Verification of Crack Width Evaluation in Fiber-Reinforced Cementitious Composite Reinforced with Various Types of Fiber-Reinforced Polymer Bars

Highlights

What are the main findings?

• The proposed crack width evaluation method is valid for various FRP–FRCC com-binations, as predictions matched test results.
• Greater rib height of FRP bars and higher fiber-volume fraction in FRCC improve bond strength and reduce crack width.

What is the implication of the main finding?

• The crack width evaluation method incorporating both bridging and bond behav-iors can be extended to various FRP–FRCC combinations, enabling broader struc-tural applications.
• It suggests that accurate bond modeling is essential for reliable crack width pre-diction, which in turn supports the development of design guidelines for FRP–FRCC composite members.

Abstract

This study aims to verify the adaptability of a crack width evaluation method for fiber-reinforced cementitious composite (FRCC) proposed by the authors to various combinations of fiber-reinforced polymer (FRP) bars and FRCCs. As this evaluation method requires bond constitutive laws between FRP bars and FRCC, bond tests between FRP and FRCCs were conducted. The FRP and FRCC combinations used in the bond tests were spiral-type CFRP and GFRP bars with PVA-FRCC, as well as strand-type CFRP bars with aramid–FRCC. The maximum bond stress tended to increase as the rib–height ratio of the spiral-type bars increased. When the rib–height ratio increased by 50%, the maximum bond stress of the CFRP and GFRP bars increased by 11% and 33%, respectively. For aramid–FRCC, the average maximum bond stress in the FRCC with a 0.25% volume fraction was 1.67 times that in mortar, and that in 0.50% was 2.01 times that in mortar. The bond constitutive laws were modeled using the trilinear model. Verifications of the method’s adaptability were conducted using tension tests on prisms made of spiral-type CFRP and GFRP bars with PVA-FRCC. As a result of the tension tests, when the FRP strain reached approximately 0.3%, the crack width was about 0.2 mm for CFRP bars and about 0.1 mm for GFRP bars. Verifications were also conducted using four-point bending tests on strand-type CFRP bar beams with aramid–FRCC. The crack width at the same FRP strain tended to become smaller as the fiber volume fraction of FRCC increased. When the FRP strain reached approximately 0.2%, the average crack width of the mortar specimen was around 0.25 mm, whereas it was about 0.15 mm in FRCC with a 0.25% volume fraction and about 0.10 mm at 0.5%. The test results for FRP strain versus crack width relationships were compared with the calculations using the crack width prediction formula. The test results and calculation results were in good agreement.

1. Introduction

Cementitious composites such as concrete demonstrate brittle behavior under tensile stress. Fiber-reinforced cementitious composite (FRCC) is a cementitious material that reduces the brittleness of cementitious composite under tension and bending. The fibers in FRCC bridge between cracks and transmit tensile force, thus inhibiting cracking and its propagation. Metallic fibers, such as steel fibers, and polymer fibers, such as polyvinyl alcohol (PVA) fibers and aramid fibers, are mixed into FRCC. When designing reinforced concrete (RC) structures, concrete is usually considered to not carry tensile stress. Thus, FRCC is expected to carry the tensile stress as it has high ductility under this form of stress. Therefore, it is important to quantitatively describe the tensile properties of FRCC. The bridging law (the tensile stress–crack width relationship) succinctly represents the tensile properties of FRCC.

FRCC bridging laws have been studied by numerous researchers [1,2]. Since the 1980s, researchers have discussed how the bridging law is characterized and controlled not only by the properties of the matrix and fiber, but also by the fiber–matrix interface [3,4]. For example, Amin et al. [5] proposed a method to derive the stress–crack width relationship for steel fiber-reinforced concrete using the prism bending test and experimentally verified its validity and effectiveness. Yang et al. [6] derived the bridging law for engineered cementitious composite (ECC) based on experiments and analyses, demonstrating its effectiveness in crack control and toughness enhancement. The authors also proposed fiber bridging laws for FRCC using PVA, polypropylene (PP), steel, and aramid fibers [7,8,9]. In these studies, the bridging law is essentially determined by modeling the pullout behavior of individual fibers and summing them up.

Steel rebar corrosion is one of the primary factors in the deterioration of RC structures. When rebar is exposed to harsh environments, expansive corrosion products are generated, which can crack the cover concrete and reduce the confinement around the rebar [10,11,12]. Fiber-reinforced polymer (FRP) bars have been proposed as an alternative form of reinforcement [13,14]. These bars are characterized by corrosion resistance and elastic behavior until rupture, and have the potential to extend the lifespan of structures. The mechanical properties of FRP bars, such as their tensile strength and elastic modulus, differ depending on the fibers used. Research has been conducted on the use of FRP bars as reinforcements, including carbon fiber-reinforced polymer (CFRP), glass fiber-reinforced polymer (GFRP), and aramid fiber-reinforced polymer (AFRP) [15,16,17].

In recent years, deterioration and rising maintenance costs for RC structures have become problems, but structures that combine FRP bars and FRCC may provide a solution. FRCC reinforced with FRP bars (FRP/FRCC) is expected to produce highly durable structures with low maintenance costs. To put FRP/FRCC elements into practical use, it is necessary to clarify their structural performance. Research has been conducted on the bending performance of CFRP/ECC and GFRP/ECC [18,19,20]; the bond performance of CFRP/ECC, GFRP/ECC, and CFRP/SHCC (strain-hardening cementitious composite) [21,22,23]; and the shear performance of GFRP/ECC [24]. Quantitative evaluations of crack widths in structures are also required to maintain their aesthetic appearance and durability. Sunaga et al. proposed a crack width prediction formula that considers the bridging law of FRCC [25]. This formula makes it possible to evaluate the crack width from rebar strain. This formula is based on the equilibrium between the sum of the fiber bridging force between cracks in the FRCC and the tensile force of the reinforcing bar, and the tensile force in the FRCC transfers through the bond stress between the FRCC and rebar. Therefore, information on both the bridging law and the bond constitutive law (the bond stress–slip relationship) is required. Takasago et al. used this formula to evaluate the crack width of members that combined a braided AFRP bar and PVA-FRCC [26]. However, there are few examples of crack width evaluations of FRP/FRCC members that combine other FRP bars with FRCC. Crack width evaluation should be performed for other FRP and FRCC combinations. In particular, there have been few studies related to the bond constitutive laws of FRPs and FRCCs that specifically aim to evaluate crack width.

This study aims to verify the adaptability of an FRP/FRCC crack width evaluation method to various combinations of FRP bars and FRCCs, following the approach used by Takasago et al. [26]. The flow of current research is shown in Figure 1. As this evaluation method requires the bond constitutive laws between FRP bars and FRCCs, bond tests between them are conducted. The FRP and FRCC combinations used in the bond tests are spiral-type CFRP and GFRP bars with PVA-FRCC, as well as strand-type CFRP bars with aramid–FRCC. The bond constitutive law is modeled using a trilinear model. The adaptability of the crack width evaluation method is verified through tension tests on prisms of spiral-type CFRP and GFRP bars with PVA-FRCC. Verification is also conducted using four-point bending tests on beams made of strand-type CFRP bars with aramid–FRCC. The crack width predictions using the proposed formula are compared with the test results.

Including crack width evaluation, the structural performance of members with various FRP and FRCC combinations can be more accurately assessed, improving design guidelines and broadening the application of these composite systems.

2. Bond Properties of FRP/FRCC

2.1. Bond Properties of Spiral-FRP/PVA-FRCC

2.1.1. Outline of Bond Test for Spiral-FRP/PVA-FRCC

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Materials

Table 1 lists the material properties of the FRP bars, and Figure 2 shows photos of these bars. Spiral-type CFRP bars (CS5, CS10, and CS15) and GFRP bars (GS5, GS10, and GS15) were used. Spiral-type bars are made by helically winding fibers of the same type as the base material around the surface of a straight FRP bar. The rib size is adjusted according to the thickness of the wound fibers. Each type of FRP bar has three rib–height ratio targets (5%, 10%, and 15%) at the time of manufacturing. The rib–height ratios shown in the table are based on actual measurements.

Table 2 shows the material properties of PVA fibers used for PVA-FRCC. Figure 3 shows a photo of the fibers. They have a diameter of 100 μm and a length of 12 mm. Table 3 shows the mix proportion and compressive properties of the FRCC. Thus, φ100 mm × 200 mm cylinder compression tests followed by JIS A 1108 [27] were conducted to obtain the FRCC’s compressive properties. All specimens were made from the same batch of FRCC. The fiber volume fraction of the FRCC is 2%.

Table 1. Material properties of spiral-type FRP bars.

ID	FRP Bar	Cross-Section (mm2)	Average Diameter (mm)	Perimeter (mm)	Rib–Height Ratio (%)	Elastic Modulus [SD] (GPa)
CS5	Spiral CFRP	102	11.4	35.9	5.7	127 [9.3]
CS10		114	12.1	37.9	8.8	111 [6.1]
CS15		129	12.8	40.3	9.2	100 [9.8]
GS5	Spiral GFRP	94	11.0	34.4	5.9	43.2 [0.8]
GS10		103	11.4	35.9	8.5	38.0 [3.8]
GS15		114	12.1	37.9	8.8	31.6 [3.0]

Note: Results of tension test followed by JIS A 1192 [28]. Three test pieces. SD: standard deviation.

Table 1. Material properties of spiral-type FRP bars.

ID	FRP Bar	Cross-Section (mm2)	Average Diameter (mm)	Perimeter (mm)	Rib–Height Ratio (%)	Elastic Modulus [SD] (GPa)
CS5	Spiral CFRP	102	11.4	35.9	5.7	127 [9.3]
CS10		114	12.1	37.9	8.8	111 [6.1]
CS15		129	12.8	40.3	9.2	100 [9.8]
GS5	Spiral GFRP	94	11.0	34.4	5.9	43.2 [0.8]
GS10		103	11.4	35.9	8.5	38.0 [3.8]
GS15		114	12.1	37.9	8.8	31.6 [3.0]

Note: Results of tension test followed by JIS A 1192 [28]. Three test pieces. SD: standard deviation.

Table 2. Material properties of PVA fibers.

Length (mm)	Diameter (µm)	Tensile Strength (MPa)	Elastic Modulus (GPa)	Density (g/cm3)
12	100	1200	28	1.3

Note: Nominal values provided by the manufacturer.

Table 2. Material properties of PVA fibers.

Length (mm)	Diameter (µm)	Tensile Strength (MPa)	Elastic Modulus (GPa)	Density (g/cm3)
12	100	1200	28	1.3

Note: Nominal values provided by the manufacturer.

Table 3. Mix proportion and compressive properties of PVA-FRCC.

Specimen	Unit Weight (kg/m3)					Compressive Strength [SD] (MPa)	Elastic Modulus [SD] (GPa)
	W	C	S	FA	PVA
CSxx or GSxx/PVA200	380	678	484	291	26	51.7 [1.7]	17.3 [0.7]

Note: W: water, C: high-early-strength Portland cement, FA: type II fly ash of Japanese Industrial Standard (JIS A 6201 [29]), S: sand under 0.2 mm in size, and PVA: PVA fiber. Three compression test pieces. SD: standard deviation.

Table 3. Mix proportion and compressive properties of PVA-FRCC.

Specimen	Unit Weight (kg/m3)					Compressive Strength [SD] (MPa)	Elastic Modulus [SD] (GPa)
	W	C	S	FA	PVA
CSxx or GSxx/PVA200	380	678	484	291	26	51.7 [1.7]	17.3 [0.7]

Note: W: water, C: high-early-strength Portland cement, FA: type II fly ash of Japanese Industrial Standard (JIS A 6201 [29]), S: sand under 0.2 mm in size, and PVA: PVA fiber. Three compression test pieces. SD: standard deviation.

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Specimens

The list of test specimens is shown in

Table 4. List of bond test specimens for spiral-FRP/PVA-FRCC.

Specimen	FRCC Fiber	FRP Bar	Diameter of FRP Bar (mm)	Bond Length (mm)	Dimensions of Cross-Section	Fiber Volume Fraction (%)	Number of Specimens
CS5/PVA200	PVA Length 12 mm Diameter 100 μm	Spiral CFRP	11.4	48	100 mm × 100 mm	2	3
CS10/PVA200			12.1				3
CS15/PVA200			12.8				3
GS5/PVA200		Spiral GFRP	11.0				3
GS10/PVA200			11.4				3
GS15/PVA200			12.1				3

, and the specimen geometry and loading method are illustrated in Figure 4. Each specimen is a rectangular block with a cross-section of 100 mm × 100 mm and a height of 120 mm; one FRP bar is arranged at the center. A steel coupler is attached to the end of the FRP bar for fixation in the testing machine’s chuck. The bond length in the test section is 48 mm, approximately four times the diameter of the bar. To prevent cone-shaped failure at the end of the bonded section, unbonded zones are provided by covering the bar. The specimens were fabricated by placing the FRP bar in the formwork of a wooden panel and casting FRCC from the direction indicated in the figure. The specimen was placed on the loading plate attached to the upper head of a 2 MN universal testing machine, and monotonic pullout loading was applied by gripping the coupler with the chuck on the lower head. To avoid restraining lateral displacement in the specimen, a Teflon sheet was inserted between it and the loading plate. The measurement items are the applied load and the free-end slip.

2.1.2. Results of Bond Test for Spiral-FRP/PVA-FRCC

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Failure modes

Figure 5 shows examples of CS5/PVA200, CS10/PVA200, and CS15/PVA200 after loading. In all specimens, cracking occurred after reaching the maximum bond stress, and as the cracks widened, the load decreased. Fiber bridging across the cracks was observed. No significant differences in the cracking behavior of the FRCC were found due to differences in the rib–height ratio. Photos of FRP bars after loading are shown in Figure 6. The primary failure mode was the pullout of the reinforcing bar from the FRCC (hereafter referred to as Pullout A). In some cases, the spiral fiber of the bar detached (marked by red circles in Figure 6), resulting in a failure mode where only the core of the reinforcing bar was pulled from the FRCC (hereafter referred to as Pullout B).

Figure 7 shows examples of GS5/PVA200, GS10/PVA200, and GS15/PVA200 after loading. The failure modes were similar to those observed in the CS series specimens. Figure 8 shows photos of the FRP bars after loading: the main failure mode was Pullout A, and one bar circled in the figure shows Pullout B.

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Bond stress–slip curve

Load-end slip is calculated by adding the elongation of the bar to the measured free-end slip, assuming that the bond stress in the test section is uniform. The bond stress is calculated by dividing the pullout load by the surface area of the bar in the bond region.

Figure 9 shows the bond stress–load-end slip curves of CS5/PVA200, CS10/PVA200, and CS15/PVA200. The curves without symbols are Pullout A specimens, and those with a double circle symbol are Pullout B specimens. In CS5/PVA200 Pullout A specimens, the bond stress gradually decreased after reaching the maximum bond stress, until the load-end slip reached approximately 5 mm. Subsequently, the bond stress rapidly decreased until the load-end slip reached about 9 mm. In the CS5/PVA200 Pullout B specimen, the bond stress gradually decreased after reaching its maximum, until the load-end slip reached approximately 10 mm. In CS10/PVA200 Pullout A specimens, the bond stress decreased once and then increased again before reaching its maximum. After reaching the maximum, the bond stress rapidly decreased until the load-end slip reached about 9 mm. In the CS10/PVA200 Pullout B specimen, the bond stress gradually decreased after reaching its maximum, until the load-end slip reached approximately 12 mm. In CS15/PVA200 specimens, the bond stress rapidly decreased after reaching its maximum until the load-end slip reached approximately 10 mm.

Figure 10 shows the bond stress–load-end slip curves of GS5/PVA200, GS10/PVA200, and GS15/PVA200. In GS5/PVA200 specimens, the bond stress gradually decreased after reaching its maximum, until load-end slip reached approximately 5 mm. Subsequently, the bond stress rapidly decreased until the load end slip reached about 10 mm. In GS10/PVA200 Pullout A specimens, the bond stress gradually decreased after reaching its maximum, until load-end slip reached approximately 5 mm. After reaching the maximum, the bond stress rapidly decreased until the load-end slip reached about 10 mm. In the GS10/PVA200 Pullout B specimen, the bond stress remained stable after reaching the maximum until the load-end slip reached approximately 5 mm. Then, the bond stress rapidly decreased until the slip reached about 10 mm. In GS15/PVA200, the bond stress rapidly decreased after reaching the maximum until the load-end slip reached approximately 10 mm. In GS15/PVA200 specimens, the bond stress rapidly decreased after reaching the maximum until the load-end slip reached approximately 10 mm.

Table 5. Bond test results for spiral-FRP/PVA-FRCC.

Specimen		Test Result
		Failure Mode	Max. Bond Stress (MPa)	Average (MPa)
CS5/PVA200	CS5/PVA200-1	Pullout A	12.8	12.7 (12.6)
	CS5/PVA200-2	Pullout A	12.4
	CS5/PVA200-3	Pullout B	12.8
CS10/PVA200	CS10/PVA200-1	Pullout A	14.6	13.5 (14.0)
	CS10/PVA200-2	Pullout B	12.6
	CS10/PVA200-3	Pullout A	13.4
CS15/PVA200	CS15/PVA200-1	Pullout A	11.4	13.0
	CS15/PVA200-2	Pullout A	14.2
	CS15/PVA200-3	Pullout A	13.4
GS5/PVA200	GS5/PVA200-1	Pullout A	10.1	9.75
	GS5/PVA200-2	Pullout A	8.45
	GS5/PVA200-3	Pullout A	10.7
GS10/PVA200	GS10/PVA200-1	Pullout A	12.8	12.5 (12.3)
	GS10/PVA200-2	Pullout A	11.8
	GS10/PVA200-3	Pullout B	12.9
GS15/PVA200	GS15/PVA200-1	Pullout A	13.6	13.0
	GS15/PVA200-2	Pullout A	12.2
	GS15/PVA200-3	Pullout A	13.3

( ): Average excluding Pullout B specimen.

lists the bond test results. The average is the mean of the maximum bond stress. The values in parentheses represent the average excluding Pullout B. The maximum bond stress tends to increase as the rib–height ratio increases. For CS series specimens, when the rib–height ratio increased from 5.7% to 8.8%, the maximum bond stress increased by 11%. For GS series specimens, when the ratio increased from 5.9% to 8.8%, the maximum bond stress increased by 33%. The difference in maximum bond stress due to failure mode is small.

2.1.3. Modeling of Bond Constitutive Law for Spiral-FRP/PVA-FRCC

The bond stress–load-end slip curve is modeled by the trilinear model (Figure 11). The curve is modeled only for the main failure mode, Pullout A. ${\tau }_{max}$ is the average of the maximum bond stress. $S_{max}$ is the average of the load-end slip at the maximum bond stress. The initial stiffness, $k_1$, is the slope of the regression line obtained by the least squares method using the experimental data up to two-fifths of the maximum bond stress. The point on the line with slope $k_1$ is defined as ${\tau }_1$ and $S_1$, such that the total area enclosed between the experimental data and the model up to the maximum bond stress becomes zero. The ultimate stiffness, $k_u$, is the slope of the straight line connecting the point of maximum bond stress to the point where the bond stress first reaches a minimum after softening, corresponding to the load-end slip, $S_{Lmin}$. A summary of each characteristic value in the model is presented in

Table 6. Characteristic values in the model for spiral-FRP/PVA-FRCC.

Specimen	${\mathit{\tau }}_1$ (MPa)	${\mathit{S}}_1$ (mm)	${\mathit{k}}_1$ (N/mm3)	${\mathit{k}}_2$ (N/mm3)	${\mathit{\tau }}_{\mathit{m}\mathit{a}\mathit{x}}$ (MPa)	${\mathit{S}}_{\mathit{m}\mathit{a}\mathit{x}}$ (mm)	${\mathit{S}}_{\mathit{L}\mathit{m}\mathit{i}\mathit{n}}$ (mm)	${\mathit{S}}_{\mathit{u}}$ (mm)	${\mathit{k}}_{\mathit{u}}$ (N/mm3)
CS5/PVA200	11.4	0.174	65.2	1.34	12.6	1.10	8.79	12.4	−1.12
CS10/PVA200	10.9	0.193	56.2	1.10	14.0	3.03	9.56	12.7	−1.45
CS15/PVA200	9.13	0.136	67.2	1.04	13.0	3.84	11.2	17.0	−0.998
GS5/PVA200	8.74	0.300	29.3	0.926	9.75	1.35	10.0	13.0	−0.893
GS10/PVA200	10.9	0.218	50.1	2.64	12.3	0.751	9.90	14.2	−0.916
GS15/PVA200	11.7	0.266	44.0	3.08	13.0	0.698	11.4	14.3	−0.960

.

Figure 12 and Figure 13 compare the test results and models for the CS and GS series specimens. The developed models represent the experimental results well.

2.2. Bond Properties of Strand-CFRP/Aramid–FRCC

2.2.1. Outline of Bond Test for Strand-CFRP/Aramid–FRCC

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Materials

Table 7. Material properties of strand-type CFRP bar.

ID	FRP Bar	Diameter (mm)	Cross-Section (mm2)	Tensile Strength (MPa)	Elastic Modulus (GPa)
CSt7	Strand CFRP	15.9	125.0	1770	150

Note: Results of tension test followed by JIS A 1192 [28] provided by the manufacturer.

shows the material properties of the FRP bar, and Figure 14 shows a photo of it. A strand-type CFRP bar (CSt7) consisting of seven thin CFRP strands is used as the reinforcement. The surface of the bar is processed to be spirally uneven to improve bond performance between the matrix.

Table 8. Material properties of aramid fibers.

Length (mm)	Diameter (µm)	Tensile Strength (MPa)	Elastic Modulus (GPa)	Density (g/cm3)
12	12	3432	73	1.39

Note: Nominal values provided by the manufacturer.

shows the material properties of the aramid fibers, and Figure 15 shows a photo of them. The fibers have a diameter of 12 μm and a length of 12 mm.

Table 9. Mix proportion and compressive properties of aramid–FRCC.

Specimen	Unit Weight (kg/m3)					Compressive Strength [SD] (MPa)	Elastic Modulus [SD] (GPa)
	W	C	S	FA	AR
CSt7/MT	380	678	484	291	0	47.6 [1.9]	18.3 [0.7]
CSt7/AR025					3.475	39.6 [1.4]	15.9 [0.6]
CSt7/AR050					6.95	39.0 [1.2]	16.3 [1.5]

Note: W: water, C: high-early-strength Portland cement. FA: type II fly ash of Japanese Industrial Standard (JIS A 6201 [29]), S: sand under 0.2 mm in size, AR: aramid fiber. Three compression test pieces. SD: standard deviation.

shows the mix proportion and compressive properties of the FRCC. The mix proportion of the matrix is the same as that of the PVA-FRCC described in Section 2.1. Thus, φ100 mm × 200 mm cylinder compression tests followed by JIS A 1108 [27] were conducted to determine the compressive properties of FRCC. The test variable is the fiber volume fraction of the FRCC. CSt7/MT indicates mortar without fibers; CSt7/AR025 and CSt7/AR050 indicate fiber volume fractions of 0.25% and 0.5%, respectively.

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Specimens

A list of test specimens is shown in

Table 10. List of bond test specimens for strand-CFRP/aramid–FRCC.

Specimen	FRCC Fiber	FRP Bar	Diameter of FRP Bar (mm)	Bond Length (mm)	Dimensions of Cross-Section	Fiber Volume Fraction (%)	Number of Specimens
CSt7/MT	Aramid Length 12 mm Diameter 12 μm	Strand CFRP	15.9	64	60 mm × 100 mm	0	3
CSt7/AR025						0.25	3
CSt7/AR050						0.5	3

, and the specimen geometry and loading method are illustrated in Figure 16. Each specimen is a rectangular block with a cross-section of 60 mm × 100 mm and a height of 120 mm; one FRP bar is arranged at the center. The cross-section dimensions correspond to the equivalent cross-sectional area multiplied by each main bar on the tensile side of the beam specimen described in Section 4. The bond length in the test section is 64 mm, approximately four times the diameter of the bar. The specimen fabrications, loading method, and equipment used for loading and measurement are the same as those used in Section 2.1.

2.2.2. Results of Bond Test for Strand-CFRP/Aramid–FRCC

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Failure modes

Figure 17 shows examples of CSt7/MT, CSt7/AR025, and CSt7/AR050 after loading. In CSt7/MT, the splitting cracks widened, and the load suddenly dropped after reaching the maximum bond stress. In CSt7/AR025 and CSt7/AR050, cracks appeared in the direction of the long side of the specimen. The cracks then widened, and the load decreased after reaching the maximum bond stress. It was confirmed that the fibers were bridged cracks and that crack widening was suppressed more in CSt7/AR050 than in CSt7/AR025.

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Bond stress–slip curve

Figure 18 shows the bond stress–load-end slip curves of CSt7/MT, CSt7/AR025, and CSt7/AR050. In CSt7/MT, the bond stress vanished after reaching its maximum. In CSt7/AR025 and CSt7/AR050, the bond stress decreased rapidly after reaching its maximum until the load-end slip reached 2.2 mm. After that, the bond stress decreased gradually, repeatedly rising and falling in 2.2 mm cycles.

Table 11. Bond test results for strand-CFRP/aramid–FRCC.

Specimen		Test result
		Failure Mode	Max. Bond Stress (MPa)	Average (MPa)
CSt7/MT	CSt7/MT-1	Splitting	5.20	5.37
	CSt7/MT-2	Splitting	5.42
	CSt7/MT-3	Splitting	5.49
CSt7/AR025	CSt7/AR025-1	Pullout	8.80	9.07
	CSt7/AR025-2	Pullout	9.27
	CSt7/AR025-3	Pullout	9.13
CSt7/AR050	CSt7/AR050-1	Pullout	10.4	10.8
	CSt7/AR050-2	Pullout	11.5
	CSt7/AR050-3	Pullout	10.7

lists the bond test results. The average maximum bond stress of CSt7/AR025 is 1.67 times that of CSt7/MT, and the average maximum bond stress of CSt7/AR050 is 2.01 times that of CSt7/MT. Therefore, the bridging effect of the fibers was confirmed.

2.2.3. Modeling of Bond Constitutive Law for Strand-CFRP/Aramid–FRCC

The bond stress–load-end slip curve was modeled using the same method as in Section 2.1.3. The ultimate slip, $S_u$, in CSt7/MT is set to 2.2 mm because test results for the softening branch were not obtained in the loading test. A summary of each characteristic value in the model is presented in

Table 12. Characteristic values in the model for strand-CFRP/aramid–FRCC.

Specimen	${\mathit{\tau }}_1$ (MPa)	${\mathit{S}}_1$ (mm)	${\mathit{k}}_1$ (N/mm3)	${\mathit{k}}_2$ (N/mm3)	${\mathit{\tau }}_{\mathit{m}\mathit{a}\mathit{x}}$ (MPa)	${\mathit{S}}_{\mathit{m}\mathit{a}\mathit{x}}$ (mm)	${\mathit{S}}_{\mathit{L}\mathit{m}\mathit{i}\mathit{n}}$ (mm)	${\mathit{S}}_{\mathit{u}}$ (mm)	${\mathit{k}}_{\mathit{u}}$ (N/mm3)
CSt7/MT	3.94	0.0264	149	17.3	5.37	0.109	-	2.2	−5.42
CSt7/AR025	6.69	0.0515	130	17.6	9.07	0.187	2.24	2.63	−3.71
CSt7/AR050	8.03	0.0597	135	18.8	10.8	0.209	2.20	3.02	−3.85

. Figure 19 compares the test results and models. The developed models represent the experimental results well.

3. Crack Width Evaluation of Spiral-FRP/PVA-FRCC Prism

3.1. Outline of Tension Test of Spiral-FRP/PVA-FRCC Prism

3.1.1. Materials

The FRP bars, PVA fibers, and mix proportion used in the experiment are the same as those described in Section 2.1.

3.1.2. Specimens

A list of test specimens is shown in

Table 13. List of tension test specimens for spiral-FRP/PVA-FRCC prism.

Specimen	FRCC Fiber	FRP Bar	Diameter of FRP Bar (mm)	Dimensions of Cross-Section	Fiber Volume Fraction (%)	Number of Specimens
CS5/PVA200	PVA Length 12 mm Diameter 100 μm	Spiral CFRP	11.4	100mm × 100mm	2	2
CS10/PVA200			12.1			2
CS15/PVA200			12.8			2
GS5/PVA200		Spiral GFRP	11.0			2
GS10/PVA200			11.4			2
GS15/PVA200			12.1			2

, and the specimen geometry and loading method are illustrated in Figure 20. The specimen is a rectangular block with a cross-section of 100 mm × 100 mm and a length of 500 mm, with one FRP bar arranged at its center. The specimens were fabricated by placing the FRP bar in the formwork of a wooden panel and casting FRCC from the direction indicated in the figure.

Steel couplers were attached to both ends of the bar to facilitate fixation in the chucks of the testing machine. To control the crack locations, slits on the specimen were provided at 100 mm intervals. These slits were made on two sides using a concrete cutter after hardening the FRCC. The slit width was 2.2 mm, and the depth was adjusted such that the cross-sectional area of the FRCC at the slit location was 60% of the full cross-section. The specimen was subjected to tensile loading by gripping the couplers at both ends with the chucks of a 2 MN universal testing machine. The measured items are as follows: the applied load, the overall deformation of the specimen at two locations using LVDTs, and the deformation at six slit locations using π-type displacement transducers. The occurrence of cracks was observed in real time through visual inspection, and the cracks were traced with a pen. Unloading was performed after no new cracks were observed and the situation had stabilized.

3.2. Results of Tension Test for Spiral-FRP/PVA-FRCC

3.2.1. Crack Observation

Figure 21 shows examples of the specimens after loading. In all specimens, the first cracking occurred from the slit position. As the load increased, multiple cracks occurred near the slit. The cracking patterns did not differ depending on the type of FRP bars and rib–height ratio.

3.2.2. Load–Overall Deformation Curve

Figure 22 shows the load–overall deformation curves. The overall deformation is the average of measured deformation divided by two LVDTs. A comparison between the CS series specimens and the GS series specimens reveals the difference in stiffness due to variations in the elastic modulus of the FRP. Significant differences were not observed due to differences in rib–height ratios.

3.2.3. FRP Strain–Crack Width Relationship

The FRP bar strain is determined by dividing the load by the cross-sectional area and the elastic modulus of the bar. The crack width is the average of the deformations measured by two π-type displacement transducers facing each other until a second crack is observed at one slit location through visual observation.

Figure 23 shows FRP strain–crack width relationships. The crack width increases significantly at around 0.08% and 0.3% of the FRP strain in CS series and GS series specimens, respectively. Significant differences in crack width at the same strain due to differences in rib–height ratios are not observed. It is also confirmed that the crack widths of the GS series specimens are smaller than those of the CS series specimens at the same strain. When the FRP strain reached approximately 0.3%, the crack width was about 0.2 mm for CS series specimens and about 0.1 mm for GS series specimens. However, due to the difference in the elastic modulus of the FRP, the tensile force is greater in the CS specimens.

3.3. Crack Width Evaluation of Spiral-FRP/PVA-FRCC Prism

3.3.1. Crack Width Prediction Formula

Sunaga et al. proposed a crack width calculation formula that makes it possible to evaluate crack width from rebar strain, as indicated in Equation (1) [30]. This formula represents the relationship between the rebar strain and the crack width of the FRCC. The rebar strain is taken as the strain at the outer position of the FRCC. The adaptability to crack width determined using the spiral-FRP/PVA-FRCC prism specimen is evaluated.

$${\epsilon }_{Load}={\displaystyle \frac{{\phi }_s}{A_c\left\{{\sigma }_{cr}-{\sigma }_{br}\left(w_{cr}\right)\right\}}}{\int }_0^{S_l}{\tau }_{x}d{S}_x+{\displaystyle \frac{1+np}{2np{E}_c}}\left\{{\sigma }_{cr}+{\sigma }_{br}\left(w_{cr}\right)\right\}$$

(1)

where ${\epsilon }_{Load}$ is the rebar strain at the load-end; ${\phi }_s$ is the perimeter of the rebar, mm; ${\sigma }_{cr}$ is the crack strength, MPa; $w_{cr}$ is the crack width, mm; ${\sigma }_{br}\left(w_{cr}\right)$ is the bridging stress of the FRCC, MPa; ${\tau }_x$ is the bond stress in the slip function, MPa; $S_l$ is the slip ($=w_{cr}/2$), mm; $E_s$ is the elastic modulus of the rebar, MPa; $E_c$ is the elastic modulus of the FRCC, MPa; $A_s$ is the cross-sectional area of the rebar, mm²; $A_c$ is the cross-sectional area of the FRCC, mm²; $n$ is the elastic modulus ratio ($=E_s/E_c$); and $p$ is the cross-section area ratio ($=A_s/A_c$).

Table 14. List of parameters for calculating spiral-FRP/PVA-FRCC prism.

Specimen	${\mathit{\phi }}_{\mathit{s}}$ (mm)	${\mathit{E}}_{\mathit{s}}$ (GPa)	${\mathit{A}}_{\mathit{s}}$ (mm2)	${\mathit{\sigma }}_{\mathit{c}\mathit{r}}$ (MPa)	${\mathit{E}}_{\mathit{c}}$ (GPa)	${\mathit{A}}_{\mathit{c}}$ (mm2)	Bridging Law
CS5/PVA200	35.9	127	102	1.72	17.3	100×100	${\sigma }_{max}={2.0k}^{0.30}$ ${\delta }_{max}=0.20k^{0.18}$ ${\sigma }_2=0.60k^{0.73}$ ${\delta }_2=0.45\mathrm{m}\mathrm{m}$ ${\delta }_u=6\mathrm{m}\mathrm{m}$ $k=0.4$
CS10/PVA200	37.9	111	114
CS15/PVA200	40.3	100	129
GS5/PVA200	34.4	43.2	94
GS10/PVA200	35.9	38.0	103
GS15/PVA200	37.9	31.6	114

lists the calculation parameters. The values in Table 1 are used for the material properties of the FRP bars. The values in Table 3 are used for the elastic modulus of the FRCC. The crack strength of the FRCC is calculated by dividing the load at which the first cracking occurred in the tension test by the cross-section area of the FRCC at the slit position. The trilinear model in Section 2.1.3 is used for the bond constitutive law. The trilinear model proposed by Ozu et al. [31] is used as the fiber bridging law for the PVA-FRCC, as shown in Figure 24. The trilinear model of the bridging law considers the orientation of the PVA fibers. The model’s parameters are provided using the orientation coefficient, k, as indicated in Table 14. As the bridging law using k = 0.4 showed good compatibility with the bending test results for the 100 mm × 100 mm cross-section notched beam used by Ozu et al. [31], k is set to 0.4.

3.3.2. Comparison Between Test Results and Calculation

Figure 25 compares the test results and calculation results. In all specimens, the test results and calculations are generally in good agreement. The slope becomes smaller when the calculated crack width reaches 0.17 mm because the bridging law exceeds the maximum stress and enters the softening branch. In this region, the crack widths obtained from the experiment are smaller than the calculated values and thus can be evaluated as being on the safe side.

4. Crack Width Evaluation of Strand-CFRP/Aramid–FRCC Beam

4.1. Outline of Bending Test for Strand-CFRP/Aramid–FRCC Beam

4.1.1. Materials

Table 15. Material properties of CFRP bars.

ID	FRP Bar	Diameter (mm)	Cross-Section (mm2)	Tensile Strength (MPa)	Elastic Modulus (GPa)
CStU (Compression side)	Strand CFRP	7.2	32.7	1770	160
CSt7 (Tension side)		15.9	125.0	1770	150

Note: Results of tension test followed by JIS A 1192 [28] provided by the manufacturer.

shows the material properties of the CFRP bars used for beam specimens. The main bars on the tension side are the same strand-type CFRP as in Section 2.2. The bars on the compression side are small, with a diameter of 7.2 mm. The same aramid fibers are used for the FRCC as in Section 2.2. Although the aramid–FRCC mix proportion is the same as in Section 2.2, a separate compression test was conducted because the mixing was performed on a different day.

Table 16. Compression test results for aramid–FRCC.

Specimen	Compressive Strength [SD] (MPa)	Elastic Modulus [SD] (GPa)
CSt7/MT	51.6 [2.3]	18.1 [1.8]
CSt7/AR025	45.5 [1.9]	17.2 [1.1]
CSt7/AR050	39.2 [0.7]	13.7 [1.8]
CSt7/AR050-C	42.8 [1.1]	15.5 [0.9]

Note: Three compression test pieces. SD: standard deviation.

shows the compression test results. The test variables of the beam specimens are the fiber volume fraction of the FRCC and the loading method. CSt7/MT, CSt7/AR025, and CSt7/AR050 are subjected to monotonic four-point bending loading, and CSt7/AR050-C is subjected to one-side cyclic four-point bending loading.

4.1.2. Specimens

Table 17. List of bending test specimens for strand-CFRP/aramid–FRCC beam.

Specimen	FRCC Fiber	FRP Bar	Diameter of FRP Bar (mm)	Fiber Volume Fraction (%)	Loading	Number of Specimens
CSt7/MT	Aramid Length 12 mm Diameter 12 μm	Strand CFRP	15.9 (tens.) 7.2 (comp.)	0	Mono.	1
CSt7/AR025				0.25	Mono.	1
CSt7/AR050				0.5	Mono.	1
CSt7/AR050-C					Cyclic	1

lists the specimens. Figure 26 shows the specimen used in the four-point bending test. Two 7.2 mm main bars are arranged on the compression side; three 15.9 mm main bars are arranged on the tension side; steel couplers are used as anchorage for the bars; the beam depth is 280 mm; and the width is 180 mm. The test section of interest is the central 280 mm portion, which represents the pure bending region. Stirrups are arranged outside the test section. The specimens were fabricated using the formworks of a wooden panel and casting FRCC from the upper side of the specimen. The loading methods are monotonic four-point bending loading and one-side cyclic four-point loading using a 2MN universal testing machine. Figure 27 shows the loading cycle history. Cyclic loading is controlled by the deflection at the loading point. Loading is conducted five times at deflections of 2mm, 4mm, 6mm, and 8mm and two times at 10mm; then, the loading is performed until the ultimate state. The measurement items are the load, deflections at two loading points using LVDTs, axial deformations on the compression and tension sides at the three sections using π-type displacement transducers, and CFRP strain on the tension side at six points using strain gauges.

4.2. Result of Bending Test of Strand-CFRP/Aramid–FRCC Beam

4.2.1. Failure Progress

Figure 28 shows the specimens after loading. In CSt7/MT, CSt7/AR025, and CSt7/AR050, the load decreased due to anchorage failure outside the test section. In CSt7/AR050-C, the FRCC collapsed on the compression side in the test section, and the load reached its maximum. After the maximum, the load gradually decreased while the deflection increased.

4.2.2. Load–Deflection Curve

Figure 29 shows the load–deflection curves. The deflection value is the average of deflections at the two loading points. In CSt7/MT, the load was unloaded once at a deflection of about 7 mm due to a malfunction in the testing machine; the load was subsequently applied again. CSt7/AR050-C maintained toughness even after the maximum load. Furthermore, a load decrease was not observed in any cycles due to cyclic loading.

4.2.3. FRP Strain–Crack Width Relationship

Figure 30 shows the FRP strain–crack width curves. The strain is the average of the measured strains at six points on the tension side main bars at the loading points. The crack width is calculated by dividing the deformation measured by each π-type displacement transducer on the tension side by the number of observed cracks in each section of the displacement transducer on the tension side at the maximum load. In monotonic loading specimens, the crack width at the same strain was smaller when the fiber volume fraction increased. When the FRP strain reached approximately 0.2%, the average crack width of CSt7/MT was around 0.25 mm; conversely, it was about 0.15 mm in CSt7/AR025 and about 0.10 mm in CSt7/AR050. The crack width of CSt7/AR050-C is larger than that of CSt7/AR050 at the same strain. Damage accumulation in the test section due to cyclic loading may have caused the cracks to expand.

4.3. Crack Width Evaluation of Strand-CFRP/Aramid–FRCC Beam

4.3.1. Crack Width Prediction Formula

The crack width calculation formula introduced in Section 3.3.1 is based on the relationship with the strain of the FRP measured outside the specimen without fiber bridging. However, in the beam specimens, FRP strain is measured within the test section, where fiber bridging exists between cracks. In addition to the crack width prediction formula mentioned in Section 3.3.1, Sunaga et al. also use the crack width calculation formula for the fiber bridging section, as indicated by Equation (2) [30]. Thus, the adaptability to crack width of the strand-CFRP/aramid–FRCC beam specimen is evaluated.

$${\epsilon }_{sl}={\displaystyle \frac{{\phi }_s}{A_c\left\{{\sigma }_{cr}-{\sigma }_{br}\left(w_{cr}\right)\right\}}}{\int }_0^{S_l}{\tau }_{x}d{S}_x+{\displaystyle \frac{{\sigma }_{br}\left(w_{cr}\right)}{E_c}}+{\displaystyle \frac{1+np}{2np{E}_c}}\left\{{\sigma }_{cr}-{\sigma }_{br}\left(w_{cr}\right)\right\}$$

(2)

where ${\epsilon }_{sl}$ is the rebar strain at the crack; the other symbols are the same as those in Equation (1).

Table 18. List of parameters for calculation of strand-CFRP/aramid–FRCC beam.

Specimen	${\mathit{\phi }}_{\mathit{s}}$ (mm)	${\mathit{E}}_{\mathit{s}}$ (GPa)	${\mathit{A}}_{\mathit{s}}$ (mm2)	${\mathit{\sigma }}_{\mathit{c}\mathit{r}}$ (MPa)	${\mathit{E}}_{\mathit{c}}$ (GPa)	${\mathit{A}}_{\mathit{c}}$ (mm2)	Bridging Law
CSt7/MT	50.0	150	125.0	1.78	18.1	60×100	${\sigma }_{max}=1.67 \mathrm{M}\mathrm{P}\mathrm{a}$ (AR025) ${\sigma }_{max}=3.34 \mathrm{M}\mathrm{P}\mathrm{a}$ (AR050) ${\delta }_{max}=1.4\mathrm{m}\mathrm{m}$ ${\delta }_u=2\mathrm{m}\mathrm{m}$
CSt7/AR025				4.44	17.2
GSt7/AR050				5.42	13.7
GSt7/AR050-C				5.30	15.5

lists the calculation parameters. The values in Table 15 are used for the material properties of the FRP bar. The values in Table 16 are used for the elastic modulus of the FRCC. The cross-sectional area of the FRCC is the equivalent cross-sectional area borne by each main bar on the tension side of the beam. The crack strength of FRCC is determined using the results of four-point bending tests on 100 mm × 100 mm × 400 mm rectangle specimens made from the same batch as the beam specimens. The trilinear model in Section 2.2.3 is used for the bond constitutive law. The results of the direct tension test of the aramid–FRCC conducted by Asayama et al. [32] are used as the fiber bridging law of the FRCC. The fibers and mix proportions of the FRCC used in that study were exactly the same as those used in the current research. The tension test results are modeled using the bilinear model, and Figure 31 shows a bilinear model of the aramid–FRCC. Table 18 also includes the values of the bilinear model’s parameters.

4.3.2. Comparison Between Test Results and Calculations

Figure 32 compares the test results and calculation results. Except for CSt7/MT, the calculated results generally agree with the experimental results. The average crack width in the test results at the same strain is smaller than the crack width determined using calculations above an FRP strain of about 0.1%.

In the beam specimens, unlike the tension test specimens with slits, the crack initiation locations are not predetermined, and therefore, some variation occurs. The proposed formula provides the maximum crack width, and the experimental value of the largest crack width among multiple cracks agrees with the calculated value.

5. Conclusions

This study aimed to verify the adaptability of a crack width prediction formula for FRCC to various combinations of FRP bars and FRCCs. Bond tests were performed for spiral-type CFRP and GFRP bars with PVA-FRCC and strand-type CFRP bars with aramid–FRCC. The adaptability of the calculations was verified through tension tests on prisms of spiral-type CFRP and GFRP bars with PVA-FRCC and four-point bending tests on strand-type CFRP bar beams with aramid–FRCC. The findings are summarized below.

• In bond tests of spiral-type CFRP and GFRP bars with PVA-FRCC, the maximum bond stress tends to increase as the rib–height ratio increases. For CFRP specimens, when the rib–height ratio increased from 5.7% to 8.8%, the maximum bond stress increased by 11%. For GFRP specimens, when the ratio increased from 5.9% to 8.8%, the maximum bond stress increased by 33%.
• In bond tests of strand-type CFRP with aramid–FRCC, the maximum bond stress tends to increase as the FRCC fiber volume fraction increases. The average maximum bond stress in FRCC with a 0.25% volume fraction is 1.67 times that of mortar, and that in 0.50% is 2.01 times that in mortar.
• The bond constitutive laws for spiral-type CFRP and GFRP bars with PVA-FRCC and strand-type CFRP with aramid–FRCC were modeled using the trilinear model.
• In tension tests of spiral-type CFRP and GFRP bars with PVA-FRCC, the crack width at the same FRP strain is smaller in the GFRP specimens than in the CFRP specimens. When the FRP strain reached approximately 0.3%, the crack width was about 0.2 mm for CFRP specimens and about 0.1 mm for GFRP specimens.
• The tension test results of the FRP strain–crack width relationship were compared with results calculated using the crack width prediction formula. The test results and the calculated crack width show good compatibility.
• In four-point bending tests of a strand-type CFRP beam with aramid–FRCC, the crack width at the same FRP strain tends to become smaller as the fiber volume fraction of the FRCC increases. When the FRP strain reached approximately 0.2%, the average crack width of the mortar specimen was around 0.25 mm, whereas it was about 0.15 mm in FRCC with a 0.25% volume fraction and about 0.10 mm in 0.5%.
• The bending test results for the FRP strain–crack width relationship were compared with results calculated using the crack width prediction formula. The test results and calculated crack width also show good compatibility.

6. Further Research

This study aimed to verify the adaptability of a crack width prediction method to combinations of spiral-type CFRP and GFRP bars with PVA-FRCC and strand-type CFRP bars with aramid–FRCC. The prediction formula includes both the bridging law of the FRCC and the bond constitutive law between the FRP bars and the FRCC. These two factors influence the predicted values; however, in terms of bond behavior, in particular, they are affected by the surface geometry and surface treatment (such as sand coating) of FRP bars. Therefore, further verification is necessary.

The bond behavior between rebar and FRCC is influenced by size effects [33,34,35]. In this study, bond tests were conducted using specimens with limited dimensions; therefore, further investigation is needed to determine whether our predictions apply to other specimen sizes. Notably, the bridging law can be verified as a material constitutive model through [33,34,35] FRCC tension tests or bending tests. Further investigations into these factors are expected to deepen our understanding of the structural applications of FRP–FRCC combinations.