# Calculation of Cracks in Partially Steel Fiber Reinforced Concrete Beams with BFRP Bars

Authors: Wei Shen; Shengping Chen; Jianbo Zhang
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

The effects of steel fiber volume rate and steel fiber concrete thickness on the crack development and crack resistance of basalt fiber reinforced polymer (BFRP) beams were investigated by flexural tests on six partially reinforced BFRP steel fiber reinforced concrete beams. The results showed that the addition of steel fibers effectively limited the extension height of the initial crack and the crack expansion after cracking, and the difference in the extension height of the crack between the partial steel fiber test beam and the full-section steel fiber test beam was small. For BFRP reinforced concrete beams partially reinforced with steel fibers, the development of the maximum crack width can be effectively suppressed under ultimate load. Its crack arresting effect is more similar to that of full-section steel fiber reinforced concrete beams with BFRP bars; it shows that adding steel fibers only in the compression zone of the beam can not only improve the load capacity of the beam but also obtain good economic benefits. According to the test results, on the basis of the bond-slip theory and the no-slip theory, a comprehensive calculation method of the crack spacing considering the bond effect between the steel bar and the concrete and the influence of the concrete protective layer is proposed. According to the calculation formula of crack width, and according to the influence of the actual plastic deformation of concrete in the tensile area of the section, the calculation formula of the normal section crack resistance of BFRP reinforced steel fiber concrete beams is carried out, and the calculation results are compared with the test results, which are in good agreement.


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## Body

## 1. Introduction

The urgency of introducing alternative steel reinforcement, the main reinforcement in traditional concrete structures, is driven by cost issues related to environmental friendliness and corrosion. This is because reinforcement bars are susceptible to rusting in extreme environments (saline areas, chemical plants, coastal zones, etc.), which seriously affects the structural bearing capacity, ductility, and durability [1]. Regarding the recent research on Reinforced Concrete (RC), many scholars have also conducted finite element analysis and various alternative models to evaluate RC elements under various loads in recent years; Shishegaran et al. [2–7] used nonlinear finite element analysis. With different alternative models, new models and different algorithms under different loads are studied, and it is found that the improved model has more advantages than the traditional one. The investigation of different RC structures and concrete’s electrical conductivity and seismic high temperature performance was studied, and experiments under different conditions were carried out [8–11].Fiber-reinforced composite (FRP) reinforcement has strong corrosion resistance and is an ideal steel replacement material [12]. The use of FRP reinforcement instead of steel can be an effective solution to this problem among the many measures to prevent corrosion of steel bars [13]. Because of its strong corrosion resistance, FRP bars are often used in environments that are prone to corrosion of steel bars (such as marine engineering, chemical plants, and other special buildings). However, the modulus of elasticity of FRP bars is much lower than that of steel bars, and the stress-strain curve does not have an obvious yield point, which is prone to brittle damage as well as large cracks and deformations, which seriously affects the application and promotion of FRP bars in concrete structural engineering [14–16]. FRP tendons have excellent characteristics such as high strength, corrosion resistance, light weight, low relaxation, easy processing, high fatigue strength, and elastic self-recovery, which have broad application prospects in engineering. However, compared to steel reinforcement, FRP bars have a lower modulus of elasticity, a linear stress-strain relationship, and no significant yield point [17],and the poor bonding performance of FRP reinforcement to concrete leads to a faster reduction in stiffness of FRP reinforced concrete beams after cracking, which results in larger deflections and cracks [18].Common commercial FRP reinforcement bars include glass fiber (GFRP bars), carbon fiber (CFRP bars), aramid fiber (AFRP bars), and basalt fiber (BFRP bars). These reinforcement bars exhibit different mechanical properties, physical appearance, and surface structure and, therefore, require extensive research. Studies have shown that the overall shear load capacity of FRP reinforced concrete beams is lower than that of conventional reinforced concrete beams under general conditions. Compared to normal reinforced beams, GFRP reinforced beams can have greater deflection and larger crack spread widths due to the low elastic modulus of GFRP bars [19]. Due to the lower modulus of elasticity of GFRP bars, beams reinforced with these bars also exhibit lower postcracking bending stiffness compared to concrete beams reinforced with normal steel bars. In a study of CFRP-reinforced concrete beams, it was shown that the modulus of elasticity of CFRP reinforcement was three to four times higher than that of GFRP reinforcement. Karayannis et al. [20] showed that the flexural stiffness of CFRP beams is higher than that of GFRP reinforced beams and that any cracking in concrete beams reinforced with CFRP reinforcement is accompanied by a sudden drop in load, due to the formation of long flexural cracks and the low reinforcement ratio of CFRP reinforcement, which leads to a local reduction in stiffness at the cracked cross section of the beam. The tensile strength of BFRP bars is more than twice that of ordinary steel bars, and the modulus of elasticity is about 1/4 of that of ordinary steel bars. The modulus of elasticity of BFRP tendons is mainly related to the content of basalt fibers, with the increase of basalt fiber content, that is, the greater the tensile modulus of elasticity of BFRP tendons. BFRP tendons are basically linear elastic deformation before damage, and there is no plastic yield step before damage occurs. The fracture is abrupt and brittle. The elongation ranged from 2.0% to 2.5% [21]. BFRP bars have significant advantages in terms of tensile strength and corrosion resistance, making them ideal for use as longitudinal tension bars. Figure 1 shows the stress-strain relationship of BFRP tendons.Figure 1
Stress-strain relationship of BFRP bars.The effectiveness of steel fibers on the flexural performance of reinforced concrete structural members has been investigated in recent years, and steel fibers can increase moment strength, ductility, damage toughness, and energy absorption capacity. The performance of SFRC beams with longitudinal reinforcement is also enhanced with an increase in the number of cracks, a decrease in crack width and height, restricted crack extension, and delayed concrete spalling [22]. It was also shown that the overall performance of reinforced concrete beams containing steel fibers was improved, with concrete specimens without fibers breaking in a brittle manner, while SFRC beams with 1% steel fibers broke in a ductile manner after spalling of the concrete [23]. Since steel fibers also cause rusting in a corrosive-prone environment, making the steel fibers in the lower tensile zone of the beams limited in their efficiency, only steel fibers are added to the compressive zone, and plain concrete is used in the tensile zone to make FRP reinforced partially steel fiber concrete beams, for which there is a lack of studies related to them. Partially incorporating steel fibers at the beam cross section can not only save the cost significantly and avoid the waste of raw materials, but also can give full play to the strengthening and toughening effect of steel fibers and FRP reinforcement, which can ensure the safety of the beam and, at the same time, significantly save cost to reduce the self-weight, and it has a good application panorama.In the practice of enhancing the serviceability of FRP reinforced concrete structures, one of the most effective practices is the combination of steel fiber concrete with FRP reinforcement [24]. The addition of steel fibers increases the ultimate compressive strain of the concrete, which allows the member to evolve from brittle to ductile and also greatly reduces the crack width of the beam [25]. Steel fibers are added to FRP reinforced concrete beams to make FRP reinforced steel fiber reinforced concrete beams, and the tensile properties of the steel fibers are used to limit the crack development in FRP reinforced concrete beams. It was shown [26] that both FRP reinforced steel fiber reinforced concrete beams and FRP reinforced partial steel fiber reinforced concrete beams with steel fibers in the tensile zone of the beam only can effectively stop the development of cracks. When rusting of steel fibers occurs in a corrosive-prone environment, it can cause problems such as durability and safety of structural beams [27]. And the current design codes also generally do not account for the tensile effect of steel fiber concrete in the tensile zone [28]. Since steel fibers can also corrode in harsh corrosive environments [29], especially when the beam is cracked in bending, the steel fibers mixed in its tensile zone are more susceptible to corrosion, which causes durability problems in structural beams [30]. However, adding steel fibers to the concrete mix to enhance the ultimate compressive strain of concrete can enhance the flexural performance of the member. Therefore, it can be considered that only steel fiber is added in the upper part of the beam in the compression zone, while ordinary concrete is used in the tension zone to form a FRP reinforced concrete beam with partial steel fiber reinforcement, that is, the use of enhanced compressive strength and toughness of the compressive zone of the beam to achieve the purpose of inhibiting the expansion of cracks, while being able to save material to improve economic efficiency.It is of great practical importance to study the positive section cracking moment of concrete beams as an important indicator of the cracking resistance of the members. At present, scholars have conducted a preliminary study on the cracking moment of the positive section of concrete stacked beams. Zhang [31] conducted flexural experiments on a stacked beam made of plain concrete in the compression zone and steel fiber concrete in the tension zone and deduced a method for calculating the positive section crack resistance of reinforced steel fiber concrete beams through the section analysis method. Karim and AhmedElRefai [32] conducted flexural experiments on stacked beams made of high-strength concrete in the compressive zone and recycled concrete with lower strength in the tensile zone and verified the feasibility of heterostrength concrete beams by the moment analysis method. This paper establishes a cracking moment calculation model for BFRP reinforced concrete beams with reference to the theory of calculating the positive section cracking resistance of reinforced concrete beams, calculates the cracking moment of BFRP reinforced concrete beams with steel fibers, and compares the calculated results with the experimental values to verify the reasonableness of the numerical calculation model. The formulae for calculating the maximum crack width of steel fiber reinforced concrete beams with partial BFRP reinforcement and full-section steel fiber reinforced concrete beams with BFRP reinforcement were established to reasonably calculate the maximum crack width of the test beams under normal use, and the calculated results were compared with the test values to verify the reasonableness of the calculation model.


## 2. Trial Overview

### 2.1. Specimen Design

Because the yield strain of FRP reinforcement is very small, brittle fracture will occur under the action of large stress. The test beam is designed as an overreinforced beam. In order to study the influence of the incorporation rate of steel fiber and the height of the steel fiber reinforced concrete layer on the deflection of the test beam, the mixing rate of steel fiber and the height of the steel fiber reinforced concrete layer were used as variables, and six sections were designed with the size of 150 mm × 300 mm × 2100 mm test beam. In the first group, the thickness of the steel fiber reinforced concrete layer on the upper part of the beam is 180 mm, and the mixing rate of steel fiber on the upper part of the beam is 0.5% (B2), 1% (B3), and 1.5% (B4). The second group was quantified with the addition rate of steel fiber 1%, and the thickness of the steel fiber reinforced concrete layer was 180 mm (B3), 210 mm (B5), and 300 mm (B6). The control group was a BFRP reinforced plain concrete beam (B1). Stirrups are arranged between the support and the loading point to prevent shear failure of the test beam during the loading process, and no erection bars are arranged in the pure bending section of the beam to eliminate the influence of erection bars. Those beams are shown in Figure 2.Figure 2
Specimen size. (a) Side view of the test beam. (b)B1 section. (c) B2/3/4 section. (d) B5 section. (e) B6 section.
(a)(b)(c)(d)(e)When the formed specimens were poured, two groups of six standard cube test blocks of 100 mm × 100 mm × 100 mm were made for each beam at the same time and were jointly cured for 28 d in the same environment to test its compressive strength.


### 2.2. Test Materials

The raw materials used for the test beam are as follows: the cement is P.042.5 grade ordinary Portland cement; the coarse aggregate is crushed stone with a particle size of 5∼20 mm; the fine aggregate is medium-coarse ordinary river sand, water agent; steel fibers are milled and wave-shaped with an aspect ratio of 37; the longitudinal bars are deep-ribbed BFRP bars with a diameter of 14 mm, and the vertical bars and stirrups are third-grade threaded steel bars with a diameter of 8 mm. The parameters of the specimens are shown in Table1.Table 1
Specimen parameters.
Specimen numberTotal water-to-binder ratio (kg/m3)Stirrup spacing (mm)Section reinforcement ratio (%)Fiber content (%)Fiber reinforced concrete layer height (mm)Destruction modeB10.428@750.7900Super-rib damageB20.428@750.790.5180Super-rib damageB30.428@750.791180Super-rib damageB40.428@750.791.5180Super-rib damageB50.428@750.791210Super-rib damageB60.428@750.791300Super-rib damageBasalt Fiber Reinforced Polymer (BFRP) is a new type of nonmetallic composite fiber reinforcement material made of basalt fiber as reinforcement material, synthetic resin as basic material, and appropriate amount of auxiliary agent, which is formed by pultrusion process and special surface treatment. Compared with ordinary steel bars, BFRP bars have the advantages of corrosion resistance, high strength, light weight, fatigue resistance, insulation, etc. It has a wide range of applications in civil engineering and can be used as an effective method to solve the problem of corrosion of steel bars. The properties of the materials are shown in Tables2 and 3.Table 2
Mechanical properties of BFRP bars.
Diameter (mm)E (GPa)Tensile strength (MPa)Rupture extension rate (%)Density (g/cm2)1452.212092.32.02Table 3
Steel fiber properties.
CategoryLength (mm)Aspect ratioDensity (g/cm2)Tensile strength (MPa)E (GPa)Milling pin type39377.8600200


### 2.3. Experiment with Loading and Testing Content

The test beam adopts the loading method of four-point bending and static load graded loading, and the measuring points are arranged as shown in Figure3.Figure 3
Lading point layout.The test beam was loaded with 5 kN per stage before cracking, and 15 kN per stage after cracking, and the load gradient was reduced when the test beam was near cracking and failure. After each level of load is loaded, it needs to be sustained for 3 minutes, and the data is collected after the data is stable. Main test contents: cracking load, ultimate load; steel fiber reinforced concrete strain in compression zone, BFRP reinforcement strain in mid-span tension zone; crack width; deflection at test beam support, loading point, and mid-span.


## 2.1. Specimen Design

Because the yield strain of FRP reinforcement is very small, brittle fracture will occur under the action of large stress. The test beam is designed as an overreinforced beam. In order to study the influence of the incorporation rate of steel fiber and the height of the steel fiber reinforced concrete layer on the deflection of the test beam, the mixing rate of steel fiber and the height of the steel fiber reinforced concrete layer were used as variables, and six sections were designed with the size of 150 mm × 300 mm × 2100 mm test beam. In the first group, the thickness of the steel fiber reinforced concrete layer on the upper part of the beam is 180 mm, and the mixing rate of steel fiber on the upper part of the beam is 0.5% (B2), 1% (B3), and 1.5% (B4). The second group was quantified with the addition rate of steel fiber 1%, and the thickness of the steel fiber reinforced concrete layer was 180 mm (B3), 210 mm (B5), and 300 mm (B6). The control group was a BFRP reinforced plain concrete beam (B1). Stirrups are arranged between the support and the loading point to prevent shear failure of the test beam during the loading process, and no erection bars are arranged in the pure bending section of the beam to eliminate the influence of erection bars. Those beams are shown in Figure 2.Figure 2
Specimen size. (a) Side view of the test beam. (b)B1 section. (c) B2/3/4 section. (d) B5 section. (e) B6 section.
(a)(b)(c)(d)(e)When the formed specimens were poured, two groups of six standard cube test blocks of 100 mm × 100 mm × 100 mm were made for each beam at the same time and were jointly cured for 28 d in the same environment to test its compressive strength.


## 2.2. Test Materials

The raw materials used for the test beam are as follows: the cement is P.042.5 grade ordinary Portland cement; the coarse aggregate is crushed stone with a particle size of 5∼20 mm; the fine aggregate is medium-coarse ordinary river sand, water agent; steel fibers are milled and wave-shaped with an aspect ratio of 37; the longitudinal bars are deep-ribbed BFRP bars with a diameter of 14 mm, and the vertical bars and stirrups are third-grade threaded steel bars with a diameter of 8 mm. The parameters of the specimens are shown in Table1.Table 1
Specimen parameters.
Specimen numberTotal water-to-binder ratio (kg/m3)Stirrup spacing (mm)Section reinforcement ratio (%)Fiber content (%)Fiber reinforced concrete layer height (mm)Destruction modeB10.428@750.7900Super-rib damageB20.428@750.790.5180Super-rib damageB30.428@750.791180Super-rib damageB40.428@750.791.5180Super-rib damageB50.428@750.791210Super-rib damageB60.428@750.791300Super-rib damageBasalt Fiber Reinforced Polymer (BFRP) is a new type of nonmetallic composite fiber reinforcement material made of basalt fiber as reinforcement material, synthetic resin as basic material, and appropriate amount of auxiliary agent, which is formed by pultrusion process and special surface treatment. Compared with ordinary steel bars, BFRP bars have the advantages of corrosion resistance, high strength, light weight, fatigue resistance, insulation, etc. It has a wide range of applications in civil engineering and can be used as an effective method to solve the problem of corrosion of steel bars. The properties of the materials are shown in Tables2 and 3.Table 2
Mechanical properties of BFRP bars.
Diameter (mm)E (GPa)Tensile strength (MPa)Rupture extension rate (%)Density (g/cm2)1452.212092.32.02Table 3
Steel fiber properties.
CategoryLength (mm)Aspect ratioDensity (g/cm2)Tensile strength (MPa)E (GPa)Milling pin type39377.8600200


## 2.3. Experiment with Loading and Testing Content

The test beam adopts the loading method of four-point bending and static load graded loading, and the measuring points are arranged as shown in Figure3.Figure 3
Lading point layout.The test beam was loaded with 5 kN per stage before cracking, and 15 kN per stage after cracking, and the load gradient was reduced when the test beam was near cracking and failure. After each level of load is loaded, it needs to be sustained for 3 minutes, and the data is collected after the data is stable. Main test contents: cracking load, ultimate load; steel fiber reinforced concrete strain in compression zone, BFRP reinforcement strain in mid-span tension zone; crack width; deflection at test beam support, loading point, and mid-span.


## 3. Analysis of Test Results

### 3.1. Crack Distribution of the Test Beam

The actual failure of BFRP reinforced concrete beams and BFRP reinforced concrete beams is shown in Figure4, and the distribution of cracks is shown in Figure 5.Figure 4
The distribution of cracks in the actual failure of the test beam. (a) BFRC1. (b) BFRC2. (c) BFRC3. (d) BFRC4. (e) BFRC5. (f) BFRC6.
(a)(b)(c)(d)(e)(f)Figure 5
Crack distribution of the test beam. (a) BFRC1. (b) BFRC2. (c) BFRC3. (d) BFRC4. (e) BFRC5. (f) BFRC6.
(a)(b)(c)(d)(e)(f)When a certain load is reached during the loading process of the test beam, a vertical initial crack first appears in the pure bending section of the test beam, and when the loading continues, an oblique crack appears in the shear span and extends from the bottom of the beam to the loading point. Continuing to load, many dendritic microcracks appeared in the pure bending section of the test beams BFRC2, BFRC3, BFRC4, BFRC5, and BFRC6 incorporating steel fibers, and with the application of the load, the sound of the steel fibers being pulled out could be heard, and then, the main crack penetrated through the test beam, the steel fiber reinforced concrete on the upper part of the beam bulged, and the test piece was damaged. When the test beam BFRC1 without steel fiber was continuously pressurized, the oblique cracks continued to expand and directly penetrated the upper part of the beam, the concrete in the compression zone was uplifted, and the specimen was completely destroyed.


### 3.2. Effect of Steel Fiber Volume Rate on Cracks

As shown in Figure6, compared with the BFRC1 test beam, with the increase of steel fiber volume rate, the limiting effect of the test beam on the crack width under normal use is not significant, which is due to the fact that the crack resistance of the member under normal use is mainly borne by the concrete and the stressing longitudinal reinforcement, while the lower tensile zone of the beam is not mixed with steel fibers, so the effect of steel fibers is not obvious. When the bearing capacity exceeds the normal service limit state, the inhibition of the crack width of the test beam becomes more obvious with the increase of the volume rate of steel fibers, which is due to the fact that the addition of steel fibers improves the strength of concrete to a certain extent, thus effectively inhibiting the width of cracks. The maximum crack widths of the test beams BFRC2, BFRC3, and BFRC4 were reduced by 19%, 23%, and 28% under ultimate load condition (165 kN), respectively, compared to the test beams without steel fibers.Figure 6
Load-maximum crack width curves under different volume ratios of steel fibers.After the test beam BFRC1 reached the cracking load, the primary crack extended substantially upward with the increase of load, and there was an obvious abrupt change point, as shown in Figure7. When the load was loaded to 50% of the ultimate load, the crack height extension rate decreased. Compared with the test beam BFRC1, the test beams BFRC2, BFRC3, and BFRC4 did not have obvious abrupt change points during the extension of the crack height, and the height of the initial crack was smaller than that of the test piece BFRC1, and it became more significant for inhibiting the extension of the crack height as the volume rate of steel fiber increased. The extension heights of the primary cracks were reduced by 28%, 39%, and 41%, respectively.Figure 7
Load-maximum crack height curves under different volume ratios of steel fibers.


### 3.3. Effect of Steel Fiber Concrete Thickness on Cracks

The limitation of the crack width by the thickness of the steel fiber concrete in the upper compression zone of the beam is shown in Figure8. The greater the thickness of the steel fiber concrete, the more obvious the limitation of the maximum crack width for the same volume rate of the steel fiber. This is due to the fact that the steel fibers in the lower part of the beam act as a bridge to limit the crack development after the beam cracking. Under the ultimate load condition (165 kN), the crack widths of the test beams BFRC3, BFRC5, and BFRC6 were reduced by 22%, 29%, and 38%, respectively.Figure 8
Load-maximum width curves of steel fiber concrete with different thicknesses.The thickness of steel fiber concrete affects the crack extension height, as shown in Figure9. Overall, the greater the thickness of steel fiber concrete, the smaller the crack extension rate. At the initial cracking, the crack heights of the test beams BFRC3, BFRC5, and BFRC6 were reduced by 39%, 46%, and 50%, respectively.Figure 9
Load-maximum crack height curves under different steel fiber concrete thicknesses.


### 3.4. Modulus of Elasticity of Steel Fiber Concrete

Figure10 shows the effect of steel fibers on the static modulus of elasticity of concrete, and it can be seen from the figure that, with the increase of steel fiber admixture rate, the static modulus of elasticity of concrete is enhanced less. The modulus of elasticity of concrete is the ratio of positive stress to linear strain in the elastic range, which is used to reflect the ability of concrete to resist compressive strain. In this paper, the cut-line modulus value corresponding to 0.5 times the ultimate stress is taken as the modulus of elasticity of steel fiber concrete.Figure 10
Effect of steel fiber admixture rate on modulus of elasticity.


### 3.5. Analysis of Ductility of BFRP Reinforced Concrete Beams with Steel Fibers

The ductility of a structure is the deformability of the structure, and from an energy point of view, ductility is defined as the loss performance of a component. For the BFRP reinforced experimental beam, the steel fiber will increase the ultimate compressive strain. Experience shows that the BFRP reinforced beam has not been broken during the beam compression failure process, so this experiment adopts the calculation method adopted by the ACI440.1R-15 [33] specification, and the design of FRP reinforced concrete beam in ACI440.1R-15 specification is controlled by its deformation satisfying the applicability, so the evaluation of the ductility of FRP reinforced concrete beam is the energy consumed in the limit state and the normal service limit state (mid-span displacement). This evaluation method has better applicability. Therefore, the ductility evaluation of FRP-reinforced concrete stacked beams can be as follows:(1)μ=EuEscr.In the formula,Eu and Escr are the energy absorbed under the limit state and normal use state, respectively.After calculation, the calculation results of this ductility coefficient are shown in Table4.Table 4
Ductility calculation results.
Specimen numberB1B2B3B4B5B6μ3.103.774.195.854.154.14The calculation results show that the addition of steel fibers has a great influence on improving the ductility index of the beam, and the ductility of the beam can be improved to different degrees. Compared with the test beamB1, the ductility of the test beams B2, B3, and B4 is increased by 22%, 35%, and 89%, and the ductility of the test beams B5 and B6 is increased by 34% and 34%. It can be seen that the improvement of ductility mainly depends on the volume fraction of steel fibers in the compression zone.


## 3.1. Crack Distribution of the Test Beam

The actual failure of BFRP reinforced concrete beams and BFRP reinforced concrete beams is shown in Figure4, and the distribution of cracks is shown in Figure 5.Figure 4
The distribution of cracks in the actual failure of the test beam. (a) BFRC1. (b) BFRC2. (c) BFRC3. (d) BFRC4. (e) BFRC5. (f) BFRC6.
(a)(b)(c)(d)(e)(f)Figure 5
Crack distribution of the test beam. (a) BFRC1. (b) BFRC2. (c) BFRC3. (d) BFRC4. (e) BFRC5. (f) BFRC6.
(a)(b)(c)(d)(e)(f)When a certain load is reached during the loading process of the test beam, a vertical initial crack first appears in the pure bending section of the test beam, and when the loading continues, an oblique crack appears in the shear span and extends from the bottom of the beam to the loading point. Continuing to load, many dendritic microcracks appeared in the pure bending section of the test beams BFRC2, BFRC3, BFRC4, BFRC5, and BFRC6 incorporating steel fibers, and with the application of the load, the sound of the steel fibers being pulled out could be heard, and then, the main crack penetrated through the test beam, the steel fiber reinforced concrete on the upper part of the beam bulged, and the test piece was damaged. When the test beam BFRC1 without steel fiber was continuously pressurized, the oblique cracks continued to expand and directly penetrated the upper part of the beam, the concrete in the compression zone was uplifted, and the specimen was completely destroyed.


## 3.2. Effect of Steel Fiber Volume Rate on Cracks

As shown in Figure6, compared with the BFRC1 test beam, with the increase of steel fiber volume rate, the limiting effect of the test beam on the crack width under normal use is not significant, which is due to the fact that the crack resistance of the member under normal use is mainly borne by the concrete and the stressing longitudinal reinforcement, while the lower tensile zone of the beam is not mixed with steel fibers, so the effect of steel fibers is not obvious. When the bearing capacity exceeds the normal service limit state, the inhibition of the crack width of the test beam becomes more obvious with the increase of the volume rate of steel fibers, which is due to the fact that the addition of steel fibers improves the strength of concrete to a certain extent, thus effectively inhibiting the width of cracks. The maximum crack widths of the test beams BFRC2, BFRC3, and BFRC4 were reduced by 19%, 23%, and 28% under ultimate load condition (165 kN), respectively, compared to the test beams without steel fibers.Figure 6
Load-maximum crack width curves under different volume ratios of steel fibers.After the test beam BFRC1 reached the cracking load, the primary crack extended substantially upward with the increase of load, and there was an obvious abrupt change point, as shown in Figure7. When the load was loaded to 50% of the ultimate load, the crack height extension rate decreased. Compared with the test beam BFRC1, the test beams BFRC2, BFRC3, and BFRC4 did not have obvious abrupt change points during the extension of the crack height, and the height of the initial crack was smaller than that of the test piece BFRC1, and it became more significant for inhibiting the extension of the crack height as the volume rate of steel fiber increased. The extension heights of the primary cracks were reduced by 28%, 39%, and 41%, respectively.Figure 7
Load-maximum crack height curves under different volume ratios of steel fibers.


## 3.3. Effect of Steel Fiber Concrete Thickness on Cracks

The limitation of the crack width by the thickness of the steel fiber concrete in the upper compression zone of the beam is shown in Figure8. The greater the thickness of the steel fiber concrete, the more obvious the limitation of the maximum crack width for the same volume rate of the steel fiber. This is due to the fact that the steel fibers in the lower part of the beam act as a bridge to limit the crack development after the beam cracking. Under the ultimate load condition (165 kN), the crack widths of the test beams BFRC3, BFRC5, and BFRC6 were reduced by 22%, 29%, and 38%, respectively.Figure 8
Load-maximum width curves of steel fiber concrete with different thicknesses.The thickness of steel fiber concrete affects the crack extension height, as shown in Figure9. Overall, the greater the thickness of steel fiber concrete, the smaller the crack extension rate. At the initial cracking, the crack heights of the test beams BFRC3, BFRC5, and BFRC6 were reduced by 39%, 46%, and 50%, respectively.Figure 9
Load-maximum crack height curves under different steel fiber concrete thicknesses.


## 3.4. Modulus of Elasticity of Steel Fiber Concrete

Figure10 shows the effect of steel fibers on the static modulus of elasticity of concrete, and it can be seen from the figure that, with the increase of steel fiber admixture rate, the static modulus of elasticity of concrete is enhanced less. The modulus of elasticity of concrete is the ratio of positive stress to linear strain in the elastic range, which is used to reflect the ability of concrete to resist compressive strain. In this paper, the cut-line modulus value corresponding to 0.5 times the ultimate stress is taken as the modulus of elasticity of steel fiber concrete.Figure 10
Effect of steel fiber admixture rate on modulus of elasticity.


## 3.5. Analysis of Ductility of BFRP Reinforced Concrete Beams with Steel Fibers

The ductility of a structure is the deformability of the structure, and from an energy point of view, ductility is defined as the loss performance of a component. For the BFRP reinforced experimental beam, the steel fiber will increase the ultimate compressive strain. Experience shows that the BFRP reinforced beam has not been broken during the beam compression failure process, so this experiment adopts the calculation method adopted by the ACI440.1R-15 [33] specification, and the design of FRP reinforced concrete beam in ACI440.1R-15 specification is controlled by its deformation satisfying the applicability, so the evaluation of the ductility of FRP reinforced concrete beam is the energy consumed in the limit state and the normal service limit state (mid-span displacement). This evaluation method has better applicability. Therefore, the ductility evaluation of FRP-reinforced concrete stacked beams can be as follows:(1)μ=EuEscr.In the formula,Eu and Escr are the energy absorbed under the limit state and normal use state, respectively.After calculation, the calculation results of this ductility coefficient are shown in Table4.Table 4
Ductility calculation results.
Specimen numberB1B2B3B4B5B6μ3.103.774.195.854.154.14The calculation results show that the addition of steel fibers has a great influence on improving the ductility index of the beam, and the ductility of the beam can be improved to different degrees. Compared with the test beamB1, the ductility of the test beams B2, B3, and B4 is increased by 22%, 35%, and 89%, and the ductility of the test beams B5 and B6 is increased by 34% and 34%. It can be seen that the improvement of ductility mainly depends on the volume fraction of steel fibers in the compression zone.


## 4. BFRP Reinforced Concrete Beams with Partial Steel Fiber Reinforcement Are Tested for Cracking

Referring to the positive section cracking test method for reinforced concrete beams, the effect of the actual plastic deformation of the concrete in the tensile zone of the cross section on the cracking resistance is reflected by the plasticity coefficient of the resisting moment of the cross section, and the positive section cracking resistance is calculated for steel fiber reinforced concrete beams with partial BFRP reinforcement and steel fiber reinforced concrete beams with full cross section of BFRP reinforcement, whose expressions are as follows:(2)Mcr=β∙γ⋅ftk⋅W0,where β is the steel fiber crack strength enhancement factor; refer to literature [34] to obtain(3)β=1+αsλsf,λsf=lfdfvf,where Mcr is the positive cross-sectional cracking moment of BFRP reinforced concrete beam; γ is the cross-sectional moment of resistance plasticity coefficient, and for rectangular cross-section, γ takes the value of 1.55; ftk is the standard value of tensile strength of concrete; αsis the tensile strength enhancement factor of steel fiber to concrete taken as 0.31; λsf is the characteristic value of steel fiber admixture, where lf is the fiber length, df is the steel fiber diameter, and vf is the steel fiber volume rate; W0 is the elastic resistance moment of the beam section to the tensile edge.The strain distribution of the test beam in the state of imminent cracking is shown in Figure9. In accordance with the calculation method for reinforced concrete beams, the elastic moment of resistance to the tensile edge of the cross section of a BFRP reinforced concrete beam with steel fibers is(4)W0=I0h−x0.Cross-sectional form-center conversion of partially reinforced steel fiber concrete beams with BFRP bars:(5)x0=αEfcbd2/2+bh−dh+d/2+Afh0αEf−1αEfcbd+bh−d+AfαEf−1.Cross-sectional form center moment of inertia:(6)I0=αEfcbd−x033+αEfcbx033+bh−dh+d2−x02+bh−d312+Afh0−x02αEf−1.BFRP reinforcement full-section steel fiber reinforced concrete beam section shape center conversion:(7)x0=αEfcbd2/2+Afh0αEf−1αEfcbh+AfαEf−1.Cross-sectional form center moment of inertia:(8)I0=αEfcbh312+αEfcbhh2−x02+Afh0−x02αEf−1,where I0 is the moment of inertia of the converted section to its form center; x0 is the distance from the form center of the converted section to the upper compressed edge of the beam; αEfcis the elastic modulus ratio of steel fiber concrete to concrete; αEf is the elastic modulus ratio of BFRP reinforcement to concrete.The modulus of elasticity of concrete is the ratio of positive stress to linear strain in the elastic range and is used to reflect the ability of concrete to resist compressive strain. In this paper, the cut-line modulus value corresponding to 0.5 times of ultimate stress is taken as the static modulus of elasticity of steel fiber concrete, and the measured results are shown in Table5 and the calculated value of cracking moment is compared with the actual cracking moment in Table 6, and the theoretical value is in good agreement with the actual value. Figure 11 shows the computational model.Table 5
Elastic modulus of SFRC at different volume rates (×104N/mm2).
SF—0.5%1%1.5%E3.403.523.553.57Table 6
Comparison of experimental and theoretical values of cracking bending moment.
Specimen numberPe (kN)P (kN)Me (kN∙m)Mcr (kN∙m)Mcr (Me)B128.8029.678.68.91.03B233.7330.3310.19.10.9B334.4032.0010.39.60.93B435.2033.6710.610.10.95B534.8733.3310.5100.95B637.6734.0011.310.20.9Note:Pe is the measured cracking, P is the theoretical cracking load load, Me is the measured bending moment in this paper, and Mcr is the theoretical bending moment in this table.Figure 11
Strain distribution of steel fiber reinforced concrete beams with BFRP bars near cracking state.


## 5. Calculation of Crack Width of BFRP Reinforced Concrete Beams with Steel Fibers

The calculation of the maximum crack width of FRP reinforced concrete beams in the code [35] intuitively reflects the tensile hardening effect of concrete. The average crack width ωm is first obtained as the difference between the strain of the longitudinal reinforcement and the concrete within the crack spacing, that is, as the product of（ε¯f−ε¯c）and（lm), calculated as follows:(9)ωm=εf¯−εc¯lm,where ωm is the average crack width; εf¯ is the average strain of longitudinal reinforcement between cracks; εc¯ is the average strain of concrete between cracks; lm is the average crack spacing.Letαc=1−ε¯c/ε¯fandε¯f=ψ⋅εf=ψσf/Ef. Then, the formula for calculating the average crack width can be written as(10)ωm=αc⋅ψ⋅σfEf⋅lm,where αc is the crack influence coefficient; ψ is the strain inhomogeneity coefficient between longitudinal bars and concrete; σf is the stress of longitudinal bars; and Ef is the modulus of elasticity of longitudinal bars.Based on the average crack widthωm, considering the inhomogeneity of the material quality and the dispersion of the crack width, for the calculation of the maximum crack width of the member, it is generally calculated by multiplying the average crack width ωm by the enlargement factor τs. Then, the maximum crack width ωmax of the test beam under short-term loading is calculated by the following formula:(11)ωmax=τsωm=τs⋅αc⋅ψ⋅σfEf⋅lm,where ωmax is the short-term maximum crack width and τs is the expansion factor.Referring to the results of the literature [35], the effect of steel fibers on the force characteristic coefficients (αc,τs) of the test beams is not obvious, so the value of 0.85 for αc and 1.66 for τs is taken in this paper.


### 5.1. The Value of the Relative Bond Characteristic Coefficientvi of BFRP Bars and Concrete

The main factor affecting the crack expansion in flexural members of concrete beams is the bond performance of longitudinal tensioned tendons to concrete. As for BFRP reinforced partial steel fiber concrete beams, since steel fibers are added only in the compression zone, the formula for calculating the average crack spacing can be followed for the average crack spacing of FRP reinforced concrete beams in the specification [35]:(12)lm=1.9c+0.08deqpte,deq=∑nidi2∑nividi,where c is the concrete protective layer thickness; deq is the equivalent diameter of longitudinal reinforcement; pte is the reinforcement rate calculated by the effective tensile concrete cross-sectional area; when pte0.01, take = 0.01; di is the nominal diameter of the i-th type of FRP bars; ni is the number of i-th type of FRP bars; and vi is the relative bonding characteristic coefficient of FRP bars; when >1.5, take 1.5; when no test data, take 0.7.The specification [35] does not give the value of the relative bond coefficient vi for each type of FRP tendons but suggests that the value of the relative bond coefficient vi is determined according to the experiment and limits the range of vi: 1.5 when vi >1.5, and 0.7 when there is no test data. It can be seen that the value of the relative bond coefficient vi for each type of FRP tendons needs to be verified.The average crack spacing measured experimentally in this paper is shown in Table7. The values of the relative bond coefficient vi between BFRP reinforcement and concrete were backcalculated for the BFRP reinforced plain concrete beams and some steel fiber reinforced concrete beams. When vi is taken as 1.1, the measured average crack spacing under ultimate load bearing capacity agrees well with the theoretical value (Table 7).Table 7
Average fracture spacing (mm).
Specimen numberBFRC1BFRC2BFRC3BFRC4BFRC5BFRC6Number of lacerations101515151416Average crack spacing12412111912112295


### 5.2. Stress of BFRP Bars at Cracks

For some of the steel fiber reinforced beams, since the lower part of the beam is not mixed with steel fibers, the beam is subjected to the main tensile stress by the BFRP tendons after cracking, and the maximum load increase from the initial cracking load to the limit state of normal use is small; therefore, the effect of steel fibers is not considered in the normal use state.From the cross-sectional moment equilibrium condition, it is obtained that(13)M=σfAfh0−x3,where M is the bending moment at the cracked section; Af is the cross-sectional area of BFRP bars; h0 is the effective height of the section. Deformation coordination equation:(14)εsfuεf=xh0−x,where εsfu is the steel fiber concrete strain in the upper part of the beam; εf is the BFRP tendon strain; x is the distance from the upper surface of the beam to the neutral axis. The stress at the BFRP reinforcement between the cracks is(15)σf=3Mεf+εsfuAfh03εf+2εsfu.For full-section steel fiber-reinforced concrete beams, the stress transfer of the steel fibers needs to be considered, as in Figure12. Therefore, for the flat section assumptions of full-section steel fiber-reinforced concrete beams with BFRP bars, the effect of the steel fibers is to be considered.Figure 12
Stress-strain of full-section steel fiber test beam under normal service condition.From the bending moment balance, it is obtained that(16)M=σfAfh0−x3+12σsftbh−xh+x3.The stress at the BFRP reinforcement between cracks in full-section steel fiber concrete beams can be obtained by associating equations (13) with (15) as follows:(17)σf=2M−σSftbh−xh+x/32Afh0−x/3.The residual tensile strength of steel fiber concrete beams after cracking was calculated using the literature [36] using the following formula:(18)σsft=13αstλsf1+f,where αst is the bond strength of steel fiber and concrete, for milling pin type steel fiber to take 1.2ftk; f is the steel fiber and concrete in the crack edge shear friction coefficient, take 1/3.


### 5.3. Intercrack Tensioned BFRP Bars Strain Unevenness Factorψ

In the code [37], FRP reinforced concrete beams follow the method of calculating the strain inhomogeneity factor for reinforced concrete beams as follows:(19)ψ=1.1−0.65ftkσfpte.Since the strain inhomogeneity coefficient of steel bars is based on a huge amount of experimental data of reinforced concrete beams, and the difference in material properties between FRP bars and steel bars makes it not applicable to calculate the strain inhomogeneity coefficient of FRP bars by following the calculation method of steel bars, the addition of steel fibers also affects the strain of BFRP bars between cracks, and in this paper, the formula for calculating the strain inhomogeneity coefficientψ is modified based on the specification [38], and the cracking moment is multiplied by a discount factor of 0.8, considering the adverse effect of concrete shrinkage in the tensile zone. Based on the data obtained from experiments, the formula for calculating the strain inhomogeneity coefficient of steel fiber concrete beams with BFRP bars is fitted as follows:(20)ψ=ω11−0.8McrM,where ω1is the coefficient, by fitting the results for BFRP reinforced concrete beams with steel fibers, and ω1is recommended to take the value of 1.4.


### 5.4. Comparison of Experimental Results

In this study, 30% and 40% ultimate bearing capacity are used as the corresponding load levels under normal use condition, respectively. The test results are compared with the theoretical calculated values (Table8). The average values of calculated/measured values, respectively, are 1.013 and 1.014, and the coefficients of variation are 0.033 and 0.035. The calculated values are in good agreement with the experimental values.Table 8
Comparison between experimental values and values calculated by the modified formula.
Specimen number0.3Mu0.4MuΔt (mm)Δe (mm)Δe/ΔtΔt (mm)Δe (mm)Δe/ΔtBFRC10.490.470.9590.740.771.041BFRC20.520.531.0190.810.831.025BFRC30.520.541.0380.890.870.978BFRC40.720.710.9861.071.121.046BFRC50.520.541.0380.860.891.035BFRC60.270.281.0370.510.490.961Note:Mu is the bending moment at ultimate load carrying capacity, Δt is the test values, and Δe is the calculated values in this table.The crack calculation formula used in this paper is based on the bond-slip theory and the no-slip theory. After a comprehensive analysis, a comprehensive calculation method of the crack spacing is proposed considering the bond effect between the steel bar and the concrete and the influence of the concrete protective layer. The crack width in the reinforced steel fiber reinforced concrete beam was predicted. The calculation results using the ACI440.1R-15 [33] and CSA S806-12 [39] calculation methods are quite different from the experimental results, and a better prediction cannot be made. This is because the calculation methods of ACI440.1R-15 [33] and CSA S806-12 [39] do not consider the limiting effect of steel fibers on the crack width, and the calculated values are generally larger than the experimental values. It can better predict the maximum crack width of BFRP reinforced steel fiber reinforced concrete beams under service load, and it has good applicability.


## 5.1. The Value of the Relative Bond Characteristic Coefficientvi of BFRP Bars and Concrete

The main factor affecting the crack expansion in flexural members of concrete beams is the bond performance of longitudinal tensioned tendons to concrete. As for BFRP reinforced partial steel fiber concrete beams, since steel fibers are added only in the compression zone, the formula for calculating the average crack spacing can be followed for the average crack spacing of FRP reinforced concrete beams in the specification [35]:(12)lm=1.9c+0.08deqpte,deq=∑nidi2∑nividi,where c is the concrete protective layer thickness; deq is the equivalent diameter of longitudinal reinforcement; pte is the reinforcement rate calculated by the effective tensile concrete cross-sectional area; when pte0.01, take = 0.01; di is the nominal diameter of the i-th type of FRP bars; ni is the number of i-th type of FRP bars; and vi is the relative bonding characteristic coefficient of FRP bars; when >1.5, take 1.5; when no test data, take 0.7.The specification [35] does not give the value of the relative bond coefficient vi for each type of FRP tendons but suggests that the value of the relative bond coefficient vi is determined according to the experiment and limits the range of vi: 1.5 when vi >1.5, and 0.7 when there is no test data. It can be seen that the value of the relative bond coefficient vi for each type of FRP tendons needs to be verified.The average crack spacing measured experimentally in this paper is shown in Table7. The values of the relative bond coefficient vi between BFRP reinforcement and concrete were backcalculated for the BFRP reinforced plain concrete beams and some steel fiber reinforced concrete beams. When vi is taken as 1.1, the measured average crack spacing under ultimate load bearing capacity agrees well with the theoretical value (Table 7).Table 7
Average fracture spacing (mm).
Specimen numberBFRC1BFRC2BFRC3BFRC4BFRC5BFRC6Number of lacerations101515151416Average crack spacing12412111912112295


## 5.2. Stress of BFRP Bars at Cracks

For some of the steel fiber reinforced beams, since the lower part of the beam is not mixed with steel fibers, the beam is subjected to the main tensile stress by the BFRP tendons after cracking, and the maximum load increase from the initial cracking load to the limit state of normal use is small; therefore, the effect of steel fibers is not considered in the normal use state.From the cross-sectional moment equilibrium condition, it is obtained that(13)M=σfAfh0−x3,where M is the bending moment at the cracked section; Af is the cross-sectional area of BFRP bars; h0 is the effective height of the section. Deformation coordination equation:(14)εsfuεf=xh0−x,where εsfu is the steel fiber concrete strain in the upper part of the beam; εf is the BFRP tendon strain; x is the distance from the upper surface of the beam to the neutral axis. The stress at the BFRP reinforcement between the cracks is(15)σf=3Mεf+εsfuAfh03εf+2εsfu.For full-section steel fiber-reinforced concrete beams, the stress transfer of the steel fibers needs to be considered, as in Figure12. Therefore, for the flat section assumptions of full-section steel fiber-reinforced concrete beams with BFRP bars, the effect of the steel fibers is to be considered.Figure 12
Stress-strain of full-section steel fiber test beam under normal service condition.From the bending moment balance, it is obtained that(16)M=σfAfh0−x3+12σsftbh−xh+x3.The stress at the BFRP reinforcement between cracks in full-section steel fiber concrete beams can be obtained by associating equations (13) with (15) as follows:(17)σf=2M−σSftbh−xh+x/32Afh0−x/3.The residual tensile strength of steel fiber concrete beams after cracking was calculated using the literature [36] using the following formula:(18)σsft=13αstλsf1+f,where αst is the bond strength of steel fiber and concrete, for milling pin type steel fiber to take 1.2ftk; f is the steel fiber and concrete in the crack edge shear friction coefficient, take 1/3.


## 5.3. Intercrack Tensioned BFRP Bars Strain Unevenness Factorψ

In the code [37], FRP reinforced concrete beams follow the method of calculating the strain inhomogeneity factor for reinforced concrete beams as follows:(19)ψ=1.1−0.65ftkσfpte.Since the strain inhomogeneity coefficient of steel bars is based on a huge amount of experimental data of reinforced concrete beams, and the difference in material properties between FRP bars and steel bars makes it not applicable to calculate the strain inhomogeneity coefficient of FRP bars by following the calculation method of steel bars, the addition of steel fibers also affects the strain of BFRP bars between cracks, and in this paper, the formula for calculating the strain inhomogeneity coefficientψ is modified based on the specification [38], and the cracking moment is multiplied by a discount factor of 0.8, considering the adverse effect of concrete shrinkage in the tensile zone. Based on the data obtained from experiments, the formula for calculating the strain inhomogeneity coefficient of steel fiber concrete beams with BFRP bars is fitted as follows:(20)ψ=ω11−0.8McrM,where ω1is the coefficient, by fitting the results for BFRP reinforced concrete beams with steel fibers, and ω1is recommended to take the value of 1.4.


## 5.4. Comparison of Experimental Results

In this study, 30% and 40% ultimate bearing capacity are used as the corresponding load levels under normal use condition, respectively. The test results are compared with the theoretical calculated values (Table8). The average values of calculated/measured values, respectively, are 1.013 and 1.014, and the coefficients of variation are 0.033 and 0.035. The calculated values are in good agreement with the experimental values.Table 8
Comparison between experimental values and values calculated by the modified formula.
Specimen number0.3Mu0.4MuΔt (mm)Δe (mm)Δe/ΔtΔt (mm)Δe (mm)Δe/ΔtBFRC10.490.470.9590.740.771.041BFRC20.520.531.0190.810.831.025BFRC30.520.541.0380.890.870.978BFRC40.720.710.9861.071.121.046BFRC50.520.541.0380.860.891.035BFRC60.270.281.0370.510.490.961Note:Mu is the bending moment at ultimate load carrying capacity, Δt is the test values, and Δe is the calculated values in this table.The crack calculation formula used in this paper is based on the bond-slip theory and the no-slip theory. After a comprehensive analysis, a comprehensive calculation method of the crack spacing is proposed considering the bond effect between the steel bar and the concrete and the influence of the concrete protective layer. The crack width in the reinforced steel fiber reinforced concrete beam was predicted. The calculation results using the ACI440.1R-15 [33] and CSA S806-12 [39] calculation methods are quite different from the experimental results, and a better prediction cannot be made. This is because the calculation methods of ACI440.1R-15 [33] and CSA S806-12 [39] do not consider the limiting effect of steel fibers on the crack width, and the calculated values are generally larger than the experimental values. It can better predict the maximum crack width of BFRP reinforced steel fiber reinforced concrete beams under service load, and it has good applicability.


## 6. Conclusion

(1)
Both steel fiber volume rate and steel fiber concrete height can inhibit the extension of crack height, and compared with BFRP reinforced plain concrete beams, BFRP reinforced concrete beams with steel fibers are smoother in the process of crack extension, and there is no obvious abrupt change point. Steel fiber volume rate has little effect on crack width for some steel fiber reinforced concrete test beams under normal use. The maximum crack width was reduced by 19%∼34% when the ultimate load was 165 kN. The thickness of steel fiber concrete can effectively inhibit the extension height of the initial crack of the test beam, and the crack height is reduced by 28%∼50% at the initial crack.(2)
Based on the measured average crack spacing of some steel fiber reinforced concrete beams compared with the theoretical value, it is recommended that the relative bond coefficientvi be taken as 1.1 for the deep-ribbed BFRP bars with 14 mm diameter.(3)
The incorporation of steel fibers provided some improvement in the crack resistance of the BFRP reinforced concrete beams, and the crack load resistance of the BFRP reinforced partially steel fiber concrete stacked beams was improved by 17% to 21% compared to the test beamB1.(4)
The influence of the plastic deformation actually existing in the concrete in the tensile zone of the cross section on the crack resistance can be calculated by converting the moment of inertia of the cross-sectional form center to the cross-sectional crack resistance moment of the steel fiber reinforced concrete beam with partial BFRP reinforcement and the steel fiber concrete beam with full section of BFRP reinforcement, and the formula of the positive cross-sectional crack resistance considering the action of steel fiber is proposed, and the calculated values are compared with the experimental values, which are in good agreement.(5)
The formulae for calculating the maximum crack width of steel fiber reinforced concrete beams with partial BFRP reinforcement and full-section steel fiber reinforced concrete beams with BFRP reinforcement in positive service condition were established, and the calculated results were in good agreement with the experimental values.



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*Source: 9158379-2022-09-02.xml*