Method for calculating spacing between side walls in fiber reinforced concrete

Abstract

The application discloses a kind of calculation methods of fiber reinforced concrete in side wall spacing, comprising the following steps: step S1, generate three-dimensional steel fiber space distribution model;FRC matrix is regarded as ideal state, its material attribute is uniformly distributed and there is no particle bubble influence;Set up the boundary of simulated concrete matrix, use Monte Carlo method to generate fiber randomly, it is randomly put into the cubic boundary space, the N fiber that is put is located in the cubic boundary and does not intersect each other;Step S2, establish side wall effect model;Based on the fiber space distribution model established in step S1, insert a plane parallel to the single side of cube as sampling surface in model;Step S3, draw P s (x) the relationship curve of x;Step S4, analysis side wall effect influence spacing;Different influence spacing is divided, with the advantages that calculation step is simple and fast, significantly improve the efficiency of calculating side wall effect influence area, higher accuracy, fit reality etc..

Description

Technical Field

[0001] This invention belongs to the field of fiber-reinforced concrete technology, specifically relating to a method for calculating the spacing between sidewalls in fiber-reinforced concrete. Background Technology

[0002] Fiber-reinforced concrete (FRC) is a new type of cement-based composite material that uses cement paste, mortar, or concrete as a matrix and non-continuous short fibers or continuous long fibers as reinforcing materials, uniformly incorporated into the concrete. Compared to ordinary concrete, FRC incorporates fibers into the concrete matrix, and the bridging effect of the fibers compensates for the inherent shortcomings of ordinary concrete, such as low tensile strength and poor toughness. Due to its excellent physical and mechanical properties, FRC has been successfully applied in numerous practical engineering projects, including bridge deck paving, pier reinforcement, and curtain walls.

[0003] In recent years, with the increasingly widespread application of FRC (Flat Composite Material), many scholars have paid more attention to studying its impact on performance. However, traditional mechanical testing has drawbacks such as high cost, long cycle time, and narrow applicability. To quickly and accurately analyze the mechanical properties of composite materials, micromechanical analysis of composite materials based on numerical simulation is an important direction for efficient analysis of the mechanical properties of FRC and structural design.

[0004] In finite element simulation, the more models there are and the closer they are to the actual structure, the more accurate the analysis results will be. In actual casting, steel fibers are randomly distributed in the FRC matrix, in large numbers and without intersecting each other. Furthermore, due to the influence of the formwork, the distribution of steel fibers in the matrix exhibits a phenomenon of more fibers in the middle and fewer at both ends. In order to realistically simulate the influence of the wall effect in FRC and establish a more realistic finite element model, it is necessary to study the wall effect model in FRC.

[0005] Existing studies on the edge-wall effect of concrete have the following main shortcomings: 1) Cutting test analysis: As described in the literature "Study on the Mechanical Properties of Flow-Induced Fiber Oriented UHPC", the specimen is cut at different locations. When cut from inside the matrix, the fiber orientation is not restricted by the mold edge-wall; when cut from the area between the specimen and the mold edge-wall, the orientation is restricted by the mold edge-wall. Studies on the edge-wall effect of steel fibers are mainly limited to traditional cutting tests, and the edge-wall effect of steel fibers is understood based on image analysis technology after cutting. Since the cutting test is a destructive test, the material cost is high and the time required is long. It is only suitable for individual tests and cannot draw general conclusions.

[0006] 2) Simulation of aggregate boundary effect: In the literature "Computer simulation of boundary effect of two-dimensional aggregate distribution", aggregates are generated by generating random numbers and distributed on a rectangular concrete section to study the influence of boundary effect on equal volume aggregate gradation and Fuller aggregate gradation. The study of concrete boundary effect mainly focuses on the boundary effect after the aggregates are randomly placed. However, in stereoscopic terms, aggregates and steel fibers show significant differences in appearance and mechanical properties. The algorithm is also limited to two dimensions, so there are errors in the boundary effect analysis. Summary of the Invention

[0007] This invention aims to provide a method for calculating the spacing between sidewalls in fiber-reinforced concrete. The formula is simple and convenient to calculate, reducing a large number of function simulation processes. It solves the problem that traditional cutting tests are limited to individual tests and lack universality due to their destructive nature, and that the large differences in appearance and mechanical properties between aggregates and steel fibers lead to errors in the analysis of sidewall effects.

[0008] Therefore, the technical solution adopted by this invention is a method for calculating the spacing between sidewalls in fiber-reinforced concrete, comprising the following steps:

[0009] Step S1: Generate a three-dimensional steel fiber spatial distribution model; treat the FRC matrix as an ideal state, with uniformly distributed material properties and no particle or bubble effects; set the cubic boundary of the simulated concrete matrix, randomly generate fibers using the Monte Carlo method, and randomly place them into the cubic boundary space. All N placed fibers are located within the cubic boundary and do not intersect each other. In the formula: l f d f V f V represents the fiber length, fiber diameter, fiber volume fraction, and the volume of the set cube, respectively.

[0010] Step S2: Establish a wall effect model; based on the fiber spatial distribution model established in step S1, insert a plane parallel to one side of the cube as a sampling surface into the model;

[0011] Step S3, draw P s The relationship curve of (x)-x; the sum of the cross-sectional areas of all fibers intersecting the sampling surface in the calculation model; the fiber volume fraction at a point in three-dimensional space can be represented by the area fraction of the surface intersecting the fibers on the sampling surface at that point. In the formula: x is the distance between the sampling surface and the side of the cube; S f (x) represents the total area of ​​all fibers intersecting the sampling surface that are cut off; S(x) represents the area of ​​the sampling surface; P s (x) represents the area fraction of steel fibers on the sampling surface; plot P with the distance x between the sampling surface and the side as the abscissa and the area fraction of steel fibers on the sampling surface as the ordinate.s The relationship curve (x)-x is used to obtain the variation law of fiber volume fraction with position within the model;

[0012] Step S4: Analyze the influence spacing of the sidewall effect; divide the steel fibers into different influence spacings, namely, the central region unaffected by the sidewall, the surface region affected by one sidewall effect, the edge region affected by two sidewall effects, and the corner region affected by three sidewall effects, so as to analyze the degree of influence of the sidewall effect on the steel fibers in each region.

[0013] As a preferred option, in step S1, the rand function built into the mathematical calculation software MATLAB is used to generate a uniformly distributed random number within the interval, which can meet the requirements of random sampling and is simple to operate.

[0014] More preferably, in step S1, the generation of a single fiber involves first randomly generating the spatial coordinates P1(x1, y1, z1) of one of its endpoints, and then randomly generating the angles α, β, and γ representing the fiber's position relative to the x-axis, y-axis, and z-axis of the three Cartesian coordinates, with values ​​ranging from [0, 2π]. This yields the position value of the single fiber in space. The operation is reasonable and simple. The distribution of the fiber in space is controlled by six degrees of freedom, therefore, six values ​​need to be automatically generated.

[0015] More preferably, in step S1, based on the fiber length l f Diameter d f Using α, β, and γ, determine the spatial coordinates (x2, y2, z2) of the other end point P2 of the fiber; determine whether the position value of a single fiber in space is within the set cube boundary. If it exceeds the boundary, the fiber needs to be regenerated to avoid the interference of fibers located outside the cube boundary with the sidewall effect analysis results.

[0016] Further preferably, in step S1, it is determined whether the newly generated single fiber located within the boundary of the set cube intersects with other existing fibers. The determination is based on whether the minimum distance between the two line segments is greater than the diameter of the fiber. If they intersect, the fiber needs to be regenerated to avoid the steel fiber area fraction on the fiber intersection sampling surface interfering with the sidewall effect analysis results.

[0017] Further preferably, in steps S1-S3, when analyzing the wall effect, 1000 three-dimensional steel fiber spatial distribution models are generated for each parameter, and the average results are studied to ensure that the change law of fiber volume fraction is a smooth curve, thus ensuring the accuracy of fiber volume fraction.

[0018] The beneficial effects of this invention are:

[0019] (1) The Monte Carlo method is used to randomly generate fibers and randomly place them into the boundary space of a cube. All N fibers placed are located within the boundary of the cube and do not intersect each other, which fully simulates the real FRC matrix. It is suitable for simulating the effect of the side wall effect on steel fibers in the FRC matrix. A plane parallel to one side of the cube is inserted as a sampling surface. The fiber area fraction is obtained by the ratio of the cross-sectional area of ​​the steel fiber on the sampling surface to the area of ​​the sampling surface. Then, the relationship curve between the fiber area fraction and its distance from the edge length is plotted to obtain the effect of the side wall effect on the steel fiber on the sampling surface. The calculation steps are simple and fast, which can significantly improve the efficiency of calculating the area affected by the side wall effect.

[0020] (2) Compared with the traditional cutting test, which has low universality, the three-dimensional fiber spatial distribution cube model has a wide range of applications. It can simulate different types of fiber-reinforced concrete matrices according to reality, and has strong universality. Compared with the computer simulation that is limited to the boundary effect of aggregate distribution in two dimensions, the influence of the sidewall effect of steel fiber in three-dimensional space is more in line with the structural design of actual engineering, resulting in higher accuracy of the sidewall effect simulation experiment and closer to reality.

[0021] (3) Plot P with the distance x between the sampling surface and the side as the abscissa and the area fraction of steel fibers on the sampling surface as the ordinate. s The relationship curve (x)-x is obtained to show the variation of fiber volume fraction with position within the model, which fully understands the internal situation of fiber-reinforced concrete matrix. Different influence intervals are divided into those not affected by the sidewall, affected by one sidewall effect, affected by two sidewall effects, and affected by three sidewall effects, which represent the degree of influence of the sidewall effect on different areas of the fiber-reinforced concrete matrix. The division is reasonable and can accurately understand the influence of the sidewall effect on the fiber-reinforced concrete matrix from multiple directions and angles.

[0022] In summary, it has advantages such as simple and fast calculation steps, significantly improved efficiency in calculating the area affected by the wall effect, higher accuracy, and close resemblance to reality. Attached Figure Description

[0023] Figure 1 This is a flowchart of the steps of the present invention.

[0024] Figure 2 A flowchart for random fiber generation.

[0025] Figure 3 This is a schematic diagram of the fiber structure generated in an ideal FRC matrix.

[0026] Figure 4 This is a schematic diagram of the fiber cutting process at the sampling surface.

[0027] Figure 5 This is a schematic diagram of the cross-section where the sampling surface intersects with the fiber.

[0028] Figure 6 For P s (x)-x relationship curve.

[0029] Figure 7 This diagram illustrates the surface region of a fiber-reinforced concrete matrix affected by one edge wall effect, the edge region affected by two edge wall effects, and the corner region affected by three edge wall effects.

[0030] Figure 8 This is a schematic diagram of the central region of a fiber-reinforced concrete matrix that is unaffected by the sidewalls. Detailed Implementation

[0031] The present invention will be further described below with reference to the embodiments and accompanying drawings:

[0032] Combination Figure 1 — Figure 8 As shown, a method for calculating the spacing between sidewalls in fiber-reinforced concrete is described, and the specific implementation steps are as follows:

[0033] Step S1: Generate a three-dimensional spatial distribution model of steel fibers; the fiber-reinforced concrete matrix is ​​regarded as an ideal state, with its material properties uniformly distributed and without the influence of particles and air bubbles.

[0034] Define the cubic boundary of the fiber-reinforced concrete matrix, randomly generate fibers using the Monte Carlo method, and randomly place them into the cubic boundary space. All N fibers placed are located within the cubic boundary and do not intersect each other.

[0035]

[0036] In the formula: l f d f V f V represents the fiber length, fiber diameter, fiber volume fraction, and the volume of the set cube, respectively.

[0037] For example, when the specimen is a cube with dimensions of 50×50×50mm, the length of the steel fiber is l f =13mm, diameter d f =0.2mm, steel fiber volume content 2%.

[0038] The number of fibers N is:

[0039]

[0040] In step S1, the rand function, which is built into the mathematical calculation software MATLAB, is used to generate a uniformly distributed random number within the interval, which can meet the requirements of random sampling.

[0041] In step S1, the generation of a single fiber involves first randomly generating the spatial coordinates P1(x1, y1, z1) of one of its endpoints, and then randomly generating the angles α, β, and γ representing the fiber's position relative to the x-axis, y-axis, and z-axis of the three Cartesian coordinates, with values ​​ranging from [0 to 2π], thereby obtaining the position value of the single fiber in space.

[0042] In step S1, based on the fiber length l f Diameter d f Using α, β, and γ, determine the spatial coordinates (x2, y2, z2) of the other end point P2 of the fiber; determine whether the position value of a single fiber in space is within the set cube boundary. If it exceeds the boundary, the fiber needs to be regenerated.

[0043] In step S1, it is determined whether the newly generated single fiber located within the boundary of the set cube intersects with other existing fibers. The determination is based on whether the minimum distance between two line segments is greater than the diameter of the fiber. If they intersect, the fiber needs to be regenerated.

[0044] To determine whether the second fiber intersects with the first fiber, the criterion is whether the minimum distance between the two line segments is greater than the fiber's diameter. The minimum distance between the two line segments is calculated as follows:

[0045] 1) Let line segments l1 and l2 represent two fibers in space, and denote the endpoints of l1 as P1 and P2; and the endpoints of l2 as Q1 and Q2; then l1 and l2 can be represented by vectors. It is represented as shown in the following formula.

[0046]

[0047]

[0048] 2) Any point on the straight line containing the two line segments is shown in the following formula.

[0049]

[0050]

[0051] Where λ1 and λ2 represent any point on the line. The problem of finding the shortest distance between two lines can be equivalent to an optimization problem with boundary conditions, as shown in the following equation.

[0052]

[0053] 3) Solve equation 5. From the minimum condition, we know that: Expanding and simplifying the formula yields a system of equations, as shown below.

[0054]

[0055]

[0056] 4) Solving the system of equations yields the values ​​of λ1 and λ2. Here, λ1 and λ2 are the feet of the perpendiculars from the lines containing l1 and l2. We then analyze whether the feet of the perpendiculars lie on line segments l1 and l2. Based on this, the calculation of the minimum distance is divided into three cases:

[0057] Case a: When λ1∈[0,1] and λ2∈[0,1], it means that the feet of the two perpendiculars are both on their respective line segments, and the minimum distance between the two lines is equal to the length of the common perpendicular.

[0058] Case b: When λ1∈[0,1] and or And λ2∈[0,1], which means that only one foot of the perpendicular lies on the line segment, and the other foot lies on the extension of the line segment. Suppose the line segment with the foot of the perpendicular on the line segment is l1, and the line segment with the foot of the perpendicular on the extension is l2. In this case, the minimum distance is equal to the distance from the endpoint of l2 that is closer to the foot of the perpendicular to the foot of the perpendicular on l1.

[0059] Case c: When and This means that both feet of the perpendiculars are located on the extensions of the line segments. In this case, the minimum distance is equal to the distance between the endpoints of the two line segments closest to the feet of the perpendiculars.

[0060] After determining the minimum distance between fibers, compare this distance with the fiber diameter. If the distance is less than the fiber diameter, it means that there is an intersection between the fibers, which should be discarded; otherwise, it means that the two fibers do not intersect.

[0061] Step S2: Establish a sidewall effect model; based on the fiber spatial distribution model established in step S1, insert a plane parallel to one side of the cube as a sampling surface into the model.

[0062] Step S3, draw P s The relationship curve of (x)-x; the sum of the cross-sectional areas of all fibers intersecting with the sampling surface in the calculation model; the fiber volume fraction at a certain point in three-dimensional space can be represented by the area fraction of the surface intersecting with the fiber on the sampling surface at that point.

[0063]

[0064] In the formula: x is the distance between the sampling surface and the side of the cube; S f (x) represents the total area of ​​all fibers intersecting the sampling surface that are cut off; S(x) represents the area of ​​the sampling surface; P s (x) represents the area fraction of steel fibers on the sampling surface.

[0065] Plot P with the distance x between the sampling surface and the side as the x-axis and the area fraction of steel fibers on the sampling surface as the y-axis. s The relationship curve (x)-x is used to obtain the variation law of fiber volume fraction with position within the model;

[0066] In steps S1-S3, when analyzing the wall effect, 1000 three-dimensional steel fiber spatial distribution models are generated for each parameter, and the average results are studied to ensure that the change law of fiber volume fraction is a smooth curve.

[0067] As the number of models increases, the fiber volume fraction exhibits a certain variation pattern. When the number of models equals 1000, the variation pattern of the volume fraction is a smooth curve. Therefore, in the analysis of the wall effect, 1000 models are generated for each parameter, and the average results are taken for study.

[0068] Step S4: Analyze the influence spacing of the sidewall effect; divide the steel fibers into different influence spacings, namely, the central region unaffected by the sidewall, the surface region affected by one sidewall effect, the edge region affected by two sidewall effects, and the corner region affected by three sidewall effects, so as to analyze the degree of influence of the sidewall effect on the steel fibers in each region.

[0069] like Figure 7 , Figure 8 As shown, the numbers on the cuboids indicate the degree of influence of the wall effect on the region. For example, the cube numbered 3 at the corner indicates that the wall effect on three sides has affected the fiber distribution in this region. Figure 7 The diagram shows a surface region affected by one sidewall effect, an edge region affected by two sidewall effects, and a corner region affected by three sidewall effects.

[0070] Figure 8 A schematic diagram of the central region unaffected by the sidewalls is shown, with the core area numbered 0. The steel fibers in this region are not affected by the sidewall effect and are nearly uniformly distributed.

Claims

1. A method for calculating the spacing between sidewalls in fiber-reinforced concrete, characterized in that, Includes the following steps: Step S1: Generate a three-dimensional steel fiber spatial distribution model; consider the fiber-reinforced concrete matrix as an ideal state, with uniformly distributed material properties and no particle or air bubble influence; set the cubic boundary of the fiber-reinforced concrete matrix, randomly generate the coordinates of the fibers using the Monte Carlo method, and randomly place them into the cubic boundary space. All N fibers placed are located within the cubic boundary and do not intersect each other. In the formula: l f d f V f V represents the fiber length, fiber diameter, fiber volume fraction, and the volume of the set cube, respectively. Step S2: Establish a wall effect model; based on the fiber spatial distribution model established in step S1, insert a plane parallel to one side of the cube as a sampling surface into the model; Step S3, draw P s The relationship curve of (x)-x; the sum of the cross-sectional areas of all fibers intersecting the sampling surface in the calculation model; the fiber volume fraction at a point in three-dimensional space can be represented by the area fraction of the surface intersecting the fibers on the sampling surface at that point. In the formula: x is the distance between the sampling surface and the side of the cube; S f (x) represents the total area of ​​all fibers intersecting the sampling surface that are cut off; S(x) represents the area of ​​the sampling surface; P s (x) represents the area fraction of steel fibers on the sampling surface; plot P with the distance x between the sampling surface and the side as the abscissa and the area fraction of steel fibers on the sampling surface as the ordinate. s The relationship curve (x)-x is used to obtain the variation law of fiber volume fraction with position within the model; Step S4: Analyze the influence spacing of the sidewall effect; divide the steel fibers into different influence spacings, namely, the central region unaffected by the sidewall, the surface region affected by one sidewall effect, the edge region affected by two sidewall effects, and the corner region affected by three sidewall effects, so as to analyze the degree of influence of the sidewall effect on the steel fibers in each region.

2. The method for calculating the spacing between sidewalls in fiber-reinforced concrete according to claim 1, characterized in that: In step S1, the rand function built into the mathematical calculation software MATLAB is used to generate a uniformly distributed random number within the interval, which can meet the requirements of random sampling.

3. The method for calculating the spacing between sidewalls in fiber-reinforced concrete according to claim 1, characterized in that: In step S1, the generation of a single fiber involves first randomly generating the spatial coordinates P1(x1, y1, z1) of one of its endpoints, and then randomly generating the angles α, β, and γ representing the fiber's angles with the Cartesian coordinate axes x, y, and z, with values ​​ranging from [0, 2π]. Since α, β, and γ are generated using random numbers, the cosine values ​​are also random; therefore, random numbers cosα, cosβ, and cosγ in the range [-1, 1] can be directly generated. Based on the fiber length l... f Diameter d f Using cosα, cosβ, and cosγ, we can find the spatial coordinates (x2, y2, z2) of the other end point P2 of the fiber, thus obtaining the numerical position of a single fiber in space.

4. The method for calculating the spacing between sidewalls in fiber-reinforced concrete according to claim 3, characterized in that: In step S1, based on the fiber length l f Diameter d f Using α, β, and γ, determine the spatial coordinates (x2, y2, z2) of the other end point P2 of the fiber; determine whether the position value of a single fiber in space is within the set cube boundary. If it exceeds the boundary, the fiber needs to be regenerated.

5. The method for calculating the spacing between sidewalls in fiber-reinforced concrete according to claim 4, characterized in that: In step S1, it is determined whether the newly generated single fiber located within the boundary of the set cube intersects with other existing fibers. The determination is based on whether the minimum distance between the two line segments is greater than the diameter of the fiber. If they intersect, the fiber needs to be regenerated.

6. The method for calculating the spacing between sidewalls in fiber-reinforced concrete according to claim 1, characterized in that: In steps S1-S3, during the analysis of the sidewall effect, 1000 three-dimensional steel fiber spatial distribution models are generated for each parameter, and the average results are studied to ensure that the variation law of fiber volume fraction is a smooth curve.

Images