Inﬂuence of Matrix Strength on Bridging Performance of Fiber-Reinforced Cementitious Composite with Bundled Aramid Fiber

: The bundled aramid ﬁber has good bond properties in the cementitious matrix, and is expected to have high bridging performance in the ﬁber-reinforced cementitious composite (FRCC). To investigate the inﬂuence of matrix strength on the bridging performance of FRCC with the bundled aramid ﬁber, the uniaxial tension test of FRCC, the pullout test for an individual ﬁber, and the calculation of bridging law are conducted with the main parameters of matrix strength and ﬁber volume fraction. From the test results, the maximum tensile load of FRCC and the maximum pullout load of an individual ﬁber increase as the matrix strength also increases. The calculation result of the bridging law considering the effect of matrix strength expresses the bridging performance of the bundled aramid ﬁber well. The calculation result also shows that the bridging strength has a linear relationship up to a compressive strength of around 50 MPa.


Introduction
The fiber-reinforced cementitious composite (FRCC) is one of the cementitious materials in which short discrete fibers with certain volume fractions are mixed into cementitious matrix to improve the brittle behavior of composites, especially in the tensile and bending field. Compared with traditional cementitious materials such as concrete and mortar, the FRCC is expected to exhibit a high performance in ductility because of the fiber bridging that transfers the tensile force through cracks [1]. In addition, the FRCC is also expected to have high durability when it is used in reinforced concrete (RC) structures, because fibers can control the crack openings in the matrix, which prohibits the penetrations of aggressive attacks to deteriorate the internal reinforcing rebars and the FRCC itself [2][3][4].
According to various studies, steel fibers or polymeric fibers such as polyethylene (PE), polyvinyl alcohol (PVA), and polypropylene (PP) fibers have been utilized in the FRCC [5]. While FRCC mixing with steel fibers commonly shows tension softening behavior after initial cracking, polymeric fibers are commonly used rather than steel fibers in the FRCC, which shows higher ductility. Aramid fibers are known as one of the polymeric fibers that have high tensile strength, durability, and heat and chemical resistance [6]. It has been reported that aramid fibers have been used for the strengthening of RC structures by the external bonding of a fiber sheet [7,8]. However, few research studies can be found concerning FRCC mixed with discrete aramid fibers [9,10]. Since a commercially provided single aramid fiber has a small diameter of 12 µm, it cannot be expected that the aramid fibers and the cementitious matrix have strong bond strengths [11]. In the case of the PVA fiber, it has been considered that the alcohol group in a PVA molecule leads to a good

Used Materials
The bundled aramid fiber shown in Figure 1 is targeted in this study. The original yarns of aramid fibers with a nominal diameter of 12 µm are twisted to form a thick individual fiber and sized so as not to unravel in the FRCC. The diameter of the bundled fiber is 0.5 mm, and the length of the chopped fiber for FRCC is 30 mm. Table 1 shows the dimensions and mechanical characteristics of the fiber. The weight of bundled aramid fibers mixed into FRCC for the volume fractions of 1% and 2% is 13.9 kg/m 3 and 27.8 kg/m 3 , respectively. dimensions and mechanical characteristics of the fiber. The weight of bundled aramid fibers mixed into FRCC for the volume fractions of 1% and 2% is 13.9 kg/m 3 and 27.8 kg/m 3 , respectively. (a) (b) (c)  Table 2 shows the three mixture proportions of the cementitious matrix applied in this study. The water/cement ratio in the three mixture proportions is varied to obtain different target compressive strengths, while the water/binder ratio is kept constant with similar fresh properties by changing the unit weight of the fly ash. A coarse aggregate is not used to investigate the fundamental characteristics of FRCC. The naming of the series (Fc24, Fc36, and Fc48) is based on the target compressive strength of the mixtures as the specified strength for the design of the structural elements. These FRCCs show self-consolidating properties, even if the volume fraction of the fiber is over 2%.

Specimens
A uniaxial tension test is conducted for FRCC rectangular prism specimens with slits. Since the unified standard or specification of the uniaxial tension test method for FRCC does not exist within the knowledge of the authors, the prism specimen shown in Figure  2 is prepared considering the ease of specimen setup and measuring the axial deformation   Table 2 shows the three mixture proportions of the cementitious matrix applied in this study. The water/cement ratio in the three mixture proportions is varied to obtain different target compressive strengths, while the water/binder ratio is kept constant with similar fresh properties by changing the unit weight of the fly ash. A coarse aggregate is not used to investigate the fundamental characteristics of FRCC. The naming of the series (Fc24, Fc36, and Fc48) is based on the target compressive strength of the mixtures as the specified strength for the design of the structural elements. These FRCCs show self-consolidating properties, even if the volume fraction of the fiber is over 2%.

Specimens
A uniaxial tension test is conducted for FRCC rectangular prism specimens with slits. Since the unified standard or specification of the uniaxial tension test method for FRCC does not exist within the knowledge of the authors, the prism specimen shown in Figure 2 is prepared considering the ease of specimen setup and measuring the axial deformation at the crack position. The specimen is a rectangular prism with two bolts (M20) embedded at both ends to transfer the tensile load. In order to control the position of the crack opening, two slits were set on both narrow sides in the middle of the specimen. The depth and width of the slit are 20 mm and 3 mm, respectively. The area of the ligament is 60 × 70 mm 2 . The slits were made by a concrete cutter after the hardening of FRCC to avoid the influence of the flow of the matrix at the casting. The notched rectangular prism specimens with slits have been generally utilized to investigate the tensile characteristics of cementitious materials, such as the FRCC, in the same way as concrete [26,27].
at the crack position. The specimen is a rectangular prism with two bolts (M20) embedded at both ends to transfer the tensile load. In order to control the position of the crack opening, two slits were set on both narrow sides in the middle of the specimen. The depth and width of the slit are 20 mm and 3 mm, respectively. The area of the ligament is 60 × 70 mm 2 . The slits were made by a concrete cutter after the hardening of FRCC to avoid the influence of the flow of the matrix at the casting. The notched rectangular prism specimens with slits have been generally utilized to investigate the tensile characteristics of cementitious materials, such as the FRCC, in the same way as concrete [26,27]. The experimental parameters are matrix strength (mixture proportion shown in Table 2, i.e., Fc24, Fc36, and Fc48) and fiber volume fraction (0%, 1%, and 2%). Therefore, nine series of specimens were determined for the uniaxial tension test. Five specimens were manufactured for each series, so a total of 45 specimens were tested. Since the fresh FRCCs have self-consolidating properties, fresh FRCCs were poured from one end of the mold and allowed to flow naturally until the mold was fully filled. The specimens were cured in the natural environment until the days of loadings.
At the same time carrying out the uniaxial tension test, a compression test was also conducted to confirm the compressive strength and elastic modulus of the FRCC. For each series, three cylinder specimens (100 × 200 mm) were tested in accordance with JIS A 1108 [28] and JIS A 1149 [29]. The cylinder specimens were also cured in the natural environment until the days of the loadings. Table 3 lists the compressive properties of each series. As shown by the results, the exact compressive strengths were much higher than the target compressive strengths due to the use of high early strength Portland cement and long curing times. The compressive strength of the three mixture proportions showed obvious differences between each other.

Loading and Measurement
A uniaxial tension test is carried out using a universal testing machine with a capacity of 2MN. Figure 3 shows the setup of the loading and measurement for the uniaxial tension The experimental parameters are matrix strength (mixture proportion shown in Table 2, i.e., Fc24, Fc36, and Fc48) and fiber volume fraction (0%, 1%, and 2%). Therefore, nine series of specimens were determined for the uniaxial tension test. Five specimens were manufactured for each series, so a total of 45 specimens were tested. Since the fresh FRCCs have self-consolidating properties, fresh FRCCs were poured from one end of the mold and allowed to flow naturally until the mold was fully filled. The specimens were cured in the natural environment until the days of loadings.
At the same time carrying out the uniaxial tension test, a compression test was also conducted to confirm the compressive strength and elastic modulus of the FRCC. For each series, three cylinder specimens (φ100 × 200 mm) were tested in accordance with JIS A 1108 [28] and JIS A 1149 [29]. The cylinder specimens were also cured in the natural environment until the days of the loadings. Table 3 lists the compressive properties of each series. As shown by the results, the exact compressive strengths were much higher than the target compressive strengths due to the use of high early strength Portland cement and long curing times. The compressive strength of the three mixture proportions showed obvious differences between each other.

Loading and Measurement
A uniaxial tension test is carried out using a universal testing machine with a capacity of 2MN. Figure 3 shows the setup of the loading and measurement for the uniaxial tension test. Since the increasing external moment caused by setup irregularity and local fracture caused by the secondary moment would be an inevitable factor in the experiment, pin-fix ends were applied at the boundaries to minimize possible effects on the results [30]. Two displacement transducers (Pi-type) were set at the middle area of 100 mm in length on both sides to measure the axial deformation at the slit position. The loading speed was set to be from 0.5 to 1 mm per minute as the head speed. Visible cracks observation and photographing were done after loading. In addition, the upper and lower parts of the specimen were forcibly pulled apart and fibers on the fracture surface were counted.
test. Since the increasing external moment caused by setup irregularity and local fracture caused by the secondary moment would be an inevitable factor in the experiment, pin-fix ends were applied at the boundaries to minimize possible effects on the results [30]. Two displacement transducers (Pi-type) were set at the middle area of 100 mm in length on both sides to measure the axial deformation at the slit position. The loading speed was set to be from 0.5 to 1 mm per minute as the head speed. Visible cracks observation and photographing were done after loading. In addition, the upper and lower parts of the specimen were forcibly pulled apart and fibers on the fracture surface were counted.  Figure 4 shows the typical failure modes of specimens. Failure modes of specimens can be mainly divided into two types: tensile failure and bending failure. The specimens of tensile failure generated obvious crack(s) throughout the slits on both sides. The specimens of bending failure generated slant cracks from one slit, and it did not penetrate the slit at the other side. The occurrence of bending failure is considered to be caused by the nonuniform distribution of the fibers in the matrix. The specimens, in which failure mode was detected to be bending failure, are not discussed in the following sections.   Figure 4 shows the typical failure modes of specimens. Failure modes of specimens can be mainly divided into two types: tensile failure and bending failure. The specimens of tensile failure generated obvious crack(s) throughout the slits on both sides. The specimens of bending failure generated slant cracks from one slit, and it did not penetrate the slit at the other side. The occurrence of bending failure is considered to be caused by the nonuniform distribution of the fibers in the matrix. The specimens, in which failure mode was detected to be bending failure, are not discussed in the following sections.   Figure 5 shows the examples of crack patterns of specimens in tensile failure. As revealed by the figure, specimens without fibers (Fc24-N, Fc36-N, and Fc48-N) only generated one crack throughout the slits. As for FRCC specimens, an obvious crack throughout slits could be observed while multiple fine cracks were generated near the slits. As for FRCC specimens, the number of cracks increases with increasing fiber volume fraction  Figure 5 shows the examples of crack patterns of specimens in tensile failure. As revealed by the figure, specimens without fibers (Fc24-N, Fc36-N, and Fc48-N) only generated one crack throughout the slits. As for FRCC specimens, an obvious crack throughout slits could be observed while multiple fine cracks were generated near the slits. As for FRCC specimens, the number of cracks increases with increasing fiber volume fraction and matrix strength.   Figure 5 shows the examples of crack patterns of specimens in tensile failure. As revealed by the figure, specimens without fibers (Fc24-N, Fc36-N, and Fc48-N) only generated one crack throughout the slits. As for FRCC specimens, an obvious crack throughout slits could be observed while multiple fine cracks were generated near the slits. As for FRCC specimens, the number of cracks increases with increasing fiber volume fraction and matrix strength.  Figure 6 shows examples of the visual appearance of fracture surfaces in Fc36 series specimens. It could be detected that several fibers on the fracture surface unraveled when they were pulled out from the matrix. The number of fibers on the fracture surface of each specimen was counted after loading. The counted number is considered in comparing the tension test results and calculation of bridging laws in Section 5. The average numbers of fibers on the fracture surface were not proportional to the expected ones given by the planned fiber volume fractions. Although the measured number of fibers was added to the mixture, inconstant fiber distribution may be observed in the case of relatively small dimensions of the specimen. they were pulled out from the matrix. The number of fibers on the fracture surface of each specimen was counted after loading. The counted number is considered in comparing the tension test results and calculation of bridging laws in Section 5. The average numbers of fibers on the fracture surface were not proportional to the expected ones given by the planned fiber volume fractions. Although the measured number of fibers was added to the mixture, inconstant fiber distribution may be observed in the case of relatively small dimensions of the specimen.  Figure 7 shows the tensile load-axial deformation relationship obtained from the uniaxial tension test. Only FRCC specimens showing tensile failure are plotted. In order to compare the tensile behavior between each series, the average curve of test results in each series is also shown in the figure as a red line. While specimens without fibers did not retain the tensile load after first cracking, the tensile load of FRCC specimens decreased gradually after the peak of a large axial deformation with a wide crack opening. In addition, FRCC specimens showed a tensile strain-hardening property that load-increases after the first crack is generated.   Figure 7 shows the tensile load-axial deformation relationship obtained from the uniaxial tension test. Only FRCC specimens showing tensile failure are plotted. In order to compare the tensile behavior between each series, the average curve of test results in each series is also shown in the figure as a red line. While specimens without fibers did not retain the tensile load after first cracking, the tensile load of FRCC specimens decreased gradually after the peak of a large axial deformation with a wide crack opening. In addition, FRCC specimens showed a tensile strain-hardening property that load-increases after the first crack is generated. specimen was counted after loading. The counted number is considered in comparing the tension test results and calculation of bridging laws in Section 5. The average numbers of fibers on the fracture surface were not proportional to the expected ones given by the planned fiber volume fractions. Although the measured number of fibers was added to the mixture, inconstant fiber distribution may be observed in the case of relatively small dimensions of the specimen.  Figure 7 shows the tensile load-axial deformation relationship obtained from the uniaxial tension test. Only FRCC specimens showing tensile failure are plotted. In order to compare the tensile behavior between each series, the average curve of test results in each series is also shown in the figure as a red line. While specimens without fibers did not retain the tensile load after first cracking, the tensile load of FRCC specimens decreased gradually after the peak of a large axial deformation with a wide crack opening. In addition, FRCC specimens showed a tensile strain-hardening property that load-increases after the first crack is generated.   Table 4 summarizes the results of the uniaxial tension test. As revealed by the results, the tensile loads at the first crack of FRCC specimens are larger than those of specimens without fibers (Fc24-N, Fc36-N, and Fc48-N), indicating that the addition of fibers has an inhibitory effect on cracks occurring. The tensile loads at the first crack also increase with the increase of the matrix strength and fiber volume fraction. The maximum loads of FRCC specimens are larger than the tensile loads at first crack, which confirms a tensile Axial deformation (mm) Tensile load (kN)   Table 4 summarizes the results of the uniaxial tension test. As revealed by the results, the tensile loads at the first crack of FRCC specimens are larger than those of specimens without fibers (Fc24-N, Fc36-N, and Fc48-N), indicating that the addition of fibers has an inhibitory effect on cracks occurring. The tensile loads at the first crack also increase with the increase of the matrix strength and fiber volume fraction. The maximum loads of FRCC specimens are larger than the tensile loads at first crack, which confirms a tensile strain-hardening property.  Figure 8 shows the relationship between maximum load and the experimental parameters, i.e., the compressive strength of FRCC and fiber volume fraction. The lines in the figure are connected by the average values of the maximum loads in each series. By comparing the specimens with the same fiber volume fraction, average maximum loads increase with the increase of compressive strength. On the other hand, by comparing the specimens with the same mixture proportion, average maximum loads increase as the fiber volume fraction increases. rameters, i.e., the compressive strength of FRCC and fiber volume fraction. The lines in the figure are connected by the average values of the maximum loads in each series. By comparing the specimens with the same fiber volume fraction, average maximum loads increase with the increase of compressive strength. On the other hand, by comparing the specimens with the same mixture proportion, average maximum loads increase as the fiber volume fraction increases.  Figure 9 shows the dimensions of the specimen for the pullout test of the individual bundled aramid fiber and the constitution of the mold. The specimen is a thin plate made of the matrix without a fiber in which an individual fiber is embedded at the center of the plate. The dimension in the plane section is 30 × 30 mm 2 , and the thickness of the plate is one of the experimental parameters. The mold consists of two acrylic plates and three rubber plates. A total of five plates are fixed by bolts so as not to cause any visible deformation of the rubber plates. An individual fiber is positioned by the holes of the upper and lower rubber plates. A cementitious matrix is poured from the injection hole and the ventilator holes function so as not to make air voids. The thickness of the specimen is varied by changing the thickness of the middle rubber plate. The dimensions of the specimen and the mold are exactly the same as those used in the authors' previous study [17].   Figure 9 shows the dimensions of the specimen for the pullout test of the individual bundled aramid fiber and the constitution of the mold. The specimen is a thin plate made of the matrix without a fiber in which an individual fiber is embedded at the center of the plate. The dimension in the plane section is 30 × 30 mm 2 , and the thickness of the plate is one of the experimental parameters. The mold consists of two acrylic plates and three rubber plates. A total of five plates are fixed by bolts so as not to cause any visible deformation of the rubber plates. An individual fiber is positioned by the holes of the upper and lower rubber plates. A cementitious matrix is poured from the injection hole and the ventilator holes function so as not to make air voids. The thickness of the specimen is varied by changing the thickness of the middle rubber plate. The dimensions of the specimen and the mold are exactly the same as those used in the authors' previous study [17].

Specimens
comparing the specimens with the same fiber volume fraction, average maximum loads increase with the increase of compressive strength. On the other hand, by comparing the specimens with the same mixture proportion, average maximum loads increase as the fiber volume fraction increases.  Figure 9 shows the dimensions of the specimen for the pullout test of the individual bundled aramid fiber and the constitution of the mold. The specimen is a thin plate made of the matrix without a fiber in which an individual fiber is embedded at the center of the plate. The dimension in the plane section is 30 × 30 mm 2 , and the thickness of the plate is one of the experimental parameters. The mold consists of two acrylic plates and three rubber plates. A total of five plates are fixed by bolts so as not to cause any visible deformation of the rubber plates. An individual fiber is positioned by the holes of the upper and lower rubber plates. A cementitious matrix is poured from the injection hole and the ventilator holes function so as not to make air voids. The thickness of the specimen is varied by changing the thickness of the middle rubber plate. The dimensions of the specimen and the mold are exactly the same as those used in the authors' previous study [17].  The experimental parameters are matrix strength (mixture proportion shown in Table 2, i.e., Fc24, Fc36, and Fc48) and the embedded length of fiber (thickness of the plate, 4 mm, 8 mm, and 12 mm). Therefore, nine series of specimens were determined for the pullout test. Five specimens were manufactured for each series, so a total of 45 specimens were basically tested. The matrix compressive strengths obtained by three cylinder specimens (φ100 × 200 mm) for each mixture proportion in the testing age (average 10day curing time) were 26.5 MPa, 37.1 MPa, and 52.6 MPa for Fc24, Fc36, and Fc48 series specimens, respectively.

Loading and Measurement
Pullout load was applied using an electronic system universal testing machine with a capacity of 200 N, as shown in Figure 10. The specimen was fixed via an adhered steel plate, and the embedded fiber was clamped directly by throwing a jig. The head speed was set to 1 mm per minute. The pullout load and head displacement were recorded. The loading and measurement methods are exactly the same as those applied in the authors' previous study [17].

Loading and Measurement
Pullout load was applied using an electronic system universal testing machine with a capacity of 200 N, as shown in Figure 10. The specimen was fixed via an adhered steel plate, and the embedded fiber was clamped directly by throwing a jig. The head speed was set to 1 mm per minute. The pullout load and head displacement were recorded. The loading and measurement methods are exactly the same as those applied in the authors' previous study [17].

Failure Pattern and Pullout Load vs. Slip Relationship
In all of the tested specimens, the fiber was pulled out from the matrix without observing a clear rupture of the fiber, as shown in Figure 11. The surface of the fiber embedded in the matrix was a little damaged. Only one specimen (Fc24-8 mm) could not be loaded due to the damage to the specimen at the detaching of the mold. Pullout load-slip curves of all the tested specimens are shown in Figure 12. The slip is calculated from the measured displacement of the loading head, subtracting the elongation of fiber outside the matrix, as in the previous study [17]. The averaged curves are calculated in each series of specimens to compare the curves between the different series

Failure Pattern and Pullout Load vs. Slip Relationship
In all of the tested specimens, the fiber was pulled out from the matrix without observing a clear rupture of the fiber, as shown in Figure 11. The surface of the fiber embedded in the matrix was a little damaged. Only one specimen (Fc24-8 mm) could not be loaded due to the damage to the specimen at the detaching of the mold. mens, respectively.

Loading and Measurement
Pullout load was applied using an electronic system universal testing machine with a capacity of 200 N, as shown in Figure 10. The specimen was fixed via an adhered steel plate, and the embedded fiber was clamped directly by throwing a jig. The head speed was set to 1 mm per minute. The pullout load and head displacement were recorded. The loading and measurement methods are exactly the same as those applied in the authors' previous study [17].

Failure Pattern and Pullout Load vs. Slip Relationship
In all of the tested specimens, the fiber was pulled out from the matrix without observing a clear rupture of the fiber, as shown in Figure 11. The surface of the fiber embedded in the matrix was a little damaged. Only one specimen (Fc24-8 mm) could not be loaded due to the damage to the specimen at the detaching of the mold. Pullout load-slip curves of all the tested specimens are shown in Figure 12. The slip is calculated from the measured displacement of the loading head, subtracting the elongation of fiber outside the matrix, as in the previous study [17]. The averaged curves are calculated in each series of specimens to compare the curves between the different series Pullout load-slip curves of all the tested specimens are shown in Figure 12. The slip is calculated from the measured displacement of the loading head, subtracting the elongation of fiber outside the matrix, as in the previous study [17]. The averaged curves are calculated in each series of specimens to compare the curves between the different series of specimens. As shown in the figure, the curves generally show two stages, i.e., the load increases lineally up to the maximum load, and decreases gradually. The pullout load becomes almost zero when the slip reaches the embedded length of the fiber. The maximum load generally increases as the matrix strength and the embedded length also increase. of specimens. As shown in the figure, the curves generally show two stages, i.e., the load increases lineally up to the maximum load, and decreases gradually. The pullout load becomes almost zero when the slip reaches the embedded length of the fiber. The maximum load generally increases as the matrix strength and the embedded length also increase.  Figure 13 shows the relationship between the maximum pullout load (Pmax) and compressive strength of the matrix (fc). The black plots show the average values of the maximum pullout loads in each series. It can be recognized that the maximum pullout loads increase as the matrix strength increases. Straight lines can be obtained by the least square method for each embedded length series of the specimens. The coefficients of the lines are different in each series. The maximum pullout load of longer embedded length specimens is highly influenced by the matrix strength. It is considered that the bond resistance of bundled fiber is due to a constant bond, along with the embedded fiber-like friction mechanism.   Figure 13 shows the relationship between the maximum pullout load (P max ) and compressive strength of the matrix (f c ). The black plots show the average values of the maximum pullout loads in each series. It can be recognized that the maximum pullout loads increase as the matrix strength increases. Straight lines can be obtained by the least square method for each embedded length series of the specimens. The coefficients of the lines are different in each series. The maximum pullout load of longer embedded length specimens is highly influenced by the matrix strength. It is considered that the bond resistance of bundled fiber is due to a constant bond, along with the embedded fiber-like friction mechanism.  Figure 14 shows the relationship between the embedded length of the fiber (lb) and the coefficients of the lines (α) shown in Figure 13. The curve shown in the figure is obtained by the regression analysis as it is expressed by the powered function of the embedded length. Finally, the maximum pullout load can be evaluated by the following Equation (1).

Evaluation of Maximum Pullout Load
where, Pmax : maximum pullout load (N); lb : embedded length of fiber (mm); fc : compressive strength of matrix (MPa).

Calculation Method of Bridging Law
The calculation method of the bridging law is exactly the same as that in the authors' previous study [17], except for the maximum pullout load of the individual fiber. In the authors' previous study, the maximum pullout load was simply given by a proportional relationship with the embedded length without considering the matrix strength. The bridging law is obtained by the summation of forces carried by the individual bridging  Figure 14 shows the relationship between the embedded length of the fiber (l b ) and the coefficients of the lines (α) shown in Figure 13. The curve shown in the figure is obtained by the regression analysis as it is expressed by the powered function of the embedded length. Finally, the maximum pullout load can be evaluated by the following Equation (1).
where, P max : maximum pullout load (N); l b : embedded length of fiber (mm); f c : compressive strength of matrix (MPa).  Figure 14 shows the relationship between the embedded length of the fiber (lb) and the coefficients of the lines (α) shown in Figure 13. The curve shown in the figure is obtained by the regression analysis as it is expressed by the powered function of the embedded length. Finally, the maximum pullout load can be evaluated by the following Equation (1).

Calculation Method of Bridging Law
The calculation method of the bridging law is exactly the same as that in the authors' previous study [17], except for the maximum pullout load of the individual fiber. In the authors' previous study, the maximum pullout load was simply given by a proportional relationship with the embedded length without considering the matrix strength. The bridging law is obtained by the summation of forces carried by the individual bridging

Calculation Method of Bridging Law
The calculation method of the bridging law is exactly the same as that in the authors' previous study [17], except for the maximum pullout load of the individual fiber. In the authors' previous study, the maximum pullout load was simply given by a proportional relationship with the embedded length without considering the matrix strength. The bridging law is obtained by the summation of forces carried by the individual bridging fibers considering the probability density function for fiber inclination angle, fiber centroidal location, snubbing effect due to fiber inclination angle, and the apparent rupture strength of the fiber. Though the bilinear model that is the same as that in the previous study for the pullout load-slip model is adapted, the maximum pullout load is given by Equation (1). The parameters for the calculation are summarized in Table 5. Table 5. Parameter input values for bridging law calculation.

Parameter Input
Cross-sectional area of individual fiber, A f (mm 2 ) 0.196 Length of fiber, l f (mm) 30 Apparent rupture strength of fiber [17], σ fu (MPa) σ fu = 1080 × e −0.667ψ Bilinear model [17] Maximum pullout load, P max (N) Crack width at P max , w max (mm) Elliptic distribution [17] Orientation intensity for x-y plane, k xy Orientation intensity for z-x plane, k zx Principle orientation angle, θ r  Figure 15 shows the comparison of tensile load-crack width curves between uniaxial tension test results and bridging law calculation results for each series of tested parameters. Tables 6-8 show the maximum tensile load obtained by the uniaxial tension test and bridging law calculation for Fc24, Fc36, and Fc48 series of specimens, respectively. fibers considering the probability density function for fiber inclination angle, fiber centroidal location, snubbing effect due to fiber inclination angle, and the apparent rupture strength of the fiber. Though the bilinear model that is the same as that in the previous study for the pullout load-slip model is adapted, the maximum pullout load is given by Equation (1). The parameters for the calculation are summarized in Table 5. Elliptic distribution [17] Orientation intensity for x-y plane, kxy Orientation intensity for z-x plane, kzx Principle orientation angle, θr  Figure 15 shows the comparison of tensile load-crack width curves between uniaxial tension test results and bridging law calculation results for each series of tested parameters. Tables 6-8 show the maximum tensile load obtained by the uniaxial tension test and bridging law calculation for Fc24, Fc36, and Fc48 series of specimens, respectively.  Tensile load is divided by the number of fibers, i.e., the counted number of fibers on the fracture surface in the case of the test results as described in Section 3.3, or the number of effective fibers [19] given by following Equation (2) in the case of the calculation. The crack width for the test results is obtained from the measured axial deformation divided by the number of cracks observed in the uniaxial tension test.

Comparison of Calculation Result with Uniaxial Tension Test Result
where, Nf : number of effective fibers; Vf : fiber volume fraction; Am : cross-sectional area of matrix; Af : cross-sectional area of individual fiber; ηf : fiber effectiveness.   Tensile load is divided by the number of fibers, i.e., the counted number of fibers on the fracture surface in the case of the test results as described in Section 3.3, or the number of effective fibers [19] given by following Equation (2) in the case of the calculation. The crack width for the test results is obtained from the measured axial deformation divided by the number of cracks observed in the uniaxial tension test.
where, N f : number of effective fibers; V f : fiber volume fraction; A m : cross-sectional area of matrix; A f : cross-sectional area of individual fiber; η f : fiber effectiveness.   As seen in Figure 15, the calculation results of bridging law generally express the results of the uniaxial tension test well. Concerning the maximum load, the tensile maximum load per fiber of the averaged test results for Fc24, Fc36, and Fc48 series of specimens are 1.07, 0.97, and 0.93 times the maximum load obtained by the calculation, respectively. Thus, it is considered that the calculated bridging law considering the effect of matrix strength is adaptable to express the bridging performance of the bundled aramid fiber.
The effect of matrix strength on the maximum tensile load is investigated based on the bridging law calculation. Figure 16 shows the relationship between compressive strength and bridging strength, which is defined as the maximum tensile load divided by the cross-sectional area of the matrix. The calculations are carried out for several compressive strengths from 20 MPa to 70 MPa with fiber volume fractions (V f ) of 1% and 2%. The calculation results show that the bridging strength shows a linear relationship up to a compressive strength of around 50 MPa. Beyond 50 MPa, the increment of bridging strength becomes small. This is due to the rupture of fiber in the calculation.

Conclusions
To investigate the influence of matrix strength on the bridging performance of the FRCC with bundled aramid fiber, the uniaxial tension test of FRCC, the pullout test for an individual fiber, and the calculation of bridging law were conducted with the main parameters of matrix strength and fiber volume fraction. The following are concluded from this study.
1. From the uniaxial tension test results, maximum tensile load increases as the compressive strength of FRCC and fiber volume fraction increases.
2. From the pullout test results, the maximum pullout load generally increases as the matrix strength and the embedded length of the fiber also increase.
3. From the pullout test results, the maximum pullout load is evaluated by matrix compressive strength and the embedded length of fiber.
4. The calculated bridging law considering the effect of matrix strength is adaptable to express the bridging performance of the bundled aramid fiber.
5. The bridging law calculation result shows that the bridging strength shows a linear relationship up to a compressive strength of around 50 MPa. Beyond 50 MPa, the increment of bridging strength becomes small due to the rupture of the fiber.
It is considered that these findings will be valuable to evaluate the tensile properties of FRCC by the matrix strength, which is generally exhibited by compressive strength in the design of structural elements, such as coupling beams, columns, seismic walls, and beam-column joints. The authors also consider that simple calculation methodologies for the tensile strength and toughness of FRCC will provide the effective use of FRCC in the structures, and these will be studied in the future.

Conclusions
To investigate the influence of matrix strength on the bridging performance of the FRCC with bundled aramid fiber, the uniaxial tension test of FRCC, the pullout test for an individual fiber, and the calculation of bridging law were conducted with the main parameters of matrix strength and fiber volume fraction. The following are concluded from this study.
1. From the uniaxial tension test results, maximum tensile load increases as the compressive strength of FRCC and fiber volume fraction increases.
2. From the pullout test results, the maximum pullout load generally increases as the matrix strength and the embedded length of the fiber also increase.
3. From the pullout test results, the maximum pullout load is evaluated by matrix compressive strength and the embedded length of fiber.
4. The calculated bridging law considering the effect of matrix strength is adaptable to express the bridging performance of the bundled aramid fiber.
5. The bridging law calculation result shows that the bridging strength shows a linear relationship up to a compressive strength of around 50 MPa. Beyond 50 MPa, the increment of bridging strength becomes small due to the rupture of the fiber.
It is considered that these findings will be valuable to evaluate the tensile properties of FRCC by the matrix strength, which is generally exhibited by compressive strength in the design of structural elements, such as coupling beams, columns, seismic walls, and beam-column joints. The authors also consider that simple calculation methodologies for the tensile strength and toughness of FRCC will provide the effective use of FRCC in the structures, and these will be studied in the future.