Experimental analysis of bending performance of secondary steel fibre-reinforced concrete and fuzzy optimisation of damage constitutive model

Four-point bending tests were conducted on SFRC of different varieties, lengths, and dosages to determine their effects on the flexural tensile strength and flexural toughness of secondary mixed concrete. Under the same length and dosage, end-hooked steel fibres had the best reinforcing effect on the flexural tensile strength. The addition of the fibres significantly improved the flexural toughness of the concrete. Except for the shear-type steel fibres, the load–deflection curves of all the specimens were complete and uninterrupted. However, owing to the combined influence of factors such as the type of steel fibre, fibre length, and fibre content, the bending performance exhibited uncertainty. Thus, the uncertain state caused by microdefects in the secondary SFRC material was regarded as the damage variable, and the damage constitutive model of the secondary SFRC under a bending load was obtained by employing the Weibull distribution. An improved fuzzy ant colony optimisation (FACO) algorithm was utilised to conduct fuzzy optimisation of the shape, scale, and position parameters of the damage constitutive model. The numerical example indicated that the constitutive-model value of the bending damage of secondary SFRC optimised by the FACO algorithm was closer to the actual value than the fitting value of the traditional ant colony optimisation (ACO) algorithm and least-squares method. Overall, the error was smaller and the algorithm was more efficient. The results indicate that the proposed model and method are effective for analysing uncertainty problems in large-volume hydraulic structure engineering.

1 Introduction

In large-volume hydraulic concrete engineering, secondary concrete is widely used because of its low heat of hydration. However, large volumes of concrete are susceptible to cracking. Incorporating fibres into the concrete matrix has become an important technical measure for effectively improving the brittle nature of concrete and controlling the crack width. Extensive research has been conducted globally on steel fibre-reinforced concrete (SFRC) with small coarse aggregates, and the corresponding specifications guide engineering practice. Relevant standards have progressively relaxed restrictions on aggregate size: from ‘not greater than 20 mm’ in the Technical Specification for Fiber Reinforced Concrete Structures (CECS 38:2004) to ‘not greater than 25 mm’ in JG/T 472–2015 Steel Fiber Reinforced Concrete.

Early research generally concluded that excessively large aggregates make it difficult for steel fibres to disperse uniformly in concrete, thereby weakening their reinforcing effect (He et al., 2022; Wu et al., 2016; Zhang et al., 2024; Liu and Li, 2025; Han et al., 2019; Asaye et al., 2025; Wang, 2022). Although research on SFRC with small coarse aggregates is abundant, studies on low-strength secondary SFRC with a maximum coarse-aggregate size exceeding 25 mm are scarce. In large-volume hydraulic structures, the bending performance of the secondary SFRC is a key indicator of its structural suitability. Bending performance tests accurately measure critical parameters, such as crack resistance and ductility, providing data support for strength design and crack control in structures such as dams and sluices. Damage constitutive models provide a theoretical basis for the precise analysis of mechanical behaviour. Therefore, research on the bending performance testing and damage constitutive modelling of secondary SFRC has profound significance for practical engineering.

Scholars have conducted substantial amounts of related global research. Huang and Li (2024) conducted experiments in which 35-mm-long steel fibres were incorporated to produce secondary SFRC with static and flexural fatigue performance equivalent to that of grade I SFRC. Elices and Rocco (2008) analysed the effects of aggregate size on the fracture energy, tensile strength, and elastic modulus of different concrete types. They obtained the values for the fracture energy and elastic modulus via notched beam tests. Wang et al. (2024) studied the effects of the coarse-aggregate size on the mechanical properties of concrete. They characterised the microscale properties of the interfacial transition zone using nanoindentation and scanning electron microscopy (SEM) and discussed the influence of the coarse-aggregate size on the compressive strength and elastic modulus. Guo et al. (2024) investigated the effects of different admixture systems on the tensile performance of recycled concrete via uniaxial tension and three-point bending tests. They found that flexural tensile tests overestimated the post-cracking performance of the material—a phenomenon that was more pronounced in recycled concrete. Pirmoradi et al. (2025) defined the particle contact and fibre orientation distributions using different probability density functions. They implemented a hierarchical multiscale constitutive model within a finite-element framework based on an incremental iterative update algorithm. The applicability of the multiscale model was validated through finite-element simulations of the failure behaviour of fibre-reinforced concrete specimens under tension, compression, and bending. Saatci et al. (2024) studied the effects of the steel fibre type and ratio on the unidirectional bending performance of HyFRC thin plates using three different steel fibre types and polyvinyl alcohol (PVA) fibres. Through an inverse analysis of crack width–stress variations in three- and four-point bending specimens, they found that the steel fibre type and ratio were consistently dominant factors across all test types for HyFRC specimens. Cui et al. (2025) prepared ultra-high-toughness concrete (UHTC) with four different nano-SiO₂ dosages. They studied the bending performance under two different forming processes (thin-layer ramming and conventional casting) via four-point bending tests. They conducted computed tomography scanning to analyse the internal fibre orientation distribution in UHTC and established a constitutive model linking the bending performance to the fibre orientation. In summary, most studies on the bending performance of secondary SFRC have been based on the fitting of experimental data. However, their conclusions may not accurately represent the inherent uncertainties in engineering applications. Furthermore, research on bending damage constitutive models is relatively limited, and consistent conclusions regarding model optimisation for engineering uncertainty have not yet been reached (Elkafrawy et al., 2025; He et al., 2004; Shi, 2005).

Therefore, this study investigated the effects of the steel fibre type, length, and volume dosage on the bending performance of secondary SFRC through four-point bending tests. Using the results, a constitutive model for the bending performance of secondary SFRC was established. An intelligent algorithm was then used to perform fuzzy optimisation of the model parameters, providing a valuable reference for engineering applications of secondary SFRC.

2 Test overview

The test used commonly employed C25 concrete as the base material. Steel fibres were incorporated into the concrete via the equal-volume replacement method, substituting coarse aggregates (Zhao, 2016a; Chen, 2016; K and M, 2023). The other mix proportion parameters remained consistent with those of the plain concrete, except for the fibres. The mixing ratios are presented in Table 1. Ordinary Portland cement (P.O. 42.5) was used. The fine aggregate used was natural river sand with a fineness modulus of 2.65. The coarse aggregate consisted of stones with sizes of 5–20 mm and 20–40 mm mixed at a 4:6 mass ratio. Grade I fly ash was used as the supplementary cementitious material. A polycarboxylate high-performance water reducer was used to improve the workability (Tran et al., 2022; Zheng et al., 2022; Dutta et al., 2018). Tap water was obtained from the Jingling University Science and Technology Laboratory.

Table 1

Table 1. Mix ratio of secondary SFRC.

Three types of steel fibres were used: a shear type and end-hooked type from Hebei Zhitai Company and a milling type from Harex Company. The parameters and test conditions are presented in Table 2, and the fibre morphologies are shown in Figure 1. Nine groups of specimens were designed for the four-point bending tests. Prismatic specimens with dimensions of 150 mm × 150 mm × 550 mm were used, with three parallel specimens per group. The four-point bending tests strictly followed the JG/T 472–2015 Steel Fibre-Reinforced Concrete specifications. An electrohydraulic servo universal testing machine was used to apply a load at a rate of 0.1 mm/min. A pre-load of 1 kN was applied before the test to eliminate the influence of support displacement (Li et al., 2018; Thomas and Ramaswamy, 2007; Liu X. et al., 2020). The loading setup is shown in Figure 2.

Table 2

Table 2. Fibre parameters and dosages.

Figure 1

Figure 1. Shapes of three different types of steel fibres. (a) Milling type. (b) End-hooked. (c) Shear type.

Figure 2

Figure 2. Four-point bending loading device.

The specimen preparation process was as follows: steel fibres, coarse and fine aggregates, cement, and other raw materials were placed in a forced-action mixer and mixed for 1 min to achieve uniformity. Subsequently, the water reducer and tap water were added, and mixing continued for another 2 min. The resulting mixture is illustrated in Figure 3. Although slight balling occurring at a fibre dosage of 2%, the steel fibres were uniformly distributed in the concrete at the other dosages, with good cohesiveness and water retention. After the mixture was placed in plastic moulds, compaction was achieved using a vibrating table. The specimens were demoulded 24 h after casting and immediately moved to a standard curing room, where they remained for 28 d at constant temperature and humidity (Conforti et al., 2019; Bao et al., 2024; Xiao et al., 2024; Fares and Bakir, 2024).

Figure 3

Figure 3. Steel fibre concrete mixture.

3 Bending performance test of secondary SFRC

3.1 Test method

Four-point bending tests were conducted using a 2000-kN electrohydraulic servo testing machine. During testing, the moulded side face of the specimen was set as the loading surface. After specimen installation, displacement sensors for measuring the mid-span deflection were installed simultaneously. Concentrated loads were applied at the third point of the span by using a displacement-controlled strategy. Before initial cracking, loading proceeded at a constant deflection increase rate of 0.1 mm/min to ensure a smooth load increment. After initial cracking, the loading rate was adjusted to 0.15 mm/min to adapt to the stress characteristics of the crack-propagation stage. The cracking locations were carefully examined during the tests. If tensile cracks appeared outside the third-point span region, the test result for that specimen was deemed invalid, and the specimen was remade and retested (Liu Z. et al., 2020; Raju et al., 2020; Sudheer and Chandra Mohan Rao, 2025).

3.2 Test results and analysis

The flexural tensile strength was calculated according to the standard formula (Equation 1) and was accurate to 0.01 MPa.

$f_{ftm}=\frac{FL}{b{h}^2}(1)$

Here, $f$_ftm represents the flexural tensile strength of the SFRC, $F$ represents the failure load of the specimen, L represents the specimen span, b represents the specimen cross-sectional width, and h represents the specimen cross-sectional height. The calculation results for each group are presented in Table 3.

Table 3

Table 3. Flexural tensile strength test results.

3.2.1 Failure characteristics of SFRC specimens

The plain concrete beam cracked during the four-point bending tests. As the load increased, the crack developed rapidly, accompanied by loud noises, leading to brittle fracture (Kang et al., 2024). After failure, the crack was narrow and straight and exhibited no propagation, as shown in Figure 4a. The addition of steel fibres significantly improved the flexural failure mode of the concrete. Specimens with lower fibre dosages formed one distinct main crack upon failure, bending and propagating upwards, as shown in Figure 4b. The specimens with higher fibre dosages developed multiple secondary cracks around the main crack after cracking at the bottom. In some cases, multiple cracking points appeared, even at the onset of bottom cracking, as shown in Figure 4c.

Figure 4

Figure 4. Failure morphologies of concrete specimens. (a) Plain concrete without fibres. (b) SFRC with 0.5% fibre volume fraction. (c) SFRC with 1.5% fibre volume fraction.

3.2.2 Load–deflection curves of SFRC

Figure 5 presents a comparison of the load–deflection curves of SFRC and plain concrete with variations in the steel fibre type, length, and dosage. As shown in Figure 5a, plain concrete specimen F0 without fibres has a relatively small ultimate flexural load. Upon reaching the ultimate load, the specimen fractured directly and exhibited brittle-failure characteristics. The shear- and milling-type SFRC did not fracture directly at the ultimate load. However, the load–deflection curves exhibited a sharp drop. At a mid-span deflection of 2 mm, the load was almost zero, indicating significant brittleness. In contrast, the load–deflection curve of end-hooked SFRC specimen D310 was full. After reaching the ultimate load, the curve decreased slowly, indicating better toughness. The comparison indicates that incorporating end-hooked steel fibres significantly improves the toughness of the SFRC, whereas shear- and milling-type fibres result in more brittle behaviour.

Figure 5

Figure 5. Four-point bending load–deflection curves. (a) Load–deflection curves for different fibre types. (b) Load–deflection curves for different fibre lengths. (c) Load–deflection curves for different fibre dosages.

In Figure 5b, the cracking loads of the end-hooked SFRC with different lengths are close to those of plain concrete. However, the ultimate loads were significantly higher than those of plain concrete. The load–deflection curves of the end-hooked SFRC were full, indicating good toughness. Compared to D310 (35 mm), D510 (50 mm) and D610 (60 mm) exhibited more significant strain hardening, suggesting that longer fibres provide better toughness. Figure 5b also shows that the ultimate load of D510 exceeds that of D610; however, the unloading stage exhibited a faster decline than that of D610. This may be because at the same dosage (1.0%), D510 had more fibres than D610. Although this can restrict crack development, the shorter fibre bond length in D510 facilitates fibre pullout. In Figure 5c, end-hooked SFRC with the same length but different dosages exhibits full load–deflection curves and good toughness. At a fibre dosage of 1.5%, the ultimate load was the highest, and a relatively long load plateau appeared, indicating optimal toughness. At a dosage of 2%, the load decreased because of balling. Therefore, the dosage of end-hooked steel fibres should not exceed 1.5%.

Through the SEM experiment, it was found that the shear-type interface transition layer had obvious defects, with local holes appearing in multiple places as shown in Figure 6a. The reason for this is that the length-to-diameter ratio of this type of fiber is relatively small, and the hardness of a single fiber is relatively high, which makes it impossible to be compacted during the pouring process, resulting in local water seepage areas and larger gaps around the fibers. As shown in Figure 6b, due to the fact that this type of fiber is relatively heavy per single fiber, it sinks and gathers during the vibration process. The gathered fibers cannot independently bear the load, causing a sudden release of local stress, which then interrupts the load transmission of the entire specimen, manifesting as a sudden drop in the load-deformation curve. Thus, the use of end-hooked steel fibres for flexural members is recommended.

Figure 6

Figure 6. Microstructure of shear-type specimens. (a) Upper side of the specimen. (b) Lower side of the specimen.

3.2.3 Analysis of flexural tensile strength results

Figure 7 shows a comparison of the flexural tensile strength (f_ftm) of SFRC with that of plain concrete, as influenced by the fibre type, length, and dosage. In Figure 7a, compared with plain concrete, the shear-type SFRC exhibits no strength increase, and its flexural strength is reduced. Incorporating milling-type and end-hooked steel fibres increased the flexural strength by 27% and 100%, respectively. As shown in Figure 7b, for the end-hooked SFRC with the same dosage (1.0%) but different lengths, the flexural strength exceeds that of plain concrete. The strength was the highest for 50-mm fibres. As shown in Table 3, the increase reaches 158%, exceeding the strength of D610 (60 mm, 1.0%), which is consistent with the splitting tensile strength trend. As shown in Figure 6c, for 60-mm end-hooked fibres, the flexural strength is the highest at a 1.5% dosage, with an increase of 194%.

Figure 7

Figure 7. Variation of concrete flexural tensile strength. (a) Flexural strength vs. fibre type. (b) Flexural strength vs. fibre length. (c) Flexural strength vs. fibre dosage.

In summary, when 35-, 50-, and 60-mm steel fibres were incorporated into low-strength secondary concrete, the compressive strength, splitting tensile strength, and flexural tensile strength of the SFRC were the highest for the 50-mm fibres. With an increase in fibre length, the strength initially increased and then decreased. The core reason why the 50-mm steel fibers perform better is that they achieve a better balance in terms of fiber quantity distribution, length-to-diameter ratio compatibility, and pull-out mechanism efficiency (Zhao, 2016b). However, the 60-mm long fibers disrupt this balance due to their own size characteristics, resulting in a decline in the reinforcing effect.

The regression statistical analysis results for the flexural tensile strength test values of the end-hooked SFRC with three fibre lengths are shown in Figure 8 (excluding the D620 data). The influence coefficient α_tm for end-hooked SFRC was 1.67, with a coefficient of variation of 0.168. The fitted α_tm value exceeds the value of α_tm = 1.13 suggested in the specification JG/T 472–2015 Steel Fiber Reinforced Concrete for a maximum coarse-aggregate size of 25 mm. Therefore, incorporating end-hooked steel fibres into low-strength secondary concrete generally increases the flexural tensile strength. The flexural strength exhibited uncertainty, and the experimental values were significantly higher than the specified values.

Figure 8

Figure 8. Linear fitting of the flexural and tensile strength of end-hooked SFRC.

3.2.4 Analysis of flexural toughness results

According to the test results and the specification JG/T 472–2015 Steel Fiber Reinforced Concrete, the equivalent initial flexural tensile strength, initial flexural toughness ratio, equivalent flexural tensile strength, and flexural toughness ratio for each group of SFRC four-point bending specimens were calculated using Equations 2–6. The results are presented in Table 4. The flexural toughness before the peak deflection was characterised by the initial flexural toughness ratio R_e,p, calculated as follows:

${\mathit{R}}_{\mathbf{e},\mathbf{p}}={\mathit{f}}_{\mathbf{e},\mathbf{p}}/{\mathit{f}}_{\mathbf{\text{ftm}}}(2)$

${\mathit{f}}_{\mathbf{e},\mathbf{p}}=\frac{\mathbf{L}{\mathbf{\Omega }}_{\mathbf{p}}}{\mathbf{b}{\mathbf{h}}^{\mathbf{2}}{\mathbf{\delta }}_{\mathbf{p}}}(3)$

where R_e,p represents the initial flexural toughness ratio; f_e,p represents the equivalent initial flexural tensile strength (MPa); δ_p represents the mid-span deflection at peak load (mm); Ω_p represents the area under the load–deflection curve up to a mid-span deflection of δ_p (N·mm); and f_ftm represents the flexural tensile strength of SFRC (MPa).

Table 4

Table 4. Flexural performance test results for the specimens.

The flexural toughness after the peak deflection was characterised by the flexural toughness ratio R_e,p, calculated as follows:

${\mathit{R}}_{\mathbf{e},\mathbf{k}}={\mathit{f}}_{\mathbf{e},\mathbf{k}}/{\mathit{f}}_{\mathbf{\text{ftm}}}(4)$

${\mathit{f}}_{\mathbf{e},\mathbf{k}}=\frac{\mathbf{L}{\mathbf{\Omega }}_{\mathbf{p},\mathbf{k}}}{\mathbf{b}{\mathbf{h}}^{\mathbf{2}}{\mathbf{\delta }}_{\mathbf{p},\mathbf{k}}}(5)$

${\mathit{\delta }}_{\mathbf{p},\mathbf{k}}={\mathbf{\delta }}_{\mathbf{k}}-{\mathbf{\delta }}_{\mathbf{p}}(6)$

where R_e,p represents the flexural toughness ratio at mid-span deflection; f_e,k represents the equivalent flexural tensile strength at mid-span deflection δ_k (MPa); Ω_p,k represents the area under the load–deflection curve from δ_p to δ_k (N·mm); δ_p,k represents the increase in mid-span deflection from δ_p to δ_k (mm); δ_k represents the given calculation mid-span deflection L/k (mm); and the k values are 300, 250, 200, and 150, respectively.

As indicated by Table 4, the incorporation of end-hooked steel fibres into the secondary concrete significantly improved its flexural toughness. Regarding the initial flexural toughness ratio, comparing J310, X310, and D310 revealed that incorporating end-hooked fibres (D310) provided the most significant improvement. The initial flexural toughness ratios of D310 (35 mm, 1.0%), D510 (50 mm, 1.0%), and D610 (60 mm, 1.0%) were close, indicating that the toughness before the peak load in end-hooked SFRC is not significantly affected by fibre length. Among D605 (60 mm, 0.5%), D610 (60 mm, 1.0%), D615 (60 mm, 1.5%), and D620 (60 mm, 2.0%), D615 had the highest initial flexural toughness indices, and D605 had the lowest. Therefore, a fibre dosage higher than 0.5%–2.0% is recommended. From the comparison of D510 (50 mm, 1.0%) and D615 (60 mm, 1.5%), a dosage of 1.0% is recommended for low-strength secondary SFRC from an economic perspective.

In Table 4, comparing J310, X310, and D310 reveals that end-hooked SFRC (D310) had a higher equivalent flexural tensile strength and flexural toughness ratio than the other two types, indicating better toughness after the peak load. Comparing D310, D510, and D610 revealed that the equivalent flexural tensile strength increased with the fibre length, whereas the bending toughness remained relatively unchanged indicating good post-peak flexural toughness for all end-hooked SFRC lengths. Comparing D605, D610, D615, and D620 for 60-mm end-hooked fibres revealed that the equivalent flexural tensile strength increased with fibre dosage up to 1.5%, and the bending-toughness values were similar within this range. The reason why the end-hooked fibers have better toughness than the shear-type and milling-type fibers lies in the fact that their hooked geometric structure significantly enhances the mechanical anchoring effect and optimizes the bonding slip behavior. This results in a more stable cooperative effect between the fibers and the matrix.

3.3 Uncertainty analysis

The flexural tensile strengths of three specimens per group are presented in Table 5 and Figure 9, revealing significant engineering uncertainty in the flexural strength. In Figure 9a, plain concrete and shear-type SFRC exhibit a larger uncertainty in the measured flexural strength. The incorporation of end-hooked and milling-type fibres resulted in a smaller uncertainty. Figure 9b shows that incorporating fibres generally reduces the uncertainty compared to plain concrete. The reduction was most significant for the 50-mm fibres, suggesting a matching relationship between the flexural strength and fibre length. In Figure 9c, the uncertainty varies considerably for the same fibre length but different dosages. This was most significant at the 2% dosage, indicating that excessive fibre content may lead to balling.

Table 5

Table 5. Flexural tensile strength test results for different specimens.

Figure 9

Figure 9. Flexural strength of three specimens per group for fibre concrete. (a) Variation for different fibre types. (b) Variation for different fibre lengths. (c) Variation for different fibre dosages.

4 Fuzzy optimisation of bending damage constitutive model for secondary SFRC

4.1 Bending damage constitutive model

Incorporating damage information into the constitutive model to reflect the stress–strain relationship from initial loading to failure is a widely accepted approach in academia. Previous studies indicated significant differences in the bending characteristics of secondary SFRC and ordinary concrete. To accurately characterise the bending performance of secondary SFRC in engineering applications, a damage constitutive model for bending was derived from the four-point bending test results using the Weibull statistical distribution. Considering the uncertain distribution of bending performance in engineering, an improved intelligent algorithm was used to optimise the parameters of this constitutive model (Colorni et al., 1996; Wang and Zhang, 2005).

Fundamentally, the discrete distribution of microdefects within a material causes an overall uncertainty. However, in damage mechanics, for simplicity, all microdefects are often treated as continuously distributed. The influence of these microdefects on the mechanical properties of the material is represented by one or more continuous internal field variables, which are known as damage variables and denoted as D. The damage variable D reflects the characteristics of the microdefects within the material, and it quantifies the degree of internal damage and degradation. To simplify the model based on Hooke’s law and the principle of strain equivalence, the concrete material is treated as an elastic body with a damage factor. The strain caused by stress σ acting on the damaged material is equivalent to the strain caused by the effective stress acting on the undamaged material. Thus, for a one-dimensional problem, the damage constitutive relationship for the secondary SFRC can be expressed as

$\mathbf{\sigma }=\mathbf{E}\mathbf{\epsilon }\left(\mathbf{1}-\mathbf{D}\right)(7)$

where σ represents the stress, E represents the elastic modulus, ε represents the strain, and D is the damage variable.

Considering the characteristics of the stress–strain curve for SFRC and the fact that concrete material strength follows a Weibull statistical distribution, it can be assumed that the material damage parameter D also follows this distribution. Therefore, the Weibull distribution function with parameters D and ε can be expressed as

$\mathbf{D}=\mathbf{1}-\mathbf{exp}\left[-{\left(\frac{\mathbf{\epsilon }-\mathbf{\gamma }}{\mathbf{\beta }}\right)}^{\mathbf{\alpha }}\right](8)$

where α is the shape parameter, β is the scale parameter, and γ is the location parameter.

According to continuum damage mechanics, the stress–strain relationship for secondary SFRC under uniaxial stress is given by Equation 9. Substituting Equation 8 into Equation 7 yields the bending damage constitutive model for secondary SFRC.

$\mathbf{\sigma }=\mathbf{1}-\mathbf{E}\mathbf{\epsilon }\mathbf{exp}\left[-{\left(\frac{\mathbf{\epsilon }-\mathbf{\gamma }}{\mathbf{\beta }}\right)}^{\mathbf{\alpha }}\right](9)$

Among them, α reflects the rate and concentration of damage evolution, and is closely related to the intensity of microcracks within the SFRC. When α is large, damage occurs rapidly and concentratedly within a relatively narrow strain range. When α is small, damage evolves slowly with the increase of strain. β reflects the ability of the SFRC to resist damage propagation. The larger β is, the wider the strain range that the material can withstand and the larger the ultimate strain will be. γ represents the minimum strain at which microdamage begins to occur. It is usually taken as a small positive value, which also reflects the existence of initial microdamage in the SFRC (Chen and Tsai, 2009; Zhang and Wu, 2010).

4.2 ACO algorithm and fuzzy improvement

ACO algorithm, which was proposed by Dorigo and colleagues in Italy in the 1990s, is an intelligent algorithm simulating the foraging behaviour of real ant colonies in nature. It is particularly suitable for stochastic search solutions to nonlinear problems.

The algorithm works as follows: Under the constraints of the objective function, each ant starts from its current ‘city’ (initial state) and moves to the next ‘city’ (a feasible solution or part of a solution) according to specific rules. During the search, each ant uses the pheromone trails left by the other ants to determine the optimal path. These trials contain heuristic information that guides the ants towards potential global solutions. This process iterates such that each ant greedily searches for feasible solutions and identifies the solution as the current best based on objective constraints. Different ants in the colony may hold different ‘best’ solutions simultaneously. Using global information feedback, the search evolves towards a global optimum.

However, ACO algorithm has drawbacks for solving large-scale problems: long convergence time, difficulty maintaining population diversity, susceptibility to local optima, and limited capability for handling fuzzy problems.

The ACO algorithm was improved as follows (Li and Zhang, 2018; Bayat, 2025):

1. During the early stages of the search, pheromone accumulation was low. To avoid premature convergence to the local optima, the difference in pheromone levels should not be overly emphasised. As pheromone trails begin to form and the iteration count increases, the difference in the pheromone levels should be gradually increased to help escape the local optima and obtain better solutions.

2. Previously, pheromone updates were based solely on the paths of the current best ants. The improved method uses a comprehensive fuzzy calculation involving the current best solution for each ant and a tour counter to determine the pheromone update amount for each ant per iteration.

The procedure of the improved fuzzy ant colony optimisation (FACO) algorithm is shown in Figure 10.

1. The iteration count is set to $nc$ = 0. Initialise the pheromone function ${\tau }_{ij}$ and pheromone increment $\Delta {{\tau }_{ij}}^k$.

2. Initialise the starting point set. Each ant moves from city $i$ to city $j$ with a probability ${P_{ij}}^k\left(t\right)$ (Equation 10) and adds city $j$ to the visited set. The next city is selected from the unvisited cities. The tour probability is given as

${{\mathrm{P}}_{\mathrm{i}\mathrm{j}}}^{\mathrm{k}}\left(\mathrm{t}\right)=\left\{\begin{array}{l}\frac{{\left[{\tau }_{\mathrm{i}\mathrm{j}}\left(\mathrm{t}\right)\right]}^{\theta }{\left[{\eta }_{\mathrm{i}\mathrm{j}}\left(\mathrm{t}\right)\right]}^{\eta }}{{\displaystyle {\sum }_{\mathrm{s}\in {\mathrm{J}}_{\mathrm{k}}\left(\mathrm{i}\right)}}{\left[{\tau }_{\mathrm{i}\mathrm{j}}\left(\mathrm{t}\right)\right]}^{\theta }{\left[{\eta }_{\mathrm{i}\mathrm{j}}\left(\mathrm{t}\right)\right]}^{\eta }},\mathrm{j}\in {\mathrm{J}}_{\mathrm{k}}\left(\mathrm{i}\right)\\ 0,j\notin J_k\left(i\right)\end{array}\right.(10)$

where ${\tau }_{\mathrm{i}\mathrm{j}}\left(t\right)$ for the pheromone after t iterations, ${\eta }_{\mathrm{i}\mathrm{j}}\left(t\right)$ is the corresponding heuristic factor for t, and $J_{\mathrm{k}}\left(i\right)$ is the set of cities that the ant can still visit. And, θ and η represent the intensity control factor of pheromones and the heuristic factor.

1. Calculate the objective-function value $Y_k\left(k=1,···,m\right)$ for each ant. The best current solution is recorded for each iteration.

2. The pheromone is updated based on a fuzzy calculation involving each ant’s current best solution and tour counter values. The updated pheromone quantity is given by Equation 11:

${\tau }_{ij}\left(t+n\right)=\rho ·{\tau }_{ij}\left(n\right)+{\stackrel{\sim }{c}}^k{\displaystyle \sum _k}\Delta {{\tau }_{ij}}^k(11)$

where $\rho \left(0<\rho <1\right)$ denotes the pheromone evaporation rate. $\stackrel{\sim }{c}$ is the fuzzy coefficient for the best solution pheromone and is calculated using Equation 12:

$\stackrel{\sim }{c}=\frac{\left|\tau \left(Q_{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}}\right)-\tau \left(Q_{\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{t}}\right)\right|}{\left|\tau \left(Q_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}\right)-\tau \left(Q_{\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{t}}\right)\right|},\stackrel{\sim }{c}\in \left[0,1\right](12)$

where $\tau \left(Q_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}\right),\tau \left(Q_{\mathrm{w}\mathrm{o}\mathrm{r}\mathrm{s}\mathrm{t}}\right),\tau \left(Q_{\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}}\right)$ represent the best, worst, and current objective-function values encountered by the ant during its tour, respectively.

1. After one iteration, reset the pheromone increment of each edge $\left(i,j\right)$ to 0, $nc\leftarrow nc+1$.

2. If $\mathrm{n}\mathrm{c}<\mathrm{n}{\mathrm{c}}_{\max }$ or the best solutions found by all ants are not identical, proceed to Step (2). Otherwise, stop iterating, and the current best solution is the global optimum.

Figure 10

Figure 10. Flowchart of the FACO algorithm.

4.3 Fuzzy optimisation of bending damage constitutive model

The improved FACO algorithm was used to invert the parameters of the bending damage constitutive model of SFRC in this study (Pa et al., 2025; Singh et al., 2025; Wen et al., 2025). A RedHat system was configured on a UNIX host, and the number of ants was set as $m=100$, with $\theta =2$, $\eta =5$, and $\rho =0.75$. Each ant was initialised with a random set of damage parameters, and the initial pheromone amount ${\tau }_{ij}$ and increment $\Delta {{\tau }_{ij}}^k$ were updated using Equation 11 to modify the parameters. The fuzzification coefficient for the current optimal solution pheromone during the tour process is denoted as $\stackrel{\sim }{c}$.

The objective function for the parameter fuzzy inversion of the bending damage constitutive model is defined as the root mean square error between the test load-deflection curve and the optimized value obtained by the FACO algorithm. Its mathematical expression is shown in Formula 13.

$\mathrm{f}=\sqrt{\frac{1}{n}{\displaystyle \sum _1^n}{\left(P_{Test,i}-P_{FACO,i}\right)}^2}(13)$

Among them, n represents the number of data points on the load-deflection curve; P_Test,i is the ith test load value; P_FACO,i is the load value at the ith point predicted by the bending damage constitutive model using parameters α, β, and γ.

After 500 iterations, according to the algorithm, the global optimal solution was the inversion result of each damage parameter, as shown in Table 6.

Table 6

Table 6. Fuzzy inversion values of parameters for the bending damage constitutive model.

4.4 Numerical verification

The core function of a damage constitutive model is to describe the intrinsic relationship between the stress and strain during loading. Key parameters (elastic modulus, peak stress, peak strain, ultimate strain, post-peak stiffness, etc.) are defined and extracted from the stress–strain curve; however, they cannot be directly obtained from the load–deflection curve. Therefore, to effectively validate the model, the load–deflection curves of the secondary SFRC four-point bending specimens must be converted into stress–strain curves (Huang et al., 2025; Ling et al., 2025).

According to strength of materials the bending moment, deflection, and maximum normal stress at the third point of the beam span are respectively expressed by the following Equations 14–16:

$M=\frac{1}{2}Pl(14)$

$f=\frac{5P{l}^3}{12El}(15)$

$\sigma =\frac{M{y}_{\mathit{max}}}{I}=\frac{M\mathrm{h}}{I}=\frac{Pl\mathrm{h}}{4I}(16)$

where M represents the bending moment at the third point of the span; f represents the deflection at the third point of the span; σ represents the maximum normal stress on the cross-section at the third point of the span; P represents the load applied to the beam; l represents the length to the loading point (span/3); E represents the elastic modulus of the beam; I represents the moment of inertia of the beam cross-section; and h represents the height of the beam cross-section.

Assume that the measuring distance of displacement is L (the effective deformation segment length from the mid-span to the support in the four-point bending test), and the total displacement measured is Δ. The strain be expressed as the average deformation rate within the measuring distance. The influence of the local deformation concentration caused by cracks on the overall strain needs to be considered. Through the relevant engineering analysis, the measured values of displacement gauges can be used to simplify and characterize the overall strain of steel fiber concrete during the continuous strain stage before cracking and the non-continuous strain stage after cracking. As shown in Equation 17.

$\epsilon =\frac{\Delta }{L}(17)$

Here, Δ represents the total displacement within the measuring section L.

Taking specimens D310, D510, D610, D615, J310 and X310 under similar conditions as examples, the experimentally obtained bending load–deflection curves were converted into stress–strain curves using Equations 16, 17. These were then compared with the optimised secondary SFRC bending damage constitutive-model FACO values, as well as the values obtained by the ACO algorithm and the least squares fitting values as shown in Figure 11.

Figure 11

Figure 11. Comparison of FACO damage model values, least-squares fitting values, traditional ant colony values and experimental values. (a) D310. (b) D510. (c) D610. (d) D615. (e) J310. (f) X310.

In Figure 11, the constitutive-model values for the secondary SFRC bending damage optimised by the FACO algorithm are closer to the experimental values than the ACO algorithm values and least-squares fitting values. This indicates that the improved model better characterises the actual bending performance of secondary SFRC in engineering. As shown in Table 7, after 500 iterations, the global optimal rate of the FACO algorithm reached 93%. This indicates that compared with the other two algorithms, it can effectively avoid local optima. The improved FACO algorithm has a convergence efficiency that is 20% higher than that of the traditional ACO algorithm and 35% higher than that of the least squares method. This clearly indicates that it converges more rapidly. Thus, the proposed model and method provide a valuable reference for large-volume hydraulic structural engineering projects.

Table 7

Table 7. Comparison of algorithm efficiency.

5 Conclusion

F1. lexural tensile strength analysis revealed that plain concrete exhibited brittle failure with indistinct cracks. The addition of fibres transformed the failure mode to ductile, with prominent mid-span cracking. Compared with ordinary concrete, when the same length and dosage of steel fibers are used, the incorporation of milling-type and hook-end respectively increases the bending strength by 27% and 100%. This shows that the hook-end steel fibers provide the best enhancement effect in terms of bending tensile strength. For a 1.0% usage of hook-end fibers, 50-mm fiber can increase the flexural strength by 158%, which can achieve the maximum improvement effect. For 60-mm hook-end fibers, a usage of 1.5% is the most ideal, and its flexural strength will increase by 194%.

2. Based on the analysis of bending toughness, it can be seen that the bending toughness of concrete containing end-hook steel fiber has been significantly improved. Except for the D605 specimen, the initial bending toughness ratios of the other end-hook steel fiber concrete specimens before the peak stress are close. This indicates that when the fiber content is greater than 0.5%, the initial bending toughness depends on the concrete itself and is not significantly related to the length and content of the end-hook steel fibers. After the peak stress, the initial bending toughness index of D615 was the highest, while that of D605 was the lowest. Therefore, it is recommended to use a fiber content of 0.5%–2.0%. The bending toughness of D615 is similar to that of D510. Using 50-mm long, 1.0% content end-hooked steel fibers is relatively cost-effective.

3. The improved FACO algorithm reduces the local optima by introducing a fuzzy coefficient for the best solution pheromone. This allows a comprehensive fuzzy calculation to determine the pheromone update amount per ant per iteration. Compared to the ACO algorithm and least square method, the improved algorithm has a lower mutation probability and is more suitable for optimising uncertain parameters in multidimensional complex problems. In addition, the improved FACO algorithm has a convergence efficiency that is 20% higher than that of the ACO algorithm and 35% higher than that of the least squares method. This clearly indicates that it converges more rapidly.

4. The uncertain state caused by microdefects in the concrete material was treated as a damage variable. A bending damage constitutive model for secondary SFRC was established using the Weibull distribution. The shape, scale, and location parameters of the damage constitutive model were optimised using the improved FACO algorithm. Numerical verification indicated that the optimised constitutive-model values were closer to the experimental values than traditional ACO algorithm values and the least-squares fitting values, and the overall error was reduced. This demonstrates that the optimised model better characterises the bending performance of the secondary SFRC in large-volume hydraulic structure engineering.

Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

Author contributions

WX: Conceptualization, Writing – original draft. QX: Software, Writing – review and editing. CY: Funding acquisition, Writing – review and editing. SL: Methodology, Writing – review and editing.

Funding

The author(s) declared that financial support was received for this work and/or its publication. This work was supported by Jiangsu Province Industry-University-Research Cooperation Project of China (grant number BY2021059, CY).

Acknowledgements

We express our sincere thanks to Chief Engineers Jiang Lin and to Lin Jian for their enthusiastic support in providing related information.

Conflict of interest

The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

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