International standards employ a variety of tests to evaluate the mechanical properties of FRC composites. The material is typically described using small beams that are subjected to flexural stress. To establish important parameters, following JCI-SF4 and ASTM C1609/C1609M a bending test on unnotched prismatic specimens, and following EN14651-20 a three-point bending test on notched specimens, was performed. Following ASTM C1609 and EN14651-2005, FRC beams were cast both with and without notches by Tarabin et al., 2023a. The cast beam was 600 mm in length, with 450 mm in clear span L, and L/3 as the width and depth of the beam which is 150 mm. The notch’s dimensions—5 mm for the opening and 25 mm for the depth—also met EN14651-2005 requirements. A three-point bending test on the casting beam was applied at the center of the beam span with two roller supports at a distance of 25 mm from the center of the support to the nearest edge of the beam. Concrete examples of FRC were created using a mix with varying steel fibers of 0.25%, 0.5%, and 0.75% percentages by volume. The study employed hook-end steel fibers with dimensions of, l f = 60 mm for length d f = 0.92 mm for diameter, and l f d f = = 65 for aspect ratio. Steel fibers were chosen because of their extensive application, remarkably high strength, excellent crack resistance, high fracture toughness, and reasonable cost. The average tensile strength of hooked steel fibers ranges between 1,000 MPa and 1,200 MPa (Choi et al., 2019). Table 1 provides a summary of the mechanical characteristics of the research, as well as residual flexural strength and flexural strength measured according to standard EN14651. However, in this study, only notch beam data was considered for NLFEA purposes to evaluate the fracture parameters of the SFRC beams.

A numerical model of the tested notch beams was developed in DIANA fea (“DIANA Finite Element Analysis Release Notes Release 10.4.” A two-dimensional-length FEM model of the tested FRC beams with different steel fiber ratios was developed. FRC beam along with loading steel plates and supports were modelled with a regular plane stress element type. To model the beam and plates, we used 2D plane stress 8-noded quadrilateral iso-parametric elements called Q8MEM. These elements have 18 degrees of freedom. Additionally, interface line elements L8IF were used between the FRC beams and the steel plates. To perform the NLFEA, the load was applied in the form of displacement at a rate of 1 mm. A Newton-Raphson iteration method was employed with a convergence tolerance of 0.001 to perform the analysis. As in 3-point bending test, a notch beam is supported by two roller supports and load at the top similar boundary conditions were given to the FEM model with roller support at one end and pin support at the other end.

A coarse mesh of size 15 mm was selected for the complete model however a fine mesh of size 4 mm was selected near the notch of the beam to get more accuracy in the result. A linear relationship between the Figures 1A, B below shows the geometry and mesh of the FRC beam model.

The DIANA material model often employs a smeared crack technique. For the cracking behavior of reinforced concrete and SFRC, DIANA offers a total strain crack model. For SFRC modeling, an official tensile curve called FRCCON has been provided for a few years. An input stress-CMOD or stress-strain relation may describe the concrete’s softening behavior when mixed with steel fibers. The Model Code 2010 and/or experiments on small bending prisms may be used to determine the parameters of this tensile curve. These experimentally-derived parameters are detailed in Table 1 for this research. The constitutive model of the FRC beam is discussed below in detail.

A smeared crack approach is used FEM modeling for the cracking behaviour in concrete. In the smeared cracking approach, a rotating crack model is adopted for the cracking propagation along the FRC. In the rotating crack model, the crack direction keeps on rotating under the load throughout the material. In the rotating crack approach as the tensile stress increases the ultimate tensile strength of concrete, a crack initiates and propagates in that direction.

For FRC’s compressive behaviour, a parabolic model as shown below is considered in this study. The parabolic model is based on the fracture energy of concrete as the er following expressions provided in Eqs 1–4, and Figure 2.

Where f c is the compressive strength of concrete G _c is the compressive fracture energy and h is element size. α c is the strain at maximum compressive strength and α u is the ultimate strain where concrete softening occurs in compression.

The above compression curve in DIANA can be explained as f = − f c 1 3 α j α c / 3  if  α c / 3 < α j ≤ 0 − f c 1 3 1 + 4 α j − α c / 3 α c − α c / 3 − 2 α j − α c / 3 α c − α c / 3 2  if  α c < α j ≤ α c / 3 − f c 1 − α j − α c α u − α c 2  if  α u < α j ≤ α c 0  if  α j ≤ α u (1)

Where α c 3 = − 1 3 f c E (2) α c = − 5 3 f c E = 5 α c 3 (3) α u = α c − 3 2 G c h f c (4) α c and α c 3   is not dependent on the element size or compressive fracture energy, and finally, the ultimate strain α u is the property at which the material softens in compression.

The residual flexural tensile strength of an SFRC material is used to characterize its tensile behaviour. This strength is obtained from either the load against the crack mouth opening displacement curve or the load versus deflection curve that is obtained from the bending test. The formulae to calculate residual flexural strength by using Eq. 5, f R , j = 3 F j l 2 b h s p 2 (5)

Where f R , j is the residual flexural strength, and the load at 0.5 mm, 1.5 mm, 2.5 mm, and 3.5 mm crack mouth opening displacement (CMOD) are termed as F j value in a typical load-CMOD curve. l , b , are the length and width of the FRC beam h s p . is the distance from the top of notch to the top of concrete beam. Depending on the class of SFRC constitutive model of SFRC materials can be described as shown below in Figure 3.

The corresponding values of the tensile stress-strain constitutive model with different steel fiber ratios were obtained from the experimental results as Tabled in Table 1. The difference of the residual flexural strength during the experiment was incorporated in DIANA fea to simulate the behavior of steel fibers in concrete. The linear mechanical properties of FRC for the non-linear FEM model are summarized in Table 2.

The supports and the loading steel plate were considered as linear material during the testing. The linear material properties are given below in Table 3.

For the validation of the above constitutive model, a compressive cylinder test model was developed in DIANA fea to validate the experimental results with the developed FEM model. For, this purpose a cylinder with material properties of the SF0.5 model was developed as shown in Figure 4. SF0.5 was modeled because in the experiment a little change in compressive strength and modulus of elasticity had been reported by Tarabin et al., 2023a among three samples of FCR. The load was applied in displacement form at a rate of 1 mm. Also, the mesh size was kept 15 mm similar to the mesh size in the original beam model. The load-displacement curve for the compressive strength cylinder of SF0.5 is shown in Figure 4B. The calculated compressive strength comes out to be 28.50 MPa which is almost equal to the experimental result of 28.79 MPa.

A mesh analysis was conducted with three different mesh sizes, i.e., 2 mm, 4 mm, and 10 mm, element size. The same mesh sizes were considered around the notch area and the mesh analysis result is shown in Figure 5. These mesh sizes were selected for the edges of the tip of the notch as that is the most sensitive area due to crack propagation and load applied. It can be seen that the selected mesh size (4 mm) has reached a displacement of 4.2 mm which is more than the CMOD value of 3.5 mm which is necessary to obtain the fracture energy of SFRC. However, in the coarse and fine mesh, the divergence in analysis occurred much earlier than the load CMOD value at 3.5 mm.

The NLFEA was conducted to evaluate the effect of steel fibers on the fracture parameters of FRC. The parameters calculated are Fracture energy and Fracture toughness. The fracture parameters are calculated by the Eqs 6–8.

1- Fracture Energy Calculation.

Fracture energy is calculated by a method described by the Japanese Concrete Institute (JCI) (“Standard: Method of test for load-displacement curve.—Google Scholar.” Accessed: Oct. 27, 2023. The main difference between RILEM and JCI methods is the control of displacement rates. RILEM considers the load point displacement (LPD) via linear variable differential transducers (LVDTs) while JCI uses crack mouth opening displacement (CMOD) via measuring the crack opening at the tip of the notch using a gauge. Researchers have shown that both LPD and CMOD are related linearly for a certain range of values. That’s why JCI methods consider a factor of 0.75 for this conversion. G F = 0.75 W o + W 1 A l i g (6)

Where W 1 is work done by weight of the sample and testing equipment, W o is the work done upto the rupture of the specimen ( N ∙ mm ) in other words it is the area under the load-CMOD curve till rupture of the sample. A l i g (mm²) is the b * h of sample where h is the depth of the beam excluding the notch depth. G F is fracture Energy (N/mm).

2- Fracture Toughness.

Fracture toughness is a measure of a material’s ability to resist an unstable fracture or crack. There are different standards and methods to predict the fracture toughness of concrete materials. However, in this study, the fracture toughness is calculated by method reported by Tada et al., 1973. K I C = 3 P L 2 t b − a 2 . π . a . F a b (7) F a b = 1 π . 1.99 − a b 1 − a b 2.15 − 3.93 a b + 2.7 a b 2 1 + 2 a b 1 − a b 3 2 (8)

Where K I C (MPa ^. m m ) is fracture toughness of FRC, P is the maximum load (N), L is the span of the beam (mm) and b is the width of the beam and a is the depth of notch (mm) and t is the thickness of the beam (mm).

The Figure 6 shows the NLFEA load CMOD curve for FRC beams with varying percentages of the steel fibers.

It is evident from the picture that steel fibers have a significant effect on the load-carrying capacity of SFRC. SFRC with a steel ratio of 0.25% is 7.94 kN while for 0.5% and 0.75% steel fiber the peak load is 18 kN and 24 kN respectively. However, steel fibers have an adverse effect on the post peak curve of the load-CMOD. After the peak load for 0.25% and 0.5% steel fibers there is less loss in the ductility of the FRC material as compared to 0.75%. This may be attributed to the lack of proper distribution of steel fibers in a 0.75% ratio. Introducing steel fibers enhances the load-carrying capability of the FRC beams. By growing the proportion of steel fibers, the load-bearing capability of FRC beams experiences a substantial enhancement. The use of steel fibers with a concentration of 0.75% resulted in achieving an optimum load-carrying capability of 24 kN. However, once the ultimate load capability is reached, the rate of decline of the 0.75% steel fiber is steeper compared to the lesser proportion of steel fibers. According to (Wang et al., 2022), the inclusion of steel fibers in rubberized self-compacting concrete resulted in a significant increase in crack propagation after cracking.

The Figure 7 shows the tensile stress curve against the CMOD. As discussed in the load-deflection curve the steel fibers have a great impact on the tensile behaviour of the FRC material. The maximum tensile strength was obtained in the 0.75% fiber with 6.6MPa, while for 0.5% and 0.25% steel fiber ratio, the tensile stresses were recorded as 4.9 MPa and 1.6 MPa respectively. Similarly in the post-peak behavior of the FRC beams a brittle attitude has been observed in the higher percentages of the steel fibers. The introduction of steel fibre in FRC resulted in a better post-peak cumulative load for stress-CMOD behaviours. The FRC mixture with 0.75% steel fiber content exhibited the most favorable post-cracking characteristics among the other mixtures. This study showcased the beneficial impact of using steel fibre in enhancing the spread of cracks and the dissipation of energy in FRC. Due to their comparatively greater elastic modulus, the dispersed steel fibers may effectively prevent the occurrence of the first fracture when subjected to tensile stress in a mixture, hence enhancing their strength (Wang et al., 2020).

The effect of steel fibers on the crack width and crack propagation is shown in Figure 8. The addition of steel fibers in the FRC has positive impact. It can be seen that the crack width at the last step of NLFEA is less with 0.75% steel fiber as compared to 0.5% and 0.25%. This is due to the high tensile strength and ductility of steel fibers as compared to surrounding matrix in FRC. The use of steel fibres improves the FRC beams’ performance to support loads. Increasing the percentage of steel fibres significantly improves the overall ability of FRC beams to support loads. The best load-carrying capacity was attained by using steel fibres at a content of 0.75%. The rate of fall of the 0.75% steel fibre, in contrast to the smaller fraction of steel fibres, is greater after the ultimate load capacity is attained. Steel fibres were added to rubberized self-compacting concrete, and found that this significantly increased the crack’s ability to propagate once it cracked (Wang et al., 2022).

The fracture parameters of the FRC beams with varying percentages of the steel fibers are summarized in Table 4.

From Table 4 it can be observed that fracture energy of the FRC beams increases with the increase in steel fiber percentage. However, there is not much difference in fracture energy between 0.5% and 0.75%. Similar behaviour can be seen in the case of fracture toughness co-efficient. The fracture values reported for beams made of FRC with varying steel fiber concentrations are consolidated in Table 4. The K _IC values were measured as 9.28 MPa mm^1/2, 21.17 MPa mm^1/2, and 28.29 MPa mm^1/2 after 28 days of curing for FRC samples containing 0.25%, 0.5%, and 0.75% steel fibers, respectively. Furthermore, the inclusion of steel fiber resulted in an improvement in both the critical crack mouth opening displacement (CMODc) and the fracture energy ( G F ). The study revealed that the initiation of fracture localization prior to reaching the ultimate load was also postponed in FRC specimens in comparison to specimens containing 0.25% steel fibers. The use of steel fibres in FRC has been shown to boost the cracking resistance by improving the cracking process zone and reducing the brittleness of the composite. In addition, the steel fibre also offers uninterrupted fracture bridges that restrict the growth of macrocracks (Wang et al., 2022). The K I C and G F of FRC might not be influenced by the dimensions of the assessed samples but rather rely solely on the fluctuations in the combination. These fluctuations can be considered as the specimen-size independent variables of FRC content (Nallathambi et al., 1987). The inclusion of steel fibers has a comparable impact on rubberized self-compacting concrete (RSCC), as reported by Wang et al. (Wang et al., 2022). Furthermore (Tarabin et al., 2023a), also reported similar findings.