Introduction

Concrete materials are widely used in the construction industry, comprising cementitious materials and aggregates, with coarse aggregate accounting for approximately 70% of the total concrete volume1. Since World War II, rapid global urbanization has led to the extensive consumption of natural resources, such as sand and stone, for constructing residential buildings and infrastructure. This has resulted in significant amounts of waste concrete2, negatively impacting the ecological environment. To address this issue, waste concrete blocks can be processed into recycled coarse aggregate (RCA), which replaces a certain percentage of natural coarse aggregate (NCA) in the production of recycled aggregate concrete (RAC). RAC represents a new type of green material that aligns with global goals for environmental protection and economic efficiency. However, the presence of microcracks within RCA and the adhesion of residual mortar on the aggregate surface lead to reduced strength and durability in RAC3,4, limiting its engineering applications.

Incorporating fibers into RAC to create fiber-reinforced recycled aggregate concrete (FRAC) can significantly enhance the overall performance of RAC5. Fibers intensify the connections between the various components of the RAC, mitigating the expansion of microcracks and improving its mechanical properties and durability. This enhancement broadens the application areas of RAC6,7. The incorporation of different types of fibers has led to significant improvements in FRAC across various aspects8. Currently, steel fibers are the most widely used among the available fiber types. However, the extensive use of steel fibers can increase project budgets without necessarily yielding better economic benefits in practical engineering applications.

Researchers often opt for silicate fibers or synthetic fibers over steel fibers, as this choice reduces the production costs of FRAC while ensuring excellent performance across various aspects. Basalt fiber (BF) is a type of silicate fiber with promising application potential, produced from natural basalt ore that is crushed and treated at 1500 °C9. BF not only exhibits strong mechanical and chemical properties but also possesses natural compatibility with cement, enhancing the concrete matrix’s compactness and homogeneity, thereby improving its toughness10,11. Zhang et al.12 demonstrated that during the mixing process of RAC, BF is divided into fine filaments and gradually disperses throughout the RAC, which improves its densification and enhances its physical properties. Alnahhal and Aljidda13 showed that the maximum damage deflection of RAC beams expands with increasing BF admixture. Yahong et al.14 conducted SEM tests on RAC and discovered that BF creates effective connections between cracks within the concrete matrix, absorbing energy from expanding cracks and reducing the likelihood of further propagation of microcracks.

Polyacrylonitrile fiber (PANF) is an organic synthetic fiber that is lightweight, weather resistant, and corrosion resistant15. Impact resistance, toughness, and fatigue life are improved to varying degrees when PANF is incorporated into concrete16,17,18,19. Deng et al.20 found that concrete beams containing PANF have higher ultimate fatigue strength than conventional beams, indicating that PANF enhances their toughness. Zeng and Tang21 investigated the enhancement of lightweight aggregate concrete (LWAC) through PANF and showed that its stress–strain relationships and load–deflection relationships have higher peaks than those of ordinary LWAC. Additionally, the relationship curves of the former show a smoother descending phase, attributed to its superior plastic deformation capacity.

The effects of incorporating single BF and PANF on the performance of RAC has been extensively investigated. However, RAC is a multiphase, non-homogeneous material, and the reinforcing effect of individually mixing a specific fiber is limited. By combining two types of fibers in RAC, each fiber can leverage its unique advantages to enhance the performance of RAC at different levels and load-bearing stages22. BF possesses higher strength and hardness, which contribute to the load-bearing capacity of concrete, while PANF is softer and improves the cracking resistance of concrete23. Mixing these two types of fibers can more effectively utilize their complementary benefits24,25,26,27,28,29. While the incorporation of BF alone can improve mechanical properties30, the incorporation of a mixture of BF and PANF into RAC results in higher flexural strength relative to BF alone31.

Currently, RAC is primarily used in infrastructure such as roads and bridges, which are mainly subjected to cyclic loads such as vehicles and trains during service life. As a result, fatigue damage may occur in concrete structures32. In order to expand the application scenarios of RAC under cyclic loads, it is essential to study their fatigue performance. In fatigue property testing of concrete materials, a common method involves applying sinusoidal wave loading to the specimen using a fatigue testing machine. The relationship between stress level and fatigue life has been discussed by many researchers, and the relationship established is referred to as the “S–N curve” or the “Wohler fatigue curve”33. Concrete is a typical material exhibiting various anisotropic properties, and small cracks formed during the preparation process can lead to significant dispersion in fatigue test results. Therefore, it is necessary to analyze fatigue life with the support of mathematical methods. The Weibull distribution provides a favorable fit for the fatigue life of concrete33,34,35. Kaur et al.36 incorporated the probability of failure into the fatigue life data of steel fiber concrete using the Weibull distribution and gave presented “Pf-S-lgN” curves for various combinations of concrete at different probabilities of failure. Saini et al.37 conducted flexural fatigue tests on self-compacted RAC and utilized the probability of failure to plot the “Pf-S–N curve” and establish a prediction model for the flexural fatigue life of self-compacted RAC, and the predicted results show that the doping of RCA reduces the fatigue life.

There is limited research on the integration of fatigue damage phenomena with sustainability concepts in the design of structures such as RAC roads and bridges. In this study, flexural strength test under static force and fatigue test under cyclic loading were conducted on concrete specimens using RCA fixed in place of 25% NCA to analyze the effect of incorporating single BF and PANF and mixed BF-PANF on the performance of RAC. Five types of concrete were prepared for this experiment, namely natural aggregate concrete (NAC), RAC, basalt fiber recycled aggregate concrete (BFRAC), polyacrylonitrile fiber recycled aggregate concrete (PANFRAC), and BF-PANF recycled aggregate concrete (BPRAC). The relationship between stress levels and fatigue life was analyzed using Weibull distribution theory based on cyclic loading fatigue tests. Considering different failure probabilities, both single logarithmic fatigue prediction equations and double logarithmic fatigue prediction equations were established for predicting the ultimate fatigue strength of FRAC. This study enhances the theoretical framework for this type of FRAC in the context of cyclic fatigue loading, providing valuable references for engineers and technicians designing FRAC applications in components predominantly subjected to flexural fatigue.

Experimental program

Raw materials

Two cementitious materials were used in this test, cement and fly ash. P.O 42.5 ordinary silicate cement was selected for cement and fly ash was class I fly ash. Fly ash replaces 15% of cement, which can increase the sustainability of RAC38,39. The detailed parameters of the cementitious materials are shown in Table 1.

Table 1 Detailed parameters of cementitious materials (wt%).
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There are two kinds of coarse aggregate are selected, namely NCA and RCA, and their particle size ranges from 5 to 25mm. The NCA uses continuously graded limestone rubble. RCA is a discarded specimen from the China Construction Commercial Concrete Station in Wuhan (original strength of C35), which is processed through crushing, sieving, cleaning, and drying. The processing flow is illustrated in Fig. 1. The particle grading situation and apparent morphology of the coarse aggregate are shown in Fig. 2, and the main physical indexes are shown in Table 2. The fine aggregate is selected from natural river sand with a maximum grain size of not more than 5 mm.

Fig. 1
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Processing flow of RCA.

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Fig. 2
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Grading curve and overview of coarse aggregate.

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Table 2 Physical indexes of coarse aggregate.
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There are two kinds of fibers used in the test, BF and PANF, whose overview is shown in Fig. 3, and physical indexes are shown in Table 3. The water used for concrete mixing is tap water from Wuhan area.

Fig. 3
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Overview of fibers.

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Table 3 Physical indexes of fiber.
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Mixing ratio design

The design strength of the specimen was C40 and the substitution rate of RCA was 25%. The ratio design was carried out according to the Chinese specification “Specification for the Design of Ordinary Concrete Proportion” (GB/T JGJ55-2011)40. There were a total of 12 sets of fit ratios in this test, namely NAC, RAC, BFRAC, PANFRAC and BPRAC. NAC and RAC were used as a reference group to provide a baseline control for the experiment. BFRAC were single doped BF group designed with three different doping levels (0.1%, 0.15% and 0.2%). PANFRAC were single doped PANF group designed with two different doping levels (0.1% and 0.15%). BPRAC was designed for mixing to incorporate BF-PANF group with five doping combinations, which are mentioned in Table 4, where B0.1P0.15 indicates 0.1% mixing of BF and 0.15% mixing of PANF.

Table 4 Mixing ratio design (kg/m3).
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Specimen preparation

The process of preparation of concrete specimens is illustrated in Fig. 4. Various raw materials were weighed according to the specifications outlined in Table 4 and water was added to the mixture in three stages. Initially, cement, fly ash, coarse aggregate, and fine aggregate were weighed and placed into the mixer for 60 s to ensure even distribution of the raw materials. Next, one-third of the mixing water was added, and the mixture was blended for an additional 60 s before introducing the fibers. Subsequently, BF and PANF were added to the mixer in a dispersed manner, along with another one-third of the mixing water, and mixed for 90 s. Finally, the remaining mixing water was added, and the mixture was blended for another 60 s. The collapse of the resulting mixture was measured and subsequently loaded into a 100 mm × 100 mm × 400 mm mold. After molding, the specimens were placed on a vibration table for thorough vibration to minimize internal porosity.

Fig. 4
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Process of concrete specimen preparation.

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After casting, the specimens were kept in a controlled environment (temperature of 20 ± 5C, relative humidity ≥ 95%) for 48h and then demolded. Once demolded, the specimens were assigned identification numbers. Following the guidelines outlined in the Chinese specification “Standard for Test Method of Physical and Mechanical Properties of Concrete” (GB/T 50081-2019)41, the specimens were then transferred to a standard curing room (temperature 20 ± 2 °C, relative humidity ≥ 95%) for 28 days to undergo flexural strength testing. To minimize the influence of the initial preparation and curing period on the test results, the specimens were required to be maintained for an additional 6 months prior to fatigue testing.

Test methods

Both flexural strength test and flexural fatigue test were conducted using specimens with dimensions of 100mm × 100mm × 400mm. Flexural strength test was carried out after the specimens were cured for 28 days. The flexural fatigue test was carried out after the specimen had been cured for 6 months.

Flexural strength test

The four-point flexural strength test (Fig. 5) follows the Chinese specification “Standard for Test Method of Physical and Mechanical Properties of Concrete” (GB/T 50081-2019)41. Comprehensive test conditions, the selected specimens are non-standard specimen. The two supports below the specimen had a span of 300 mm and were 50 mm from the end of the specimen. Distance between the applied load points on the top of the specimen is 100mm and the distance between the supports on each side is 100mm. The prismatic specimens are placed on the MTS microcomputer-controlled electronic pressure testing equipment, the position of the press bracket is adjusted so that it is aligned with the position of the specimen and can be loaded, recording of peak loads. The four-point bending flexural strength is calculated using the following Eq. (1):

$$f_{{\text{f}}} = 0.85\frac{Fl}{{bh^{2} }}$$
(1)

where ff is flexural strength, MPa; F is failure load, kN; l is the span between the support, mm; b is the specimen cross-section width, mm; h is the specimen cross-section height, mm; non-standard specimen size coefficient of 0.85.

Fig. 5
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Flexural strength test.

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Flexural fatigue test

The specimen placement and loading locations for the flexural fatigue test are identical to those used in the four-point flexural strength test. The flexural fatigue test was conducted using fatigue testing machine manufactured by MTS and the test procedure is illustrated in Fig. 6. In this test, force-controlled cyclic loading was employed, as depicted in Fig. 7. This loading method closely simulates the effects of vehicle loading on pavements42, with a loading frequency set at 10 Hz. The failure load in the flexural strength test served as the basis for the loading parameters of the flexural fatigue test, which was executed at different stress levels (0.6, 0.7, and 0.8) for each group of specimens. To better simulate the action of vehicle loads on concrete pavements, the fatigue load cycle ratio (r = Lmin/Lmax) was set to 0.1, and the relationship between the above parameters satisfies Eq. (2). Before to start the test, a pressure of 1 kN was applied to the specimen; this pressure was then released, and the process was repeated until the displacement change was controlled within 1 mm. The fatigue test cannot continue indefinitely without specimen failure. Therefore, an upper limit of 2 × 106 cyclic loadings is established in the control software. If the specimen remains intact after reaching 2 × 106 cycles, its fatigue life is recorded as 2 × 106. If the specimen fails before this point, the testing machine program will automatically stop43 and record the cycle count at that time, designating it as the fatigue life.

$$S = \frac{{L_{\max } }}{F}$$
(2)
Fig. 6
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Test process of flexural fatigue.

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Fig. 7
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Schematic of cyclic loading.

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Results

The flexural strength test results

The variation in specimen flexural strength and fiber reinforcement coefficient for each group of specimens is shown in Fig. 8. The flexural strength of RAC is 8.9% lower than that of NAC. The fiber reinforcement coefficient is derived from the FRAC group compared to the RAC group, indicating that the incorporation of fibers into the RAC can enhance its flexural strength to varying degrees. The flexural strength of PANFRAC was improved by 16.82% and 10.68% compared to RAC when the incorporation of PANF alone was 0.1% and 0.15%, respectively. Conversely, the flexural strength of BFRAC exhibited an increasing and then decreasing trend with the increase of single-doped BF doping. Specifically, the flexural strength of BFRAC was enhanced by 8.18%, 12.27%, and 6.14%, respectively, compared with that of RAC when the BF doping was 0.1%, 0.15%, and 0.2%. The observed results can be attributed to the strong adhesion of PANF and BF to the cement mortar44. A suitable dosage of PANF and BF can work synergistically with the concrete matrix to inhibit the propagation of small cracks within the concrete, thus improving the flexural strength of FRAC. However, as the PANF and BF dosage increased, the fibers may not disperse uniformly within the concrete matrix, resulting in defects within the concrete and reducing the flexural strength of FRAC45.

Fig. 8
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Flexural strength and fiber reinforcement coefficient.

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In this study, a mixture of BF and PANF was blended into RAC, and the flexural strength of BPRAC was significantly enhanced compared to RAC, with the enhancement ranging from 9.77% to 26.59%. The flexural strength of the doping combination of 0.1% BF + 0.15% PANF was 5.57 MPa, the highest value in the BPRAC group and among all test specimens. The optimal BPRAC is improved by 26.59%, 8.37% and 12.75% compared to RAC, optimal PANFRAC and optimal BFRAC respectively. From the fiber reinforcement coefficient, it is evident that both single doped fibers and hybrid fibers have significant enhancement effect on the flexural strength of RAC, and the enhancement effect of suitable doped hybrid fibers is more obvious. PANF is a fiber with a low modulus of elasticity and high flexibility. In the early stages of concrete subjected to external forces, it can effectively absorb energy, dispersing stress at the crack tip to reduce internal stress concentration. BF, on the other hand, is a fiber with a high modulus of elasticity and high tensile strength. Even when concrete reaches its ultimate destructive strength in the later stages of loading, BF can continue to bear some external forces until detachment or rupture. When BPRAC is subjected to flexural damage, an appropriate dosage of PANF and BF can enhance ductility at different stages of loading and at various structural levels, thereby improving the flexural strength.

The flexural fatigue test results

The peak cyclic load parameters for the fatigue test could be obtained on the basis of the flexural test as shown in Table 5. The fatigue life of NAC, RAC, BFRAC, PANFRAC and BPRAC were obtained as shown in Table 6. The average fatigue life of the specimens is illustrated in Fig. 9, while some of the failure specimens are also demonstrated. Concrete is a composite material with non-uniformity, causing concrete specimens to have varying fatigue life at the same stress level46. As indicated in Table 6 that the fatigue life of NAC, RAC, BFRAC, PANFRAC and BPRAC decreases with increasing stress levels. Figure 9 shows that the average fatigue life of RAC is the shortest at the same stress level, whereas the average fatigue life of both single and mixed fiber concrete is longer than that of RAC, and the incorporation of fibers significantly increases the average fatigue life of RAC. The fibers positively interact within the concrete matrix and work together with the matrix to withstand the external cyclic loads, consume energy, and slow down the expansion of small cracks inside the specimens, thus improving the flexural fatigue life of concrete.

Table 5 Peak cyclic load parameters.
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Table 6 Fatigue life.
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Fig. 9
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Flexural fatigue test.

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Discussion

Fatigue life analysis

Among the fatigue life analysis methods, fitting the fatigue life of concrete based on the probability of failure to derive the lognormal and Weibull distributions is commonly employed47. The lognormal probability density function has the horizontal axis as its asymptote, which means that the survival rate reaches 100% only when lgN approaches infinity46, which is not consistent with the fatigue test results. In contrast, the Weibull probability density function demonstrates a minimum safe lifetime. Utilizing the Weibull distribution allows for the determination of both the safe life and the minimum safe life of concrete under flexural fatigue, effectively modeling various states in engineering48. In summary, this study utilized the Weibull distribution to analyze the flexural fatigue life distribution of the specimens.

The distribution pattern of the flexural fatigue life of each group of specimens can be represented by the probability density function of the three-parameter Weibull distribution49, as in Eq. (3):

$$f\left( N \right){ = }\frac{b}{{N_{a} - N_{0} }}\left[ {\frac{{N - N_{0} }}{{N_{a} - N_{0} }}} \right]^{b - 1} \exp \left[ { - \left( {\frac{{N - N_{0} }}{{N_{a} - N_{0} }}} \right)^{b} } \right]\left( {N_{0} \le N < \infty } \right)$$
(3)

where N is the fatigue life, b is the shape parameter, Na is the characteristic life parameter, and N0 is the minimum life parameter.

The distribution function of the Weibull distributed random variable (Np) can be derived from Eq. (3) as in Eq. (4):

$$F\left( {N_{p} } \right) = P\left( {N < N_{p} } \right) = 1 - \exp \left[ { - \left( {\frac{{N_{p} - N_{0} }}{{N_{a} - N_{0} }}} \right)^{b} } \right]$$
(4)

Then the cumulative function of reliability (P) is shown in Eq. (5):

$$P\left( {N > N_{p} } \right) = 1 - P\left( {N < N_{P} } \right) = \exp \left[ { - \left( {\frac{{N_{P} - N_{0} }}{{N_{a} - N_{0} }}} \right)^{b} } \right]$$
(5)

The minimum fatigue life of the concrete specimen is zero, so that N0 = 0, and the probability density function of the three-parameter Weibull distribution simplifies to a two-parameter one, as in Eq. (6):

$$f\left( N \right) = \frac{b}{{N_{a} }}\left[ {\frac{N}{{N_{a} }}} \right]^{b - 1} \exp \left[ { - \left( {\frac{N}{{N_{a} }}} \right)^{b} } \right]\left( {0 \le N < \infty } \right)$$
(6)

Then the reliability can be simplified to the following Eq. (7):

$$P\left( {N > N_{p} } \right) = 1 - P\left( {N < N_{P} } \right) = \exp \left[ { - \left( {\frac{{N_{P} }}{{N_{a} }}} \right)^{b} } \right]$$
(7)

Taking the logarithm twice on both sides of Eq. (7) to obtain the following Eq. (8):

$$\ln \left[ {\ln \left( \frac{1}{P} \right)} \right] = b\ln N_{p} - b\ln N_{a}$$
(8)

\(Y = \ln \left[ {\ln \left( \frac{1}{P} \right)} \right]\), \(X = \ln N_{p}\), \(C = b\ln N_{a}\), Eq. (8) is simplified to the following Eq. (9):

$$Y = bX - C$$
(9)

Equation (9) allows testing the hypothesis that the data obtained from the test obey a two-parameter Weibull distribution. According to the data obtained from the regression analysis, the hypothesis is accepted if X and Y have a favorable linear relationship. The reliability corresponding to the fatigue life of concrete specimens can be expressed by the following Eq. (10):

$$P = 1 - \frac{i}{n + 1}$$
(10)

where n is the total number of samples at a single stress level, and i is the serial number of specimens arranged from smallest to largest fatigue life.

Figure 10 illustrates the linear regression curves of the fatigue life of each group of specimens at different stress levels, and the estimated of the characteristic parameters of the two-parameter Weibull distribution are shown in Table 7. The correlation coefficients (R2) listed in Table 7 show that all the values are greater than 0.85, which indicates that the linear regression curves in Fig. 10 demonstration a strong correlation, and also verifies that the fatigue life of the specimens all obey the two-parameter Weibull distribution.

Fig. 10
figure 10

Linear regression plot with two-parameter Weibull distribution.

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Table 7 Estimation results of the characteristic parameters of the Weibull distribution.
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The shape parameters reflect the discrete type of fatigue life of concrete samples; specifically, larger shape parameters correspond to smaller dispersion. However, research on this topic is limited, and the effects of factors such as stress level, loading frequency, and fiber admixture on the shape parameters of concrete specimens remain unclear. Kaur et al.50 found that the shape parameter decreases as the stress level increases, suggesting a gradual increase in the discrete pattern of fatigue life. Notably, this trend appears independent of factors like RCA substitution rate, fiber doping, and mineral content. According to Table 7, there is no significant relationship between fiber doping or fiber type and the shape parameters of the specimens, which aligns with the findings of Kaur et al. Additionally, the shape parameters do not show a significant correlation with stress level, indicating a need for further investigation into the shape parameters of the specimens.

Fatigue equation

The fatigue life of concrete specimens has a large discreteness. To objectively reflect the fatigue life trends of these specimens, various equations have been proposed to predict their flexural fatigue strength. One commonly used method is the Wholer equation, which predict the flexural fatigue strength of natural concrete based on the relationship between stress level and fatigue life51. Ballinger43 developed the Wholer equation to evaluate and predict the flexural fatigue strength of natural concrete based on the relationship between stress level and fatigue life. Wholer equation is not only suitable for evaluating the fatigue strength of plain concrete, but can also be applied to fiber-reinforced concrete. Additionally, Bawa and Singh49 utilized “S–N curve” to predict the flexural fatigue strength of self-compacting concrete with mixed fibers. The Wholer equation can be expressed as follows Eq. (11):

$$S = P + Q\lg (N)$$
(11)

where P and Q are test coefficients.

Vesic and Saxena52 chose another model for the fatigue equation, such as Eq. (12). This equation satisfies the extreme boundary condition where the fatigue life tends to infinity as the stress level approaches zero50.

$$S = E\left( N \right)^{ - F}$$
(12)

where E and F are test coefficients. Equation (13) is obtained by taking the logarithm of both sides of Eq. (12):

$$\lg \left( S \right) = \lg \left( E \right) - F\lg \left( N \right)$$
(13)

However, the single and double logarithmic fatigue equations in the traditional sense have only 50% survival probability and cannot be applied in practical engineering. Therefore, it is necessary to introduce an additional parameter, the probability of failure, and to establish the single and double logarithmic fatigue equations with different failure probabilities to explore the effect of failure probability on the fatigue performance pattern of concrete.

The fatigue life of the concrete specimens all obeyed a two-parameter Weibull distribution, and the probability of failure can be expressed by the following Eq. (14):

$$N = N_{a} \left| {\ln \left( {1 - P_{f} } \right)} \right|^{\frac{1}{b}}$$
(14)

where Pf is the probability of failure.

The fatigue life of each group of concrete specimens at different failure probabilities was calculated by bringing the relevant parameters in Table 7 into Eq. (14)53, and the results are shown in Table 8.

Table 8 The fatigue life at different failure probabilities.
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The regression coefficients P and Q of the single logarithmic fatigue equations for concrete specimens at different failure probabilities can be fitted to the data in Table 8 by linear regression with Eq. (11). In the same way, the regression coefficients for the double logarithmic fatigue equation can be obtained using Eq. (13). The calculated results are shown in Table 9.

Table 9 The regression coefficients at different failure probabilities.
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Table 9 shows that NAC, RAC, BFRAC, PANFRAC, and BPRAC, for different failure probabilities at each stress level, with the exception of a few combinations (the NAC group with a failure probability of 0.05, the B0.2P0 and B0.1P0.15 groups with a failure probability of 0.05–0.1, and the B0.15P0.15 group with a probability of failure of 0.05), remaining combinations have correlation coefficients above 0.9. The above results can show that the regression fitting of the fatigue life of the concrete specimens of this test using single logarithmic equations and double logarithmic equations under different failure probabilities has a well linear relationship, and the fatigue life prediction results are accurate and reliable.

The coefficients of the single and double logarithmic fatigue equations for NAC, RAC, BFRAC, PANFRAC and BPRAC at different failure probabilities can be obtained using the fitting results in Table 9, which provide a theoretical reference for fatigue life prediction. The probability of failure for each group of specimens was 0.1, and the linear regression curves for the single logarithmic and double logarithmic equations are shown in Figs. 11 and 12. The larger values of parameters P and E in Table 9, the higher linear regression plot, the better fatigue performance of specimens. The parameters Q and F in Table 9 exhibit the slopes of the linear regression plots, indicating the sensitivity of the fatigue life of specimens to the change in stress. It can be observed from Figs. 11a and 12a that the linear regression curves of RAC are lower and flatter than those of NAC, indicating that the substitution of 25% NCA by RCA reduces the fatigue life of the specimens and the sensitivity of the fatigue life to the change in stress. It can be observed from Figs. 11b and 12b that the absolute values of the height and slope of the linear regression curves gradually increase as the fibers are doped, and the curves become steeper and steeper, which implies that the doping of BF and PANF into RAC improves the fatigue life of BFRAC and PANFRAC, but the fatigue life is more sensitive to the change of stress. The P and E values of PANFRAC showed the highest enhancement of 16.72% and 25.48% compared to RAC when the PANF doping was 0.1%. The BPRAC mixed with BF and PANF showed the highest enhancement in P and E values of 16.16% and 24.23% for B0.2P0.1 as compared to RAC. According to the linear regression curve with a probability of failure of 0.1, the fatigue life of B0.2P0.1 is 0.56% lower than that of single-doped PANFRAC with a doping level of 0.1%, but the fatigue life of B0.2P0.1 is 4.18% less sensitive to stress changes.

Fig. 11
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The single logarithmic equation linear regression plot at a failure probability of 0.1

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Fig. 12
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The double logarithmic equation linear regression plot at a failure probability of 0.1

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The ultimate fatigue strength of a specimen can be calculated using the single logarithmic fatigue equation and the double logarithmic fatigue equation. The ultimate fatigue strength is the maximum fatigue stress that the specimen can withstand for a given number of cycles and is usually expressed using a percentage of the static flexural strength47. It is generally accepted that when the number of cycles reaches 2 million times and still not destroyed, it is judged that the specimen has failed or to reach the maximum number of cycles. The ultimate flexural fatigue strengths of specimens are shown in Table 10, assuming that N is 2 million, obtained by substituting N into the single logarithmic fatigue equation and double logarithmic fatigue equation, combined with the coefficients from Table 9.

Table 10 The ultimate fatigue strength of specimens.
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As can be seen from Table 10, the fatigue strength of RAC corresponding to 2 million cycles is slightly lower compared to NAC. BF and PANF doping in RAC can increase the ultimate fatigue strength to varying degrees. The ultimate fatigue strength of PANFRAC was increased by 1.11% and 1.01% over RAC when the PANF doping was 0.1% and 0.15%. The ultimate fatigue strength of BFRAC was increased by 5.47%, 5.64% and 1.17% over RAC when the BF doping was 0.1%, 0.15% and 0.2%. When the content of BF is 0.1% and the content of PANF is 0.1% and 0.15%, the enhancement effect for the ultimate fatigue strength is better than the doping of BF and PANF alone, which is 7.47% and 12.19% higher than RAC. However, mixing 0.15% BF with 0.1% and 0.15% PANF into RAC slightly decreased the ultimate fatigue strength by 3.12% and 4.94%, respectively. B0.1P0.15 has the best ultimate fatigue strength when the failure probability is 0.1. The ultimate fatigue strengths obtained from Table 10 are plotted as bar graphs as shown in Fig. 13, and all the predictions of the double logarithmic fatigue equations are higher than those of the single logarithmic fatigue equations. When the maximum value of the applied stress is less than the ultimate fatigue strength, the specimen is able to reach the upper limit of 2 million cycles without damage.

Fig. 13
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Comparison of ultimate fatigue strength.

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Conclusions

In this study, flexural strength and flexural fatigue properties of RAC are investigated with hybrid BF-PANF. Following are the important outcomes of this research:

The flexural strength of RAC is 8.9% lower than that of NAC, incorporating single BF and PANF and hybrid BF-PANF into RAC can effectively increase its flexural strength. Compared with RAC, the flexural strength increased by 12.27% and 16.82%, respectively, with the optimal dosage of 0.15%BF and 0.1%PANF. The optimal mixing is 0.1%BF + 0.15%PANF, and the flexural strength was increased by 26.59% compared to RAC. The hybrid fibers showed the most significant increase in flexural strength of RAC.

The flexural fatigue life of concrete specimens decreases with the increase of stress level, and RAC has the shortest flexural fatigue life at the same stress level, and the incorporation of fibers can improve the flexural fatigue life of RAC. Mixing the optimal dosage of 0.1% BF + 0.15% PANF increases the flexural fatigue life by 111.86% and 595.48% compared to NAC and RAC at a stress level of 0.7, respectively.

Based on the two-parameter Weibull distribution theory, the flexural fatigue life of each group of concrete specimens was verified at stress levels of 0.6, 0.7 and 0.8. The results show that the R2 of each group of specimens is greater than 0.85, the fitting correlation is favorable, and the flexural fatigue life of the specimens obeys the two-parameter Weibull distribution.

Based on the two-parameter Weibull distribution theory, the fatigue equations under different failure probabilities are derived by single logarithmic method and double logarithmic method for flexural fatigue life fitting, and the correlation coefficients R2 are all greater than 0.85, with high prediction accuracy.

The failure probability was 0.1, and the ultimate fatigue strength of the specimens was predicted under the condition of 2 million cycles, and the prediction result of the double logarithmic method was better than that of the single logarithmic method. The predicted results show that the ultimate fatigue strength of BPRAC mixed with 0.1% BF + 0.15% PANF is the largest.

This study investigated the effect of different admixtures of BF and PANF on RAC. The flexural strength and flexural fatigue properties were optimized when the BF admixture was 0.1% and the PANF admixture was 0.15%. The fatigue life was accurately predicted using the double logarithmic method, providing a better estimation of the fatigue strength. Based on these findings, the authors suggest that modifying the combination of fiber types and increasing the number of specimens at each stress level could result in a more rigorous study.