Steel Fibre Reinforced Concrete Meso-Scale Numerical Analysis

Concrete is a heterogeneous composite consisting of aggregate, cement paste, and void. Steel fibre reinforced concrete (SFRC) has been widely studied experimentally and numerically in recent decades. .e fibre geometry model program generated by a secondary development ANSYS programwas exported tomidas FEA for analysis..e constitutive concrete model adopts the total strain crack model of concrete. A steel fibre bond slip is considered in an equivalent manner using the von Mises model. .e results of the three-dimensional meso-scale numerical analysis method agree well with the experimental values of steel fibre concrete beams.


Introduction
Concrete is a typical heterogeneous composite consisting of aggregate, cement paste, and void. is heterogeneity can be amplified by the addition of fibres. When added to concrete, steel fibres not only significantly improve its postcracking tensile resistance and toughness, but its impact resistance, fatigue resistance, and durability are also greatly improved [1][2][3][4][5][6]. In addition to testing, meso-scale modelling provides a way to choose materials and fibres. Meso-scale concrete simulation is utilised for evaluation of chloride penetration [7][8][9], carbonation [10][11][12], corrosion [13][14][15], elevated temperature [16], and aggregate shapes [17,18]. More importantly, the mechanical analysis of concrete can be performed through meso-scale simulation. To explore the mechanical properties of steel fibre reinforced concrete (SFRC), numerous experimental and numerical studies have been performed in this area.
For the SFRC force analysis, scholars mainly research and summarise the static [19,20] and dynamic loads [21][22][23]. Yasmin et al. [19,20] analysed postcracking material behaviour by static load testing and simulation. is study investigated the applicability of a recently proposed multiscale model with discrete and explicit representation of steel fibres to obtain the SFRC postcracking parameters.
Under the action of a dynamic load, Xu Z. et al. [21] presented a numerical simulation of dynamic impact tests on SFRC specimens to study the dynamic material properties of SFRC using LS-DYNA and investigated the dynamic failure behaviour of SFRC material under impact loading at different strain rates. A 3D numerical model has been proposed by Fang and Zhang [22] to investigate the mechanical properties of SFRC material under intense dynamic loading. Numerical analysis showed that steel fibres in concrete could improve the elastic modulus and the peak strain of SFRC material at a high strain rate. Hao et al. [23] investigated the influence of the shape of aggregates on numerical prediction in meso-scale modelling of SFRC materials with spiral fibres under dynamic splitting tension.
Many studies [24][25][26][27] have compared the experimental results with the results of finite element analysis (FEA) and found that the simulated results strongly agree with the experiment. Reliability analyses of SFRC corbels based on stochastic finite elements were performed for the first time by Gulsan [24]. Research on SFRC mechanical properties involved experiments by Sun et al. (2017) [25], where ANSYS finite element software of two SFRC T-beams and one concrete T-beam was used. Vítor [26] simulated the behaviour of SFRC. Jingu [27] established a finite element model distributed-force approach that simulated the transfer of crack bridging forces over the fibre embedment lengths and their associated effects on cracking behaviour. e experimental and numerical results prove that numerical analysis is an important tool for SFRC research. e use of finite element software can make it easier to study the SFRC interface [28][29][30] and derive an SFRC numerical constitutive model [31][32][33], secondary development, and application [34]. In 2010, Radtke [28] proposed to build a model using the concept of discrete fibre. e SFRCs are modelled by applying discrete forces to a background mesh to analyse the behaviour of the fibre-matrix interface. Pros et al. [29] applied the numerical immersed boundary approach and a phenomenological meso-model for concretefibre interaction. Liang and Wu [30] modelled fibre and concrete separately and linked them with slide line contact to truly reflect the interfacial behaviour of fibre and mortar. Guillermo [31] (2012) used the discrete crack formula to simulate the constitutive, mesoscopic, and macroscopic observations of the SFRC fracture behaviour. Qi et al. [32] proposed a constitutive model, including a bilinear model for compression and a drop-down model for tension. e finite element model established by Luís [33] applied a constitutive damage model.
For the concrete mechanical analysis of meso-scale simulation, this study focuses on (1) the establishment of random fibres and (2) the relationship between fibres and the matrix. is study used ANSYS to generate a three-dimensional geometric program of randomly distributed fibres; then, the geometric model was imported into midas FEA for nonlinear analysis and compared with the existing flexural test and direct tensile test. e von Mises model is adopted to represent the equivalent of bond slippage of steel fibre.

Generation of Geometric Model of Randomly
Distributed Fibres e SFR in the SFRC three-dimensional meso-scale numerical model should be uniformly and randomly distributed in the concrete matrix.
is study uses ANSYS parametric design language (APDL) to conduct secondary development of ANSYS and write a fibre parameterisation program that generates linear fibres in specified cubes, which can quickly and effectively generate geometric models of randomly distributed fibres in a specified cube. e fibre geometry model is then imported into a professional finite element software for analysis. e basic modelling scheme of the fibre geometry model program is as follows.
Step 1. Generation of fibre origin point K i.
In the global coordinate system, within the specified cube space {0, X 0 ; 0, Y 0 ; 0, Z 0 }, the starting point K i of the fibre is generated. Use a random function to generate the X coordinate X i , Y coordinate Y i , and Z coordinate Z i of the starting point K i of the fibre and satisfy 0 ≤ X i ≤ X 0 , 0 ≤ Y i ≤ Y 0 , and 0 ≤Z i ≤ Z 0 .
Step 2. Establish and rotate a local coordinate system at ki.
Step 2-1. At the fibre starting point K i , establish a local coordinate system K i X′Y′Z′.
Step 2-2. Switch the local coordinate system KiX′Y′Z′ to the current coordinate system. Keep the origin (Ki) of the local coordinate system KiX′Y′Z′ unchanged, and rotate the X′ axis, Y′ axis, and Z′ axis of the local coordinate system KiX′Y′Z′ at random angles; Step 3. Generation of fibre end point K j .
Step 3-1. e fibre end point K j is generated at the point (l f , 0, 0) of the rotated local coordinate system K i X′Y′Z′, where l f is the length of the fibre.
Step 3-2. Switch the local coordinate system to the global coordinate system. In the global coordinate system, extract the coordinates of the fibre end point K j .
Step 4. Generation of a fibre. Connect line K i and K j with line to get a randomly distributed fibre, as shown in Figure 1.
Step 5. Write Step 1-Step 4 as subroutines or macrofiles and repeat N calls to generate N fibres. It is noteworthy that when K j exceeds the range of the substrate in Step 3-2, if both K i and K j are deleted, the starting point K i and end point K j of the fibre are regenerated from Step 1, instead of regenerating end point K j of fibre from Step 2-2, which would have resulted in less fibre distribution at the edge of the matrix, leading to uneven fibre distribution in the matrix. e following are the examples of fibre generation with different fibre volume ratios. e size of the substrate is 300 mm × 150 mm × 150 mm. According to the abovementioned method, the fibre volume ratio V f of the fibre geometric model is 0.5%, 1.0%, 1.5%, 2.0%, 2.5%, and 3.0%. Because the geometric characteristics of the fibres used are the same, the number of fibres increases as the volume content increases. e fibre generation results are shown in Figure 2.
e orientation and distribution of fibres govern the performance of fibre reinforced concrete and play a major role in fracture results that are used to evaluate the structural competence of the composite material [35]. Figure 2 clearly indicates that as the fibre volume ratio V f increases from 0.5% to 3.0%, the fibre distribution becomes increasingly dense. Regardless of their number, the fibres can be guaranteed to be randomly and uniformly distributed in the matrix.  Advances in Civil Engineering 3

3.1.
Modelling. e geometric model of the fibre generated by ANSYS is exported in the Initial Graphics Exchange Specification (IGES) format, which can be simulated in professional finite element software. is study uses midas FEA to calculate and analyse the SFRC structure.
Firstly, import the geometric model of steel fibre in IGES format and then define the steel fibre as rebar elements. e steel fibre was modelled using embedded rebar elements that added the stiffness of the reinforcement to the parent elements. e basic properties of steel fibre include position information, shape information, and physical properties, without degrees of freedom. e midas FEA can realise the automatic coupling of steel and concrete, so an SFRC mesoscale analysis model is established. Figure 3 shows the threedimensional meso-scale numerical model of an SFRC beam established in midas FEA. In the model, elastic blocks are added at the support and displacement loading to prevent stress concentration.

Constitutive Relation of Concrete.
Concrete crack analysis methods include discrete crack model and smeared crack model. e advantage of the discrete crack model is that it can specifically simulate the discontinuity caused by the crack, bond slip between steel bar, and concrete. However, the accuracy of the analysis is greatly affected by the input parameters, and the finite element modelling is relatively complicated. e smeared crack model assumes that the local cracks are distributed in a certain range. It is generally used on reinforced concrete structures with a large number of steel bars. Its advantage is that the modelling is relatively simple.
e total strain crack model in the smeared crack model can be used in midas FEA [36], whose advantages are as follows: (1) crack distribution is convenient to show and the crack unit does not separate at the crack position; (2) the crack direction changes with the direction of the main strain, only the normal stress is generated on the crack surface, and the calculation process is simpler.
e total strain crack model has many satisfactory examples in analysing concrete structures [37,38]. e parabolic model of the total strain crack model is a commonly used model for the constitutive relationship of concrete under compression. e parabolic model was based on the fracture energy theory [39]. As shown in Figure 4, the model is determined by three parameters: concrete compressive strength f c , fracture energy G c , and characteristic element length h. e calculation formula of G c is shown in the following equation: In equation (1), f cmo is the benchmark average compressive strength, and its value is 10 N/mm 2 . G co is related to the maximum aggregate size, and the corresponding relationship is listed in Table 1.
e peak strain ε 0 corresponding to the concrete compressive strength fc is shown in the following equation: Characteristic element length h can be obtained by equation (3) according to the ultimate compression strain at the softening stage ε cu , ε 0 , and G c : Tensile models of concrete in total strain crack models such as constant, elastic, brittleness, linear, exponential, Hordijk, and multilinear models are provided in midas FEA.
is study uses a multilinear model. When the concrete (e) (f) Advances in Civil Engineering tensile strength is exceeded, softening occurs according to a user-defined polyline, as shown in Figure 5.

Bond Slip of Concrete and Steel
Fibre. Studies show that the primary form of fibre failure is the steel fibre being pulled out of the matrix rather than breaking. at is, the strength required to pull out the steel fibre is lower than the yield strength of the steel fibre. e steel fibres strength of the pulled out can be obtained by testing or correctly simulating the bond slip between the fibre and the matrix. However, both experiments and numerical values are very difficult to achieve. e bond slip of steel fibre is complex. erefore, this study adopts nominal yield strength f y equivalent bond slip between the fibre and the matrix. Based on the abovementioned reasons, in the numerical analysis process of this study, it is not necessary to consider the bond-slip behaviour between steel fibre and concrete. In Section 3.4, the idea and method of nominal yield strength f y equivalent bond slip between the fibre and the matrix will be described in detail.

Nominal Yield Strength of Steel
Fibre. Studies show that the primary form of fibre failure is it being pulled out rather than broken, which results from the high tensile yield strength of steel fibre and tests. e bond slip of steel fibre after tension is a very complex process. Many scholars have performed research work in this field [40][41][42][43][44]. Timon [43] considered the bond behaviour through the discrete crack model. Timon [44] used the bond model developed by Rabczuk and Belytschko to study the cohesive crack method of reinforced concrete structures.
Studies [41,[45][46][47][48][49] indicated that there are six different stages in the bond slip process of hooked-end steel fibre ( Figure 6 When the full debonding phase is reached (point c), the bond between the fibre and the matrix cannot resist the pullout load and the bond between the fibre and the matrix is destroyed and the fibre reaches the critical pullout status and the peak load at point c.
In the analysis of the three-dimensional meso-scale numerical SFRC beam model, it is difficult to simulate the bond slip of steel fibre. erefore, the von Mises model is    proposed as an equivalent to the bond slip of the steel fibre, as shown in Figure 7, where the nominal yield strength f y of the von Mises model steel fibre is where P u is the maximum pullout force of steel fibre (N); A f is the sectional area of steel fibre (mm 2 ). e basis and advantages of simplifying the bond slip constitutive model of fibre to an ideal elastic-plastic model are as follows: (1) e maximum pullout force of the bond slip of steel fibre is consistent with the nominal yield strength f y of steel fibre. erefore, the ultimate bearing capacity of the SFRC structure is also consistent; (2) If the bond slip model is used, the actual geometric shape and surface shape of steel fibre should be considered. However, in this study, the ideal elastic-plastic constitutive model is used instead of the fibre bond slip curve. In the program, only the nominal yield strength f y of the steel fibre is input, which avoids the modelling of the fibre shape in the finite element program and is very convenient and fast; (3) In this study, the bond slip constitutive model of steel fibre is simplified as an ideal elastic-plastic model. If the bond slip curve of steel fibre is to be closer, the multilinear tensile model can also be used to simulate it; (4) For different types of fibre, the maximum pullout force of steel fibre is different, and the nominal yield strength f y is also different, which can be obtained by testing or consulting related research.  Figure 8. e characteristics of the three fibres used in the test are shown in Table 2. Among them, ZP305 and RC80/60BN are conventional strength fibres, which are widely used in traditional SFRC. e RC80/30BP fibre has a high yield strength of 2300 MPa.   6 Advances in Civil Engineering ere are 5 groups of tests, each group has 4 test beams. e primary parameters of the test beams are shown in Table 3: e method described in Section 3.1 is used to build the three-dimensional meso-scale numerical models of SFRC beams. Displacement load value is −3.05 mm. e established finite element model is shown in Figure 3.

Stress of Steel Fibre.
e three-dimensional SFRC meso-scale numerical model is composed of a concrete matrix and randomly distributed steel fibres. erefore, the analysis result can intuitively obtain the stress state and role    Advances in Civil Engineering of the steel fibre during the loading process. e following uses the M_1 group as an example. M_1 group uses RC80/60BN type steel fibre and the fibre volume ratio is 0.75%. As shown in Figure 8(a), four characteristic points are selected during the loading process to observe the process of steel fibre stress changes. At point a, the microcracks in the concrete gradually expanded, and the concrete showed a certain plasticity. At this moment, the corresponding vertical load value was approximately 33.9 kN, which is close to the maximum bearing capacity of the plain concrete beam (approximately 33.8 kN). Because steel fibres limit the effect of crack propagation, the specimen can continue to bear the load. Figure 10(a) shows that the maximum tensile stress and maximum compressive stresses of the steel fibre are 39.4 MPa and −44.2 MPa, respectively. At this moment, the stress level of the steel fibre is low; in addition, the fibre stress is consistent with the stress of the concrete beam, with the upper part under compression and the lower part under tension.
When the ultimate bearing capacity is reached (point b), the stress of the steel fibre increases rapidly, which can be obtained from Figure 10 e loading mechanism uses vertical displacement. As the vertical displacement further increases, more steel fibres participate in bearing the load. e two main cracks formed under the two loading points. Due to the action of the steel fibres, the SFEC beam shows good ductility, as shown in point c and point d in Figure 10.
According to the analysis, the stress development process of the steel fibre in the SFRC beam can be clearly displayed, and the steel fibre stress at each stage and location can be viewed.

Crack.
In the meso-scale numerical model of the SFRC beam, concrete adopts the total strain crack model, which can well reflect the mechanical properties of concrete, including the process of crack generation and development, and output crack width. Figure 11 shows the crack development of the beam at point b. At this time, the maximum crack width of the beam is 0.024 mm, which indicates that the steel fibre has a good constraint on the crack development.

Meso-Scale Numerical Analysis of Direct Tensile Test.
DINH [50] also carried out the direct tensile test of steel fibre concrete beam. e direct tensile member is a dog-boneshaped member, and the MTS testing machine is used for tensile loading. e member size and test device are shown in Figure 12.
ere are 3 groups of tests, each group has 4 test beams. e primary parameters of the test beams are shown in Table 4: Using the modelling method and material constitutive model described in Section 3, the randomly distributed steel fibre geometry model generated in ANSYS is imported into midas FEA. e bottom end of the model is fixed by the boundary conditions, and a    Figure 13. Load-displacement curves of meso-scale numerical tensile analysis and experiment shown in Figure 14.
e results of experiment and meso-scale numerical analysis fit well, the peak loads almost coincide, and the softening stage is also well simulated.

e Influence of the Random Distribution of Steel Fibres on the Results.
e fibres generated in the matrix at different times have different random distributions. Whether the difference in the random distribution of steel fibres will have a big impact on the calculation results needs further research.
Taking the M_1 test group in the flexural test as an example, three groups of steel fibre models with different distributions were generated, and the results of load-displacement curves were compared. e results show that the load-displacement curves are almost completely consistent. e abovementioned research shows that although the fibre distributions generated at different times are different, the overall fibre distribution is random and uniformly distributed, so it has little effect on the calculation results.

e Influence of Element Division Size on the Result.
Whether the element division size of meso-scale numerical analysis model has a significant impact on the calculation results requires further study.
is paper proposes to divide the element size according to the reference fibre length. Select the specimen M-1 in Section 4.1 for the element size impact analysis. A total of 5 M-1 meso-analysis models with different element sizes have been established. e element sizes are 1/6 times, 1/3 times, 1/2 times, 2/3 times, and 1 times the length of steel fibre, respectively. e load-displacement curve of 5 M-1 meso-analysis models is shown in Figure 15. It can be seen that different element sizes have a certain impact on the results. e overall trend and bearing capacity of the calculation results are basically the same. However, after careful comparative analysis, it can be obtained that when the element size is    Advances in Civil Engineering 13 greater than 1/2 of the fibre size, after the maximum bearing capacity point a, there is no obvious descending segment (point b). At the same time, there is also a certain bearing capacity difference at strengthening stage (point c). erefore, in order to ensure the accuracy of the meso-analysis results, it is recommended to divide the element size according to 1/2 times the length of the steel fibre or smaller.

Summary and Conclusion
Unlike other studies, this study uses the von Mises model to consider the bond slip behaviour of steel fibres in an equivalent manner in a microlevel SFRC beam model and simulation results are in good agreement with the experimental results. e main conclusions are as follows: (1) e steel fibre generation program developed by ANSYS is fast and efficient, and the generated steel fibres are uniformly and randomly distributed within the matrix range, and the entire concrete matrix is covered. (2) e SFRC meso-scale model combines the concrete matrix with randomly distributed steel fibres. e concrete uses a total strain crack model, which can reflect the mechanical characteristics of concrete, especially the occurrence and development of cracks. e steel fibre is defined as a rebar element, which can realise the automatic coupling between the steel fibre and the concrete substrate.
(3) e bond-slip characteristics of steel fibres are equivalent with an ideal elastoplastic model, which can quickly realise modelling and calculation, and can well reflect the mechanical characteristics of SFRC strength and ductility; (4) e SFRC meso-scale model can well reflect the stress changes of steel fibres during loading. e structural failure mechanism can be explained based on the changes in steel fibre stress at the meso level.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest regarding the publication of this article.

Authors' Contributions
X. W. Xue, M. Z. Wu, Y. J. Cui, and Z. W. Li performed the methodologies, supervised the study, and reviewed and edited the article. J. W. Wu performed the methodologies and investigation and wrote the original draft. X. W. Xue will be responsible for the contact work of the paper in the future.