Effects of the addition of short straight steel fibers on ...

Material constants of the composite

The volume proportion of steel fibers (Vf), which was 0.75% or 1.5% in batches B and C, resulted in an increase of 9.6% and 21% in compressive strength (fc) compared to batch A (control), respectively. The coefficient of variation of fc strength of the individual batches was 5.0%; 12.7% and 2.9%, respectively (Table 2). The results obtained are confirmed by studies that report an increase in the compressive strength of high-strength fiber-reinforced concrete from 10 to 20% with the addition of fibers 0&#;2%6,32,33.

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Selected statistics of strength distribution fc and strains ε0 are shown in a box plot (Fig. 6). These are the positions of the arithmetic mean, median, first and third quartiles, outlying values, maximum and minimum values (including the median). The outlying observation visible on the graph (79.9 MPa&#;sample B2), was further confirmed with the Grubbs statistical test (in the remainder of the paper, regression analyses and σ&#;ε comparisons were performed excluding the outlier). Based on the standard34, samples from batch A were assigned a concrete grade of C90/105, whereas those from batches B and C&#;were assigned a grade of C100/115.

Box plot of: compressive strength ((a); strain εc measured with LVDT sensors ((b), strain gauges ((c) and DIC method ((d).

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In order to determine the significance of differences between the mean values of the compressive strength tests of each batch, one-way analysis of variance (ANOVA) was performer using the Statistica software, taking the fiber-reinforcement ratio (Vfλ) as a qualitative predictor. The assumptions of the ANOVA tests in the form of normality of distribution within groups and homogeneity of variances were met. The normality of distribution is shown in Fig. 7 and confirmed by the Shapiro&#;Wilk test (test value p&#;=&#;0.13 and is greater than the accepted level of significance α&#;=&#;0.05). Homogeneity of variance was checked with Levene&#;s test (p&#;=&#;0.62&#;>&#;α). The ANOVA analysis of variance test shows that the p value is lower than the accepted level of significance, so there is a basis for rejecting the null hypothesis. This means that mean compressive strength in batches A, B and C is significantly different (Table 3). Analogous analyses and conclusions apply to the differences between the mean values of different batches as regards: strain ε0, ultrasonic wave velocity v and toughness ratio TR (Table 3). In all cases, p&#;<&#;&#;<&#;α was obtained with the exception of differences in mean strain ε0DIC (p slightly greater than α).

Normality diagram of compressive strength (explanation: \({y}_{i}={(x}_{i}-\overline{x})/s\); \({\widetilde{y}}_{i}={\phi }^{-1}({u}_{i})\); \({p}_{i}=(3i-1)/(3n+1)=\phi ({u}_{i})\); \({x}_{i}\)&#;compressive strength results, ordered in ascending order; \(\overline{x}\)&#; arithmetic mean; s&#;standard deviation; \(\phi ({u}_{i})\)&#;standard normal distribution cumulative distribution function for compressive strength; \({p}_{i}\)&#;cumulative frequencies; i&#;index in ascending series; n&#;number of observations).

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The addition of fibers increased the strain ε0 corresponding to the maximum compressive stress (this was confirmed by all strain measurement methods). For batch B, the increase was 7&#;12%, while for batch C it was 14&#;19%.

Figure 8a shows the results of compressive strength and density of the composite depending on the degree of fiber reinforcement (Vfλ) and the regression equation Eq. (2) with the determination coefficient R2&#;=&#;0.79. The standard error of the slope coefficient was 2.62. Based on t-Student statistic, for the assumed level of significance, a significantly statistical structural parameter of the model was obtained (p&#;<&#;&#;<&#;α).

$${f}_{c}=20.54{V}_{f}\lambda +{f}_{c}{\prime}$$

(2)

where \({f}_{c}{\prime}\)&#;compressive strength of HSC without fibers.

Compressive strength (f(c) and density (ρ) as a function of: degree of fiber reinforcement ((a) and ultrasonic wave velocity ((b); (red color indicates outliers that were not included in the regression analysis).

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Figure 8b shows the relationship between compressive strength and ultrasonic wave velocity results (also the relationship for composite density). A linear equation was proposed to estimate the compressive strength as a function of wave speed Eq. (3) with R2&#;=&#;0.75. The standard error of estimators of the equation Eq. (3) was 0. for the slope coefficient and 70.3 for the intercept (statistically significant model parameters).

$${f}_{c}=0.v-358.8$$

(3)

The regression line shows a general trend that an increase in the wave speed of the sample corresponds to its higher compressive strength. The use of the ultrasonic method can be useful for evaluating the strength of HSC concretes reinforced with short steel fibers. Despite the good fit of the data to models 2 and 3, it is important to keep in mind their limitations in application, since they were determined from experimental data obtained according to the adopted test method. The use of polynomial instead of linear approximation in models 2 and 3 increases the value of the coefficient of determination by about 5%, but the model parameters are statistically insignificant.

Figure 9 shows the dependence of strain ε0LVDT and toughness ratio TR on the degree of fiber reinforcement Vfλ and the corresponding regression equations Eq. (4 i 5), for which the coefficient R2 was 0.69 and 0.54. The parameters of the models are statistically significant (standard errors of the values of the slope coefficients were 0.066 and 0.).

$${\varepsilon }_{0}={\varepsilon }_{0}{\prime}+0.66{V}_{f}\lambda$$

(4)

$$TR=TR{\prime}+\mathrm{0,023}{V}_{f}\lambda$$

(5)

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where \({\upvarepsilon }_{0}^{\mathrm{^{\prime}}}\)&#;strains of concrete without fibers in [&#;], \(TR{\prime}\)&#;toughness ratio of concrete without fibers.

Strains εLVDT corresponding to maximum compressive stress and toughness ratio (TR) as a function of the degree of fiber reinforcement (red color indicates outliers that were not included in the regression analysis).

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Stress&#;strain relationship

Analyzing the results from Fig. 4, the course of σ&#;ε curves may be divided into three phases. Phase I&#;approximately linear-elastic σ&#;ε relationship (the slope of the curves of all samples remains practically the same when the stresses are not large); phase II&#;nonlinear σ&#;ε relationship before reaching the maximum stress σmax (the development of internal micro-damage of the composite and its non-elastic properties are the reason for the nonlinear shape of the graph at higher stress) and phase III&#;after exceeding the stress σmax, the curves drop rapidly, with the rate of descent for samples with fibers being different and slightly smaller than those of concrete without fibers. The fiberless concrete samples underwent explosive destruction after the σmax stress was reached (Fig. 5), while fiber-reinforced ones showed the ability to carry a small residual stress (Fig. 4), which was preceded by a sharp drop in stress (with a marked difference in the case of samples C5 and C6). In other words, the energy-absorbing capacity of fiber-reinforced samples compared to samples without fibers is relatively low and less than the difference observed in studies of similar subject matter5,8,11,21,22,29.

It should be noted that in literature mentioned above, other than short straight fibers for concrete were used (hooked, corrugated). Despite the satisfactory agreement of the results, i.e. σ-ε curves over the entire load range for the three strain measurement methods (for example, for selected samples they are shown in Fig. 10), it is worth noting that the characteristics of σ-ε curves of compressed samples strongly depend on the structure of dispersed fiber reinforcement, which is described by mechanical efficiency. It determines the interaction of individual fibers in the concrete, enabling it to inhibit cracking. Mechanical efficiency depends on: the slenderness (λ) and shape of the fibers, the volumetric proportion of the fibers in the composite (Vf), the spatial distribution of the fibers in the concrete, and the adhesion of the fibers to the cement matrix, resulting from adhesion, friction and mechanical anchorage35. The adhesion of smooth, straight fibers to the matrix is due only to adhesion and friction, while fibers deformed in the process (e.g. hooked) are additionally conditioned by mechanical anchorage36. This observation provides a better understanding of the differences in the rate of descent of σ-ε curve parts in the results from our own research and those from the literature.

σ-ε relationships in compression obtained from LVDT sensors, strain gauges and the DIC method for samples: (a) A4; (b) B4 and (c) C4.

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In order to confirm the above observations, a comparative analysis of the σ-ε dependence of fiber-reinforced concrete in compression was performed. Table 4 contains selected analytical models describing σ-ε relationships. These models are mostly based on the previously proposed model developed by Carreira and Chu37 presented by Eq. (6).

$$\sigma ={f}_{c }\frac{\beta \left(\frac{\varepsilon }{{\varepsilon }_{0}}\right)}{\beta -1+{\left(\frac{\varepsilon }{{\varepsilon }_{0}}\right)}^{\beta }}$$

(6)

where σ and ε&#;stresses and strains of the composite, respectively; β&#;material constant; fc&#;compressive strength of the composite; ε0&#;strains corresponding to the maximum compressive stress.

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It should be noted that the models in Table 4 were developed based on tests of concrete modified with steel fibers, which were not short and straight in shape, unlike the fibers used for concrete in this study. Part of the models, i.e. Someh and Saeki38, Nataraja et al.22, Baros et al.10 and Ou et al.21, were specified for plain concrete (fc&#;<&#;50 MPa). However, due to the intention of conducting the broadest possible comparative analyses, they were decided to be considered.

Experimental values of the parameters fc&#; and ε0 (arithmetic mean of the results of the series, see Table 2) were used for the analysis&#;for some models also the parameter fc. It was assumed that for the strain ε0 the equation ε0&#;=&#;εLVDT occurs. Figure 11 shows the σ&#;ε curves from our own tests, as well as the curves described by the equations in Table 4. Comparing the experimental and theoretical results, it can be seen that all the considered σ&#;ε models overestimate the σ stresses in the descending phase of the curve, hence to describe the σ&#;ε relationship of HSC reinforced with short fibers, our own analytical model Eq. (7) was proposed, which is also a modification of the proposal by Carreira i Chu37.

$$\sigma ={f}_{c }\frac{\beta \left(\frac{\varepsilon }{{\varepsilon }_{0}}\right)}{\beta -1+{\left(\frac{\varepsilon }{{\varepsilon }_{0}}\right)}^{\beta }}\left\{\begin{array}{c}\beta ={\beta }_{1} gdy \frac{\varepsilon }{{\varepsilon }_{0}} \le 1\\ \beta ={\beta }_{2} gdy \frac{\varepsilon }{{\varepsilon }_{0}}>1\end{array}\right.$$

(7)

where β1, β2&#;material constants of the ascending and descending parts of the σ-ε curve, respectively; other designations used as in Eq. (6); fc and ε0 defined respectively by Eq. (3 i 4).

Comparisons of σ-ε relationships of HSC reinforced with short steel fibers inn compression based on analytical models and own experimental tests: (a) A-series (without fibers); (b) B-series&#;Vfλ&#;=&#;0,49; (c) C-series&#;Vfλ&#;=&#;0,98 (sample designations 1; 2; 3; 4; 5; 6, see Table 2;).

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Using the Levenberg&#;Marquardt estimation method, β parameters of the model (7) were estimated. For HSC with the degree of fiber reinforcement Vfλ&#;=&#;0.49 (batch B) parameters β1 and β2 are 5.12 i 14.76, respectively&#;with a standard error 0.032 and 8.89. In turn, for the composite with the degree of fiber reinforcement Vfλ&#;=&#;0.98 (batch C) parameters β1 and β2 have values of 6.84 and 5.75 (standard errors&#;0.045 and 0.418). In the description of the σ-ε relationship of HSC the structural parameters were set as β1&#;=&#;11.17 (standard error 0.07) and β2&#;=&#;0. All β parameters of the model are statistically significant except for the β2 parameter for batch B, for which a relatively large standard error was observed. This is most likely due to the large scatter in the results from our own tests, as well as the small test sample (the descending part of the σ-ε curve, which is described as points in Fig. 11b and c). This state of affairs was due to the limitation of the possibility of adopting a high sampling frequency of the results and the abrupt manner in which the B-series specimens were destroyed after exceeding the limit load during the test. In other words, the relatively low content of straight, short steel fibers in the concrete slightly affected the occurrence of residual stress in the composite&#;to a much lesser extent than in the case of the results of the C-series specimens). Comparing the theoretical dependencies of σ&#;ε with experimental results, it can be concluded that the proposed model approximates the actual σ&#;ε curves quite well, as shown in Fig. 11. Mean absolute percentage error of the model for the results in batch B and C is 7.1% and 8.4%. The error is larger if the results of the model of the falling part of the σ&#;ε curve are compared (22.6% and 19.9%, respectively). Figure 12 shows the relationship between the parameters β1 and β2, and the degree of fiber reinforcement, and presents the regression equations of these parameters Eq. (8). The regression equations extend the state of knowledge in the subject matter21, however, due to the number of observations, it is not possible to evaluate them statistically. It is worth noting that extending the study to a larger number of sample series, differing in steel fiber content, would make it possible to generalize the conclusions regarding the fit of the proposed model to the data.

$$\begin{aligned} \beta_{1} = & 16.35(V_{f} \lambda )^{2} - 20.39V_{f} \lambda + 11.17 \\ \beta_{2} = & - 0.57(V_{f} \lambda )^{2} - 1.17V_{f} \lambda + 3.15 \\ \end{aligned}$$

(8)

Parameters β1 i β2 as a function of the degree of fiber reinforcement.

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