The Effect of Fiber Volume Fraction on Fiber Distribution in Steel Fiber Reinforced Self-Compacting Concrete

Abstract

This paper investigates the effect of fiber volume fraction on fiber distribution in steel fiber reinforced self-compacting concrete (SFRSCC) through experiments and numerical simulations. Three types of SFRSCC beam specimens with different fiber volume fractions (0.3%, 0.6%, and 0.9%) were cut to expose the steel fibers. The number and the orientation angle of the steel fibers on the beam sections were determined by image analysis techniques. Fiber density, fiber segregation coefficient, fiber dispersion coefficient and fiber orientation coefficient were applied to evaluate fiber distribution on the beam sections. Based on the experimental data, numerical models simulating the pouring process of fresh SFRSCC were established to analyze the overall fiber distribution in the specimens. The results show that the distribution state of the fibers on the beam sections is not random and uniform, which is correlated to the fiber volume fraction. Due to the variable rheological properties, a greater fiber volume fraction causes better fiber uniformity, lower fiber segregation and worse fiber alignment on the beam sections. Meanwhile, the numerical results show that the distribution law of fibers along the length direction of the specimens is almost independent of the fiber volume fraction. In addition, increasing the fiber volume fraction results in the increase of the average angle of the fiber orientation in the specimens. The results can provide a reference for optimizing the fiber distribution in the concrete matrix.

1. Introduction

Steel fiber reinforced self-compacting concrete (SFRSCC) [1,2,3,4,5] combines the advantages of steel fiber reinforced concrete (SFRC) and self-compacting concrete (SCC). Specifically, the addition of steel fibers can enhance the mechanical properties of the concrete matrix [6,7,8,9,10,11], and SCC can fill the formwork without vibration. In recent years, SFRSCC has increasingly replaced ordinary concrete in various engineering structures, especially in bridges, roads and military engineering. During the SFRSCC pouring process, fiber distribution state changes with the flow of SCC. The final state of the fiber distribution has a significant influence on the mechanical properties of SFRSCC. Therefore, the fiber distribution in SCC is one of the important concerns in the field of SFRSCC [12,13,14].

The fiber distribution in SCC is not completely uniform and random, but is affected by various material and pouring factors, such as pouring methods [15,16,17,18], specimen size [19,20,21], rheology of fresh concrete [22,23] and roughness of formwork walls [24]. Due to the variation in material and pouring factors, SFRSCC or SFRC specimens with the same size and concrete mix may exhibit significantly different mechanical properties. Doyon-Barbant et al. [25] poured a large slab using fresh SFRC and found that the tensile and flexural behavior of the specimens were significantly affected by fiber orientation, while the shear behavior of specimens was mainly related to the fiber density. Torrijos et al. [26] compared the post-cracking responses of SFRSCC beam specimens fabricated by central pouring, tube pouring and vertical pouring methods. They found that the maximum residual strength of the beam specimens poured by the tube was higher than that of the specimens poured by center and vertical methods. This is because the tube pouring method caused more fibers to be distributed on the cracked planes. Swamy [27] pointed out that the reinforcement efficiency of the fibers decreased significantly when the fibers and tensile stress changed from an aligned state to a random state. Therefore, it is important to investigate the influence of pouring factors on fiber distribution to improve the mechanical properties of SFRSCC.

So far, several researchers have studied the effects of material and pouring factors on the fiber distribution in fresh concrete through experiments and numerical simulations. Kang et al. [17] analyzed the influence of pouring method on fiber orientation and distribution in ultra-high-performance fiber-reinforced concrete (UHPFRC) and found that the flexural behavior of the UHPFRC poured by the parallel method was better than that of the transverse type, because the parallel pouring method can increase the fiber density and optimize the fiber alignment in the UHPFRC specimens. Švec et al. [24] poured fresh SFRSCC into three different types of formworks to evaluate the impact of formwork surface on fiber distribution and flexural performance of specimens. The experimental results showed that for the smooth formwork surfaces, the fibers near the bottom formwork were more susceptible to the flow pattern of the SCC, and the flexural performance of beam specimens from different positions of the slab exhibited significant discreteness. Moreover, they simulated the pouring process of fresh SFRSCC in the slab using the lattice Boltzmann method (LBM) and confirmed the above conclusions. Boulekbache et al. [23] and Jasiūnienė et al. [22] investigated the effect of the rheological properties of SCC on fiber distribution. The results showed that changes in the rheological properties of SCC can lead to differences in fiber distribution in the specimens. Kang et al. [28] analyzed the impact of shear flow and radial flow on fiber orientation in UHPCC by solving Jeffery’s equation. They reported that as the flow distance increased, the fibers tended to be aligned with the flow direction in shear flow, and the fibers tended to be perpendicular to the flow direction in radial flow. In addition to the above factors, fiber volume fraction is an indirect factor affecting fiber distribution. Changing fiber volume fraction leads to variation in the rheological properties of fresh concrete, which in turn affects the fiber distribution in fresh concrete. Raju et al. [20] found that increasing the fiber volume fraction did not always increase the flexural strength of SFRSCC beam specimens due to differences in fiber distribution. Therefore, studying the effect of fiber volume fraction on fiber distribution plays an important role in optimizing the mechanical properties of SFRSCC. Although some references mention the effect of fiber volume fraction on fiber distribution in concrete matrices, studies on the effect of fiber volume fraction on fiber distribution in SFRSCC are still limited. At present, the quantitative effect of fiber volume fraction on fiber distribution in SCC is not clear.

Therefore, this study aims to provide insight into the influence of fiber volume fraction on fiber distribution in SFRSCC. Nine groups of the four-point bending test were performed on SFRSCC beam specimens with three fiber volume fractions (0.3%, 0.6% and 0.9%). The number and the orientation angle of the steel fibers on the beam sections were determined by image analysis techniques. Several fiber coefficients were used to analyze fiber distribution and orientation at different fiber volume fractions. Numerical models verified by experimental results were conducted to simulate the pouring process of SFRSCC specimens. Furthermore, the overall fiber distribution in the specimens was analyzed. The results of this study can provide a reference for optimizing fiber distribution and improving mechanical properties of SFRSCC.

2. Experiment Program

In this section, nine SFRSCC beam specimens with three fiber volume fractions (0.3%, 0.6% and 0.9%) were cast to perform the four-point bending tests and obtain the fiber distribution on the beam sections of the specimen.

2.1. Materials and Experimental Methods

The ingredients of the SCC mixtures include cement, fly ash, river sand, gravels, tap water and superplasticizer. The basic properties of the above materials and mix design of the SCC mixtures can be found in our previous study [29]. The mix design of the SCC mixtures consisted of optimization of the aggregate skeleton and optimization of paste. The mixing proportions of the SCC mixture are listed in

Table 1. Mixing proportions of SCC mixture (Unit: kg/m³).

Cement	Fly Ash	Sand	Coarse Aggregate	Water	Superplasticizer
428	107	783	784	198	11.77

. S-0.3, S-0.6 and S-0.9 mixtures were prepared with three volume fractions (0.3%, 0.6% and 0.9%) of hooked-end steel fibers.

Table 2. Characteristics of steel fibers.

Fiber Type	Length (mm)	Diameter (mm)	Aspect Ratio	Density (kg/m3)	Tensile Strength (MPa)
Hooked-end	35	0.75	47	7850	1106

shows the characteristics of steel fibers, and Figure 1 shows the hooked-end steel fibers used in the experiment. The symbol S-x means that the volume fraction of 0.x% of fibers is added to the plain SCC. The workability of the fresh concrete mixtures is shown in

Table 3. Workability of different fresh concrete mixtures.

Mixtures	Slump-Flow Diameter (mm)	V-Funnel (s)
SCC	755	8.8
S-0.3	725	11.3
S-0.6	680	12.6
S-0.9	650	14.5

. The center pouring method was used to fabricate nine SFRSCC beams specimens with dimensions 100 mm × 100 mm × 400 mm. During the pouring process, the fresh concrete filled the formwork by its own gravity and was not vibrated. After specimen curing, the flexural performance of the hardened SFRSCC beam specimens was obtained by the four-point bending tests, and the results of flexural properties can be found in the Ref [29]. Figure 2 shows the beam specimens and setups of the four-point bending tests.

After the four-point bending tests, the specimens were cut with a rock cutter. The rock cutter and the positions of the beam sections are shown in Figure 3. As can be observed, two transverse sections and one longitudinal section were used to evaluate the fiber distribution and orientation. It should be noted that the transverse section $T{C}_{crack}$ was set at a distance of 20 mm from the crack planes to avoid the effect of fiber pullout. Then, the sandpaper and grinders were used to polish the beam sections to increase the contrast between the fibers and the concrete matrix. Finally, the number and inclined angle of the fibers on the beam sections were determined by image analysis techniques. The image analysis techniques used include grayscale of image, binarization of image, identification of fiber sections and fitting of fiber sections (Figure 4).

Based on the fiber distribution data on the beam sections, the effect of fiber volume fraction on fiber distribution was comprehensively evaluated by several fiber coefficients.

The fiber density $d_n$ can be calculated by Equation (1)

$$d_n=\frac{N_f}{A_c}$$

(1)

where $N_f$ and $A_c$ represent the number of fibers and the area of the beam sections, respectively.

Assuming that the fiber distribution is random and uniform in the specimens, the theoretical value of fiber density $d_{n_{3D}}$ on the beam sections is expressed by the following equation [30,31]:

$$d_{n_{3D}}=0.5\frac{v_f}{A_f}$$

(2)

where $A_f$ represents the cross-sectional area of the fibers and $v_f$ represents the fiber volume fraction of SFRSCC.

Due to the flow of fresh concrete and the density difference, the horizontal and vertical distribution of the fibers along the specimens is not uniform. Therefore, the degree of fiber segregation in the vertical direction is evaluated by the fiber segregation coefficient ${\kappa }_{seg}$, which is expressed as the following equation [32]:

$${\kappa }_{seg}=\frac{{\displaystyle \sum _{i=1}^{N_f}{\overline{d}}_i}}{h{N}_f}$$

(3)

where $h$ represents the height of the beam sections and ${\overline{d}}_i$ represents the distance from the center of the ith fiber to the top of the beam sections. When ${\kappa }_{seg}$ is in the range of 0.5 to 1, it means that the overall distribution of fibers is close to the upper half of the beam sections. The schematic diagram of the calculation of the segregation coefficient ${\kappa }_{seg}$ is shown in Figure 5.

The uniformity of fiber distribution on the beam sections is evaluated by the fiber dispersion coefficient ${\alpha }_{dis}$, which can be expressed as the following equation [33]:

$${\alpha }_{dis}=\exp [-\sqrt{\frac{{\displaystyle \sum {(x_i-\overline{x})}^2}}{N_u}/\overline{x}}]$$

(4)

where $x_i$ represents the number of fibers in the ith unit, $\overline{x}$ represents the average value of the number of fibers in all units, and $N_u$ is the number of the units. The fiber distribution coefficient ${\alpha }_{dis}$ tends to be 1 for uniform distribution of fibers or 0 for severely uneven distribution of fibers.

The fiber orientation angle $\theta$ (Figure 6) between the axial direction of the fiber and the normal direction of the beam sections is expressed by the following equation [34]:

$$\theta =\arccos (\frac{d_{fi}}{l})$$

(5)

where $d_{fi}$ represents the fiber diameter and $l$ represents the major axis length of the fiber section.

According to the orientation angle $\theta$ of each fiber on the beam sections, the fiber orientation coefficient ${\eta }_{\theta }$ is expressed by the following equation [34]:

$${\eta }_{\theta }=\frac{1}{N_f}{\displaystyle \sum _{i=1}^{N_f}\cos {\theta }_i}$$

(6)

The fiber orientation coefficient ${\eta }_{\theta }$ tends to be 1 for a fiber orientation perpendicular to the beam sections or 0 for a fiber orientation parallel to the beam sections.

Another fiber orientation coefficient ${\alpha }_f$ can be expressed by the following equation [35]:

$${\alpha }_f=\frac{N_f{A}_f}{v_f}$$

(7)

The orientation coefficients ${\eta }_{\theta }$ and ${\alpha }_f$ both have a range of 0 to 1.

2.2. Results and Discussion

2.2.1. Fiber Density

Table 4. Fiber densities on the beam sections of each specimen, fibers/cm².

Specimen Name	${\mathit{d}}_{{\mathit{n}}_{3\mathit{D}}}$	${\mathit{d}}_{{\mathit{n}}_{\left|\right|}^{\mathit{e}}}$	${\mathit{d}}_{{\mathit{n}}_{\left|\right|}^{\mathbf{c}}}$	${\mathit{d}}_{{\mathit{n}}_{\perp }}$
S-0.3	0.34	0.39	0.56	0.31
S-0.6	0.68	0.71	1.05	0.65
S-0.9	1.02	1.06	1.51	0.98

shows the values of fiber density for each specimen. The symbols $d_{n_{\left|\right|}^e}$ and $d_{n_{\left|\right|}^{\mathrm{c}}}$ refer to the fiber density for the transverse sections $T{C}_{end}$ and $T{C}_{crack}$ (Figure 3), respectively. The symbol $d_{n_{\perp }}$ refers to fiber density on the longitudinal section $LC$ (Figure 1). The fiber density on the cutting plane increases with the increasing of fiber volume fraction, independent of the location of the beam sections. Theoretically, when the fiber volume fraction increases from 0.3% to 0.6%, the fiber density on the beam sections should be doubled. However, the increments of the fiber density on the beam sections are not equal to the theoretical increments. For example, the values of $d_{n_{\left|\right|}^e}$ in S-0.6 and S-0.9 are 82.1% and 171.8% higher than that in S-0.3, respectively.

Regardless of the fiber volume fraction, the value of $d_{n_{\left|\right|}^{\mathrm{c}}}$ is significantly higher than the theoretical fiber density $d_{n_{3D}}$, which indicates that the fibers are not random and uniform in the SFRSCC specimens. The difference between the values of $d_{n_{\left|\right|}^{\mathrm{c}}}$ and $d_{n_{3D}}$ can be attributed to two factors. First, high fluidity of SCC leads to fibers gradually becoming parallel to the flow direction of fresh concrete. In addition, the beam section $T{C}_{crack}$ is close to the concrete pouring point, where fibers tend to accumulate. Second, the fibers near the formwork walls tend to align with the axial direction of the specimens. Martinie and Roussel [36] reported that regions less than half the fiber length from the formwork walls are affected by the wall effect. Therefore, the above factors result in a significant increase in the probability of fibers intersecting the beam section $T{C}_{crack}$. In contrast, for SFRSCC with three fiber volume fractions, the values of $d_{n_{\left|\right|}^e}$ are slightly larger than the values of $d_{n_{3D}}$. This is mainly because the tendency of fiber orientation to be parallel to the beam axis is weakened significantly due to the restriction of the end formwork walls and the flow direction of SCC, which cause a decrease in the probability of intersection of the fibers with the beam sections. As expected, the values of $d_{n_{\perp }}$ are slightly less than the values of $d_{n_{3D}}$, since the main flow direction of SFRSCC is along the length of the specimens.

2.2.2. Fiber Segregation and Uniformity

Table 5. Fiber segregation coefficient ${\kappa }_{seg}$ and dispersion coefficient ${\alpha }_{dis}$ of the beam sections for each specimen.

Specimen Name	$\mathit{T}{\mathit{C}}_{\mathit{e}\mathit{n}\mathit{d}}$		$\mathit{T}{\mathit{C}}_{\mathit{c}\mathit{r}\mathit{a}\mathit{c}\mathit{k}}$
	${\mathit{\kappa }}_{\mathit{s}\mathit{e}\mathit{g}}$	${\mathit{\alpha }}_{\mathit{d}\mathit{i}\mathit{s}}$	${\mathit{\kappa }}_{\mathit{s}\mathit{e}\mathit{g}}$	${\mathit{\alpha }}_{\mathit{d}\mathit{i}\mathit{s}}$
S-0.3	0.59	0.27	0.56	0.38
S-0.6	0.57	0.34	0.54	0.39
S-0.9	0.56	0.37	0.53	0.41

presents the results of fiber segregation coefficient ${\kappa }_{seg}$ and dispersion coefficient ${\alpha }_{dis}$. Since the values of ${\kappa }_{seg}$ and ${\alpha }_{dis}$ are mainly influenced by gravitational forces and concrete flow, only the results for the transverse sections are given in Table 5. The values of ${\kappa }_{seg}$ are 6% to 18% higher than 0.5, which indicates that the steel fibers exhibit different degrees of vertical segregation during the flow of SCC and the fibers are mainly distributed in the middle and lower parts of the specimens. Moreover, the values of ${\kappa }_{seg}$ decrease with increasing fiber volume fraction. This can be attributed to the fact that increasing fiber content causes an increase in the cohesiveness of the fresh SFRSCC, and the increase in the cohesiveness of fresh SFRSCC may cause a decrease in the degree of fiber segregation. Similarly, Hosseinpoor et al. [37] found that the vertical segregation of aggregates in the paste decreased with increasing viscosity of the fresh SCC.

On the other hand, the values of ${\alpha }_{dis}$ increase with increasing fiber volume fraction, indicating that the distribution of fibers on the beam sections tends to be more uniform with increasing fiber volume fraction. As the fiber volume fraction increased from 0.3% to 0.9%, the values of ${\alpha }_{dis}$ on the planes $T{C}_{end}$ and $T{C}_{crack}$ increased by 6.8% and 35.8%, respectively. This can also be attributed to the increase of the plastic viscosity of SFRSCC due to the increase in the fiber volume fraction. Jasiūnienė et al. [22] adjusted the plastic viscosity of SFRSCC by using zeolite, and the experimental results showed that the fiber distribution on the beam sections was more uniform with the increase of plastic viscosity of SCC. The results of their experiments are similar to those in this study. In addition, the values of ${\alpha }_{dis}$ on the plane $T{C}_{crack}$ are larger than those on the plane $T{C}_{end}$ due to the vertical segregation of the fibers.

Figure 7 shows the fiber distribution on the beam sections. The black dots represent the fibers, and the red dashed line represents the average value of the vertical coordinates of the fibers. According to Table 5 and Figure 7, the degree of fiber segregation on the plane $T{C}_{end}$ is more severe than that on the plane $T{C}_{crack}$. For the same fiber volume fraction, the value of ${\kappa }_{seg}$ on the plane $T{C}_{end}$ is 5.4% to 5.7% higher than that on the plane $T{C}_{crack}$. This result indicates that dynamic segregation of fibers occurs during the flow of SFRSCC. The plane $T{C}_{end}$ is further from the pouring point than the plane $T{C}_{crack}$, so the dynamic segregation of fibers on the plane $T{C}_{end}$ is more severe.

2.2.3. Fiber Orientation

Table 6. Fiber orientation coefficients ${\eta }_{\theta }$ and ${\alpha }_f$ of the beam sections for each specimen.

Specimen Name	$\mathit{T}{\mathit{C}}_{\mathit{e}\mathit{n}\mathit{d}}$		$\mathit{T}{\mathit{C}}_{\mathit{c}\mathit{r}\mathit{a}\mathit{c}\mathit{k}}$		$\mathit{L}\mathit{C}$
	${\mathit{\alpha }}_{\mathit{f}}$	${\mathit{\eta }}_{\mathit{\theta }}$	${\mathit{\alpha }}_{\mathit{f}}$	${\mathit{\eta }}_{\mathit{\theta }}$	${\mathit{\alpha }}_{\mathit{f}}$	${\mathit{\eta }}_{\mathit{\theta }}$
S-0.3	0.57	0.72	0.83	0.80	0.46	0.70
S-0.6	0.52	0.69	0.78	0.78	0.48	0.73
S-0.9	0.52	0.68	0.74	0.76	0.48	0.70

shows the results of the fiber orientation coefficients ${\alpha }_f$ and ${\eta }_{\theta }$ calculated by Equations (6) and (7). It can be seen that the values of ${\alpha }_f$ and ${\eta }_{\theta }$ are different. This is because the coefficients ${\alpha }_f$ and ${\eta }_{\theta }$ are calculated based on the orientation angle $\theta$ and the number of fibers $N_f$, respectively. The values ${\alpha }_f$ and ${\eta }_{\theta }$ on the planes $T{C}_{end}$ and $T{C}_{crack}$ decrease with increasing fiber volume fraction. For the plane $T{C}_{end}$, the values of ${\alpha }_f$ and ${\eta }_{\theta }$ in S-0.9 are 8.8% and 5.6% lower than those in S-0.3, respectively. For the plane $T{C}_{crack}$, the values of ${\alpha }_f$ and ${\eta }_{\theta }$ in S-0.9 specimens are 10.8% and 5.0% lower than those in S-0.3 specimens, respectively. These results indicate that the flowability of SFRSCC is closely related to the fiber orientation. Increasing the fiber volume fraction significantly weakens the flowability of SFRSCC, resulting in a weaker ability of fibers to align with the flow direction of fresh concrete. In contrast, the values of ${\alpha }_f$ and ${\eta }_{\theta }$ on the plane $LC$ show a slight variation. The values of ${\alpha }_f$ are closer to the coefficient corresponding to the 3D randomly distributed state (0.5), while the values of ${\eta }_{\theta }$ are slightly larger than the coefficient corresponding to the 2D randomly distributed state (0.637) [30].

In addition, the values of ${\alpha }_f$ and ${\eta }_{\theta }$ on the plane $T{C}_{crack}$ are larger than those on the plane $T{C}_{end}$, which indicates that the distance between the beam section and the pouring position influences the values of ${\alpha }_f$ and ${\eta }_{\theta }$. Figure 8 shows the binary images of the beam sections $T{C}_{crack}$ and $T{C}_{end}$ at three fiber volume fractions. It can be seen that the number of elliptical sections of fibers on the plane $T{C}_{crack}$ is less than that on the plane $T{C}_{end}$, which illustrates that the fibers on the plane $T{C}_{crack}$ tend to be perpendicular to the beam sections.

3. Numerical Simulation

In this section, the pouring process of SFRSCC specimens with three fiber volume fractions (0.3%, 0.6% and 0.9%) was simulated numerically. The overall fiber distribution in the specimens was further analyzed.

3.1. Numerical Method

In order to evaluate the overall fiber distribution inside the SFRSCC specimens, a numerical method proposed by Ref. [38] is used to simulate the pouring process of SFRSCC. In this method, fresh SFRSCC is treated as a liquid phase of the concrete and a solid phase of the fibers. Specifically, the fresh concrete is considered as a non-Newtonian fluid, and its rheological behavior is defined by the Bingham model. The steel fibers are considered as the rigid bodies undergoing only translation and rotation. Then, the software CFX 18.2 and Matlab code are used to simulate the motion of steel fibers in fresh SCC.

Considering the effect of the rheological behavior of fresh SFRSCC on fiber motion, the plastic viscosity ${\mu }_{SF}$ and yield stress ${\tau }_y$ in the Bingham model are determined by Equations (8) and (9) [39,40], respectively.

$${\mu }_{SF}={\mu }_C\left\{(1-v_f)+\frac{\pi v_f{l}_d^2}{3\ln (2l_d)}\right\}$$

(8)

$${\tau }_y=\frac{\rho g{H}^2}{2R}$$

(9)

where ${\mu }_C$ represents plastic viscosity of plain concrete, $l_d$ represents the aspect ratio of steel fibers, $\rho$ represents the density of fresh SFRSCC, $H$ represents the final concrete height and $R$ represents the final concrete radius.

The process of the numerical method [38] can be briefly summarized as follows:

• Based on the test parameters, the pouring process of concrete is simulated to obtain the concrete velocity field.
• Three-dimensional random steel fibers (Figure 9) are generated above the inlet of the numerical model.
• According to the concrete velocity field, the motion of fibers immersed in fresh SCC can be determined at each time step.
• The coordinate information of the fibers in the computational domain is used to determine the fiber distribution.

3.2. Numerical Settings

According to the experiments in Section 2.1, a numerical model of the specimen (Figure 10) was established via the software CFX. The velocity of the inlet was 0.3 m/s.

Table 7. Rheological properties of fresh concrete.

Concrete Mixtures	Plastic Viscosity (Pa·s)	Yield Stress (Pa)
S-0.3	19.2	22.3
S-0.6	30.7	34.1
S-0.9	42.4	38.7

presents the rheological properties of fresh SFRSCC mixtures calculated by Equations (8) and (9). Then, the pouring process of the SFRSCC specimens with three fiber volume fractions was simulated by the above numerical method. In these simulations, the time step was 0.01 s, and the total computation time was 8.34 s. The steel fibers with volume fractions of 0.3%, 0.6% and 0.9% were simulated by approximately 750, 1490 and 2240 rigid bodies, respectively. Other settings of the numerical model can be found in the Ref [38].

3.3. Results and Discussion

3.3.1. Pouring Process of SFRSCC and Fiber Distribution

For the S-0.6, Figure 11 shows the states of concrete and steel fibers at two different instances. It is seen that the distribution state of the fibers changes with the variation of the concrete. Due to the limitation of the formwork walls, the fibers near the formwork walls have a tendency to align with the formwork walls. In contrast, the fibers in the middle of the formwork are more or less random. As the distance between the fibers and the end formwork walls decreases, the fibers tend to be distributed obliquely upward. A similar fiber distribution state was also observed by Zhou et al. [21]. Furthermore, Figure 12 shows the final fiber distribution for the SFRSCC mixtures with three fiber volume fractions (0.3%, 0.6% and 0.9%).

3.3.2. Fiber Distribution and Orientation

The experimental and numerical results are compared in

Table 8. Experimental and numerical results of fiber density.

Specimen Name	$\mathit{T}{\mathit{C}}_{\mathit{e}\mathit{n}\mathit{d}}$		Error (%)	$\mathit{T}{\mathit{C}}_{\mathit{c}\mathit{r}\mathit{a}\mathit{c}\mathit{k}}$		Error (%)	$\mathit{L}\mathit{C}$		Error (%)
	Exp	Num		Exp	Num		Exp	Num
S-0.3	0.39	0.36	7.7	0.56	0.53	5.4	0.31	0.34	9.8
S-0.6	0.71	0.67	5.6	1.06	1.05	0.1	0.65	0.69	6.2
S-0.9	1.06	0.97	8.5	1.51	1.59	5.3	0.98	1.01	3.1

and

Table 9. Experimental and numerical results of fiber orientation coefficient ${\eta }_{\theta }$.

Specimen Name	$\mathit{T}{\mathit{C}}_{\mathit{e}\mathit{n}\mathit{d}}$		Error (%)	$\mathit{T}{\mathit{C}}_{\mathit{c}\mathit{r}\mathit{a}\mathit{c}\mathit{k}}$		Error (%)	$\mathit{L}\mathit{C}$		Error (%)
	Exp	Num		Exp	Num		Exp	Num
S-0.3	0.72	0.70	2.8	0.80	0.84	5.0	0.70	0.73	4.3
S-0.6	0.69	0.69	0	0.78	0.81	3.8	0.73	0.74	1.4
S-0.9	0.68	0.66	2.9	0.75	0.78	4.0	0.70	0.72	2.9

. The symbols Exp and Num represent the values obtained from the experiments and numerical simulations, respectively. The errors of fiber density and fiber orientation coefficient are less than 10%, which proves the accuracy of the numerical method.

Figure 13 shows the distribution of fibers along the length direction of the specimens with different fiber volume fractions. Considering that the number of fibers varies with the fiber volume fractions, the number of fibers in each interval is normalized. The number of fibers around the concrete pouring point is slightly higher than that near both ends of the specimen. This can be attributed to the accumulation of fibers near the concrete pouring point. In general, the distribution law of fibers along the length direction of specimens is almost independent of the fiber volume fraction.

Figure 14 shows the probability density distribution of the orientation angle $\theta$ at different fiber volume fractions. In order to eliminate the impact of the end formwork walls on the fiber orientation, only the fibers in the middle region of the specimens (50–350 mm), as shown in Figure 14, are considered. The fiber orientation histogram for the S-0.3 specimens is slightly skewed left, the fiber orientation histogram for the S-0.6 specimens is more or less centered and the S-0.9 specimens exhibit a slightly right-skewed orientation distribution. This indicates that the average values of $\theta$ increase with an increase of the fiber volume fraction, which is caused by the variation of the rheological properties of fresh SFRSCC mixtures. The fresh mixtures with the lower rheological parameters may have better flowability, and thus the angles of the fibers immersed in the mixture are smaller.

4. Conclusions

In this study, the effect of fiber volume fraction on fiber distribution and orientation in SFRSCC was investigated and analyzed by experimental and numerical methods. The following conclusions can be drawn:

• The values of fiber density $d_{n_{\left|\right|}^{\mathrm{c}}}$ are obviously larger than the theoretical value of fiber density $d_{n_{3D}}$, which is mainly owing to the high fluidity of SCC and the wall effect. In contrast, the values of fiber density on the planes $T{C}_{end}$ and $LC$ are close to the theoretical value $d_{n_{3D}}$.
• As the fiber volume fraction increases, the fiber segregation coefficient ${\kappa }_{seg}$ decreases, the fiber dispersion coefficient ${\alpha }_{dis}$ increases and the fiber orientation coefficients ${\alpha }_f$ and ${\eta }_{\theta }$ on the transverse planes both decrease.
• The numerical results agree well with the experimental results, which can prove the accuracy of the numerical method.
• The distribution law of fibers along the length direction of specimens is almost independent of the fiber volume fraction. On the other hand, the probability density distribution of fiber orientation angle $\theta$ changed from left-skewed to right-skewed as the fiber volume fraction increased from 0.3% to 0.9%.

The change of fiber volume fraction leads to the difference in fiber orientation in self-compacting concrete, which affects the mechanical properties of SFRSCC. However, this study only analyzed the effect of fiber volume fraction on fiber distribution in standard beam specimens. In future work, the effect of fiber volume fraction on fiber distribution in full-size beams or slabs should be further studied.