# Analysis of Flexural Bearing Capacity and Failure Mode of Precast LSFRC Pavement Slab

Authors: Zi-Jian Wang; Yi-Xuan Ban; Wen-Yu Luo; Han-Xiu Fan; Bin Zhang; Li-Ming Wu; Si-Chang Wang
Journal: Advances in Materials Science and Engineering
 (2022)

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## Abstract

To promote practical use of the top-and-bottom layered steel fiber reinforced concrete (LSFRC) plates, an appropriate constitutive model is selected using ABAQUS software to analyze the bearing capacity, failure pattern, and deformation of the LSFRC plates. An LSFRC plate model is established for the analysis and calculation. Meanwhile, a test was carried out to measure the bearing capacity of 10 LSFRC plates. The results show that, after going through the elastic stage, crack growth, and failure stage, when the maximum crack width reaches 0.1 mm, the load of the LSFRC plates can be used as the ultimate bearing load. When failure appears to be bending the LSFRC plates, the neutral axis will shift upwards, showing a pseudoplastic failure. There is a larger tensile stress area after the first crack appears at the bottom of the steel fiber concrete. The change before and after the maximum load is relatively gentle. After becoming plastic, the plate witnesses a rapid crack growth on its bottom. With similar results from the test and numerical simulation, small errors can be minimized by further analysis to provide a more accurate model for the subsequent bearing capacity analysis of the LSFRC plates.


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## Body

## 1. Introduction

Concrete is widely used in all types of construction work in China. However, over the last few years, its strength and durability have not caught up with the stricter requirements. Therefore, concrete with higher strength is in great demand. When cracks appear in the concrete, steel fiber shows a bridging function, which inhibits the tension and expansion of cracks. Steel fiber reinforced concrete (SFRC) has better mechanical properties than normal concrete. Since its creation, SFRC has been widely studied by scholars, especially for its mechanical properties and deformation capacity [1–3]. The results show that steel fiber not only improves the tensile strength [4], shear strength [5], and bending strength [6] of the concrete but also increases its toughness [7] and resistance [8], so SFRC shows better fatigue performance [9] and durability [10, 11]. However, the bearing capacity of the SFRC plate, which determines whether it can replace conventional concrete plates at the construction site, needs more attention.Currently, many researchers focus on the size of precast concrete plates. But they have ignored the relatively low bearing capacity and cracking of conventional precast concrete plates, which may not meet requirements at the construction site. Thus, it is necessary to boost the bearing capacity of precast concrete plates. This has become a hot topic in research. Khan et al. [11] mixed basalt steel fiber-calcium carbonate whisker into fly ash concrete (BSC-FAC), and studied how to improve the compressive strength, toughness index (ɳ) and performance coefficient index (ξ) of BSC-FAC by adding basalt steel fiber-calcium carbonate whisker. They analyzed the controllable range of its parameters and the changing trend of materials through an analytical fitting model of the stress-strain curve. In further research, Khan et al. [12] and others studied the mechanical properties of new hybrid fiber concrete based on calcium carbonate whisker-steel fiber-basalt fiber concrete with different basalt fiber contents and obtained stress-strain curves and load-deflection curves. To reveal the interface bonding mechanism of calcium carbonate whisker, basalt fiber, and steel fiber, they analyzed the obtained data using a scanning electron microscope (SEM). Cao and Khan [13] tested the properties of steel, polyvinyl alcohol, and calcium carbonate whisker mixed fiber reinforced cement-based composites (SPCHyFRCC) with different mixing ratios, fiber lengths, and contents by using a single-degree-of-freedom hydraulic shaking table. They analyzed its mechanical properties and found that the material could provide crack propagation resistance under dynamic load. Xie et al. [14] studied the bending behavior, fiber distribution characteristics and compression response of calcium carbonate whisker (CW) modified steel-PVA hybrid fiber reinforced cement-based composites (CW-SPTRCC) through a three-point bending test, image processing technology, and a uniaxial compression test. They established the compressive stress-strain constitutive model of CW-SPFRCC and found that CW can provide crack resistance and a filling effect on the microscopic scale to improve the bending performance of SPRCC. Xie et al. [15] determined the tensile strength (ft) and fracture toughness (KIC) of a cement-based composite (MHFRCC) by a three-point bending test based on the boundary effect model. Xie predicted the fracture failure principle of the MHFRCC structure, established the structural fracture failure zone with a 15% change, and revealed the multiscale strengthening mechanism in the cement matrix. Zhang et al. [16] performed image analysis. The experimenter improved the vibration effect of the device, which significantly increased the fluidity of the slurry. In this way, the effectiveness of the steel fiber arrangement was ensured. As the orientation of fibers after steel fiber arrangement was parallel to the casting direction, this method not only improved the ultimate tensile and bending strength of UHPC but also significantly enhanced the crack resistance and ductility of UHPC. Ali et al. [17] studied and used different quality steel fiber (WSF) and fly ash (FA) from waste tires and analyzed various basic mechanical properties of HSC. The findings demonstrate that WFS can significantly improve the tensile property, while FA has good chloride penetration resistance, and WSF and FA can significantly improve the acid corrosion resistance of HSC. Ali [18] also found that adding fly ash and hook steel fiber to recycled concrete can improve its strength and durability. The results show that adding steel fiber and fly ash to recycled concrete can greatly enhance its mechanical properties and acid corrosion resistance. Globally, academics are very enthusiastic about using various fibers to enhance the performance of concrete. Steel fiber reinforced concrete, a novel material in the research of the aforementioned scholars, has emerged as the top option for prefabricated roads on construction sites due to its exceptional performance in terms of construction difficulty and saving costs.Extensive experiments show that steel fiber concrete can greatly increase the bearing capacity of precast concrete plates. But adding many steel fibers to the concrete is both difficult and expensive. Scholars found that the top-and-bottom distribution for LSFRC plates can reduce the cost, but related studies are few, especially those for practical engineering. A number of researchers have simulated various models of layered steel fiber reinforced concrete in software and provided experimental basic ideas on how to solve the practical application of layered steel fiber reinforced concrete. Fan et al. [19] used ANSYS finite element software to change the influence of structural parameters of road pavement on the stress of precast slab. They found that the elastic modulus of the soil subgrade was inversely proportional to the settlement of the precast slab. When more steel fibers are added, the deflection of the pavement slab and tensile stress at the bottom of the slab decreased linearly. You [20] studied how the continuous gradient distribution of steel fibers will influence the mechanical properties of fiber-reinforced and cement-based materials. Ying-Bo and Zhe-Anda [21] studied fatigue defects in LSFRC and defined their degree. Huang used cubic polynomials to perform regression for the strain-cycle ratio equation and defect-cycle ratio equation, for which the correlation coefficient is close to 1. Using ANSYS finite element software, Fan et al. [22] built a finite element calculation model of layered steel fiber reinforced concrete pavement (LSFRCP) and studied how the slab length and thickness, and elastic modulus of base concrete would change under the action of driving load and temperature. The main ideas of LSFRC pavement design were presented. To enhance the bearing performance of bending components, Yi et al. [23] paved a surface layer with high density of steel fibers at the concrete bottom. The bending fatigue damage of such local high-density SFRC was very close to ductile damage. This property is more often seen in ductile materials than in plain concrete or conventional SFRC. Lu and He [24] applied SFRC to the tensile area of one-third portion of the beam. As a result, the local SFRC composite beam attained a 47.6% higher cracking load, a 13.2% higher yield load, and an 11.5% higher ultimate load than that of conventional concrete. Peng and Jiao [25] based on three groups of concrete ratios with different steel fiber distribution patterns, tested the mechanical properties of layered steel fiber concrete, steel fiber concrete, and plain concrete with the same matrix ratio. Peng [26] studied the mechanical properties of composite steel fiber reinforced concrete (CSFRC) and LSFRC, evaluated their advantages and disadvantages, and established a formula to calculate the tensile stress of both materials.The majority of the aforementioned researchers have used the curve model of steel fiber reinforced concrete to analyze the mechanical properties. A few of them consider the basic model of steel fiber reinforced concrete or integrate them into a whole model. Although this model is suitable to study the mechanical properties of two combined, it is still unclear how steel fiber functions in concrete and how it affects the mechanical properties of concrete. Therefore, it is crucial to analyze the working performance of one of the two materials while also analyzing the interaction between the two materials. Separate modeling can effectively solve this problem. Analytical failure model and numerical model are the main tools to study the LSFRC pavement slab, but there is a lack of corresponding laboratory tests to verify them. Therefore, in this paper, separate model data is established by ABAQUS, and the ultimate flexural capacity of LSFRC slabs and their simulated failure modes are obtained by analyzing the flexural capacity parameters of LSFRC slabs. This study carries out the flexural capacity tests of 10 LSFRC slabs and analyzes the crack modes, bending deformation characteristics of the slab bottom, and ultimate flexural capacity of LSFRC slabs. The significance of its engineering practice is also discussed. The outcomes of the analysis and discussion can offer theoretical and data support for the design and building of LSFRC pavement slabs.


## 2. Numerical Simulation of the LSFRC Plates

To evaluate the bearing performance of LSFRC plates and identify appropriate parameters for practical engineering, this paper uses a constitutive model from the ABAQUS software to carry out numerical simulations and analyze the bearing capacity, failure pattern, and deformation of the LSFRC plates.


### 2.1. Model Building

LSFRC is a kind of CSFRC made of layered steel fiber concrete and concrete. When establishing the finite element model, the property difference between steel fibers and concrete should be fully taken into account.


#### 2.1.1. Constitutive Model of the Concrete

The plastic damage model in ABAQUS software is used to reflect the constitutive relationship of the concrete. The model can simulate concrete and other brittle materials. Generally, the model is affected by tension and compression and can successfully control the equivalent plastic deformation on both the yield surface and the failure surface. Moreover, the concrete’s yield and flow in the macroscopic deformation are consistent with that of the hardened ductile materials.(1) Damage and stiffness degradation. Under uniaxial load, the effective plastic strain rate is as follows:(1)ε˜˙tpl=ε˜˙11pl,ε˜˙tpl=−ε˜˙11pl,where ε˜˙tpl represents the plastic strain rate in the axis.The axis stress-strain relation can be transformed into the stress-plastic strain curve as follows:(2)σt=σtε˜tpl,ε˜˙11pl,θ,fi,σc=σcε˜cpl,ε˜˙11pl,θ,fi,where σt denotes the affected variable of temperature;fi (I = 1,2, …) represents other effect variables.Figure1 shows the softening section of the stress-strain curve for unloading at one point of the concrete. The damages of elastic stiffness are different in the tensile and compressive tests. But as the strain grows, both tension and compression will aggravate the damage.Figure 1
Concrete elastic-plastic damage model. (a) Stretch. (b) Compress.
(a)(b)The concrete damage response is measured by the axial damage coefficients, where the functions of plastic strain, temperature, and field variables reflect in the changes the two factors above.The concrete damage response is measured by two axial damage coefficientsdt and dc. They are the functions of plastic strain, temperature, and field variable.(3)dt=dtε˜tpl,θ,fi0≤dt≤1,dc=dcε˜cpl,θ,fi0≤dc≤1.Let’s assume that the initial elastic modulus of the concrete isE0. When the concrete is under axial loads, the tensile and compressive stress-strain expressions are as follows:(4)σt=1−dt×E0×εt−ε˜tpl,σc=1−dc×E0×εc−ε˜cpl.In the axial tension test, the specimen produces a vertical crack. The crack continues to grow with higher equivalent stress as the cross-sectional area decreases. The crack of the specimen along the loading direction appears under a certain axial compressive load, but the damage is smaller than that under an axial tensile load. The effective cross-sectional area rapidly declines when the specimen is under great compression. The yield surface and failure surface are determined by the effective axial tension and compression:(5)σ˜t=σt1−dt=E0εt−ε˜tpl,σ˜c=σc1−dc=E0εc−ε˜cpl.(2) Concrete stress-strain curve. Figure 2 shows the constitutive relation curve of the concrete, where the compressive strength fck and tensile strength ft are the test results, which are 41.9 MPa and 7.2 MPa, respectively; the elastic modulus is 32,500 MPa, with a Poisson’s ratio of 0.17. The SFRC’s compressive strength fck is 43.4 MPa and the tensile strength ft is 8.1 MPa; its elastic modulus is 34,200 MPa, with the Poisson’s ratio of 0.17.Figure 2
Uniaxial stress-strain curve for concrete. (a) Under pressure. (b) Pulling.
(a)(b)


#### 2.1.2. Constitutive Relation of Steel Fiber

During modeling, two 3D solid models are established separately. And the steel fibers are regarded as microrebars to be bound to the concrete without considering their bonding performance. An SFRC microstructure model is established and an appropriate damage failure criterion and loading method are selected to ensure good convergence during numerical simulation.The stress-strain relationship of steel fiber is basically consistent with that of rebar. The bilinear kinematic hardening plasticity (BKIN) model is adopted to describe the constitutive relation of steel fiber [27]. At the hardening stage, the slope is set at 0.01 times of the steel fiber’s elastic modulus, namely, E=0.01E0, where E0 is the initial modulus. Figure 3 shows the steel fiber’s stress-strain curve, where the initial elastic modulus is 210 GPa, with the Poisson’s ratio of 0.27 and the tensile strength of 1,300 MPa.Figure 3
Steel fiber stress-strain curve.


#### 2.1.3. Boundary Conditions

After building the solid model, we exert constraints at the of the model’s production and loading to generate solutions. To align with the test, articulated constraints are applied at the two lower ends of the support and DOF freedom constraints are applied to the rigid pad. A displacement load is also applied to the model. To align with the test, the load on the pad is passed down to the plate section to prevent stress concentration. At the loading speed of 0.02 mm/s, LSFRC plates may rotate or slip away from the supports. Considering this phenomenon, this paper has established between the LSFRC plate support and the loading plate. Based on “hard contact” and the tangential friction theory, the final friction coefficient is 0.6.


#### 2.1.4. Stochastic Distribution Model of Fiber Microstructure

Through simulation, researchers [28, 29] have established the micromodels of fiber cement mortar and cement concrete materials and studied how they will respond to static and dynamic load. This paper further explores the 3D random distribution of steel fibers in concrete. It generates an algorithm to help ABAQUS, the finite element software, reflect the mechanical structure and microscopic model of SFRC.Based on the Monte-Carlo algorithm,Xn + 1 = R (X1, X2,…, Xn) in the recursion formula is used to generate the random value, where R is the recursion function. The new random value Xn + 1 is derived from the initial value (X1, X2,…, Xn). Using the initial value (X1, X2,…, Xn) and the recursion function R, the algorithm identifies the random sequence {Xn + 1}. The random sequence {Xn + 1} does not necessarily meet the functions of randomness and independence, so it is also known as the pseudo-random sequence. However, when the random numerical library is small, the pseudorandom sequence barely meets the randomness criteria. Under the guidance of the pseudorandom number generation principle, we write the program in Python and generate the pseudorandom sequence with ABAQUS.In this paper, steel fibers are assumed to be flat columns and their random characteristics are presented by their random locations and angles. Fiber lengthL, diameter D, volume fraction, and matrix volume V are used to determine the required number of fibers. The random distribution algorithm of steel fibers can be written in two steps:(1)
The random sequence is generated by Rand () function;(2)
The required number of steel fibers (N) is obtained;(6)N=VρVVρ=4VρVπd2l,WhereVρ is the volume of single steel fiber.(3)
Two factors should be considered when placing steel fibers into the concrete. One is to ensure that steel fibers are in random directions. Let’s suppose the original direction and new direction of steel fibers are represented byF_ original and F_ new and that a, b, and c are slopes around the X, Y, and Z axes. Thus, the algorithm for random directions of steel fibers is as follows [27, 29].(7)F_final=F_originalcosbcosccosbsinc−sinbsinasinbcosc−cossincsinasinbsinc+cosacoscsinacosbcosasinbcosc+sinasinccosasinbsinc−sinacosccosacosb.The other is to ensure that steel fibers are in random locations. Let’s suppose thatF_ Random (Xr, Yr, Zr), F_ new (Xn, Yn, Zn), and F_ original (Xo, Yo, Zo) indicate the random location, the new location, and the original location of a steel fiber in the concrete, respectively. Thus, the algorithm for generating the random location of steel fibers is as follows.(8)Xn=Xo+Xr;Yn=Y0+Yr;Zn=Zo+Zr.(4)
Based on the steps above, the steel fibers are generated in the concrete and they shall not exceed both ends of the concrete.After the boundary conditions, random location and direction algorithm are determined, the first steel fiber is generated in the concrete. Thenth steel fiber generated should have no intersection with the (n − 1)th steel fiber or beyond the two ends of the concrete. To meet this requirement, the vertical distance between the center of any two steel fibers should be no less than diameter D. After the last steel fiber is generated, the data of the random locations and directions of all steel fibers are exported [27, 29].During the modeling, the steel fiber and the concrete are regarded as two distinctive units that interact with each other. This makes the results more accurate. Figures4 and 5 show the schematic diagrams of modeling and the model.Figure 4
Schematic diagram of discrete modeling. (a) Steel fiber generation. (b) Model structure.
(a)(b)Figure 5
Schematic diagram of integral modeling.


### 2.2. Analysis of the Results

#### 2.2.1. Failure Pattern

There are three stages of SFRC failure: (1) the formation of microcracks, (2) crack growth, and (3) the specimen losing stability. In the numerical simulation, the damage index is added to SFRC’s elastic stiffness matrix to simulate its characteristics. The characteristic is that the unloading stiffness of the concrete decreases as the damage increases. This paper shows that when the matrix damage factor reaches 0.5, the concrete can be considered ineffective [30].Figure6 shows an irregular crack penetrating the LSFRC plate, with many microcracks around it. During the cracking, the steel fibers in the concrete exert bonding forces. As a result, although many irregular small cracks appear on both sides of the main crack, the LSFRC plate has not obviously deformed.Figure 6
Comparison of damage patterns between simulation results and test results.


#### 2.2.2. Deformation Analysis

According to Figure7, the curve has a brief downward trend when going up. After analyzing the cracking, it can be found that when the LSFRC plate reaches the yield limit, steel fibers hinder the crack growth on both sides of the plate. However, when steel fibers are separated from the concrete, the deflection curve rises again. Steel fibers improve the resistance and toughness of the concrete, resulting in a smoothly declining curve and good plasticity of the concrete. The results also show that the steel fiber plays an important bridging role in the macroscale viscoelastic property of the concrete.Figure 7
Deflection simulation values across the plate span.


## 2.1. Model Building

LSFRC is a kind of CSFRC made of layered steel fiber concrete and concrete. When establishing the finite element model, the property difference between steel fibers and concrete should be fully taken into account.


### 2.1.1. Constitutive Model of the Concrete

The plastic damage model in ABAQUS software is used to reflect the constitutive relationship of the concrete. The model can simulate concrete and other brittle materials. Generally, the model is affected by tension and compression and can successfully control the equivalent plastic deformation on both the yield surface and the failure surface. Moreover, the concrete’s yield and flow in the macroscopic deformation are consistent with that of the hardened ductile materials.(1) Damage and stiffness degradation. Under uniaxial load, the effective plastic strain rate is as follows:(1)ε˜˙tpl=ε˜˙11pl,ε˜˙tpl=−ε˜˙11pl,where ε˜˙tpl represents the plastic strain rate in the axis.The axis stress-strain relation can be transformed into the stress-plastic strain curve as follows:(2)σt=σtε˜tpl,ε˜˙11pl,θ,fi,σc=σcε˜cpl,ε˜˙11pl,θ,fi,where σt denotes the affected variable of temperature;fi (I = 1,2, …) represents other effect variables.Figure1 shows the softening section of the stress-strain curve for unloading at one point of the concrete. The damages of elastic stiffness are different in the tensile and compressive tests. But as the strain grows, both tension and compression will aggravate the damage.Figure 1
Concrete elastic-plastic damage model. (a) Stretch. (b) Compress.
(a)(b)The concrete damage response is measured by the axial damage coefficients, where the functions of plastic strain, temperature, and field variables reflect in the changes the two factors above.The concrete damage response is measured by two axial damage coefficientsdt and dc. They are the functions of plastic strain, temperature, and field variable.(3)dt=dtε˜tpl,θ,fi0≤dt≤1,dc=dcε˜cpl,θ,fi0≤dc≤1.Let’s assume that the initial elastic modulus of the concrete isE0. When the concrete is under axial loads, the tensile and compressive stress-strain expressions are as follows:(4)σt=1−dt×E0×εt−ε˜tpl,σc=1−dc×E0×εc−ε˜cpl.In the axial tension test, the specimen produces a vertical crack. The crack continues to grow with higher equivalent stress as the cross-sectional area decreases. The crack of the specimen along the loading direction appears under a certain axial compressive load, but the damage is smaller than that under an axial tensile load. The effective cross-sectional area rapidly declines when the specimen is under great compression. The yield surface and failure surface are determined by the effective axial tension and compression:(5)σ˜t=σt1−dt=E0εt−ε˜tpl,σ˜c=σc1−dc=E0εc−ε˜cpl.(2) Concrete stress-strain curve. Figure 2 shows the constitutive relation curve of the concrete, where the compressive strength fck and tensile strength ft are the test results, which are 41.9 MPa and 7.2 MPa, respectively; the elastic modulus is 32,500 MPa, with a Poisson’s ratio of 0.17. The SFRC’s compressive strength fck is 43.4 MPa and the tensile strength ft is 8.1 MPa; its elastic modulus is 34,200 MPa, with the Poisson’s ratio of 0.17.Figure 2
Uniaxial stress-strain curve for concrete. (a) Under pressure. (b) Pulling.
(a)(b)


### 2.1.2. Constitutive Relation of Steel Fiber

During modeling, two 3D solid models are established separately. And the steel fibers are regarded as microrebars to be bound to the concrete without considering their bonding performance. An SFRC microstructure model is established and an appropriate damage failure criterion and loading method are selected to ensure good convergence during numerical simulation.The stress-strain relationship of steel fiber is basically consistent with that of rebar. The bilinear kinematic hardening plasticity (BKIN) model is adopted to describe the constitutive relation of steel fiber [27]. At the hardening stage, the slope is set at 0.01 times of the steel fiber’s elastic modulus, namely, E=0.01E0, where E0 is the initial modulus. Figure 3 shows the steel fiber’s stress-strain curve, where the initial elastic modulus is 210 GPa, with the Poisson’s ratio of 0.27 and the tensile strength of 1,300 MPa.Figure 3
Steel fiber stress-strain curve.


### 2.1.3. Boundary Conditions

After building the solid model, we exert constraints at the of the model’s production and loading to generate solutions. To align with the test, articulated constraints are applied at the two lower ends of the support and DOF freedom constraints are applied to the rigid pad. A displacement load is also applied to the model. To align with the test, the load on the pad is passed down to the plate section to prevent stress concentration. At the loading speed of 0.02 mm/s, LSFRC plates may rotate or slip away from the supports. Considering this phenomenon, this paper has established between the LSFRC plate support and the loading plate. Based on “hard contact” and the tangential friction theory, the final friction coefficient is 0.6.


### 2.1.4. Stochastic Distribution Model of Fiber Microstructure

Through simulation, researchers [28, 29] have established the micromodels of fiber cement mortar and cement concrete materials and studied how they will respond to static and dynamic load. This paper further explores the 3D random distribution of steel fibers in concrete. It generates an algorithm to help ABAQUS, the finite element software, reflect the mechanical structure and microscopic model of SFRC.Based on the Monte-Carlo algorithm,Xn + 1 = R (X1, X2,…, Xn) in the recursion formula is used to generate the random value, where R is the recursion function. The new random value Xn + 1 is derived from the initial value (X1, X2,…, Xn). Using the initial value (X1, X2,…, Xn) and the recursion function R, the algorithm identifies the random sequence {Xn + 1}. The random sequence {Xn + 1} does not necessarily meet the functions of randomness and independence, so it is also known as the pseudo-random sequence. However, when the random numerical library is small, the pseudorandom sequence barely meets the randomness criteria. Under the guidance of the pseudorandom number generation principle, we write the program in Python and generate the pseudorandom sequence with ABAQUS.In this paper, steel fibers are assumed to be flat columns and their random characteristics are presented by their random locations and angles. Fiber lengthL, diameter D, volume fraction, and matrix volume V are used to determine the required number of fibers. The random distribution algorithm of steel fibers can be written in two steps:(1)
The random sequence is generated by Rand () function;(2)
The required number of steel fibers (N) is obtained;(6)N=VρVVρ=4VρVπd2l,WhereVρ is the volume of single steel fiber.(3)
Two factors should be considered when placing steel fibers into the concrete. One is to ensure that steel fibers are in random directions. Let’s suppose the original direction and new direction of steel fibers are represented byF_ original and F_ new and that a, b, and c are slopes around the X, Y, and Z axes. Thus, the algorithm for random directions of steel fibers is as follows [27, 29].(7)F_final=F_originalcosbcosccosbsinc−sinbsinasinbcosc−cossincsinasinbsinc+cosacoscsinacosbcosasinbcosc+sinasinccosasinbsinc−sinacosccosacosb.The other is to ensure that steel fibers are in random locations. Let’s suppose thatF_ Random (Xr, Yr, Zr), F_ new (Xn, Yn, Zn), and F_ original (Xo, Yo, Zo) indicate the random location, the new location, and the original location of a steel fiber in the concrete, respectively. Thus, the algorithm for generating the random location of steel fibers is as follows.(8)Xn=Xo+Xr;Yn=Y0+Yr;Zn=Zo+Zr.(4)
Based on the steps above, the steel fibers are generated in the concrete and they shall not exceed both ends of the concrete.After the boundary conditions, random location and direction algorithm are determined, the first steel fiber is generated in the concrete. Thenth steel fiber generated should have no intersection with the (n − 1)th steel fiber or beyond the two ends of the concrete. To meet this requirement, the vertical distance between the center of any two steel fibers should be no less than diameter D. After the last steel fiber is generated, the data of the random locations and directions of all steel fibers are exported [27, 29].During the modeling, the steel fiber and the concrete are regarded as two distinctive units that interact with each other. This makes the results more accurate. Figures4 and 5 show the schematic diagrams of modeling and the model.Figure 4
Schematic diagram of discrete modeling. (a) Steel fiber generation. (b) Model structure.
(a)(b)Figure 5
Schematic diagram of integral modeling.


## 2.1.1. Constitutive Model of the Concrete

The plastic damage model in ABAQUS software is used to reflect the constitutive relationship of the concrete. The model can simulate concrete and other brittle materials. Generally, the model is affected by tension and compression and can successfully control the equivalent plastic deformation on both the yield surface and the failure surface. Moreover, the concrete’s yield and flow in the macroscopic deformation are consistent with that of the hardened ductile materials.(1) Damage and stiffness degradation. Under uniaxial load, the effective plastic strain rate is as follows:(1)ε˜˙tpl=ε˜˙11pl,ε˜˙tpl=−ε˜˙11pl,where ε˜˙tpl represents the plastic strain rate in the axis.The axis stress-strain relation can be transformed into the stress-plastic strain curve as follows:(2)σt=σtε˜tpl,ε˜˙11pl,θ,fi,σc=σcε˜cpl,ε˜˙11pl,θ,fi,where σt denotes the affected variable of temperature;fi (I = 1,2, …) represents other effect variables.Figure1 shows the softening section of the stress-strain curve for unloading at one point of the concrete. The damages of elastic stiffness are different in the tensile and compressive tests. But as the strain grows, both tension and compression will aggravate the damage.Figure 1
Concrete elastic-plastic damage model. (a) Stretch. (b) Compress.
(a)(b)The concrete damage response is measured by the axial damage coefficients, where the functions of plastic strain, temperature, and field variables reflect in the changes the two factors above.The concrete damage response is measured by two axial damage coefficientsdt and dc. They are the functions of plastic strain, temperature, and field variable.(3)dt=dtε˜tpl,θ,fi0≤dt≤1,dc=dcε˜cpl,θ,fi0≤dc≤1.Let’s assume that the initial elastic modulus of the concrete isE0. When the concrete is under axial loads, the tensile and compressive stress-strain expressions are as follows:(4)σt=1−dt×E0×εt−ε˜tpl,σc=1−dc×E0×εc−ε˜cpl.In the axial tension test, the specimen produces a vertical crack. The crack continues to grow with higher equivalent stress as the cross-sectional area decreases. The crack of the specimen along the loading direction appears under a certain axial compressive load, but the damage is smaller than that under an axial tensile load. The effective cross-sectional area rapidly declines when the specimen is under great compression. The yield surface and failure surface are determined by the effective axial tension and compression:(5)σ˜t=σt1−dt=E0εt−ε˜tpl,σ˜c=σc1−dc=E0εc−ε˜cpl.(2) Concrete stress-strain curve. Figure 2 shows the constitutive relation curve of the concrete, where the compressive strength fck and tensile strength ft are the test results, which are 41.9 MPa and 7.2 MPa, respectively; the elastic modulus is 32,500 MPa, with a Poisson’s ratio of 0.17. The SFRC’s compressive strength fck is 43.4 MPa and the tensile strength ft is 8.1 MPa; its elastic modulus is 34,200 MPa, with the Poisson’s ratio of 0.17.Figure 2
Uniaxial stress-strain curve for concrete. (a) Under pressure. (b) Pulling.
(a)(b)


## 2.1.2. Constitutive Relation of Steel Fiber

During modeling, two 3D solid models are established separately. And the steel fibers are regarded as microrebars to be bound to the concrete without considering their bonding performance. An SFRC microstructure model is established and an appropriate damage failure criterion and loading method are selected to ensure good convergence during numerical simulation.The stress-strain relationship of steel fiber is basically consistent with that of rebar. The bilinear kinematic hardening plasticity (BKIN) model is adopted to describe the constitutive relation of steel fiber [27]. At the hardening stage, the slope is set at 0.01 times of the steel fiber’s elastic modulus, namely, E=0.01E0, where E0 is the initial modulus. Figure 3 shows the steel fiber’s stress-strain curve, where the initial elastic modulus is 210 GPa, with the Poisson’s ratio of 0.27 and the tensile strength of 1,300 MPa.Figure 3
Steel fiber stress-strain curve.


## 2.1.3. Boundary Conditions

After building the solid model, we exert constraints at the of the model’s production and loading to generate solutions. To align with the test, articulated constraints are applied at the two lower ends of the support and DOF freedom constraints are applied to the rigid pad. A displacement load is also applied to the model. To align with the test, the load on the pad is passed down to the plate section to prevent stress concentration. At the loading speed of 0.02 mm/s, LSFRC plates may rotate or slip away from the supports. Considering this phenomenon, this paper has established between the LSFRC plate support and the loading plate. Based on “hard contact” and the tangential friction theory, the final friction coefficient is 0.6.


## 2.1.4. Stochastic Distribution Model of Fiber Microstructure

Through simulation, researchers [28, 29] have established the micromodels of fiber cement mortar and cement concrete materials and studied how they will respond to static and dynamic load. This paper further explores the 3D random distribution of steel fibers in concrete. It generates an algorithm to help ABAQUS, the finite element software, reflect the mechanical structure and microscopic model of SFRC.Based on the Monte-Carlo algorithm,Xn + 1 = R (X1, X2,…, Xn) in the recursion formula is used to generate the random value, where R is the recursion function. The new random value Xn + 1 is derived from the initial value (X1, X2,…, Xn). Using the initial value (X1, X2,…, Xn) and the recursion function R, the algorithm identifies the random sequence {Xn + 1}. The random sequence {Xn + 1} does not necessarily meet the functions of randomness and independence, so it is also known as the pseudo-random sequence. However, when the random numerical library is small, the pseudorandom sequence barely meets the randomness criteria. Under the guidance of the pseudorandom number generation principle, we write the program in Python and generate the pseudorandom sequence with ABAQUS.In this paper, steel fibers are assumed to be flat columns and their random characteristics are presented by their random locations and angles. Fiber lengthL, diameter D, volume fraction, and matrix volume V are used to determine the required number of fibers. The random distribution algorithm of steel fibers can be written in two steps:(1)
The random sequence is generated by Rand () function;(2)
The required number of steel fibers (N) is obtained;(6)N=VρVVρ=4VρVπd2l,WhereVρ is the volume of single steel fiber.(3)
Two factors should be considered when placing steel fibers into the concrete. One is to ensure that steel fibers are in random directions. Let’s suppose the original direction and new direction of steel fibers are represented byF_ original and F_ new and that a, b, and c are slopes around the X, Y, and Z axes. Thus, the algorithm for random directions of steel fibers is as follows [27, 29].(7)F_final=F_originalcosbcosccosbsinc−sinbsinasinbcosc−cossincsinasinbsinc+cosacoscsinacosbcosasinbcosc+sinasinccosasinbsinc−sinacosccosacosb.The other is to ensure that steel fibers are in random locations. Let’s suppose thatF_ Random (Xr, Yr, Zr), F_ new (Xn, Yn, Zn), and F_ original (Xo, Yo, Zo) indicate the random location, the new location, and the original location of a steel fiber in the concrete, respectively. Thus, the algorithm for generating the random location of steel fibers is as follows.(8)Xn=Xo+Xr;Yn=Y0+Yr;Zn=Zo+Zr.(4)
Based on the steps above, the steel fibers are generated in the concrete and they shall not exceed both ends of the concrete.After the boundary conditions, random location and direction algorithm are determined, the first steel fiber is generated in the concrete. Thenth steel fiber generated should have no intersection with the (n − 1)th steel fiber or beyond the two ends of the concrete. To meet this requirement, the vertical distance between the center of any two steel fibers should be no less than diameter D. After the last steel fiber is generated, the data of the random locations and directions of all steel fibers are exported [27, 29].During the modeling, the steel fiber and the concrete are regarded as two distinctive units that interact with each other. This makes the results more accurate. Figures4 and 5 show the schematic diagrams of modeling and the model.Figure 4
Schematic diagram of discrete modeling. (a) Steel fiber generation. (b) Model structure.
(a)(b)Figure 5
Schematic diagram of integral modeling.


## 2.2. Analysis of the Results

### 2.2.1. Failure Pattern

There are three stages of SFRC failure: (1) the formation of microcracks, (2) crack growth, and (3) the specimen losing stability. In the numerical simulation, the damage index is added to SFRC’s elastic stiffness matrix to simulate its characteristics. The characteristic is that the unloading stiffness of the concrete decreases as the damage increases. This paper shows that when the matrix damage factor reaches 0.5, the concrete can be considered ineffective [30].Figure6 shows an irregular crack penetrating the LSFRC plate, with many microcracks around it. During the cracking, the steel fibers in the concrete exert bonding forces. As a result, although many irregular small cracks appear on both sides of the main crack, the LSFRC plate has not obviously deformed.Figure 6
Comparison of damage patterns between simulation results and test results.


### 2.2.2. Deformation Analysis

According to Figure7, the curve has a brief downward trend when going up. After analyzing the cracking, it can be found that when the LSFRC plate reaches the yield limit, steel fibers hinder the crack growth on both sides of the plate. However, when steel fibers are separated from the concrete, the deflection curve rises again. Steel fibers improve the resistance and toughness of the concrete, resulting in a smoothly declining curve and good plasticity of the concrete. The results also show that the steel fiber plays an important bridging role in the macroscale viscoelastic property of the concrete.Figure 7
Deflection simulation values across the plate span.


## 2.2.1. Failure Pattern

There are three stages of SFRC failure: (1) the formation of microcracks, (2) crack growth, and (3) the specimen losing stability. In the numerical simulation, the damage index is added to SFRC’s elastic stiffness matrix to simulate its characteristics. The characteristic is that the unloading stiffness of the concrete decreases as the damage increases. This paper shows that when the matrix damage factor reaches 0.5, the concrete can be considered ineffective [30].Figure6 shows an irregular crack penetrating the LSFRC plate, with many microcracks around it. During the cracking, the steel fibers in the concrete exert bonding forces. As a result, although many irregular small cracks appear on both sides of the main crack, the LSFRC plate has not obviously deformed.Figure 6
Comparison of damage patterns between simulation results and test results.


## 2.2.2. Deformation Analysis

According to Figure7, the curve has a brief downward trend when going up. After analyzing the cracking, it can be found that when the LSFRC plate reaches the yield limit, steel fibers hinder the crack growth on both sides of the plate. However, when steel fibers are separated from the concrete, the deflection curve rises again. Steel fibers improve the resistance and toughness of the concrete, resulting in a smoothly declining curve and good plasticity of the concrete. The results also show that the steel fiber plays an important bridging role in the macroscale viscoelastic property of the concrete.Figure 7
Deflection simulation values across the plate span.


## 3. Bending Test of the LSFRC Plates

### 3.1. Test Purpose

To evaluate the bending failure bearing capacity of the LSFRC plates, 10 LSFRC plates are made and locally loaded based on the requirements [31, 32]. Next, the results of the failure pattern and ultimate bearing capacity of the LSFRC plates are analyzed. To test the numerical simulation results, the stress distribution is compared with the test data and the specimen failure pattern.


### 3.2. Test Overview

#### 3.2.1. Design of the LSFRC Plates

LSFRC plates adopt a water-binder ratio of 0.42, sand content of 40%, fly ash dosage of 20%, steel fiber dosage of 0.6%, and a SFRC layer thickness of 2 mm. Ten 500 mm large LSFRC plates are designed with the structure shown in Figure8.Figure 8
Structural construction diagram of LSFRC panels (in: mm).


#### 3.2.2. Production of the LSFRC Plates

The top and bottom layers of the LSFRC plates are made of SFRC. Before concrete pouring, to control the layer thickness we use a marker to highlight the thickness in the mold interior. Figure9 shows the production steps of the LSFRC plates.Figure 9
Test plate production process. (a) Template making. (b) Breadboard pouring. (c) The test board pouring is complete. (d) LSFRC plate forming.
(a)(b)(c)(d)


### 3.3. Loading Scheme

#### 3.3.1. Measuring Points

A total of 8 measuring points are distributed on the LSFRC plates: 5 resistance strain gauges, with model No. Of BX120-60AA, are placed at the bottom to test the bottom strain; 2 resistance strain gauges are put at 1/8 of the plate sides to record the lateral strain; 1 displacement sensor is set at 1/4 span of the plate bottom along the loading direction to record the maximum vertical deformation of the plates; and 2 displacement gauges are put on the plate’s top to record the true vertical displacement of the plates during loading. Figure10 shows the arrangement of the measurement points.Figure 10
Schematic layout of the test class measurement points. (a) Substation site arrangement. (b) The top measurement point arrangement of the breadboard. (c) The bottom test point of the test plate is arranged live.
(a)(b)(c)


#### 3.3.2. Loading of the LSFRC Plates

The stepwise loading-unloading method and the displacement loading are applied to the LSFRC plates as shown in Figure11. Before the plate cracks, the specimen is loaded at 0.05 mm per step. When the plate’s bearing capacity declines, the width of the transverse tensile crack is recorded. When the bearing capacity drops to less than 70% of the peak load, the plate is regarded as failing.Figure 11
Test loading.


### 3.4. Test Result Analysis

#### 3.4.1. Failure Pattern

According to Figure12, during the cracking load, many microcracks, which do not directly run through the plates, appear on the lateral sides. Even though microcracks will spread across the plates, microcracks will not greatly undermine the bearing capacity of the LSFRC. However, as the loading continues, the cracks expand, and a large crack appears on the bottom of the plate, with many fibers being pulled out. The loading stops when it is up to 70% of the ultimate load. At this time, the top of the plate is not completely broken, showing very good ductility.Figure 12
Test plate damage pattern. (a) Side cracks. (b) Cracks in the bottom of the plate.
(a)(b)


#### 3.4.2. Load-Deflection Curve

According to Figure13, three stages can be observed from the beginning of loading to the failure of the LSFRC plates: elastic stage, crack growth, and failure stage.Figure 13
Load deflection curve.Elastic stage: this stage has a small initial load and nearly a straight line on the curve. The deflection in the plate increases slowly, but the whole load-deflection curve rises quickly. As the load continues to increase, the first crack appears at the bottom, and the stiffness decreases. But the steel fibers are not separated from the concrete, so the bearing section is still under tensile stress. As the load increases, deformation becomes faster. The concrete in the tensile area is partially out of work, and steel fibers slip away. The stresses on the concrete are jointly borne by the steel fibers. Therefore, after cracking, the deflection of the component does not change much, and the load-deflection curve is still nearly a straight line. The first downward trend in the curve marks the end of the elastic stage.Crack growth: after the initial crack appears, steel fibers in the microcracks show a remarkable bridging function. The plates begin the plastic deformation, and the deflection increases rapidly again. After a second rapid rise in the load-deflection curve, the growth slows for the first time. As the load increases, the crack becomes wider. Some short-cut fibers are pulled off or out, and the bonds between the steel fibers and the concrete are gradually destroyed. At this time, the load is mainly borne by the unpulled short-cut fibers, and the curve shows a continuous rise with fluctuations. As the deformation continues, the stress is mainly borne by the grid. When the plates’ bearing capacity begins to decrease, the crack growth comes to an end.Failure stage: at this stage, no new cracks appear and the bearing capacity of the plates begins to decrease. The load-deflection curve falls rapidly, and the fine crack gradually expands to a large crack. With more deflection, the width of the main crack rapidly increases, and the sectional area of the crack is constantly pulled off. The load is slowly reduced until the plate loses its bearing capacity.


#### 3.4.3. Load-Strain Curve

During the test, three strain-measuring points are distributed in the midspan of each LFRC plate bottom. They are to measure the midspan, the 1/4 span, and the side of the plate. Figure14 shows the load-strain curve of every LFRC plate.Figure 14
Load-strain curve.According to Figure14, the LSFRC plates follow a ductile failure. After the first crack appears at the bottom of SFRC, the upward shift of the neutral axis increases the tensile stress area and eventually leads to pseudoplastic failure. Although there are many microcracks around the crack on the neutral axis, thanks to the bridging action of steel fibers, the plate still has a certain bearing capacity. The bearing capacity does not fluctuate violently or decrease dramatically before and after the load reaches the limit of the plate. Therefore, steel fibers can effectively limit crack growth. After entering the plastic stage, the crack at the plate bottom develops rapidly, and the midspan strain increases until it exceeds the effective range of the strain gauge.


#### 3.4.4. Comparison between the Results of Numerical Simulation and Test

With finite element software, the comparison between the results of numerical simulation and testing is shown in Table1.Table 1
Comparison between the results of numerical simulation and test.
Project nameUltimate loadMidspan deflectionTest values36.821.51Simulated values38.291.98


#### 3.4.5. Comparison of the Deformation

According to Figure15, the fitting curves of the results of the test and numerical simulation are close, with a global error of about 6%. There are three reasons for the small values of the test. (1) When simulating the ultimate bearing capacity of the LSFRC plates, there is a lack of consideration of the concrete in the compression area, resulting in a larger midspan deflection. (2) The constitutive relation derived by the empirical formula of ABAQUS may deviate from the actual stress-strain relation of the concrete. (3) During the laboratory test, the steel fibers are randomly placed on the concrete during the pouring. Therefore, they cannot directly reflect the evenly distributed steel fibers at the plate bottom of the model.Figure 15
Comparison of simulated and measured LSFRC mid-span deflection values.In summary, steel fibers improve the crack resistance and toughness, giving the concrete good plasticity and making a smooth downward trend. The model results are consistent with the laboratory test. A separate model is used to reflect the failure stage of the LSFRC plates. According to the results of the test and numerical simulation, the model can simulate the possible slip between steel fibers and the concrete in a clear and accurate way.


## 3.1. Test Purpose

To evaluate the bending failure bearing capacity of the LSFRC plates, 10 LSFRC plates are made and locally loaded based on the requirements [31, 32]. Next, the results of the failure pattern and ultimate bearing capacity of the LSFRC plates are analyzed. To test the numerical simulation results, the stress distribution is compared with the test data and the specimen failure pattern.


## 3.2. Test Overview

### 3.2.1. Design of the LSFRC Plates

LSFRC plates adopt a water-binder ratio of 0.42, sand content of 40%, fly ash dosage of 20%, steel fiber dosage of 0.6%, and a SFRC layer thickness of 2 mm. Ten 500 mm large LSFRC plates are designed with the structure shown in Figure8.Figure 8
Structural construction diagram of LSFRC panels (in: mm).


### 3.2.2. Production of the LSFRC Plates

The top and bottom layers of the LSFRC plates are made of SFRC. Before concrete pouring, to control the layer thickness we use a marker to highlight the thickness in the mold interior. Figure9 shows the production steps of the LSFRC plates.Figure 9
Test plate production process. (a) Template making. (b) Breadboard pouring. (c) The test board pouring is complete. (d) LSFRC plate forming.
(a)(b)(c)(d)


## 3.2.1. Design of the LSFRC Plates

LSFRC plates adopt a water-binder ratio of 0.42, sand content of 40%, fly ash dosage of 20%, steel fiber dosage of 0.6%, and a SFRC layer thickness of 2 mm. Ten 500 mm large LSFRC plates are designed with the structure shown in Figure8.Figure 8
Structural construction diagram of LSFRC panels (in: mm).


## 3.2.2. Production of the LSFRC Plates

The top and bottom layers of the LSFRC plates are made of SFRC. Before concrete pouring, to control the layer thickness we use a marker to highlight the thickness in the mold interior. Figure9 shows the production steps of the LSFRC plates.Figure 9
Test plate production process. (a) Template making. (b) Breadboard pouring. (c) The test board pouring is complete. (d) LSFRC plate forming.
(a)(b)(c)(d)


## 3.3. Loading Scheme

### 3.3.1. Measuring Points

A total of 8 measuring points are distributed on the LSFRC plates: 5 resistance strain gauges, with model No. Of BX120-60AA, are placed at the bottom to test the bottom strain; 2 resistance strain gauges are put at 1/8 of the plate sides to record the lateral strain; 1 displacement sensor is set at 1/4 span of the plate bottom along the loading direction to record the maximum vertical deformation of the plates; and 2 displacement gauges are put on the plate’s top to record the true vertical displacement of the plates during loading. Figure10 shows the arrangement of the measurement points.Figure 10
Schematic layout of the test class measurement points. (a) Substation site arrangement. (b) The top measurement point arrangement of the breadboard. (c) The bottom test point of the test plate is arranged live.
(a)(b)(c)


### 3.3.2. Loading of the LSFRC Plates

The stepwise loading-unloading method and the displacement loading are applied to the LSFRC plates as shown in Figure11. Before the plate cracks, the specimen is loaded at 0.05 mm per step. When the plate’s bearing capacity declines, the width of the transverse tensile crack is recorded. When the bearing capacity drops to less than 70% of the peak load, the plate is regarded as failing.Figure 11
Test loading.


## 3.3.1. Measuring Points

A total of 8 measuring points are distributed on the LSFRC plates: 5 resistance strain gauges, with model No. Of BX120-60AA, are placed at the bottom to test the bottom strain; 2 resistance strain gauges are put at 1/8 of the plate sides to record the lateral strain; 1 displacement sensor is set at 1/4 span of the plate bottom along the loading direction to record the maximum vertical deformation of the plates; and 2 displacement gauges are put on the plate’s top to record the true vertical displacement of the plates during loading. Figure10 shows the arrangement of the measurement points.Figure 10
Schematic layout of the test class measurement points. (a) Substation site arrangement. (b) The top measurement point arrangement of the breadboard. (c) The bottom test point of the test plate is arranged live.
(a)(b)(c)


## 3.3.2. Loading of the LSFRC Plates

The stepwise loading-unloading method and the displacement loading are applied to the LSFRC plates as shown in Figure11. Before the plate cracks, the specimen is loaded at 0.05 mm per step. When the plate’s bearing capacity declines, the width of the transverse tensile crack is recorded. When the bearing capacity drops to less than 70% of the peak load, the plate is regarded as failing.Figure 11
Test loading.


## 3.4. Test Result Analysis

### 3.4.1. Failure Pattern

According to Figure12, during the cracking load, many microcracks, which do not directly run through the plates, appear on the lateral sides. Even though microcracks will spread across the plates, microcracks will not greatly undermine the bearing capacity of the LSFRC. However, as the loading continues, the cracks expand, and a large crack appears on the bottom of the plate, with many fibers being pulled out. The loading stops when it is up to 70% of the ultimate load. At this time, the top of the plate is not completely broken, showing very good ductility.Figure 12
Test plate damage pattern. (a) Side cracks. (b) Cracks in the bottom of the plate.
(a)(b)


### 3.4.2. Load-Deflection Curve

According to Figure13, three stages can be observed from the beginning of loading to the failure of the LSFRC plates: elastic stage, crack growth, and failure stage.Figure 13
Load deflection curve.Elastic stage: this stage has a small initial load and nearly a straight line on the curve. The deflection in the plate increases slowly, but the whole load-deflection curve rises quickly. As the load continues to increase, the first crack appears at the bottom, and the stiffness decreases. But the steel fibers are not separated from the concrete, so the bearing section is still under tensile stress. As the load increases, deformation becomes faster. The concrete in the tensile area is partially out of work, and steel fibers slip away. The stresses on the concrete are jointly borne by the steel fibers. Therefore, after cracking, the deflection of the component does not change much, and the load-deflection curve is still nearly a straight line. The first downward trend in the curve marks the end of the elastic stage.Crack growth: after the initial crack appears, steel fibers in the microcracks show a remarkable bridging function. The plates begin the plastic deformation, and the deflection increases rapidly again. After a second rapid rise in the load-deflection curve, the growth slows for the first time. As the load increases, the crack becomes wider. Some short-cut fibers are pulled off or out, and the bonds between the steel fibers and the concrete are gradually destroyed. At this time, the load is mainly borne by the unpulled short-cut fibers, and the curve shows a continuous rise with fluctuations. As the deformation continues, the stress is mainly borne by the grid. When the plates’ bearing capacity begins to decrease, the crack growth comes to an end.Failure stage: at this stage, no new cracks appear and the bearing capacity of the plates begins to decrease. The load-deflection curve falls rapidly, and the fine crack gradually expands to a large crack. With more deflection, the width of the main crack rapidly increases, and the sectional area of the crack is constantly pulled off. The load is slowly reduced until the plate loses its bearing capacity.


### 3.4.3. Load-Strain Curve

During the test, three strain-measuring points are distributed in the midspan of each LFRC plate bottom. They are to measure the midspan, the 1/4 span, and the side of the plate. Figure14 shows the load-strain curve of every LFRC plate.Figure 14
Load-strain curve.According to Figure14, the LSFRC plates follow a ductile failure. After the first crack appears at the bottom of SFRC, the upward shift of the neutral axis increases the tensile stress area and eventually leads to pseudoplastic failure. Although there are many microcracks around the crack on the neutral axis, thanks to the bridging action of steel fibers, the plate still has a certain bearing capacity. The bearing capacity does not fluctuate violently or decrease dramatically before and after the load reaches the limit of the plate. Therefore, steel fibers can effectively limit crack growth. After entering the plastic stage, the crack at the plate bottom develops rapidly, and the midspan strain increases until it exceeds the effective range of the strain gauge.


### 3.4.4. Comparison between the Results of Numerical Simulation and Test

With finite element software, the comparison between the results of numerical simulation and testing is shown in Table1.Table 1
Comparison between the results of numerical simulation and test.
Project nameUltimate loadMidspan deflectionTest values36.821.51Simulated values38.291.98


### 3.4.5. Comparison of the Deformation

According to Figure15, the fitting curves of the results of the test and numerical simulation are close, with a global error of about 6%. There are three reasons for the small values of the test. (1) When simulating the ultimate bearing capacity of the LSFRC plates, there is a lack of consideration of the concrete in the compression area, resulting in a larger midspan deflection. (2) The constitutive relation derived by the empirical formula of ABAQUS may deviate from the actual stress-strain relation of the concrete. (3) During the laboratory test, the steel fibers are randomly placed on the concrete during the pouring. Therefore, they cannot directly reflect the evenly distributed steel fibers at the plate bottom of the model.Figure 15
Comparison of simulated and measured LSFRC mid-span deflection values.In summary, steel fibers improve the crack resistance and toughness, giving the concrete good plasticity and making a smooth downward trend. The model results are consistent with the laboratory test. A separate model is used to reflect the failure stage of the LSFRC plates. According to the results of the test and numerical simulation, the model can simulate the possible slip between steel fibers and the concrete in a clear and accurate way.


## 3.4.1. Failure Pattern

According to Figure12, during the cracking load, many microcracks, which do not directly run through the plates, appear on the lateral sides. Even though microcracks will spread across the plates, microcracks will not greatly undermine the bearing capacity of the LSFRC. However, as the loading continues, the cracks expand, and a large crack appears on the bottom of the plate, with many fibers being pulled out. The loading stops when it is up to 70% of the ultimate load. At this time, the top of the plate is not completely broken, showing very good ductility.Figure 12
Test plate damage pattern. (a) Side cracks. (b) Cracks in the bottom of the plate.
(a)(b)


## 3.4.2. Load-Deflection Curve

According to Figure13, three stages can be observed from the beginning of loading to the failure of the LSFRC plates: elastic stage, crack growth, and failure stage.Figure 13
Load deflection curve.Elastic stage: this stage has a small initial load and nearly a straight line on the curve. The deflection in the plate increases slowly, but the whole load-deflection curve rises quickly. As the load continues to increase, the first crack appears at the bottom, and the stiffness decreases. But the steel fibers are not separated from the concrete, so the bearing section is still under tensile stress. As the load increases, deformation becomes faster. The concrete in the tensile area is partially out of work, and steel fibers slip away. The stresses on the concrete are jointly borne by the steel fibers. Therefore, after cracking, the deflection of the component does not change much, and the load-deflection curve is still nearly a straight line. The first downward trend in the curve marks the end of the elastic stage.Crack growth: after the initial crack appears, steel fibers in the microcracks show a remarkable bridging function. The plates begin the plastic deformation, and the deflection increases rapidly again. After a second rapid rise in the load-deflection curve, the growth slows for the first time. As the load increases, the crack becomes wider. Some short-cut fibers are pulled off or out, and the bonds between the steel fibers and the concrete are gradually destroyed. At this time, the load is mainly borne by the unpulled short-cut fibers, and the curve shows a continuous rise with fluctuations. As the deformation continues, the stress is mainly borne by the grid. When the plates’ bearing capacity begins to decrease, the crack growth comes to an end.Failure stage: at this stage, no new cracks appear and the bearing capacity of the plates begins to decrease. The load-deflection curve falls rapidly, and the fine crack gradually expands to a large crack. With more deflection, the width of the main crack rapidly increases, and the sectional area of the crack is constantly pulled off. The load is slowly reduced until the plate loses its bearing capacity.


## 3.4.3. Load-Strain Curve

During the test, three strain-measuring points are distributed in the midspan of each LFRC plate bottom. They are to measure the midspan, the 1/4 span, and the side of the plate. Figure14 shows the load-strain curve of every LFRC plate.Figure 14
Load-strain curve.According to Figure14, the LSFRC plates follow a ductile failure. After the first crack appears at the bottom of SFRC, the upward shift of the neutral axis increases the tensile stress area and eventually leads to pseudoplastic failure. Although there are many microcracks around the crack on the neutral axis, thanks to the bridging action of steel fibers, the plate still has a certain bearing capacity. The bearing capacity does not fluctuate violently or decrease dramatically before and after the load reaches the limit of the plate. Therefore, steel fibers can effectively limit crack growth. After entering the plastic stage, the crack at the plate bottom develops rapidly, and the midspan strain increases until it exceeds the effective range of the strain gauge.


## 3.4.4. Comparison between the Results of Numerical Simulation and Test

With finite element software, the comparison between the results of numerical simulation and testing is shown in Table1.Table 1
Comparison between the results of numerical simulation and test.
Project nameUltimate loadMidspan deflectionTest values36.821.51Simulated values38.291.98


## 3.4.5. Comparison of the Deformation

According to Figure15, the fitting curves of the results of the test and numerical simulation are close, with a global error of about 6%. There are three reasons for the small values of the test. (1) When simulating the ultimate bearing capacity of the LSFRC plates, there is a lack of consideration of the concrete in the compression area, resulting in a larger midspan deflection. (2) The constitutive relation derived by the empirical formula of ABAQUS may deviate from the actual stress-strain relation of the concrete. (3) During the laboratory test, the steel fibers are randomly placed on the concrete during the pouring. Therefore, they cannot directly reflect the evenly distributed steel fibers at the plate bottom of the model.Figure 15
Comparison of simulated and measured LSFRC mid-span deflection values.In summary, steel fibers improve the crack resistance and toughness, giving the concrete good plasticity and making a smooth downward trend. The model results are consistent with the laboratory test. A separate model is used to reflect the failure stage of the LSFRC plates. According to the results of the test and numerical simulation, the model can simulate the possible slip between steel fibers and the concrete in a clear and accurate way.


## 4. Discussion on Potential Applications

### 4.1. Economic Analysis of Pavement Structure

Taking this test site as an example, on the basis of cast-in-place ordinary concrete, steel fiber reinforced concrete designed by substitution method is applied to LSFRC pavement, and the economy of LSFRC pavement is compared with that of cast-in-place concrete pavement. The cast-in-place concrete pavement with a thickness of 0.26 m, a cast-in-place width of 4 m and a length of 4 m requires 4.16 m3 of concrete, which costs 1414.4 yuan according to the current commercial concrete price. The LSFRC pavement slab is 0.2 m thick, 4 m wide, and 4 m long, which requires 3.2 m3 concrete. Based on a 0.6% steel fiber volume ratio, it takes 150.4 kg of steel fiber to replace coarse aggregate, and it costs 1824.96 yuan in total. Therefore, paving a 4 m (width) ∗ 4 m (length) fabricated steel fiber reinforced concrete pavement only costs about 410.56 yuan, but the construction time can be shortened by 20–30 days. Furthermore, steel fiber reinforced concrete can improve the flexural and tensile performance of pavement, and it can be reused on multiple sites. We should take into account more than just its economics when promoting the use of fabricated steel fiber reinforced concrete pavement. A comprehensive comparison should be made from other perspectives, such as the treatment of waste concrete blocks, environmental pollution, and pavement repair.


### 4.2. Comprehensive Social Benefit Analysis

In addition to its advantage in speeding up construction, LSFRC pavement slab is a recyclable, environmentally friendly material that produces less construction waste. With its novel type of pavement structure, the LSFRC pavement slab can support sustainable development, which is favored by the construction industry. If it can be supported by the government and the law, the LSFRC pavement slab can develop rapidly. First, more research on fabricated road slabs is required to promote fabricated steel fiber reinforced concrete in China. Second, policies should be made to convey the supportive stance from the government, such as increasing the cost of treating construction waste, encouraging the construction of precast plants, helping concrete pavement slab prefabrication enter the industrial process stage, increasing the number of concrete precast plants on an annual basis, and promoting the use of LSFRC pavement slabs on construction sites. This will serve as the foundation for the gradual development and widespread use of LSFRC pavement slabs at various construction sites.


## 4.1. Economic Analysis of Pavement Structure

Taking this test site as an example, on the basis of cast-in-place ordinary concrete, steel fiber reinforced concrete designed by substitution method is applied to LSFRC pavement, and the economy of LSFRC pavement is compared with that of cast-in-place concrete pavement. The cast-in-place concrete pavement with a thickness of 0.26 m, a cast-in-place width of 4 m and a length of 4 m requires 4.16 m3 of concrete, which costs 1414.4 yuan according to the current commercial concrete price. The LSFRC pavement slab is 0.2 m thick, 4 m wide, and 4 m long, which requires 3.2 m3 concrete. Based on a 0.6% steel fiber volume ratio, it takes 150.4 kg of steel fiber to replace coarse aggregate, and it costs 1824.96 yuan in total. Therefore, paving a 4 m (width) ∗ 4 m (length) fabricated steel fiber reinforced concrete pavement only costs about 410.56 yuan, but the construction time can be shortened by 20–30 days. Furthermore, steel fiber reinforced concrete can improve the flexural and tensile performance of pavement, and it can be reused on multiple sites. We should take into account more than just its economics when promoting the use of fabricated steel fiber reinforced concrete pavement. A comprehensive comparison should be made from other perspectives, such as the treatment of waste concrete blocks, environmental pollution, and pavement repair.


## 4.2. Comprehensive Social Benefit Analysis

In addition to its advantage in speeding up construction, LSFRC pavement slab is a recyclable, environmentally friendly material that produces less construction waste. With its novel type of pavement structure, the LSFRC pavement slab can support sustainable development, which is favored by the construction industry. If it can be supported by the government and the law, the LSFRC pavement slab can develop rapidly. First, more research on fabricated road slabs is required to promote fabricated steel fiber reinforced concrete in China. Second, policies should be made to convey the supportive stance from the government, such as increasing the cost of treating construction waste, encouraging the construction of precast plants, helping concrete pavement slab prefabrication enter the industrial process stage, increasing the number of concrete precast plants on an annual basis, and promoting the use of LSFRC pavement slabs on construction sites. This will serve as the foundation for the gradual development and widespread use of LSFRC pavement slabs at various construction sites.


## 5. Conclusion

In this study, ABAQUS finite element software was used to establish a separate model, and the bending bearing capacity of the model under static load and the crack shape when the model is damaged are observed. On this basis, ten LSFRC slabs are poured, and their actual bending bearing capacity is tested in laboratory tests, and the failure shape, load-deflection curve, and load-strain curve of the specimens are examined. The conclusions are as follows:(1)
When steel fiber is considered in the model, during the cracking of the model slab, a through crack appears when the slab is destroyed. The adhesive force of concrete attached to steel fiber leads to many irregular tiny cracks on both sides of the main crack, but the slab has no obvious deformation.(2)
When steel fiber is used in the model, the deflection-load curve of the simulated slab reveals that it has good crack resistance, toughness, and shows good plasticity. Steel fiber plays an important role in bridging the macroscopic viscoelastic properties of concrete.(3)
During the loading of the test plate, there are tiny cracks on the side of the plate, but there is no penetration. The main crack appears at the bottom of the plate, where many fibers are pulled out. The upper part of the test plate is only partially broken, showing good ductility.(4)
The test plate can be divided into three stages from loading to plate failure: the elastic stage, the crack development stage, and the failure stage. The steel fiber has a certain bearing capacity, which can effectively control crack development, and it belongs to ductile failure. Even after the cracks appear for the first time in the slab, it still meets the standard of practical engineering.(5)
The fitting curves of the test and numerical simulation are very close, with a global error of about 6%. There are three reasons to explain the difference. (1) When simulating the ultimate bearing capacity of the LSFRC plates, there is a lack of consideration of the concrete in the compression area, resulting in a larger mid-span deflection. (2) The constitutive relation derived by the empirical formula of ABAQUS may deviate from the actual stress-strain relation of the concrete. (3) During the laboratory test, the steel fibers are randomly placed on the concrete during the pouring. Therefore, they cannot directly reflect the evenly distributed steel fibers at the plate bottom of the model.Two recommendations are made for the future: first, engineers should seriously consider the influence of the vibration of the fabricated steel fiber reinforced concrete pavement slab on its performance. Second, in the field strain detection of the assembled steel fiber reinforced concrete pavement slab, other performances of the LSFRC pavement slab, such as joint damage, should be tested and studied.



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*Source: 2528085-2022-09-29.xml*