, MPa
 f _{ f } :

Flexural strength of plain concrete, MPa
 f _{ sp } :

Splitting strength of plain concrete, MPa
 h :

Beam height, mm
 l _{ f } :

Fiber length, mm
 l _{ f } /d _{ f } :

Fiber aspect ratio
 η _{ o } :

Fiber orientation factor
 ρ :

Tensile reinforcement ratio, %
 SFRC:

Steel fiber reinforced concrete
 τ :

Fibermatrix interfacial bond strength, MPa
 V/M :

External shear to moment ratio
 V _{ c } :

Shear resistance of the concrete, MPa
 v _{ f } :

Fiber volume in percentage
 V _{ EXP } :

Experimental shear strength, MPa
 V _{ ANA } :

Predicted shear strength, MPa
 V _{ u } :

Ultimate shear strength, MPa
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Sustained financial support by the Natural Sciences and Engineering Research Council (NSERC) of Canada has been critical for conducting this research.
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Islam, M.S., Alam, S. Principal Component and Multiple Regression Analysis for Steel Fiber Reinforced Concrete (SFRC) Beams. Int J Concr Struct Mater 7, 303–317 (2013). https://doi.org/10.1007/s4006901300597
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DOI: https://doi.org/10.1007/s4006901300597