Study on bending failure and crack characteristics in ductile fiber‐reinforced concrete beams

In this study, we designed six polypropylene fiber‐reinforced concrete (PP‐FRC) beams and three reinforced concrete (RC) beams for experimental studies. Then, we analyzed the effects of incorporating PP fibers on the cracking performance of PP‐FRC beams from the perspectives of load‐displacement response, crack distribution, crack depth, crack width, crack spacing, and deformation coordination. The research results showed that the cracking load and number of cracks were positively correlated with the volume ratio of PP fibers, while the maximum crack width, average crack spacing, and average crack depth were negatively correlated with the volume ratio of PP fibers. Due to the bridging effect of fibers, the PP‐FRC beam body and reinforcement could deform in a coordinated manner within a specific loading range after cracking, which increased the deformation coordination ability between the body and reinforcement. In this study, the cracking mechanism of PP‐FRC beams was investigated. Firstly, the formula for calculating the cracking moment of PP‐FRC beams was derived from the perspective of equivalent bending tensile strength. Then, based on the fiber bridging law and the bond strength between PP‐FRC and reinforcement, theoretical formulas were proposed to predict the average crack spacing and maximum crack width of PP‐FRC beams. Finally, through comparison, it was found that the proposed formulas could be used for characteristic crack prediction.

][3] To overcome the brittleness and crack sensitivity of concrete, a large number of scholars have tried to enhance the mechanical properties of ordinary concrete with various fibers such as steel, polyvinyl alcohol (PVA), polyethylene (PE), and polypropylene (PP). 4iber-reinforced concrete (FRC) is a multiphase heterogeneous cementitious material composed of cement, coarse and fine aggregates, and admixtures, reinforced with fiber materials.6][7][8] In recent decades, a large number of scholars have studied different types of FRC materials.Ríos et al. 9 showed that the ultimate tensile strength of the steel fiber matrix was increased by a factor of 1.96 with the inclusion of the appropriate volume fraction, and ultimate tensile strains of 5%-8% were achieved.Yu et al. 10 studied the tensile behavior of FRC mixed with PE fibers, the ultimate stress of the mixture reached an amazing 17.84 Mpa, and the ultimate strain was higher than the strain capacity of steel FRC in 7.98%-11.98%.Pan's study yielded a matrix tensile ultimate stress of 3.51-4.39Mpa and an ultimate strain capacity of 2.61%-4.46%with the addition of an appropriate amount of PVA fibers. 11Dong et al. 12 found that PVA fibers can significantly improve the crack morphology, peak tensile strength, and strain of textile-reinforced mortar.Subsequently, the effect of seismic strain rate on PVA fiber-modified textilereinforced mortar was studied, 13 and the results showed that the crack strength and peak strength of the PVA fibermodified textile-reinforced mortar increased to 4.0-7.2 and 10.1-17.3Mpa, respectively.Yew et al. 14 investigated the effect of PP fibers on the mechanical properties of concrete, and the results showed that PP FRC increased splitting tensile strength by 8%-33% and flexural strength by 6.0%-7.3%.At the material level, the presence of disordered fibers in FRC increases the energy required for crack expansion during the microscopic and macroscopic cracking of the concrete body by shedding, sliding, pulling out, or fracturing the disordered fibers, thus achieving a crack arresting effect. 15Chen et al. 16 analyzed the fiber reinforcement mechanism from the perspective of energy conversion by experimentally measuring the fracture state and waveform variation law of different fiber concrete specimens and verified the above fiber crack arrest mechanism.
At the structural level, ordinary RC structures in the cracked section can no longer withstand tensile stress, the reinforcement takes on the full load of the cracked section, and the crack's width increases without restriction as the load increases. 17For structures reinforced with fiber-infused concrete, the tensile strength of the concrete is enhanced to improve the structure's ability to withstand external loads.Additionally, in FRC beams, the reinforcement is continuous and directional, while the fiber has a similar distribution to that of disordered reinforcement.The fiber and reinforcement share the cracking section, and the bridging of the fiber enhances the coordination of deformation between the reinforcement and the substrate.The bridging of the fiber reduces stress concentration in the cracking section, inhibiting the tendency for crack width and length to continue developing. 18Li et al. 19 proposed a micromechanical model for the bridging law of steel FRC, and Amin and Gilbert 20 derived a theoretical method applicable to the calculation of crack widths in FRC flexural members based on its consideration of the fiber bridging effect and the interaction between FRC and steel reinforcement.Regarding flexible fibers, Deng et al. 21found that PVA-FRC can still provide tensile force after cracking.Based on the tensile constitutive and force equilibrium conditions of PVA-FRC, the ultimate flexural bearing capacity of PVA-FRC beams was predicted.Furthermore, the bridging law derived by Ozu et al. 22 from the experimental results of PVA-FRC can be expressed through a trilinear model with trilinear control points dependent on the flexible fibers' strength and elongation.This bridging law exhibits superior agreement with the results of uniaxial tensile experiments.Zhang et al. 23 proposed theoretical formulas applicable to calculate the average crack spacing, average crack width, and maximum crack width of PVA-FRC beams based on the flexible fiber bridging law studied by Ozu et al. 22 Although PP and PVA fibers are classified as flexible fibers, the study 11,14 has shown that they are distinct materials and exhibit different fiber reinforcement mechanisms.The current theoretical equations used to compute crack width and spacing in S-FRP and PVA-FRC beams do not apply to PP-FRC beams.
PP fiber is typically viewed as a secondary reinforcement material for concrete.Meanwhile, PP-FRC offers several benefits, including low cost, lightweight, ease of mixing, and significant enhancement of concrete's mechanical properties; it has been extensively employed in critical projects. 24,25In recent years, despite the extensive experimental studies on PP-FRC beams conducted by numerous scholars, [26][27][28][29] the focus has primarily been on the durability of PP-FRC beams.Even when the impact of PP fibers on crack resistance is considered, it is only observable in the macroscopic behavior of the loaddeflection curve.The bending cracking behavior of PP-FRC beams entails a series of intricate mechanical issues, including the PP fiber bridging law and the interplay between PP-FRC and steel reinforcement.The manifestation at the macroscopic level is reflected in the width and spacing of cracks following beam cracking, and few scholars have delved into this series of inquiries.
In this study, the mechanical properties of PP-FRC were first evaluated experimentally to determine the optimum fiber volume fraction.Subsequently, the RC with the optimal fiber volume fraction was selected as the research subject, while semi-parametric and nonparametric PP-FRC were employed as control groups.For experimental analysis, six fiber-reinforced and three ordinary concrete beams were prepared.Finally, the impact of PP fiber volume parameters on the mechanical properties of PP-FRC beams was analyzed, considering factors such as load-displacement response, crack distribution, crack depth, crack width, and crack spacing.

| Material properties
High-strength concrete has a high usage rate in high-rise buildings and bridge engineering, 30 and existing research [31][32][33] has confirmed the improvement effect of PP fiber on the mechanical properties of high-strength concrete.There is relatively little research on experimental beams, so the concrete design of this experimental beam is C55 strength grade.PP-FRC consists of ordinary silicate cement (PO 42.5), fly ash (Class F Type I), mineral powder, mechanism sand (Type I, maximum particle size 4.75 mm), gravel (particle size between 5 and 20 mm), PP fiber, water, and water reducing agent.Table 1 presents the mix proportions for PP-FRC, and Table 2 shows the basic mechanical properties of PP fibers.Many scholars [32][33][34][35][36][37][38] have studied the variation of PP-FRC flexural strength and tensile strength with fiber volume fraction and obtained different results as shown in Figure 1.When the fiber volume fraction is appropriate (0.18%-0.45%),PP fibers have a significant improvement effect on the crack resistance of concrete.However, in most cases, when the PP fiber volume fraction exceeds a certain limit (about greater than 0.5%), the overall effect of fibers on concrete is negative.Scholars attribute this weakening phenomenon to the agglomeration of high-content fibers, where clustered fibers T A B L E 1 Mixing ratios of polypropylene fiber-reinforced concrete (kg/m 3 ).such as pores are distributed inside the concrete and weaken the cross-sectional strength.Therefore, the improvement effect of PP fiber on concrete is not directly proportional to the fiber volume fraction.Different concrete mix proportions and strength levels result in different mechanical properties under different PP fiber volume fractions.To determine the optimal parameter for PP fiber in this experiment, PP-FRC was trial mixed starting from parameter 0%.During the mixing process, it was observed that a significant number of fiber agglomerates became visible to the naked eye when the fiber content reached 0.7%.Figure 2b,c shows on-site images of fiber clusters.Accordingly, the study established seven parametric levels with a gradient of 0%-0.6%, resulting in 21 short beams.Each short beam had a cross-sectional length and width of 100 mm and a span of 400 mm, and three specimens were cast for each parametric level to conduct fourpoint bending experiments.Figure 4a shows PP-FRC's flexural strength under different PP fiber volume fractions, with the PP fiber volume fraction increase, the flexural strength of PP-FRC initially increases and then decreases.The peak flexural strength of PP-FRC, reaching 7.3 MPa, is achieved at a fiber volume fraction of 0.4%.

Cement
Considering the experimental funding issue, the volume ratio range of PP fibers was narrowed down.The research objects were selected as even-numbered content levels ranging from 0% to 0.6%, and further experimental analysis was conducted on four even-numbered control groups.Six standard 150 mm cubic specimens were cast for each group (a total of 24 specimens), and after a 28-day curing period in a standard curing room, splitting tensile strength and compressive strength tests were carried out.Figure 3 shows the on-site experimental setup for the basic mechanical properties of PP-FRC.Figure 4 and Table 3 present the mechanical performance test results of PP-FRC.Three specimens 400 mm in length were made for each rebar to test the relevant mechanical properties in the uniaxial tensile test.Table 4 lists the specific mechanical parameter values of the steel bars.

| Specimen design
The experimental results indicated that the optimal mechanical performance was obtained at a PP fiber volume fraction of 0.4%.The study selected PP-FRC with a volume parameter of 0.4% as the research object, with non-participating fibers (0%) and semi-participating fibers (0.2%) PP-FRC as control groups.Nine beams (3*0%, 3*0.2%, and 3*0.4%) were designed for the experiment to compare the effect of the PP fiber volume fraction on the bending cracking of concrete beams.Some researchers 23,39 use the reinforcement ratio as a variable for analysis when studying the bending cracking mechanism of FRC beams.On the other hand, the calculation methods for the maximum crack width and crack spacing of RC beams based on the bond-slip theory and empirical methods in the Chinese standard (GB/T 50010-2010) 40 have become quite mature.The purpose of this experiment is to investigate the influence of PP fiber volume fraction on crack propagation, crack width, and crack spacing during bending failure of RC beams.Therebefore, in this experiment, ribbed steel bars with a diameter of 12 mm were selected for tensile reinforcement, smooth round steel bars with a diameter of 8 mm were selected for compressive reinforcement, and hoops with a diameter of 8 mm and a spacing of 100 mm were selected for shear stress in the beam section to reduce the shear failure of the experimental beam.The reinforcement ratio of tensile steel bars in this experiment is taken as a fixed value of 1.41% when meeting the requirements for checking reinforced beams.
Figure 5 shows the detailed structure of the experimental beams and the loading positions.The experimental beams are named with initials C and P for RC and PP-FRC beams, respectively.The middle number indicates the PP fiber volume fraction, and the last indicates the experimental beam number.For example, P-0.2-2 indicates the second experimental beam of PP-FRC beam with 0.2% volume reference.

| Experimental procedure
The concentrated load for the four-point bending experiments in this study was provided by a 500 t electro-hydraulic servo testing machine, and a steel beam was used as the distribution beam.A 30 t pressure transducer and a 100 mm   percentage meter were used to measure the load and displacement, respectively, and strain gauges were used to test the strain of the steel and concrete in the pure bending section.This experiment adopts a static monotonic mode of loading, with an increment of 2 kN applied before the initial crack appears, and an increment of 4 kN applied after the crack appears.After each level of loading is completed, hold the load at constant force for 3 min, and measure and record relevant data.During the experimental process, the crack width of the experimental beam was measured using the BJZJ-LF crack width measuring instrument (measuring range: 0.02 mm), and the corresponding load of the crack width was recorded.After the crack appeared, the position and direction of the crack were plotted on the experimental beam.The loading device is shown in Figure 6, and the crack width gauge is shown in Figure 7.

| Load-deflection curve analysis
The load-deflection curves in the span of the experimental beam are shown in Figure 8.The load-deflection curve is divided into three stages:(a) Before the beam cracking, the whole beam is in the elastic stage at this time; (b) Cracking of the beam to yielding of the tensile reinforcement; (c) Yielding of the tensile reinforcement.In the third stage, the deflection increases rapidly, and the load changes slightly, while the PP fiber crack arresting effect is mainly reflected in the pre-yielding period of the reinforcement, and the corresponding beam after the steel yielding is not the focus of this study.
The average cracking load of three RC beams is 18.42 kN, while the average cracking load of three PP-FRC beams with a PP fiber volume fraction of 0.2% is 20.24 kN, which is 9.88% higher than that of RC beams, and the average cracking load of three PP-FRC beams with PP fiber volume fraction of 0.4% is 22.52 kN which is 22.25% higher than that of RC beams.The initial cracking strength indicates that incorporating PP fibers in the tensile zone can significantly increase the initial cracking load of RC beams.
In addition, the PP-FRC samples exhibit higher displacement response with increasing PP fiber volume fraction when the reinforcement reaches yield load.While PP-FRC has been observed to enhance the ultimate tensile strain and toughness of concrete, it is essential to note that this improvement may lead to a slight reduction in the overall stiffness of the beam.As measured experimentally, the ultimate loads of PP-FRC beams and RC beams show no significant difference, suggesting a limited capacity enhancement effect of PP fibers on the ultimate load capacity of beams.

| Crack extension analysis
Figure 9 illustrates the distribution of cracks resulting from four-point bending damage in RC and PP-FRC beams.The experimental beam was marked with load values corresponding to the crack extension process.The crack distribution map reveals that the density of cracks in the PP-FRC beam is higher than that in the RC beam.The whole crack expansion process of RC and PP-FRC beams is roughly the same in the following five stages: (a) Before initiating cracks, the beam is in the elastic regime, exhibiting a linear relationship between the applied load and deflection.(b) Upon reaching the cracking strain under load, either RC or PP-FRC beam exhibits the formation of at least one randomly distributed vertical microcrack at the bottom of the pure bending region or its vicinity, which is the weakest location.(c) As the load increases, the number of cracks in the pure bending region of the beam increases to a certain extent and then reaches a steady state, while the spacing between cracks tends to stabilize.(d) After the number of cracks in the purely curved section reaches saturation, the cracks propagate vertically, and the crack depth stabilizes at a certain load level.(e) Under continued loading, the primary crack initiates and propagates towards the compression zone, causing either the crushing of concrete in the compression zone or the yielding of reinforcement in the tension zone, ultimately leading to the failure of the experimental beam.

| Crack width analysis
Figure 10 presents three beams' average maximum crack width values with identical fiber volume fractions at various load levels.M is the bending moment corresponding to the loading, and M u is the ultimate bearing capacity corresponding to the bending moment.At an applied load level of 0.3, the RC beam specimen experiences cracking with a corresponding crack width of 0.03 mm.Moreover, the resulting crack widths are nearly equivalent for PP fiber volume fractions of 0.2% and 0.4%, measuring at 0.02 mm.Upon initial cracking, the disordered distribution of PP fibers in PP-FRC beams remains relaxed under small deformations due to the lower elastic modulus of the fibers compared to that of concrete.At this stage, the crack resistance effect can be achieved only by increasing the system's energy consumption.However, as the entire system does not reach the limit value of energy consumption, the resulting crack widths for different PP-FRC fiber volume fractions are nearly identical.This conclusion is consistent with the findings of Chen et al. 16 Subsequently, the crack width of both RC and PP-FRC beams exhibited a linear increase with the progression of loading levels.When the load rating reaches 0.8 the crack width of RC beams increased to nearly 0.25 mm; as the load increased, the crack width abruptly increased to 0.56 mm upon yielding the reinforcement.
For PP-FRC beams with a fiber volume fraction of 0.2%, the maximum crack width was 0.18 mm at a load rating of 0.8, representing a reduction of 28.0% compared to RC beams.However, the crack width after reinforcement yielding was 0.54 mm, indicating no significant reduction from the RC beams.For PP-FRC beams with a fiber volume fraction of 0.4%, the maximum crack width was 0.16 mm at a load rating of 0.8, representing a reduction of 36.0%compared to RC beams.The crack width after yielding reinforcement was 0.22 mm, indicating a reduction of 60.7% compared to RC beams.When the loading level exceeds 0.8, PP-FRC beams with a fiber volume fraction of 0.2% exhibit a sudden increase in cracking similar to RC beams.This is due to the relatively small number of fibers in the crack section, which results in the pulling off of PP fibers under high-level loads.At this point, the mechanical properties of PP-FRC beams are similar to those of RC beams.On the other hand, PP-FRC beams with a PP fiber volume fraction of 0.4% have twice the number of fibers in the crack section compared to 0.2%, resulting in a continued linear increase in crack width.The maximum crack width study results indicate that the bridging effect of PP fibers in PP-FRC can significantly limit crack extension and expansion.It is suggested that using PP-FRC with a higher fiber volume fraction can effectively enhance the crack resistance of beams under high-level loads.

| Analysis of the number, height, and spacing of cracks
The variation in the number of cracks in PP-FRC beams with different volume ratios under different load ratings is depicted in Figure 11a.It is observed that the number of cracks in PP-FRC beams is significantly higher than that in RC beams under the same load rating.Furthermore, until the loading level reaches 0.7, there is no significant difference in the number of cracks between PP-FRC beams, with a fiber volume fraction of 0.2% and 0.4%.However, after the loading level exceeds 0.7, the number of cracks in PP-FRC beams with a fiber volume fraction of 0.4% is more than that of a fiber volume fraction of 0.2%.The increase in tensile strength and ultimate tensile strain of PP-FRC compared to regular concrete is attributed to PP fibers, which enhance the deformation coordination between PP-FRC and reinforcement, resulting in a multi-crack damage pattern in the matrix.Figure 11b displays the average crack spacing of each specimen at different load levels.Upon comparing the average crack spacing with varying fiber volume fractions, it is observed that an increase in the PP fiber volume fraction leads to a significant reduction in the initial crack spacing, along with an improvement in the uniformity of the crack spacing.This phenomenon can be attributed to pores in the ordinary concrete matrix, which causes the first crack in the purely bending section under external load to emerge from the weakest position.This results in a random cracking location and a large dispersion of crack spacing.The introduction of PP fiber fills the void defects present in the concrete and enhances the homogeneity of the concrete.Furthermore, the variation curve of crack spacing tends to be consistent with the increase in the PP fiber volume fraction.
Analyzing the crack spacing curve, we find that at least two cracks appear in the experimental beam when the load rating M=M u reaches 0.3.The addition of PP fibers reduces the average crack spacing when 0:3 ≤ M=M u ≤ 0:8, but the effect weakens with an increasing load rating, and the difference between the average crack spacing of RC and PP-FRC beams gradually decreases with increasing load rating; When the load rating M=M u exceeds 0.8, the crack spacing develops steadily to a constant value.The average crack spacing of specimens C-0, P-0.2, and P-0.4 is 74.45 mm, 71.62 mm, and 68.53 mm, respectively, and the addition of PP fiber reduces the average crack spacing of the final damage of the experimental beam by 3.8% and 8.0%, respectively, indicating that the final crack spacing of PP-FRC beams decreases with the increase of PP fibers.
The average crack depth of the specimens at different load levels is shown in Figure 11c.The average initial crack depth of RC beams and PP-FRC beams is 71.4 mm (0%), 47.2 mm (0.2%), and 19.1 mm (0.4%).PP-FRC beams with a fiber volume fraction of 0.2% (0.4%) have reduced the average initial crack depth compared to RC beams by 32.9% (68.5%); the average crack depth at complete failure is 127.7 mm (0%), 113.8 mm (0.2%), and 113.1 mm (0.4%).PP-FRC beams with a fiber volume fraction of 0.2% (0.4%) have reduced the average crack depth compared to RC beams by 10.8% (11.4%).It indicates that the cracking-inhibiting effect of PP fiber is mainly reflected in the early stage of cracking, while the cracking-inhibiting effect in the late stage of cracking is not significant.PP-FRC exhibits superior tensile strength and toughness compared to RC.When subjected to the same load level, PP-FRC beams exhibit smaller crack depths than RC beams, which is advantageous for mitigating crack propagation.the plane section assumption.An analysis of the strain differences in the tension zone of concrete under varying load levels showed that the strain dispersion of concrete in the tension zone of RC beams is relatively significant.However, adding PP fibers improves the uniformity of concrete strain under tension.Under load, strain gauge damage was observed at the bottom of the mid-span of the RC beam at 48 kN, whereas the strain gauges at the bottom of the mid-span of the PP-FRC beam were damaged at loads of 56 kN (0.2%) and 64 kN (0.4%), respectively.Therefore, adding PP fibers to the tension zone can enhance the bearing capacity of mid-span cracking.The effect of improving tensile uniformity and increasing bearing capacity is better with a higher volume ratio of PP fibers.

| Deformation coordination analysis
Figure 13 shows a comparison of the strains between the tension reinforcement and the PFRC (RC) at the same level.In the elastic phase, the concrete and reinforcement strains converge with increasing load for both the RC and PP-FRC beams.Furthermore, after cracking, the strain of the beam and the strain of the reinforcement remain close within a specific range.However, with the increase of load, the existing balance of displacement coordination is soon disrupted, and when a specific ultimate load is reached, the concrete and the reinforcement exhibit a large deformation discoordination between them.
The ultimate deformation-coordination load of RC beams is 32 kN.However, when PP-FRC beams with a fiber volume fraction of 0.2% (0.4%) were tested, the deformation-coordination ultimate load increased to 48 kN (56 kN), representing an improvement of 50% and 75% over RC beams, respectively.This study demonstrates that even after cracking, the bridging of PP fibers in the matrix can effectively transfer tensile stresses, leading to increased deformation coordination between the matrix and the reinforcement.

| Bending cracking load
Regarding the calculation of the cracking moment, Fang et al. 41 established the formula for the cracking moment of concrete beams with different fibers by introducing the coefficient "k" for the degree of development of plastic deformation.It is assumed that the relationship between the ultimate tensile strain ε tu and the peak strain ε to when the fiber concrete beam cracks satisfies: The strain distribution of concrete in the tensile zone at cracking is shown in Figure 14b.As depicted in Figure 14c, the stress state within the tensile zone can be partitioned into two distinct regions, namely the elastic and plastic zones, based on the flat section analysis.The equation for cracking of FRC beams based on the coefficient of development of plastic deformation k can be derived from the force equilibrium condition: where W sk is the plastic resistance moment of the section, and f t is the axial tensile strength of the concrete.
The absence of a well-defined correlation between the ultimate and peak tensile strains in the context of bending-induced concrete cracking for varying fiber types and concrete compositions poses significant challenges in accurately determining W sk .In this investigation, the four-point bending damage of concrete materials is computed under the assumption of elastic behavior.The corresponding stress distribution is illustrated in Figure 14d.
The equivalent bending tensile strength f t,eq of the concrete beam at cracking is expressed in Equation (3): Further: where W 0 is the cross-sectional resistance moment of the tensile edge according to the elastic material; b is the section width; h is the section height; M crk is the cracking bending moment of the bending member.Let: where f λ p À Á is the fiber influence function.γ is the plastic equivalence factor.For concrete specimens without steel reinforcement, γ ¼ f t,f =f t , and f t,f is the nominal flexural tensile strength of the concrete.f t is the concrete axial tensile strength, the measured values are shown in Table 3. α is the section restriction factor, usually taken as 0.775-1. 42For members whose section height is less than the width (e.g., plates) take 1.In this study, the experimental beam section height is greater than the width, so 0.775 is taken.We computed the value of γ for non-FRC by inputting the data from Table 3, which resulted in γ = 1.928.Furthermore, a linear regression analysis is performed using the non-FRC γ value as the baseline, and the obtained result is denoted as f λ p À Á .Finally, we can observe the following: Substituting Equation ( 6) into Equation ( 4), Equation ( 7) is derived:

| Crack spacing
Xu et al. 43 and Bai et al. 39 modeled steel and PVA fibers to resist cracking from the bending perspective of the beam.Zhang et al. 23 approximated the four-point bending pure bending section as a uniaxial tensile model according to the relevant specification 40 and simplified the model calculation for calculating PVA fiber crack resistance by correcting the calculated results with coefficients.
The force model for bending cracking can be observed in Figure 15.When PP-FRC reaches cracking strain under load, at least crack randomly appears at the bottom of the purely curved section or nearby weakest location.This study assumes that the experimental beam is under small deformation.The present study assumes that the tension direction in the PP-FRC beam aligns with the x-axis, with the first crack located at the origin (x = 0) on the x-axis.The distance between the first and subsequent cracks is represented by the length denoted as "l" (where "l" denotes the absolute value of the distance).
Experimental beam in the 0-l interval, the reinforcement makes the deformation coordinated with the deformation of PP-FRC by the bond stress between the steel and concrete.When the relative slip between the reinforcement and the concrete reaches the PP-FRC cracking strain, the concrete is subjected to the maximum tensile stress, and the concrete stress located at l is maximum and reaches the cracking strength.Beyond this equilibrium state, the beam can produce new cracks.
In RC beams, the concrete in the cracked section no longer contributes to the load-bearing capacity, and the reinforcement solely carries the tensile stress.On the other hand, in PP-FRCC beams, the bridging effect of the fibers in the cracked section prevents the complete withdrawal of the concrete from the load-bearing capacity, resulting in the concrete still bearing some of the tensile stress.
The force equilibrium relationship between the observed cracks and the predicted cracks in the beam segment is expressed by Equation (8).
where σ s0 and σ sl denote the reinforcement stresses at the existing crack and the predicted crack location, A s is the area of the tensile reinforcement, A c denotes the area of the concrete in the tensile zone A c ¼ 2:5b hÀ h 0 ð Þ, h 0 is the adequate depth of the PP-FRC beam, σ br denotes the fiber bridging stress at the cracked section, and σ cr denotes the PP-FRC cracking strength.
The force balance relationship between existing cracks and predicted cracks within the steel bars is shown in Equation (9).
where μ denotes the perimeter of the reinforcement, d denotes the diameter of the reinforcement, l denotes the distance between the predicted crack and the existing crack, τ x denotes the bond between the reinforcement and the PP fiber concrete.The calculation formula is given in the literature. 44,45ubstituting Equation ( 9) into Equation ( 8), Equation ( 10) is derived.10), simplify to get l.
The average crack spacing l m is ca.1.5 times the predicted crack spacing 40 :

| Crack width
The average crack width (w m ) can be defined as the difference between the average strain of the reinforcement and the PP-FRC tension between the two cracks.The formula is calculated as Equation (13).
where ε sm and ε cm denote the average strain of the reinforcement and PP-FRC, ε sm ¼ φε s ¼ φσ s =E s , σ s is the reinforcement stress; E s is the modulus of elasticity of the reinforcement; and φ denotes the coefficient of reinforcement strain inhomogeneity within the crack spacing, the meaning of which represents the contribution of the reinforcement to PP-FRC cracking control.The bridging effect of the fibers enhances the deformation coordination between the matrix and the reinforcement and improves the contribution of the reinforcement to the reduction of the crack width.According to GB50010-10 40 the PP-FRC inhomogeneity coefficient φ is expressed as Equation (14).
where S 1 is the coefficient, σ cr is the tensile stress of PP-FRC, and ρ te is the effective reinforcement ratio.The stress and strain distribution at the cracking surface of the PP-FRC beam under the load is shown in Figure 16.
Pure bending section within the bending moment M for a fixed value, the location of the equivalent force under pressure to take the moment balance calculation formula is as follows: Simplify to get σ s where z is the distance of the neutral axis of the cracked section from the bottom of the beam.
where α Es is the ratio of the modulus of elasticity of the reinforcement to the concrete, let: Stress-strain relationship of polypropylene fiber-reinforced concrete beam cracking model.
Then, Equation ( 13) can be rewritten as: Maximum crack width w max :

| Fiber bridging law
Regarding the bridging law of FRC, Kanakubo 46 introduced the multifunctional, multilinear, trilinear, bilinear, elastic-plastic, and rigid-plastic models, and Ozu et al. 22 studied the flexible fiber bridging law can be incorporated into a trilinear model.The trilinear model is shown in Figure 17.The control points in the graph depend on the fiber strength and elongation of the flexible fibers.Equations ( 21)-( 24) are expressed as a function of fiber strength "r," δ tu which is related to elongation as a constant value.When the crack width is half the fiber length, the fiber is completely pulled out. 22max ¼ 0:20r 0:18 In the present study, PP-FRC beams were subjected to loading up to the yielding stage of the tensile reinforcement with fiber bridging at the main crack, as illustrated in Figure 18.During this process, the crack width was within the range limit of the crack gauge, which was 2 mm less than half of the fiber length of 6 mm.It was observed that the PP fiber did not fail under such conditions.Subsequently, a simplified bilinear model (considering only at δ 2 ) was employed to describe the characteristic points of the calculated bridging law during the experimental beam cracking phase.The key to this model is to determine the constant "r" value related to the bridging stress of fibers.The bridging stress of FRC can be directly obtained through uniaxial tensile tests 47,48 or indirectly calculated through prism bending tests. 49As shown in Figure 19, the stress-displacement curve of PP-FRC is a combination of the stress-displacement curve of ordinary concrete and the bridging stress-displacement curve of PP fibers.Taking the experimentally measured flexural strength of ordinary concrete as a reference, the bridging stress of PP fibers can be calculated by the difference between the flexural strength of PP-FRC and the flexural strength of ordinary concrete.
Combining the experimental data obtained in Table 3, when the fiber volume fraction of PP-FRC is 0.4%, the flexural strength difference between PP-FRC and ordinary concrete is 1.3 MPa.By substituting this value into Equation ( 21), we can calculate that "r" is equal to 0.238.Additionally, it is postulated that the bridging effect of a PP-FRC beam with a PP fiber volume fraction of 0.2% is equivalent to 0.5 times that of PP-FRC with a volume fraction of 0.4%.The parameters of the bilinear model utilized in this study are depicted in Figure 20.

| Comparative analysis of experimental theories
To validate the suitability of the cracking moment calculation expressed in Equation ( 7), 17 sets of bending cracking data were compiled.This compilation included four sets of experimental data for REBAE-PP-FRC beams, 41 four for GFRP-PP-FRC beams, 50 and an additional nine sets of experimental data obtained in the present study.The anticipated value of the cracking moment M crk,pr was evaluated utilizing Equation ( 9) and subsequently juxtaposed with the corresponding experimental value M crk,ex .A graphical representation of the comparison results is presented in Figure 21.The experimental findings reveal that the average M crk,pr to M crk,ex ratio of REBAE-PP-FRC reinforced beams is 1.07, accompanied by a standard deviation of 0.11 and a coefficient of variation of 0.1.This signifies that the predictive capability of Equation ( 7) for the cracking load of REBAE-PP-FRC concrete bending is commendable, and the dispersion is negligible.The average M crk,pr to M crk,ex ratio of GFRP-PP-FRC beams is 1.17, accompanied by a standard deviation of 0.2 and a coefficient of variation of 0.17.Despite the simulation effect being inferior to that of REBAE-PP-FRC-reinforced beams, the outcomes indicate that Equation (7) can offer insights for forecasting the cracking moment of GFRP-PP-FRC beams.
To verify the rationality of the average crack spacing calculation model Equations ( 13) and ( 14), and the maximum crack width calculation model, as presented in Equation ( 22), for PP-FRC (RC) beams, experimental and calculated values were compared.The results of this comparison are presented in Table 5, with the experimental values being the average of three PP-FRC (RC) beams for each parameter.
The anticipated mean crack spacing for load classes 0.6 and 0.7 exhibited certain discrepancies from the experimentally determined mean crack spacing due to the dependency of Equation ( 11) on fiber bridging force.The loading process of RC beams without the presence of fiber bridging force resulted in imprecise prediction outcomes.The values obtained from the test, utilizing the mean crack spacing and maximum crack width formulas established in this study, agreed with the computed values for PP-FRC beams.Hence, the proposed calculation formulas for mean crack spacing and mean crack width in this study are rational and can effectively reflect the cracking mechanism of PP-FRC beams.This study aimed to analyze the bending cracking behavior of PP-FRC beams under concentrated load.The conclusions were as follows: 1. Incorporating an appropriate volume ratio of PP fibers improves concrete's tensile and flexural strength, with a maximum increase of 23.5% and 22.0%, respectively.However, this improvement is accompanied by a reduction in compressive strength compared to conventional concrete.2. PP-FRC materials effectively enhance the cracking behavior of concrete beams.Compared to RC beams, PP-FRC beams exhibit a distinct cracking pattern characterized by multiple cracks.The incorporation of PP fibers results in an increase of 9.9%-22.3% in cracking strength and 33.3%-66.7% in the number of cracks.Additionally, the maximum crack width, average crack spacing, and average crack depth are negatively correlated with the volume ratio of PP fibers, decreasing by 28.0%-36%, 3.8%-8.0%,and 10.8%-68.5%,respectively.These results suggest that including PP fibers in concrete effectively mitigates the development of cracks.3. Adding PP fibers improves the concrete's homogeneity under tension and its load-bearing capacity for cracking.The PP-FRC and reinforcement deformation were coordinated within the specific loading range after cracks appeared, and the deformation coordination limit load PP-FRC increased by 50.0%-75.0%compared to RC. PP fibers increased the deformation coordination between the matrix and reinforcement after cracking as the bridging matrix of PP fibers continued to work.4. The axial tensile strength of PP-FRC was not precisely proportional to the cracking moment of PP-FRC beams.In this study, a formula for predicting the cracking moment of PP-FRC beams was derived from the perspective of equivalent bending tensile strength, and the results showed that the proposed formula could predict the cracking moment well. 5. Theoretical equations are proposed to predict PP-FRC beams' average crack spacing and maximum crack width based on the fiber bridging law and the bonding force between PP-FRC and reinforcement; the calculated results showed a high agreement between the predicted and experimental values.

F I G U R E 2
Physical properties of polypropylene fiber concrete.(a) 12 mm chopped polypropylene fiber, (b) Uneven mixing of polypropylene fiber concrete, (c) Agglomeration of polypropylene fiber concrete.F I G U R E 3 Basic mechanical property test of polypropylene fiber-reinforced concrete.(a) Bending resistance test, (b) Split tensile performance test, (c) Single-axis compression performance test.

T A B L E 3
Mean values of mechanical properties of polypropylene fiber-reinforced concrete.

T A B L E 4 4
Properties of the steel bars.Mechanical properties of polypropylene fiber-reinforced concrete.(a) Flexural strength, (b) Tensile strength, (c) Compressive strength.

F
I G U R E 5 Loading position diagram and geometric details of the experimental beam.F I G U R E 6 Schematic diagram of the field loading device.F I G U R E 7 Crack width gauge.

Figure 12
Figure12depicts the strain distribution diagram of the mid-span section for varying loads.The linear strain distribution for each specimen at the mid-span section satisfies

F I G U R E 1 1
Cracking index of the experimental beam with different fiber parameters: (a) Number of cracks, (b) Average crack spacing, (c) Average crack depth.F I G U R E 1 2 Average value of strain distribution along the depth of the specimen in the mid-span section.(a) C-0, (b) P-0.2, (c) P-0.4.

1 3F I G E 1 4
Comparison of Steel Bars and polypropylene fiber-reinforced concrete strain at the same height.(a) C-0, (b) P-0.2, (c) P-0.4.Strain stress distribution in concrete beam sections.

5
Four-point bending force model of polypropylene fiber-reinforced concrete beam.

tuF
I G U R E 1 7 Tri-linear model for bridging law.

F I G U R E 1 8
Fiber bridging action at crack locations.F I G U R E 1 9 Decomposition of polypropylene fiber-reinforced concrete (PP-FRC) stress-displacement curve.F I G U R E 2 0 Fiber bridging action at crack locations.F I G U R E 2 1 Comparison of cracking moment between test values and theoretical values.
Mechanical properties of polypropylene fibers.
F I G U R E 1 Changes in mechanical properties of polypropylene fiber-reinforced concrete.(a) Flexural strength, (b) Splitting tensile strength.