Method for calculating the orientation distribution of steel fibers in fiber-reinforced concrete

Abstract

The present invention provides a method for calculating the orientation distribution of steel fibers in fiber-reinforced concrete. By statistically analyzing the orientation probability and orientation coefficient of steel fibers in the matrix of fiber-reinforced concrete, the orientation distribution of fibers is studied, solving the problems of low efficiency and complex process in calculating the orientation distribution of steel fibers in fiber-reinforced concrete in traditional tests. It establishes a three-dimensional fiber spatial distribution cube model to randomly distribute steel fibers in the three-dimensional space model, uses the included angle formula to calculate the orientation angle of steel fibers, and obtains the orientation distribution law of steel fibers in concrete by statistically analyzing the orientation probability. Through simple and convenient formula calculations, this method can obtain the orientation law of fiber distribution in the concrete matrix, while reducing a large number of experimental operations and calculation processes, and improving the operation efficiency.

Description

Technical Field

[0001] The present invention belongs to the technical field of construction engineering, and particularly relates to a method for calculating the orientation distribution of steel fibers in fiber-reinforced concrete. Background Art

[0002] Concrete is one of the main building materials, which has the advantages of rich and cheap raw materials, simple construction technology, high compressive strength and good durability, and is widely used in engineering practice. However, the disadvantages of traditional concrete, such as high self-weight, low tensile strength and poor toughness, greatly limit the development of large-scale projects. To make up for these defects, fiber-reinforced concrete came into being.

[0003] The improvement of the mechanical properties of concrete by fibers is mainly reflected in three aspects: strengthening effect, toughening effect and crack resistance effect. When fibers are incorporated, the tensile strength, flexural toughness, ultimate strain and elastic modulus of concrete are generally improved. Usually, when the fiber orientation is parallel to the tensile direction of concrete, the fibers can exert the strengthening and toughening effects to the greatest extent; while when the fiber orientation is perpendicular to the tensile stress direction of concrete, the fibers basically have no strengthening and toughening effects. When steel fibers are incorporated into concrete, the fiber orientation shows a random orientation. However, research shows that the utilization efficiency of steel fibers with random orientation in terms of strengthening and toughening effects only accounts for 30%. Therefore, studying the orientation distribution of fibers in the concrete matrix is of great significance for the realization of fiber orientation in concrete.

[0004] Based on this, the prior art uses numerical simulation methods to establish a three-dimensional spatial distribution model of fiber concrete, and draws the orientation probability curve and orientation coefficient curve by statistically analyzing the orientation probability of steel fibers in the three-dimensional spatial model, so as to explore the orientation distribution law of steel fibers in the concrete matrix and lay a theoretical foundation for the realization of fiber orientation in concrete.

[0005] The existing research on the calculation of the orientation angle and orientation probability of fibers in fiber-reinforced concrete mainly includes:

[0006] (1) "UHPC Fiber Orientation Method and Its Influence on Tensile Performance"

[0007] As described in the literature, the image processing method is to slice and photograph a fiber-reinforced concrete specimen to obtain the cross-sectional image information, then perform binary processing on the image information by computer, extract the cross-sectional shape of the fibers on the UHPC cross-section, calculate the inclination angle of the fibers, and finally obtain the fiber orientation distribution on the entire cross-section.

[0008] (2) "Preparation of UHPC Based on Fiber Distribution and Its Penetration Performance Research"

[0009] The X-CT scanning method uses spiral CT imaging technology and X-ray computed tomography imaging technology in imaging as test means to achieve two-dimensional scanning to three-dimensional reconstruction of images, so as to observe the three-dimensional distribution of steel fibers in the concrete matrix. This method was used in the literature.

[0010] (3) "Distribution Law of Steel Fibers in Concrete Matrix and Its Relationship with Toughness"

[0011] The electromagnetic induction method is based on the differences in the electromagnetic properties (conductivity, permittivity, permeability) of the medium. By observing and studying the law of the alternating electromagnetic field's spatial distribution or its change over time, it achieves the observation goal of observing the fiber orientation in the concrete matrix. This method was used in the literature.

[0012] The above existing technologies have the following defects respectively:

[0013] Existing technology 1: "UHPC Fiber Orientation Method and Its Influence on Tensile Performance"

[0014] Defect 1: Time-consuming and many processes

[0015] The processing of the image analysis method has to go through multiple procedures: the pouring and curing of fiber-reinforced concrete take a huge amount of time. On this basis, slicing and photographing are carried out, and then the computer is used to binarize the pictures. Only after extracting the cross-sectional image information can the fiber orientation distribution be calculated. Generally speaking, the image analysis method has many processes and complex steps, and the efficiency of calculating the steel fiber orientation angle is very low, seriously restricting the practical application of the method for calculating the fiber orientation in concrete.

[0016] Defect 2: Unable to accurately reflect the actual orientation distribution of steel fibers

[0017] The image analysis method is a destructive method. During the process of cutting concrete specimens, it is inevitable to have a certain impact on the actual orientation distribution of steel fibers, and the orientation distribution of single or a few specimens cannot accurately reflect the actual orientation distribution of steel fibers.

[0018] Existing technology 2: "Preparation of UHPC Based on Fiber Distribution and Its Penetration Performance Research"

[0019] Defect 1: Unable to calculate the orientation distribution with a relatively high fiber volume fraction

[0020] The X-CT scanning method is applicable to small specimens with a small fiber volume fraction. When the specimen and the fiber volume fraction are small, the X-CT scanning method can clearly show the overall orientation distribution of fibers in the matrix. However, when the specimen and the fiber volume fraction are large, first, it is impossible to scan large-volume specimens, and second, the reconstruction process with a relatively high fiber volume fraction has high requirements for the computer and a large burden on the operation speed.

[0021] Disadvantage 2: Limited by the instrument device, the scanning cost is high

[0022] The X-CT scanning method relies on the helical CT imaging technology and X-ray computed tomography imaging technology in imaging. Both of these two technologies require instruments with certain configurations to achieve. Moreover, the diameter and length of the fiber are relatively small, and the volume of the fiber is correspondingly small. High clarity is required during scanning. In addition, the cost of one scan is also high.

[0023] Prior art 3: "Distribution Law of Steel Fibers in Concrete Matrix and Its Relationship with Toughness"

[0024] Disadvantage 1: Limited to indoor research and only for specimens with specific shapes

[0025] From Figure 1 the electromagnetic induction device in, the electromagnetic induction method can only measure the orientation distribution of some small specimens, and there are certain requirements for the shape and size of the specimens. Therefore, the electromagnetic induction method cannot be used as a widespread method for calculating the fiber orientation distribution in fiber-reinforced concrete.

[0026] Disadvantage 1: There is a certain difference from the actual fiber orientation distribution

[0027] Considering the magnetic interference between the external magnetic field and the steel fiber, there is a certain difference between the measured steel fiber orientation distribution by the electromagnetic induction method and the actual orientation distribution. Summary of the Invention

[0028] In response to the need to explore the orientation distribution law of steel fibers in the concrete matrix proposed by the prior art, as well as the defects and deficiencies of the existing technical solutions, the present invention uses a computer to simulate the three-dimensional spatial distribution form of fiber-reinforced concrete, and proposes a method for calculating the orientation distribution of steel fibers in fiber-reinforced concrete. On the basis of generating a three-dimensional spatial distribution model, the included angle formula model is used to calculate the orientation angle and orientation coefficient of the steel fiber, and the orientation probability is statistically obtained to obtain the orientation distribution law of the steel fiber in the concrete. In this way, the calculation rate of the steel fiber orientation distribution in fiber-reinforced concrete is improved, the steps that need to be processed by various instruments after casting and curing fiber-reinforced concrete in actual tests are reduced, and the calculation of the steel fiber orientation distribution in fiber-reinforced concrete is simplified.

[0029] Based on the established fiber spatial distribution model, the present invention proposes a method for calculating the fiber orientation angle in a three-dimensional space model. Furthermore, by statistically analyzing the orientation probability and orientation coefficient of steel fibers in a fiber-reinforced concrete matrix, the fiber orientation distribution is studied, solving the problems of low efficiency and complex process in calculating the steel fiber orientation distribution in fiber-reinforced concrete in traditional tests. By establishing a three-dimensional fiber spatial distribution cube model, the steel fibers are randomly distributed in the three-dimensional space model. The included angle formula is used to calculate the fiber orientation angle, and the orientation distribution law of steel fibers in concrete is obtained by statistically analyzing the orientation probability. Through simple and convenient formula calculations, this method can obtain the orientation law of fiber distribution in the concrete matrix, while reducing a large number of experimental operations and calculation processes, improving the operation efficiency.

[0030] The technical solution adopted by the present invention to solve its technical problems is as follows:

[0031] A method for calculating the steel fiber orientation distribution in fiber-reinforced concrete, characterized by comprising the following steps:

[0032] Step S1: Generate a three-dimensional steel fiber spatial distribution model

[0033] The fiber spatial distribution model is established by using the large-scale mathematical calculation software MATLAB, and the geometric position information of the fibers is determined by the Monte Carlo method; wherein, the fiber-reinforced concrete matrix is regarded as an ideal state, and its material properties are uniformly distributed without the influence of particles and bubbles;

[0034] Step S2: Calculate the fiber orientation angle and draw the orientation probability curve;

[0035] Step S3: Calculate the orientation coefficient and draw the orientation coefficient probability curve.

[0036] Furthermore, step S1 specifically includes the following steps:

[0037] Step S1.1: Generate random numbers by the Monte Carlo method:

[0038] When using the Monte Carlo method for fiber data simulation, random sampling is required, that is, a large number of random numbers are generated by a specific method, and the built-in function rand of MATLAB is used to achieve it; assume that the matrix is a cube with side length a, and the x-axis coordinate of one end point of a certain steel fiber is x1. Since the position of the fiber in space is in a uniform distribution state, the relationship between x1 and the side length a is expressed by formula (1), where R is the random number generated by the rand function:

[0039] a·R = x1 (1)

[0040] Step S1.2: Assign the random numbers to (x1, y1, z1):

[0041] The distribution of a single cylindrical steel fiber in space is controlled by six degrees of freedom, namely the spatial coordinates (x1, y1, z1) and (x2, y2, z2) of the two end points P1 and P2 of the fiber

[0042] Therefore, three random numbers R1, R2, and R3 are generated using the rand function, i.e.:

[0043] R1 = rand(1), x1 = a·R1;

[0044] R2 = rand(1), y1 = a·R2;

[0045] R3 = rand(1), z1 = a·R3;

[0046] The spatial coordinates of one end point of a single fiber are randomly generated as P1(x1, y1, z1);

[0047] Step S1.3: Calculate the spatial coordinates (x2, y2, z2) of the other end point of the fiber:

[0048] Randomly generate α, β, and γ, which represent the angles between the fiber and the three Cartesian coordinate axes respectively, with a value range of [0, 2π]; generate random numbers cosα, cosβ, and cosγ in the range [-1, 1];

[0049] Based on the length l of the fiber f and diameter d f and cosα, cosβ, and cosγ, calculate the spatial coordinates (x2, y2, z2) of the other end point P2 of the fiber;

[0050] Step S1.4: Determine whether the fiber exceeds the boundary:

[0051] Determine whether the newly generated fiber exceeds the boundary. The judgment criterion is that the two end points of the cylindrical steel fiber must be within the matrix, i.e., x1, y1, z1; x2, y2, z2 are all greater than 0 and less than a;

[0052] If it exceeds the boundary of the cubic space, repeat steps S1.2 and S1.3; if it does not exceed, proceed to the next step;

[0053] Step S1.5: Generate another fiber:

[0054] Repeat steps S1.2, S1.3, and S1.4 to generate the second fiber;

[0055] Step S1.6: Determine whether the fibers intersect:

[0056] Determine whether the two fibers intersect. The judgment basis is whether the minimum distance between the two line segments is greater than the diameter of the fiber; the calculation method of the minimum distance between the two line segments is as follows:

[0057] 1) Denote the line segments \(l_1\) and \(l_2\) representing two fibers in space. Denote the two endpoints of \(l_1\) as \(P_1\) and \(P_2\); and the two endpoints of \(l_2\) as \(Q_1\) and \(Q_2\). Then \(l_1\) and \(l_2\) are represented by vectors as shown in Equations (2) and (3):

[0058]

[0059] 2) Any point on the straight lines where the two line segments are located is represented by Equations (4) and (5):

[0060]

[0061] where \(\lambda_1\) and \(\lambda_2\) respectively represent any point on the straight line, \(0\leqslant\lambda_1\leqslant1\), \(0\leqslant\lambda_2\leqslant1\);

[0062] Therefore, the problem of solving the shortest distance between two straight lines is transformed into an optimization problem with boundary conditions, that is, to find the shortest distance between two line segments, which is transformed into finding the minimum value of the square of the distance between two line segments, as shown in Equation (6):

[0063]

[0064] 3) Solve Equations (4) and (5) respectively. According to the minimum value condition, it can be known that: Expand and simplify the formula to obtain a system of equations, as shown in Equations (7) and (8):

[0065]

[0066] 4) Solve the system of equations to obtain the values of \(\lambda_1\) and \(\lambda_2\). At this time, \(\lambda_1\) and \(\lambda_2\) are the feet of the perpendiculars on the straight lines where \(l_1\) and \(l_2\) are located. Analyze whether the feet of the perpendiculars are located on the line segments \(l_1\) and \(l_2\). Further, based on this, the calculation of the minimum distance is divided into three cases:

[0067] Case 1: When \(\lambda_1\in[0,1]\) and \(\lambda_2\in[0,1]\), it means that both feet of the perpendiculars are located on their respective line segments. Then the minimum distance between the two straight lines is equal to the length of the common perpendicular;

[0068] Case 2: When \(\lambda_1\in[0,1]\) and or and \(\lambda_2\in[0,1]\), it means that only one foot of the perpendicular is located on the line segment, and the other foot of the perpendicular is located on the extension of the line segment. Assume that the line segment on which the foot of the perpendicular is located is \(l_1\), and the line segment on which the foot of the perpendicular is on the extension is \(l_2\). At this time, the minimum distance is equal to the distance from the endpoint on \(l_2\) closer to the foot of the perpendicular to the foot of the perpendicular on \(l_1\);

[0069] Case 3: When and It is explained that both feet of perpendiculars are located on the extension lines of the line segments. At this time, the minimum distance is equal to the distance between the endpoints on the two line segments that are close to the feet of perpendiculars;

[0070] 5) After obtaining the minimum distance between the fibers, compare this distance with the fiber diameter. If the distance is less than the fiber diameter, it indicates that there is an intersection between the fibers, and this fiber needs to be discarded. Return to step S1.1 to regenerate new fibers. Otherwise, it indicates that the two fibers do not intersect;

[0071] Step S1.6: Generate N fibers

[0072] Repeat steps S1.2 - S1.5 to continue generating new fibers until the number of fibers reaches the preset quantity N, that is, n = N; among them, the quantity N of steel fibers can be calculated through the fiber volume ratio, as shown in Equation (9):

[0073]

[0074] In the formula: l f 、d f 、V f 、V respectively represent the length of the fiber, the diameter of the fiber, the volume admixture of the fiber, and the volume of the cube.

[0075] Furthermore, step S2 specifically includes the following steps:

[0076] Step S2.1: Calculate the orientation angle and orientation probability of the fibers

[0077] Calculate the orientation of each fiber and the angle θ between this fiber and the principal stress direction through the coordinate information of each fiber. As shown in Equation (10), round the calculated angle values. Divide all the angle values into 90 angle values from 1 to 90° according to Equation (10):

[0078]

[0079] In the formula, θ i represents the included angle with an angle of i°, where i is an integer, i = 1, 2, 3... 90°; N θi represents the number of fibers with an included angle of θ i ; N represents the total number of fibers in the model; P θi then represents the probability value of the fibers with an included angle of θ i ; represents the average included angle value between all steel fibers and the principal stress direction in the matrix;

[0080] Step S2.2: Draw the orientation probability curve

[0081] Statistically model the orientation angle of steel fibers and plot the orientation probability curve showing how the orientation probability value changes with the orientation angle.

[0082] Furthermore, in step S2.2, perform statistical analysis on multiple models generated for each set of parameters to determine the orientation probability curve showing how the final orientation probability value changes with the orientation angle.

[0083] Furthermore, step S3 specifically includes the following steps:

[0084] Step S3.1: Calculate the fiber orientation coefficient:

[0085] According to composite material theory, the stress of fiber-reinforced concrete is represented by Equation (13);

[0086]

[0087] According to Equation (13), the mechanical properties of fiber-reinforced concrete can be divided into two parts: the matrix and the fibers. In the equation, σ fc , σ m , and τ are respectively the tensile strength of the entire fiber-reinforced concrete material, the tensile strength of the matrix itself when the fiber volume fraction is 0, and the average bond strength between the steel fibers and the matrix; V f , l f , and d f are respectively the volume fraction, length, and diameter of the steel fibers; while η θ , η β , and η l are coefficients related to the steel fibers, where η θ is the fiber orientation coefficient, which is determined by the cosine value of the angle between the orientation of the steel fibers and the principal stress direction;

[0088] The calculation of the orientation coefficient of a single fiber is shown in Equation (14). When all the steel fibers in the model are parallel to the principal stress direction, the fiber orientation coefficient η θ = 1.0. When all the steel fibers in the model are perpendicular to the principal stress direction, the fiber orientation coefficient η θ = 0; the overall fiber orientation coefficient in the model is calculated after converting the orientation probability curve. Convert the angle values on the abscissa of the orientation probability curve into cosine values to obtain the probability curve of the orientation coefficient, and integrate the envelope area of this curve to obtain the expected value, which is the average orientation coefficient, i.e., the fiber orientation coefficient, as shown in Equation (15):

[0089]

[0090] where, l f is the length of the fiber; μ i is the orientation coefficient of a single fiber with an angle of i ° ; is the average orientation coefficient; N is the total number of fibers; is the orientation coefficient value of μ i probability of steel fibers;

[0091] Step S3.2: Plot the orientation coefficient curve of the fibers:

[0092] Statistically analyze the orientation coefficient of the steel fibers in the model, and plot the orientation coefficient curve with the orientation angle as the abscissa and the orientation coefficient probability as the ordinate.

[0093] One of the key design points of the present invention is to propose a method applicable to calculating the orientation probability of steel fibers in fiber-reinforced concrete (corresponding to steps S1 and S2).

[0094] Use the mathematical software MATLAB to generate random numbers through the Monte Carlo method to obtain the spatial coordinates of the two endpoints of the steel fibers, corresponding to steps S1.1, S1.2, and S1.3. Simulate the three-dimensional spatial distribution model of the fibers on the computer to achieve the random placement of the fibers, corresponding to steps S1.4 to S1.7. Calculate the orientation of each fiber and the angle between the fiber and the principal stress direction according to the generated coordinates of the steel fibers, statistically analyze the orientation angle of the steel fibers in the model, and plot the orientation probability curve of the orientation probability value changing with the orientation angle, corresponding to steps S2.1 and S2.2.

[0095] Another key design point of the present invention is to propose the calculation of the orientation coefficient of steel fibers applicable to the three-dimensional space model and the plotting of the orientation coefficient probability curve, corresponding to steps S3.1 and S3.2.

[0096] Compared with the prior art, the present invention and its preferred solutions have the following advantages:

[0097] Advantage 1: The simulation method can significantly improve the efficiency of calculating the fiber orientation distribution in fiber-reinforced concrete.

[0098] When using the prior art solutions, in actual tests, whether it is the destructive test using the image analysis method or the non-destructive test using the electromagnetic induction method or the X-CT scanning method, a large number of tests are required to draw certain conclusions, which consumes a lot of manpower and material resources. In contrast, the technical solution of the present invention reduces the actual casting and curing process and the subsequent processing process, solves the problems of long time consumption and many processes in the prior art solutions. Especially in the prediction of the mechanical properties of the model, using this method or the idea of this method is very likely to improve the success rate of the test. Therefore, using this method is simple and efficient, and saves time and cost at the same time.

[0099] Advantage 2: The technical solution is not limited by experimental instruments, the size and shape of specimens.

[0100] When adopting the existing technical solutions, in the electromagnetic induction method, it is often only possible to be completed indoors, and there are specific requirements for the size and shape of concrete specimens, with a small applicable range; in the X-CT scanning method, the cost of a single scanning test is high, and there are also high requirements for the computer during the reconstruction process, which is more suitable for specimens with a small fiber volume fraction. The technical solution of the present invention is not limited by experimental instruments, specimen size and shape.

[0101] Advantage three: The technical solution can better reflect the actual situation of the random and disordered distribution of fibers in the matrix.

[0102] When adopting the existing technologies, in the image processing method, the concrete specimen is cut, and some image losses will occur during the post-processing binarization process; in the electromagnetic induction method, the magnetism between the external magnetic field and the steel fibers will interfere with each other. The technical solution of the present invention avoids these problems. The three-dimensional space distribution model of steel fibers established by the computer can better reflect the actual situation of the random and disordered distribution of fibers in the matrix and can play a good auxiliary role in engineering. Brief Description of the Drawings

[0103] The following further details the present invention in conjunction with the drawings and specific embodiments:

[0104] Figure 1 It is a schematic diagram of the existing technology fiber orientation research method;

[0105] Figure 2 It is a schematic diagram of the overall process of the embodiment of the present invention;

[0106] Figure 3 It is a flowchart of the random generation of three-dimensional space steel fibers in the embodiment of the present invention;

[0107] Figure 4 It is an analysis diagram of the steel fiber orientation angle in the embodiment of the present invention;

[0108] Figure 5 It is a schematic diagram of the fiber orientation angle probability distribution curve in the embodiment of the present invention;

[0109] Figure 6 It is a schematic diagram of the spatial distribution model of fiber-reinforced concrete in the embodiment of the present invention;

[0110] Figure 7 It is a schematic diagram of the fiber orientation angle probability distribution curve in the embodiment of the present invention;

[0111] Figure 8 It is a schematic diagram of the fiber orientation coefficient probability distribution curve in the embodiment of the present invention. Specific Embodiments

[0112] To make the features and advantages of the present invention more obvious and understandable, specific embodiments are hereinafter given and described in detail as follows:

[0113] It should be noted that the following detailed description is illustrative and is intended to provide further explanation of the present application. Unless otherwise specified, all technical and scientific terms used in this specification have the same meaning as commonly understood by those of ordinary skill in the technical field to which the present application pertains.

[0114] It should be noted that the terms used herein are merely for describing specific embodiments and are not intended to limit the exemplary embodiments according to the present application. As used herein, unless the context clearly indicates otherwise, the singular forms are also intended to include the plural forms. In addition, it should be understood that when the terms "comprising" and / or "including" are used in this specification, they specify the presence of features, steps, operations, devices, components, and / or combinations thereof.

[0115] As Figure 2 shown, the specific design steps of the method for calculating the fiber orientation angle in the three-dimensional space model provided by this embodiment are as follows:

[0116] Step S1: Generate a three-dimensional steel fiber spatial distribution model

[0117] Generally, a fiber spatial distribution model can be established through the large-scale mathematical calculation software MATLAB, and the Monte Carlo method is used to determine the geometric position information of its fibers. Among them, the fiber-reinforced concrete matrix is regarded as an ideal state, and its material properties are uniformly distributed without the influence of particles and bubbles.

[0118] Step S1.1: Generate random numbers by the Monte Carlo method

[0119] When using the Monte Carlo method for fiber data simulation, random sampling is required, that is, a large number of random numbers are generated by a certain specific method. The MATLAB built-in function rand is used to implement this function. The rand function can generate uniformly distributed random numbers in the interval [0, 1], meeting the requirements of random sampling. Taking the spatial distribution of cylindrical steel fibers as an example, assuming that the matrix is a cube with side length a, and the x-axis coordinate of one end point of a certain steel fiber is x1. Since the position of the fiber in space is in a uniform distribution state, the relationship between x1 and the side length a can be expressed by Equation (1), where R is the random number generated by the rand function.

[0120] a·R = x1 (1)

[0121] Step S1.2: Assign the random numbers to (x1, y1, z1)

[0122] The distribution of a single cylindrical steel fiber in space is controlled by 6 degrees of freedom, that is, the spatial coordinates (x1, y1, z1) and (x2, y2, z2) of the two end points P1 and P2 of the fiber

[0123] Therefore, three random numbers R1, R2, and R3 are generated using the rand function, that is

[0124] R1 = rand(1) <![CDATA[x1 = a·R1]]> R2 = rand(1) <![CDATA[y1 = a·R2]]> R3 = rand(1) <![CDATA[z1 = a·R3]]>

[0125] Therefore, the spatial coordinates of one end point of a single fiber are randomly generated as P1(x1, y1, z1).

[0126] Step S1.3: Obtain the other end point (x2, y2, z2) of the fiber

[0127] Randomly generate α, β, and γ, which respectively represent the angles between the fiber and the three Cartesian coordinate axes (i.e., the x-axis, y-axis, and z-axis), and the value range is [0, 2π]. Since α, β, and γ are generated by random numbers, their cosine values are also random. Therefore, random numbers cosα, cosβ, and cosγ in the range of [-1, 1] can be directly generated.

[0128] Based on the length l of the fiber f and diameter d f and cosα, cosβ, and cosγ, obtain the spatial coordinates (x2, y2, z2) of the other end point P2 of the fiber.

[0129] Step S1.4: Determine whether the fiber exceeds the boundary

[0130] Determine whether the newly generated fiber exceeds the boundary. The judgment criterion is that both end points of the cylindrical steel fiber must be within the matrix.

[0131] If it exceeds the boundary of the cubic space, repeat Step S1.2 and Step S1.3. If not, proceed to the next step.

[0132] Step S1.5: Generate another fiber

[0133] Repeat Step S1.2, Step S1.3, and Step S1.4 to generate the second fiber.

[0134] Step S1.6: Determine whether the fibers intersect

[0135] Determine whether the two fibers intersect. The judgment basis is whether the minimum distance between the two line segments is greater than the diameter of the fiber. The calculation method of the minimum distance between the two line segments is as follows:

[0136] 1) Denote the line segments l1 and l2 representing the two fibers in space. Denote the two end points of l1 as P1 and P2; and those of l2 as Q1 and Q2. Then l1 and l2 can be represented by vectors as shown in Equations (2) and (3).

[0137]

[0138]

[0139] 2) Any point on the straight lines where the two line segments are located can be represented by equations (4) and (5).

[0140]

[0141] Among them, λ1 and λ2 respectively represent any point on the straight line, 0 ≤ λ1 ≤ 1, 0 ≤ λ2 ≤ 1,

[0142] Therefore, the problem of solving the shortest distance between two straight lines can be transformed into an optimization problem with boundary conditions, that is, to find the shortest distance between two line segments, which can be transformed into finding the minimum value of the square of the distance between two line segments, as shown in equation (6).

[0143]

[0144] 3) Solve equations (4) and (5) respectively. According to the minimum value condition, it can be known that: Expanding and simplifying the formula can obtain a system of equations, as shown in equations (7) and (8).

[0145]

[0146] 4) Solving the system of equations can obtain the values of λ1 and λ2. At this time, λ1 and λ2 are the perpendicular feet on the straight lines where l1 and l2 are located. Analyze whether the perpendicular feet are located on the line segments l1 and l2. Further, based on this, the calculation of the minimum distance is divided into 3 cases:

[0147] Case 1: When λ1 ∈ [0, 1] and λ2 ∈ [0, 1], it means that both perpendicular feet are located on their respective line segments, then the minimum distance between the two straight lines is equal to the length of the common perpendicular.

[0148] Case 2: When λ1 ∈ [0, 1] and or and λ2 ∈ [0, 1], it means that only one perpendicular foot is located on the line segment and the other perpendicular foot is located on the extension of the line segment. Assume that the line segment on which the perpendicular foot is located is l1, and the line segment on which the perpendicular foot is on the extension is l2. At this time, the minimum distance is equal to the distance from the endpoint on l2 that is closer to the perpendicular foot to the perpendicular foot on l1.

[0149] Case 3: When and it means that both perpendicular feet are located on the extensions of the line segments. At this time, the minimum distance is equal to the distance between the endpoints on the two line segments that are closer to the perpendicular feet.

[0150] 6) After obtaining the minimum distance between the fibers, compare this distance with the fiber diameter. If this distance is less than the fiber diameter, it means that there is an intersection between the fibers and it needs to be discarded. Otherwise, it means that the two fibers do not intersect.

[0151] Step S1.6: Generate N fibers

[0152] Repeat steps S1.2 to S1.6 to continue generating new fibers until the total number of fibers reaches the preset quantity N, i.e., n = N. Among them, the quantity N of steel fibers can be calculated by Equation (9):

[0153]

[0154] In the formula: l f and d f and V f and V respectively represent the length of the fiber, the diameter of the fiber, the volume fraction of the fiber, and the volume of the cube.

[0155] Among them, the specific process of randomly generating fibers in space based on MATLAB software is as Figure 3 shown:[[]]

[0156] Step S2: Calculate the fiber orientation angle and draw the orientation probability curve

[0157] Step S2.1: Calculate the fiber orientation angle and orientation probability

[0158] Due to the random distribution characteristics of steel fibers in UHPC, the orientation of the fibers is not completely parallel to the principal stress direction. This randomness of orientation will greatly affect the exertion of the bridging effect of steel fibers. Therefore, analyzing the fiber orientation law in the matrix can more comprehensively understand the strengthening mechanism of steel fibers on UHPC. Figure 4 It is the analysis diagram of the steel fiber orientation angle. It is not difficult to find that after the three-dimensional space model is established, the random and disordered distribution of fibers in the matrix conforms to the actual fiber distribution situation.

[0159] Through the coordinate information of each fiber, the orientation of each fiber and the included angle θ between the fiber and the principal stress direction can be calculated, as shown in Equation (10). For the convenience of statistical analysis, the calculated included angle values are rounded. According to Equation (10), all the included angle values are divided into 1 to 90°, a total of 90 angle values.

[0160]

[0161] In the formula, θ i represents the included angle with an angle of i°, where i is an integer, i = 1, 2, 3... 90°; N θi then represents the number of fibers with an included angle of θ i ; N represents the total number of fibers in the model; P θi then represents the probability value of the fibers with an included angle of θ i ; represents the average included angle value between all the steel fibers in the matrix and the principal stress direction.

[0162] Step S2.2: Plot the orientation probability curve

[0163] Statistically analyze the orientation angles of steel fibers in the model and plot the orientation probability curve showing how the orientation probability value changes with the orientation angle. Since the analysis results of a single model are relatively discrete, in order to obtain a convergent orientation probability curve, it is necessary to analyze multiple models generated under the conditions of each parameter. Figure 5 Shows the changes in the fiber orientation probability curve for different numbers of models.

[0164] Step S3: Calculate the orientation coefficient and plot the orientation coefficient probability curve

[0165] Step S3.1: Calculate the orientation coefficient of the fibers

[0166] According to composite material theory, the stress of fiber-reinforced concrete can be expressed by Equation (13).

[0167]

[0168] As can be seen from Equation (13), the mechanical properties of fiber-reinforced concrete can be divided into two parts: the matrix and the fibers. In the equation, σ fc , σ m , and τ are the tensile strength of the overall fiber-reinforced concrete material, the tensile strength of the matrix itself when the fiber volume fraction is 0, and the average bond strength between the steel fibers and the matrix, respectively; V f , l f , and d f are the volume fraction, length, and diameter of the steel fibers, respectively; while η θ , η β , and η l are coefficients related to the steel fibers. Among them, η θ is the fiber orientation coefficient, which is determined by the cosine value of the angle between the orientation of the steel fibers and the principal stress direction.

[0169] The calculation of the orientation coefficient of a single fiber is shown in Equation (14). When all the steel fibers in the model are parallel to the principal stress direction, the fiber orientation coefficient η θ = 1.0. When all the steel fibers in the model are perpendicular to the principal stress direction, the fiber orientation coefficient η θ = 0. The overall fiber orientation coefficient in the model is calculated by converting the orientation probability curve in Figure 5 . Convert the angle values on the abscissa of the orientation probability curve in Figure 5 into cosine values to obtain the probability curve of the orientation coefficient. Integrate the envelope area of this curve to obtain the expected value, and the average orientation coefficient, that is, the fiber orientation coefficient, can be obtained, as shown in Equation (15).

[0170]

[0171] wherein, l f is the length of the fiber; μ i is the orientation coefficient of a single fiber with an included angle of i ° ; is the average orientation coefficient; N is the total number of fibers; is the probability of steel fibers with an orientation coefficient value of μ i .

[0172] Step S3.2: Plot the orientation coefficient curve of the fibers

[0173] Statistically analyze the orientation coefficients of the steel fibers in the model, and plot an orientation coefficient curve with the orientation angle as the abscissa and the orientation coefficient probability as the ordinate.

[0174] Based on the above design, the following further introduces the solution of this embodiment through a specific example:

[0175] Example description: The specimen is a cube with dimensions of 50×50×50 mm, the fiber volume fraction is 2%, and the length l f of the steel fibers is 13 mm, the diameter d f is 0.2 mm, and the steel fiber volume content is 2%.

[0176] From Equation (9), the number of cylindrical steel fibers in this matrix can be obtained as:

[0177]

[0178] Step S1: Generate a three-dimensional spatial distribution model of steel fibers

[0179] According to the above example information, use MATLAB to randomly place the steel fibers and generate a spatial distribution model of the steel fibers, as shown in Figure 6 .

[0180] Step S2: Calculate the orientation angle of the fibers and plot the orientation probability curve

[0181] Based on the fiber spatial distribution model established in Step S1, respectively place n = 1, 10, 100, 1000 into the fiber spatial distribution model, and use the calculated results as the fiber orientation probability distribution curve, as shown in Figure 6As shown in the figure. It can be seen from the figure that when the number of models is larger, the probability curve of steel fibers becomes more regular. When n = 1000, starting from 0°, as the angle value increases, the corresponding probability value also gradually increases. When the angle is 52.41°, its probability value reaches the maximum. When the angle exceeds 52.41°, the probability value then begins to decrease to a certain value. It should be noted that in models with different lengths, diameters, and volume fractions of steel fibers, the shape and trend of the orientation probability curve do not change significantly, and the angle with the maximum probability value is approximately 52.41°.

[0182] In addition, the average included angle That is, the mathematical expectation of all included angle values in the model, which can be calculated according to Equation (11), as Figure 7 In it, the average orientation angle is 59.13°.

[0183] Step S3: Calculate the orientation coefficient and draw the probability curve of the orientation coefficient

[0184] The orientation coefficient and average orientation coefficient of steel fibers are calculated from Equations (13) and (14). Convert the angle values of the abscissa of the orientation probability curve in Figure 6 to cosine values, so as to obtain the probability curve of the orientation coefficient, as Figure 8 shown.

[0185] For this calculation example:

[0186] When adopting the technical solution of the present invention: This algorithm sets the number of fiber models to 1, 10, 100, and 1000 copies, and counts the orientation angles and orientation coefficients of steel fibers in the cubic concrete matrix under different numbers of fiber models. By drawing the orientation probability curve and the orientation coefficient curve, it can be seen that in the three-dimensional space model, although the fibers are randomly distributed in disorder, when the number of steel fibers reaches a certain value, its orientation probability curve shows a smooth curve, and the orientation angles and orientation coefficients are concentratedly distributed at a certain value.

[0187] When adopting the existing technical solution: The existing solution conducts destructive or non-destructive tests on the matrix test blocks on the basis of the fiber-reinforced concrete specimens formed by casting and curing, which is limited by instruments, the size and shape of the test blocks.

[0188] Therefore, for this example, compared with the existing technical solutions, the technical solution of the present invention reduces the time and cost of casting and curing fiber-reinforced concrete, is not limited by the size and shape of the specimens, and is even less restricted by the instrumentation. Moreover, it also reduces the time for calculating the fiber orientation angle. In addition, the technical solution of the present invention simulates the three-dimensional spatial distribution model of the fibers by computer, calculates the orientation angle and orientation coefficient of the steel fibers using a simple model formula, and obtains the orientation distribution law of the fibers in the concrete matrix. This shows that the technical solution of the present invention can significantly improve the calculation efficiency of the fiber orientation distribution, and greatly save time cost, labor cost, and material cost.

[0189] Those skilled in the art should understand that the embodiments of the present application can be provided as methods, systems, or computer program products. Therefore, the present application can take the form of an entirely hardware embodiment, an entirely software embodiment, or an embodiment combining software and hardware aspects. Moreover, the present application can take the form of a computer program product implemented on one or more computer-usable storage media (including but not limited to disk storage, CD-ROM, optical storage, etc.) that contain computer-usable program code.

[0190] The present application is described with reference to the flowcharts and / or block diagrams of methods, devices (systems), and computer program products according to the embodiments of the present application. It should be understood that each flow and / or block in the flowchart and / or block diagram, as well as the combination of flows and / or blocks in the flowchart and / or block diagram, can be implemented by computer program instructions. These computer program instructions can be provided to the processor of a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing devices to generate a machine, so that the instructions executed by the processor of the computer or other programmable data processing devices generate means for implementing the functions specified in Figure 1 one or more of the processes Figure 1 or multiple processes and / or blocks

[0191] These computer program instructions can also be stored in a computer-readable memory that can direct a computer or other programmable data processing device to work in a specific manner, so that the instructions stored in the computer-readable memory generate a manufactured article including instruction means, and the instruction means implements the functions specified in Figure 1 one or more of the processes Figure 1 or multiple processes and / or blocks

[0192] These computer program instructions can also be loaded onto a computer or other programmable data processing device, so that a series of operation steps are executed on the computer or other programmable device to generate a computer-implemented process. Thus, the instructions executed on the computer or other programmable device provide means for implementing the functions specified in Figure 1steps of one process or multiple processes and / or boxes Figure 1 steps of functions specified in one box or multiple boxes.

[0193] As described above, it is only a preferred embodiment of the present invention, and is not a limitation to the present invention in other forms. Any person skilled in the art may use the disclosed technical content to make changes or modifications into equivalent embodiments with equivalent changes. However, any simple modification, equivalent change and modification made to the above embodiments based on the technical essence of the present invention without departing from the technical solution content of the present invention still belong to the protection scope of the technical solution of the present invention.

[0194] The present invention is not limited to the above best implementation mode. Anyone can obtain other various forms of methods for calculating the orientation distribution of steel fibers in fiber-reinforced concrete under the inspiration of the present invention. All equal changes and modifications made according to the scope of the invention application of the present invention shall fall within the coverage scope of the present invention.

Claims

1. A method for calculating the orientation distribution of steel fibers in fiber-reinforced concrete, characterized in that Including the following steps: Step S1: Generate a three-dimensional steel fiber spatial distribution model Establish a fiber spatial distribution model through the large-scale mathematical calculation software MATLAB, and use the Monte Carlo method to determine the geometric position information of the fibers; among them, the fiber-reinforced concrete matrix is regarded as an ideal state, and its material properties are evenly distributed without the influence of particles and bubbles; Step S2: Calculate the fiber orientation angle and draw the orientation probability curve; Step S3: Calculate the orientation coefficient and draw the orientation coefficient probability curve; Step S3 specifically includes the following steps: Step S3.1: Calculate the orientation coefficient of the fiber: According to the composite material theory, the stress of fiber-reinforced concrete is expressed by Equation (13); According to Equation (13), the mechanical properties of fiber-reinforced concrete are divided into two parts: the matrix and the fibers. In the equation, σ fc , σ m , and τ are the tensile strength of the fiber-reinforced concrete material as a whole, the tensile strength of the matrix itself when the fiber volume fraction is 0, and the average bond strength between the steel fibers and the matrix, respectively; V f , l f , and d f are the volume fraction, length, and diameter of the steel fibers, respectively; and η θ , η β , and η l are coefficients related to the steel fibers. Among them, η θ is the fiber orientation coefficient, which is determined by the cosine value of the angle between the orientation of the steel fibers and the direction of the principal stress; The orientation coefficient calculation of a single fiber is shown in Equation (14). When all steel fibers in the model are parallel to the principal stress direction, the fiber orientation coefficient η θ = 1.

0. When all steel fibers in the model are perpendicular to the principal stress direction, the fiber orientation coefficient η θ = 0. The overall fiber orientation coefficient in the model is calculated after converting the orientation probability curve. The angular values of the abscissa of the orientation probability curve are converted into cosine values to obtain the probability curve of the orientation coefficient. The expected value is obtained by integrating the envelope area of this curve, and the average orientation coefficient, that is, the fiber orientation coefficient, is obtained as shown in Equation (15): where l f is the length of the fiber; μ i is the orientation coefficient of a single fiber with an included angle of i°; is the average orientation coefficient; N is the total number of fibers; is the probability that the orientation coefficient value is μ i for steel fibers; Step S3.2: Draw the orientation coefficient curve of the fiber: Statistically analyze the orientation coefficient of the steel fibers in the model, and draw an orientation coefficient curve with the orientation angle as the abscissa and the orientation coefficient probability as the ordinate.

2. The method for calculating the orientation distribution of steel fibers in fiber-reinforced concrete according to claim 1, wherein Step S1 specifically includes the following steps: Step S1.1: Generate random numbers by the Monte Carlo method When using the Monte Carlo method for fiber data simulation, random sampling is required, that is, a large number of random numbers are generated by a certain specific method, and the built-in function rand of MATLAB is used to achieve it; assume that the matrix is a cube with side length a, and the x-axis coordinate of one end point of a certain steel fiber is x1. Since the position of the fiber in space is in a uniform distribution state, the relationship between x1 and the side length a is expressed by Equation (1), where R is the random number generated by the rand function: a·R = x1 (1) Step S1.2: Assign the random numbers to (x1, y1, z1): The distribution of a single cylindrical steel fiber in space is controlled by 6 degrees of freedom, that is, the spatial coordinates (x1, y1, z1) and (x2, y2, z2) of the two end points P1 and P2 of the fiber Therefore, three random numbers R1, R2, and R3 are generated by the rand function, that is: R1 = rand(1), x1 = a·R1; R2 = rand(1), y1 = a·R2; R3 = rand(1), z1 = a·R3; Randomly generate the spatial coordinates of one end point of a single fiber as P1(x1, y1, z1); Step S1.3: Calculate the other end point (x2, y2, z2) of the fiber: Randomly generate α, β, and γ, which represent the angles between the fiber and the three Cartesian coordinate axes respectively, and the value range is [0, 2π]; generate random numbers cosα, cosβ, and cosγ in the range of [-1, 1]; Based on the length l of the fiber f , the diameter d f and cosα, cosβ, cosγ, the spatial coordinates (x2, y2, z2) of the other end point P2 of the fiber are obtained; Step S1.4: Determine whether the fiber exceeds the boundary: Determine whether the newly generated fiber exceeds the boundary. The judgment criterion is that the two end points of the cylindrical steel fiber must be within the matrix, that is, x1, y1, z1; x2, y2, z2 are all greater than 0 and less than a; If it exceeds the cube space boundary, repeat Step S1.2 and Step S1.3; if it does not exceed, proceed to the next step; Step S1.5: Generate another fiber: Repeat Step S1.2, Step S1.3, and Step S1.4 to generate the second fiber; Step S1.6: Determine whether the fibers intersect: To determine whether two fibers intersect, the judgment basis is whether the minimum distance between two line segments is greater than the diameter of the fiber; the calculation method of the minimum distance between two line segments is as follows: 1) Denote the line segments \(l_1\) and \(l_2\) representing two fibers in space. Denote the two endpoints of \(l_1\) as \(P_1\) and \(P_2\); and the two endpoints of \(l_2\) as \(Q_1\) and \(Q_2\). Then \(l_1\) and \(l_2\) are represented by vectors as shown in Equations (2) and (3): 2) Any point on the straight lines where the two line segments are located is represented by equations (4) and (5): where λ1 and λ2 respectively represent any point on the straight line, 0 ≤ λ1 ≤ 1, 0 ≤ λ2 ≤ 1; Therefore, the problem of solving the shortest distance between two straight lines is transformed into an optimization problem with boundary conditions, that is, to find the shortest distance between two line segments, which is transformed into finding the minimum value of the square of the distance between two line segments, as shown in equation (6): 3) Solve equations (4) and (5) respectively. According to the minimum condition, it can be known that: Expand and simplify the formula to obtain a system of equations as shown in equations (7) and (8): 4) Solve the system of equations to obtain the values of λ1 and λ2. At this time, λ1 and λ2 are the feet of the perpendiculars on the straight lines where l1 and l2 are located. Analyze whether the feet of the perpendiculars are located on the line segments l1 and l2. Further, based on this, the calculation of the minimum distance is divided into three cases: Case 1: When λ1 ∈ [0, 1] and λ2 ∈ [0, 1], it means that both feet of the perpendiculars are located on their respective line segments, then the minimum distance between the two straight lines is equal to the length of the common perpendicular; Case 2: When λ1 ∈ [0, 1] and or and λ2 ∈ [0, 1], it indicates that only one foot of the perpendicular lies on the line segment and the other foot of the perpendicular lies on the extension of the line segment; assume that the line segment on which the foot of the perpendicular lies is l1 and the line segment on which the foot of the perpendicular lies on the extension is l2. At this time, the minimum distance is equal to the distance from the end point closer to the foot of the perpendicular on l2 to the foot of the perpendicular on l1; Case 3: When and it indicates that both feet of the perpendiculars are located on the extension lines of the line segments. In this case, the minimum distance is equal to the distance between the endpoints on the two line segments that are closer to the feet of the perpendiculars; 5) After obtaining the minimum distance between the fibers, compare this distance with the fiber diameter. If this distance is less than the fiber diameter, it means that there is an intersection between the fibers, and this fiber needs to be discarded, and return to step S1.1 to regenerate new fibers. Otherwise, it means that the two fibers do not intersect; Step S1.6: Generate N fibers Repeat steps S1.2 - S1.5 to continue generating new fibers until the number of fibers reaches the preset quantity N, that is, n = N; among them, the number of steel fibers N is calculated through the fiber volume ratio, as calculated by equation (9): where: l f , d f , V f , V represent the length of the fiber, the diameter of the fiber, the volume fraction of the fiber, and the volume of the cube, respectively.

3. The method for calculating the orientation distribution of steel fibers in fiber-reinforced concrete according to claim 2, characterized in that, Step S2 specifically includes the following steps: Step S2.1: Calculate the orientation angle and orientation probability of the fibers Calculate the orientation of each fiber and the included angle θ between the fiber and the principal stress direction through the coordinate information of each fiber, as shown in equation (10). Round the calculated included angle values, and divide all the included angle values into 90 angle values from 1 to 90° according to equation (10): Where, θ i represents the included angle of i°, where i is an integer, i = 1, 2, 3... 90°; represents the included angle of θ i of the number of fibers; N represents the total number of fibers in the model; then represents the probability value of the fiber with the included angle of θ i ; represents the average included angle value between all steel fibers in the matrix and the principal stress direction; Step S2.2: Draw the orientation probability curve Statistically analyze the orientation angles of the steel fibers in the model, and draw the orientation probability curve showing the change of the orientation probability value with the orientation angle.

4. The method for calculating the orientation distribution of steel fibers in fiber-reinforced concrete according to claim 3, characterized in that, In step S2.2, analyze multiple models generated under the conditions of each parameter to determine the final orientation probability curve showing the change of the orientation probability value with the orientation angle.

Images