Anchor rod frame beam deformation and internal force calculation method based on transfer matrix method

Abstract

The invention discloses an anchor rod frame beam deformation and internal force calculation method based on a transfer matrix method, which comprises the following steps of: establishing a rectangular coordinate system by taking the left end point of a longitudinal beam in any frame beam as an original point, the direction of the longitudinal beam as an x axis and the direction vertical to the mounting plane of the frame beam as a y axis; solving a deflection curve differential equation of the beam based on a Winkler elastic foundation theory to obtain the deflection of the longitudinal beam; superposing the deflection increased due to the action of the anchoring force to obtain a deflection general solution of the i-th section of longitudinal beam under the action of the anchoring force; on the basis of the deflection general solution of the i-th longitudinal beam, according to the differential relationship among the deflection, the corner, the bending moment and the shearing force of any section on the i-th longitudinal beam, obtaining a corner, bending moment and shearing force expression of any section on the i-th longitudinal beam; and the deflection, the corner, the bending moment and the shearing solution of any position of the N sections of beams are calculated through a transfer matrix method. The method can calculate and solve the deformation and internal force of any section of frame beam, improves the accuracy and efficiency, and is more suitable for actual engineering scenes.

Description

Technical Field

[0001] The present invention belongs to the technical field of geotechnical engineering, and particularly relates to a method for calculating the deformation and internal force of an anchor rod frame beam based on the transfer matrix method. Background Technique

[0002] The anchor rod (cable) frame beam is widely used in high slope support projects and plays an important role in the protection and treatment of slope diseases. The anchor rod (cable) frame beam system consists of three parts: the frame beam, the anchor rod (cable), and the reinforced rock and soil mass, which can effectively prevent the deformation and failure of the slope body. In recent years, the research on the action mechanism and failure mode of the anchor rod (cable) frame beam has been relatively complete. For the stability effect and failure form of the anchor rod (cable) frame beam on the slope, researchers have demonstrated in detail the action mechanism and failure mechanism of the anchor rod (cable) frame beam structure through large-scale shaking table tests, dynamic (static) centrifugal model tests, numerical simulation software and other tests and methods.

[0003] However, although the application of the anchor rod (cable) frame beam is very extensive, the current design of the anchor rod (cable) frame beam is mainly based on on-site construction experience, lacking a theoretical calculation method for the anchor rod (cable) frame beam. The design of the anchor rod (cable) frame beam mainly includes the calculation of the internal force and deformation of the frame beam. Most of the existing theoretical calculations for the internal force of the anchor rod (cable) frame beam are to analyze and model individual beam segments for solution, which cannot be applied to the calculation of different slope soil supports or variable cross-section beams, lacking consideration of the influence of material differences of different beam segments on deformation and internal force, and not conforming to the situation of mutual influence in the connection of multi-segment beams in actual engineering.

[0004] Therefore, there is an urgent need to propose a simple and effective method to achieve reliable calculation of the overall internal force and deformation of the anchor rod (cable) frame beam. Summary of the Invention

[0005] The purpose of the embodiments of the present invention is to provide a method for calculating the deformation and internal force of an anchor rod frame beam based on the transfer matrix method, which can calculate and solve the deformation and internal force of any section of the frame beam, improve the accuracy and efficiency, and better adapt to the actual engineering scenario.

[0006] To solve the above technical problems, the technical solution adopted by the present invention is a method for calculating the deformation and internal force of an anchor rod frame beam based on the transfer matrix method, including the following steps:

[0007] S1. Establish a rectangular coordinate system with the left end point of the longitudinal beam in any frame beam as the origin, the longitudinal beam direction as the x-axis, and the direction perpendicular to the installation plane of the frame beam as the y-axis;

[0008] S2. Solve the differential equation of the deflection curve of the beam based on the Winkler elastic foundation theory to obtain the deflection of the longitudinal beam; superimpose the increased deflection due to the anchoring force to obtain the general solution of the deflection of the i-th longitudinal beam under the action of the anchoring force;

[0009] Based on the general solution of the deflection of the i-th longitudinal beam, obtain the expressions of the rotation angle, bending moment, and shear force of any cross-section on the i-th longitudinal beam according to the differential relationship between the deflection, rotation angle, bending moment, and shear force of any cross-section on the longitudinal beam;

[0010] S3. Based on the expressions of the rotation angle, bending moment, and shear force of any cross-section on the i-th longitudinal beam, calculate the solutions of the deflection, rotation angle, bending moment, and shear force at any position of the N-section beam by the transfer matrix method.

[0011] Further, in S2, solve the differential equation of the deflection curve of the beam based on the Winkler elastic foundation theory to obtain the deflection w i (x)0 of the longitudinal beam, and the expression is as follows:

[0012]

[0013] where x is the abscissa, and λ iy represents the deformation coefficient of the i-th longitudinal beam, E iy represents the elastic modulus of the i-th longitudinal beam; I iy represents the moment of inertia of the cross-section of the i-th longitudinal beam; k iy represents the coefficient of subgrade reaction; b iy represents the width of the longitudinal beam; C 4i-3 、C 4i-2 、C 4i-1 、C 4i are the undetermined constants of the longitudinal beam, which are solved through the boundary conditions of the beam, and the four undetermined constants of the same section of the beam are the same.

[0014] Further, in S2, the general solution of the deflection w i (x) p of the i-th longitudinal beam under the action of the anchoring force is shown in the following formula:

[0015]

[0016] where u(x - x pi ) represents the Heaviside function, and the value is as follows:

[0017]

[0018] where P i represents the anchoring force acting on the longitudinal beam, is the position where the anchoring force on the i-th longitudinal beam acts on the beam; ch is the hyperbolic cosine function, and sh is the hyperbolic sine function.

[0019] Furthermore, in S2, the differential relationships among the deflection, rotation angle, bending moment, and shear force of any cross-section on the longitudinal beam are as follows:

[0020]

[0021] where w i (x) represents the deflection of the i-th longitudinal beam under the action of the anchoring force, and θ i (x) represents the rotation angle of the i-th longitudinal beam under the action of the anchoring force; M i (x) represents the bending moment of the i-th longitudinal beam under the action of the anchoring force; Q i (x) represents the shear force of the i-th longitudinal beam under the action of the anchoring force.

[0022] Furthermore, in S2, the expressions for the rotation angle, bending moment, and shear force of any cross-section on the i-th longitudinal beam are respectively:

[0023]

[0024] where θ i (x) represents the rotation angle of the i-th longitudinal beam under the action of the anchoring force; M i (x) represents the bending moment of the i-th longitudinal beam under the action of the anchoring force; Q i (x) represents the shear force of the i-th longitudinal beam under the action of the anchoring force.

[0025] Furthermore, S3 includes the following steps:

[0026] S31. The deflection, rotation angle, bending moment, and shear force of any cross-section on the i-th longitudinal beam are expressed in matrix form as follows:

[0027]

[0028] where [A i represents the coefficient matrix before the undetermined constants C 4i-3 , C 4i-2 , C 4i-1 , C 4i , represents the influence matrix of the anchoring force P i on the deflection, rotation angle, bending moment, and shear force of the i-th longitudinal beam;

[0029] S32. Take the left endpoint x i and the right endpoint x i+1 on the i-th longitudinal beam and substitute them into the matrix in S31 respectively. Since C 4i-3 , C 4i-2 , C 4i-1 , C 4i are the same in the same beam segment, the transfer relationship between the left and right ends of the i-th longitudinal beam is as follows:

[0030]

[0031] Among them, [T i = [A i [A i+1 -1 , T i is an intermediate parameter; [A i+1 -1 represents the inverse matrix of the coefficient matrix before C i+1 , C 4i-3 , C 4i-2 , C 4i-1 , C 4i at the right endpoint x respectively represent the influence matrices of the anchoring force on the longitudinal beam deflection, rotation angle, bending moment, and shear force at the left endpoint x i , the right endpoint x i+1 ;

[0032] S33. Since the frame beam is rigidly connected at the beam segment connection, the deflection, rotation angle, bending moment, and shear force at the end of the i-th beam segment are equal to those at the starting point of the (i + 1)-th beam segment; therefore, the relationship between the starting point and the end point of each beam segment is substituted into the matrix of S32, and then the starting point matrix of the next beam segment is successively substituted into the end point matrix of the previous beam segment to achieve the transfer from the starting point of the first beam segment to the end point of the N-th beam segment.

[0033] Further, the transfer relationship from the starting point of the first beam segment to the end point of the N-th beam segment in the S33 is as follows:

[0034]

[0035] Among them, according to the Heaviside function existing in the matrix When , the function value is always 0. Therefore, at the starting point,

[0036] At the starting point position of the first beam segment, the boundary conditions are M1(x1) = 0 and Q1(x1) = 0; at the end of the N-th beam segment, the boundary conditions are M N+1 (x N+1 ) = 0 and Q N+1 (x N+1 ) = 0; thus, we get:

[0037]

[0038] Among them, is an element of the matrix , is 's inverse matrix; is the matrix ​​For the elements, the numerical values of w1(x1) and θ1(x1) are obtained by solving.

[0039] Substitute the values of w1(x1) and θ1(x1) into the transfer relationship from the starting point of the first beam segment to the end point of the Nth beam segment, and then the values of w N+1 (x N+1 ) and θ N+1 (x N+1 ) can be obtained, that is, the deflection, rotation angle, bending moment, and shear force solutions at any position in the Nth beam segment are obtained.

[0040] The beneficial effects of the present invention are as follows:

[0041] In the present invention, the analytical solutions of the internal forces and deformations of independent beam segments are established through the Winkler elastic foundation theory. Then, according to the boundary conditions at the beam ends, the transfer matrix method is used to obtain the expressions of the deflection, rotation angle, bending moment, and shear force of multiple connected beam segments. Combining the boundary conditions at both ends of the multiple beam segments, they can be solved. Among them, the elastic modulus E of each beam segment can be assigned according to the actual situation, and for a variable cross-section beam, the cross-sectional area is replaced in the form of a function. Finally, the rationality of the solutions is verified through numerical simulation and centrifuge test models. On this basis, the deformation and internal force variation laws of the bolt (cable) frame beam under different anchoring forces, foundation reaction coefficients, frame beam cross-sectional dimensions, adjacent anchoring point spacings, and frame beam cantilever lengths are studied respectively.

[0042] Compared with the prior art, the present invention has high accuracy and high efficiency; it is more suitable for actual engineering scenarios. Considering the differences in beam segments of the frame beam (linear change in cross-section, parameter changes in beam segments caused by differences in the materials of each beam segment), the calculation results have higher accuracy. The differences in each beam segment are reflected in the deformation and internal force equations in matrix form, and then the deformation and internal force equations of adjacent beam segments are integrated through the connection method (boundary conditions) between each beam segment, so as to realize the integration of the deformation and internal force equations of N independent beams into a deformation and internal force equation with the specificity of each beam segment. Through the Matlab programming software, the deformation and internal force solutions of any segment of the frame beam can be conveniently solved. Description of the Drawings

[0043] In order to more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the following will briefly introduce the drawings required for the description of the embodiments or the prior art. Obviously, the following drawings are only some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained without creative efforts based on these drawings.

[0044] Figure 1 It is a simplified structural schematic diagram of the bolt frame beam in the embodiment of the present invention.

[0045] Figure 2It is the front view of the simplified structure of the anchor rod frame beam in the embodiment of the present invention.

[0046] Figure 3 It is the simplified diagram of the longitudinal beam of the anchor rod frame beam in the embodiment of the present invention.

[0047] Figure 4 It is the physical diagram of the support of the anchor rod frame beam in the embodiment of the present invention.

[0048] Figure 5 It is the schematic diagram of the cross-sectional dimensions of the slope model in the embodiment of the present invention.

[0049] Figure 6 It is the schematic diagram of the slope model in the embodiment of the present invention.

[0050] Figure 7 It is the schematic diagram of the frame beam model in the embodiment of the present invention.

[0051] Figure 8 It is the comparison diagram of the bending moment of the solution of the numerical simulation, the centrifuge model test and the embodiment of the present invention.

[0052] Figure 9 It is the comparison diagram of the shear force of the solution of the numerical simulation and the embodiment of the present invention.

[0053] Figure 10 It is the change diagram of the deflection, rotation angle, bending moment and shear force of the beam under different anchoring forces solved in the embodiment of the present invention.

[0054] Figure 11 It is the change diagram of the deflection, rotation angle, bending moment and shear force of the beam under different foundation reaction coefficients solved in the embodiment of the present invention.

[0055] Figure 12 It is the change diagram of the deflection, rotation angle, bending moment and shear force of the beam under different cross-sectional dimensions solved in the embodiment of the present invention.

[0056] Figure 13 It is the change diagram of the deflection, rotation angle, bending moment and shear force of the beam under different anchor rod spacings solved in the embodiment of the present invention.

[0057] Figure 14 It is the change diagram of the deflection, rotation angle, bending moment and shear force of the beam under different cantilever lengths solved in the embodiment of the present invention. Specific embodiments

[0058] Next, the technical solutions in the embodiments of the present invention will be clearly and completely described in conjunction with the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all of the embodiments. All other embodiments obtained by those of ordinary skill in the art based on the embodiments of the present invention without creative efforts shall fall within the protection scope of the present invention.

[0059] Embodiment A method for calculating the deformation and internal force of an anchor rod frame beam based on the transfer matrix method, comprising the following steps:

[0060] S1. Establish a coordinate system.

[0061] The Winkler elastic foundation theory is adopted to analyze the stress and deformation of the anchor rod (cable) frame beam. The following basic assumptions are made during the solution process: The soil mass behind the slope and the concrete frame beam are homogeneous elastic media. The action of the anchor rod on the frame beam is mainly vertical force, and the axial force is ignored. The beam body of the frame beam is in close contact with the soil mass behind the slope without separation and slip. The longitudinal beams are regarded as independent continuous beams. The torsional effect of the frame beam is not considered.

[0062] See Figures 1 to 3 , the anchor rod force P acts on the intersection point of the longitudinal beams (i.e., the anchoring point). Figure 3 In, taking the starting point o of the frame beam as the origin, the direction perpendicular to the frame beam (i.e., perpendicular to the ground) as the y-axis, and the longitudinal direction of the longitudinal beam of the frame beam as the x-axis. The distance from the free end of the beam to the nearest anchoring point is the cantilever length, denoted by Z, and the distance between adjacent anchoring points is denoted by Y.

[0063] S2. Solve the general solutions of the deflection, rotation angle, bending moment, and shear force of the beam.

[0064] S21. For the i-th longitudinal beam, according to the differential equation of the deflection curve of the beam and the Winkler elastic foundation theory, we have:

[0065]

[0066] where i represents the number of segments of the beam, i = 1, 2, 3... N, and N represents the total number of segments; E iy represents the elastic modulus of the i-th longitudinal beam; I iy represents the moment of inertia of the cross-section of the i-th longitudinal beam; k iy represents the foundation reaction coefficient at the i-th longitudinal beam; b iy represents the width of the i-th longitudinal beam; x is the abscissa in the xoy coordinate system established above.

[0067] Taking the general solution of Equation (1) gives:

[0068]

[0069] where w i (x)0 represents the deflection of the beam obtained according to the differential equation of the deflection curve and the Winkler elastic foundation theory; the deformation coefficient of the i-th longitudinal beam C 4i-3 、C 4i-2 、C 4i-1 、C 4iare the undetermined constants that appear when solving Equation (1). Each beam segment has 4 undetermined constants, which can be solved through the boundary conditions of the beam, and the four undetermined constants of the same beam segment are the same. x represents the position on the longitudinal beam. The operation relationship between λ iy and x in Equation (2) is a multiplication operation.

[0070] Consider the influence of the anchoring force on the beam deflection: When is the position where the anchoring force acts on the longitudinal beam (the abscissa of the anchoring point). Since the anchoring force P i the increased deflection is:

[0071]

[0072] where ch and sh are the hyperbolic cosine and hyperbolic sine functions; w i (x) p is the deflection of the i-th longitudinal beam segment caused by the anchoring force P i acting; represents the Heaviside function, and its value is as follows:

[0073]

[0074] Combining Equations (2) and (3), the general solution w i (x) of the deflection of the i-th longitudinal beam segment under the action of the anchoring force can be obtained:

[0075]

[0076] where w i (x) represents the deflection of the i-th longitudinal beam segment under the action of the anchoring force, C 4i-3 、C 4i-2 、C 4i-1 、C 4i represent the undetermined constants of the i-th longitudinal beam segment; P i represents the anchoring force on the i-th longitudinal beam segment; is the position where the anchoring force on the i-th longitudinal beam segment acts on the beam.

[0077] S24. Differential relationships among the deflection, rotation angle, bending moment, and shear force at any cross-section on the longitudinal beam:

[0078]

[0079] where θ i (x) represents the rotation angle of the i-th longitudinal beam segment under the action of P i ; M i (x) represents the bending moment of the i-th longitudinal beam segment under the action of P i ; Q i (x) represents the shear force of the i-th longitudinal beam segment under the action of P i acting.

[0080] Substituting equation (5) into equation (6), we can solve the expressions of rotation angle, bending moment and shear force of any section on the i-th longitudinal beam:

[0081]

[0082] S3. Calculate the deflection, rotation, bending moment and shear force solutions at any position of the N-segment beam by the transfer matrix method.

[0083] S31. After obtaining the independent solutions of the deflection, rotation, bending moment and shear force of the i-th longitudinal beam, the transfer matrix method is used to transfer the deflection, rotation, bending moment and shear force between each beam. According to equations (5), (7), (8) and (9), the deflection, rotation, bending moment and shear force of the i-th longitudinal beam are expressed in the form of a matrix as follows:

[0084]

[0085] Among them, [A i ] represents the undetermined constant C 4i-3 , C 4i-2 , C 4i-1 , C 4i The coefficient matrix before Indicates the anchoring force P i The influence matrix on the deflection, rotation, bending moment and shear force of the i-th longitudinal beam. [A i ] are expressed as follows:

[0086]

[0087] S32, take the two endpoints of the i-th longitudinal beam: the left endpoint x i , right endpoint x i+1 , respectively substituted into formula (10), since C 4i-3 , C 4i-2 , C 4i-1 , C 4i Similarly, the transfer relationship between the left and right ends of the i-th longitudinal beam can be given as follows:

[0088]

[0089] Among them, [T i ]=[A i ][A i+1 ] -1 , T i is an intermediate parameter. i+1 ] -1 Represents the right endpoint x i+1 C 4i-3 , C 4i-2 , C 4i-1 , C4i The inverse matrix of the coefficient matrix before Indicates the influence matrix of the anchoring force on the longitudinal beam deflection, rotation angle, bending moment, and shear force at the left endpoint x i , the right endpoint x i+1 . It reflects the action of the anchoring force at the two endpoints x i , x i+1 .

[0090] S33. Since the frame beam is rigidly connected at the beam segment connection, the deflection, rotation angle, bending moment, and shear force at the end of the i-th beam are equal to those at the starting point of the (i + 1)-th beam. Therefore, the relationship between the starting point and the endpoint of each beam can be substituted into Equation (11), and then the starting point matrix of the next beam is successively substituted into the endpoint matrix of the previous beam to achieve the transfer from the starting point of the first beam to the endpoint of the N-th beam. The relationship is as follows:

[0091]

[0092] Among them, according to the Heaviside function existing in this matrix When x < x p , the function value is always 0. Therefore, at the starting point,

[0093] Is the transfer matrix from the starting point of the first beam to the endpoint of the N-th beam.

[0094] At the starting point of the first beam, there are boundary conditions M1(x1) = 0 and Q1(x1) = 0; at the end of the N-th beam, there are boundary conditions M N+1 (x N+1 ) = 0 and Q N+1 (x N+1 ) = 0. Therefore, through Equation (12), we get:

[0095]

[0096] Among them, Is the element of the matrix ; Is the element of the matrix ; Is 's inverse matrix.

[0097] Is the influence matrix of the anchoring force on the deflection, Are respectively the element in the first column of the third row and the element in the first column of the fourth row of this matrix. By solving , the values of these two elements can be obtained.

[0098] The numerical values of \(w_1(x_1)\) and \(\theta_1(x_1)\) can be obtained by solving from Equation (13). Similarly, by substituting the values of \(w_1(x_1)\) and \(\theta_1(x_1)\) into Equation (12), the values of \(w\) N+1 (x N+1 ) and \(\theta\) N+1 (x N+1 ) can be obtained. Thus far, the deflection, rotation angle, bending moment, and shear force solutions at any position in the N-section beam can be obtained by substituting different x values into Equation (12).

[0099] In the embodiment of the present invention, the transfer matrix method is combined with the solution of the deflection of the frame beam, and the deflection equations of each beam section are connected in the form of a transfer matrix. The parameters of the beam section and the connection method (rigid connection) between the beam sections need to be considered. Specifically, it is reflected in Formulas (10) and (11). The obtained deflection, rotation angle, bending moment, and shear force equations are transformed into matrix form, and the equations of the N-section beam are connected and summarized into Formula (11).

[0100] Verification of technical effects:

[0101] The method of the embodiment of the present invention is compared with the internal force results obtained from the existing numerical simulation and the centrifuge shaking table model test, as follows:

[0102] Numerical simulation method:

[0103] Refer to Figure 4 , and a numerical simulation calculation is carried out on the anchor frame beam model in an engineering example. This engineering example is located in the slope section of K87+391~K87+565 of the Gui (Yang) - Huang (Ping) Expressway, with a length of about 174 meters.

[0104] Model establishment. The rock formations of the slope of this project are distributed from top to bottom as red clay, strongly weathered dolomite, and moderately weathered dolomite. There are a total of four levels of slopes, and the slope ratios of the slopes from top to bottom are 1:1.25, 1:1, 1:0.75, and 1:0.75 respectively. The size of the numerical simulation slope model is the actual size of the slope. In order to eliminate the influence of excessive boundaries, it extends horizontally 10m backward at the highest point of the slope, and in order to ensure consistency with the actual situation, a slope of 1:5 is set; at the bottom of the first-level slope, it extends 31m (the width of the road surface and the central isolation belt) along the road surface direction and 16m downward. The frame beam is arranged on the third-level slope, and the cross-sectional size of the slope model is as Figure 5 shown.

[0105] A three-dimensional slope model is established in the numerical simulation software ABAQUS. The lengths of the model in the x, y, and z directions are 81.25 m, 49.8 m (the highest point is 51.8 m when adding the top extension part), and 9 m respectively. The red clay, strongly weathered dolomite, and moderately weathered dolomite of the slope model components are all established using the solid extrusion structure. For the slope support component, the frame beam, first draw a sketch in CAD and then import it into ABAQUS, and then extrude 0.4 m (the actual thickness of the frame beam) to form a solid structure. The longitudinal beam spacing of the frame beam is set to 3 m. Select two complete longitudinal beams and half of the longitudinal beams on both sides to ensure that the thickness of the model is 9 m, and the height of the frame beam is 0.3 m. The length of the anchor rod is 11.5 m and the diameter is 32 mm, and the solid element structure is also adopted. The boundary conditions are as follows: the displacements in the x direction are constrained on the front and back sides of the slope, the displacements in the z direction are constrained on the left and right sides, the displacements in the x, y, and z directions are constrained at the bottom of the slope, and the slope surface is a free surface. The mesh division is as follows: the slope body part of the slope is divided by the method of C3D10M (modified 10-node tetrahedral element), and the anchor rod and the frame beam are divided by the method of C3D8R (8-node linear hexahedral element with reduced integration control). As Figures 6 - 7 shown.

[0106] Selection of constitutive model and determination of physical parameters. The selected rock and soil layers and physical and mechanical parameters for this numerical simulation calculation are comprehensively determined by referring to the geological exploration report of this slope and the physical and mechanical properties of rock and soil in similar slope projects. The specific values are shown in Table 1. The Mohr-Coulomb model is adopted for the constitutive model, and the Mohr-Coulomb criterion is adopted for the yield criteria of each layer of material. The linear elastic model is adopted for the support structure.

[0107] Table 1 Calculation parameters of soil structure, anchor rod and frame

[0108]

[0109] Verification of the solution: By establishing a section plane in the local coordinate system, and then using the "view section" function to extract the internal force of the structure, the numerical simulation results are obtained. Compare the numerical simulation results, the results of the centrifuge shaking table model test and the results calculated by the embodiments of the present invention. The comparison results are as Figures 8 - 9 shown. Since only the bending moment data is extracted in the centrifuge shaking table model test, only the numerical simulation results and the results calculated by the embodiments of the present invention are compared in the shear force comparison diagram.

[0110] In Figures 8 - 9Among them, three anchoring forces are extracted from the numerical simulation. This is because in the construction of the numerical simulation model, a 3×3 framework is selected for research, and the anchoring forces on the longitudinal beams of each layer are extracted respectively, which is equivalent to three 9m longitudinal beams, and then solved using the formula of the embodiment of the present invention. The data of the centrifugal model test is from a published paper (Huang Zhiyi. Research on the internal force calculation and centrifugal model test of prestressed anchor cable frame beams [D]. Southwest Jiaotong University, 2014.), and the embodiment of the present invention only calculates according to the parameters in its paper.

[0111] From the comparison results, it can be seen that the numerical simulation solutions, the centrifugal model test data and the calculation results of the embodiments of the present invention are basically in agreement, the change trends match, and the errors are small, which proves the rationality of the derivation results of the present invention.

[0112] The method of the embodiment of the present invention analyzes the force formula of the frame beam, and based on the existing equations, further obtains the equation of the coordinated deformation of the frame beam with the soil body and the anchor rod in the high slope support. Through this equation, the parameters of the slope body can be obtained through on-site survey, and then the deformation and force analysis of the anchor rod frame beam support system can be carried out systematically, which has more convenient operability compared with the numerical simulation and the centrifugal model test.

[0113] Carry out parameter research on the anchor rod frame beam:

[0114] Based on the solution of the embodiment of the present invention, consider the influence mechanism of the changes of several key parameters such as the anchoring force, the foundation reaction coefficient, the cross-sectional area, the spacing between adjacent anchor rods and the cantilever length in the anchor rod frame beam support system on the deformation and internal force of the anchor rod frame beam body.

[0115] Influence of the anchoring force P on the force and deformation of the beam:

[0116] See Figure 10 , according to the anchoring force P = 345kN in the numerical simulation, different anchoring forces (0.25P, 0.50P, 0.75P, 1.0P, 1.25P, 1.5P) are selected to observe their influence on the beam. It can be seen that: as the anchoring force increases, the deflection, rotation angle, bending moment and shear force of the beam increase simultaneously; the bending moment and shear force corresponding to 0m and 9m on the beam are zero respectively, which conforms to the characteristics of a cantilever beam; at the anchoring force action points (1.5m, 4.5m, 7.5m), the bending moments of the beam under different anchoring forces reach the peak values respectively, and the shear force shows a sudden change phenomenon, and the absolute value of the sudden change is equal to the action of the anchoring force at this point.

[0117] Influence of the foundation reaction coefficient k on the force and deformation of the beam:

[0118] Select different foundation reaction coefficients according to the actual geological survey results. Figure 11It reflects the influence of different foundation reaction coefficients. It can be seen that the foundation reaction coefficient has a significant impact on the beam deflection. As the foundation reaction coefficient increases, the maximum deflection of the beam decreases from 20 mm to 4 mm. The deflection of the beam also decreases as the foundation reaction coefficient increases. The change in the foundation reaction coefficient has little effect on the beam moment and shear force.

[0119] Influence of cross-sectional dimension S on the force and deformation of the beam:

[0120] See Figure 12 , according to the cross-sectional dimension b×h = 0.3 m×0.4 m (beam width×beam height) of the frame beam in the numerical simulation, 5 different cross-sectional dimensions are selected. It can be seen that the cross-sectional dimension has a significant impact on the beam deflection and rotation angle, both of which decrease as the cross-sectional dimension increases. When the cross-sectional dimensions of b×h = 0.2 m×0.2 m and b×h = 0.6 m×0.6 m are selected, the maximum deflections of the beam are 19 mm and 5 mm respectively, with a difference of 14 mm. The absolute maximum value of the bending moment on the beam increases as the cross-sectional dimension increases, from 95 kN·m to 120 kN·m. The cross-sectional dimension has little effect on the shear force. When the cross-sectional dimension of b×h = 0.2 m×0.2 m is selected, the deflection and rotation angle of the beam change significantly, while when the cross-sectional dimensions of b×h = 0.4 m×0.4 m, b×h = 0.5 m×0.5 m, and b×h = 0.6 m×0.6 m are selected, the changes in the deflection and rotation angle are relatively small.

[0121] Influence of the spacing Y between adjacent anchor bolts on the force and deformation of the beam:

[0122] See Figure 13 , according to the spacing Y = 3 m between adjacent anchor bolts in the actual project, 10 different spacings between adjacent anchor bolts are selected to analyze the frame beam. It can be seen that when the spacing between adjacent anchor bolts of the frame beam changes from 1.5 m and 6 m to 3.5 m, the overall displacement change amplitude of the frame beam gradually becomes coordinated, and compared with that at 1.5 m, the maximum displacement on the frame beam is reduced by about 50%. In the change of the bending moment on the frame beam, when the spacing between adjacent anchor bolts changes from 1.5 m and 6 m to 3.0 - 3.5 m, the maximum value of the bending moment on the frame beam reaches the minimum. From this analysis, it can be seen that changing the spacing between adjacent anchor bolts has a greater impact on the displacement and rotation angle of the frame beam, and at the same time, it can also improve the bending moment distribution of the frame beam body. In this demonstration project, the recommended spacing between adjacent anchor bolts is 3.0 m - 3.5 m.

[0123] Influence of the change of the cantilever length Z and the spacing Y between adjacent anchor bolts on the force and deformation of the beam:

[0124] See Figure 14, according to the cantilever length of 1.5 m in the actual project, 10 different cantilever lengths are selected this time. The adjacent anchor rod spacing Y changes with the change of the cantilever length to ensure that the total length of the frame beam remains unchanged. By analyzing the deformation and stress of the frame beam, it can be seen that: during the change of the cantilever length from 0 to 2.25 m, the maximum displacement of the frame beam first decreases from 26 mm (Z = 0 m) to 10 mm (Z = 1.5 m), and then increases from 10 mm (Z = 1.5 m) to 14 mm (Z = 2.25 m); when the cantilever length is 1.5 m, the overall change range of the displacement and rotation angle of the frame beam tends to be stable; during the change of the cantilever length, the maximum bending moment on the frame beam reaches the minimum value of 78 kN·m when the cantilever length is 1.0 m, which is about 66% lower than the maximum bending moment value of 229 kN·m when the cantilever length is 0 m, and about 43.5% lower than the maximum bending moment value of 138 kN·m when the cantilever length is 2.25 m. It can be obtained from the analysis that in this demonstration project, the cantilever section has a great influence on the stress and deformation of the frame beam; when the cantilever length is 1.0 m, the stress on the frame beam reaches the best state.

[0125] Figures 10 - 13 It shows that the method of the embodiment of the present invention directly simplifies the deflection, rotation angle, bending moment, and shear force equations of N beams into one equation each, and the parameters in the equations can be adjusted in the programming software, and then the influence of the parameters on the stress of the frame beam can be intuitively observed through the way of drawing. And it can realize the modification of the size, cross-sectional area, and cantilever length of the frame beam, and then intuitively observe the change of the internal force of the frame beam through the way of drawing, so as to provide the frame beam data with the best stress mode for the project.

[0126] Each embodiment in this specification is described in a related manner. The same or similar parts between each embodiment can be referred to each other, and the differences between each embodiment and other embodiments are emphasized. In particular, for the system embodiment, since it is basically similar to the method embodiment, the description is relatively simple, and the related parts can be referred to the partial description of the method embodiment.

[0127] The above description is only the preferred embodiment of the present invention and is not intended to limit the protection scope of the present invention. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention are included in the protection scope of the present invention.

Claims

1. A calculation method for the deformation and internal force of an anchor rod frame beam based on the transfer matrix method, characterized in that It includes the following steps: S1. Establish a rectangular coordinate system with the left end point of the longitudinal beam in any frame beam as the origin, the longitudinal beam direction as the x-axis, and the direction perpendicular to the installation plane of the frame beam as the y-axis; S2. Solve the differential equation of the deflection curve of the beam based on the Winkler elastic foundation theory to obtain the deflection of the longitudinal beam; Superimpose the increased deflection due to the action of the anchoring force to obtain the general solution of the deflection of the i-th section of the longitudinal beam under the action of the anchoring force; Based on the general solution of the deflection of the i-th section of the longitudinal beam, obtain the expressions of the rotation angle, bending moment, and shear force of any section on the i-th section of the longitudinal beam according to the differential relationship between the deflection, rotation angle, bending moment, and shear force of any section on the longitudinal beam; S3. Based on the expressions of the rotation angle, bending moment, and shear force of any section on the i-th section of the longitudinal beam, calculate the solutions of the deflection, rotation angle, bending moment, and shear force at any position of the N-section beam by the transfer matrix method.

2. The method for calculating the deformation and internal force of the anchor rod frame beam based on the transfer matrix method according to claim 1, characterized in that In S2, based on the Winkler elastic foundation theory, the differential equation of the deflection curve of the beam is solved to obtain the expression of the deflection w i (x)0 as follows: where x is the abscissa, and λ iy represents the deformation coefficient of the i-th longitudinal beam, E iy represents the elastic modulus of the i-th longitudinal beam; I iy represents the moment of inertia of the cross-section of the i-th longitudinal beam; k iy represents the coefficient of subgrade reaction under the i-th longitudinal beam; b iy represents the width of the longitudinal beam; C 4i-3 、C 4i-2 、C 4i-1 、C 4i are undetermined constants of the longitudinal beam, which are solved through the boundary conditions of the beam, and the four undetermined constants of the same section of the beam are the same.

3. The method for calculating the deformation and internal force of an anchor rod frame beam based on the transfer matrix method according to claim 2, wherein, In S2, the deflection w i i (x) of the i-th longitudinal beam under the action of the anchoring force p p The general solution is shown in the following formula: Among them, represents the Heaviside function, and its values are as follows: Among them, P i represents the anchoring force acting on the longitudinal beam, is the position where the anchoring force on the i-th section of the longitudinal beam acts on the beam; ch is the hyperbolic cosine function, and sh is the hyperbolic sine function.

4. The calculation method for the deformation and internal force of the anchor rod frame beam based on the transfer matrix method according to claim 1, characterized in that, In S2, the differential relationship between the deflection, rotation angle, bending moment, and shear force of any section on the longitudinal beam is: Among them, w i (x) represents the deflection of the i-th longitudinal beam under the action of the anchoring force, and θ i (x) represents the rotation angle of the i-th longitudinal beam under the action of the anchoring force; M i (x) represents the bending moment of the i-th longitudinal beam under the action of the anchoring force; Q i (x) represents the shear force of the i-th longitudinal beam under the action of the anchoring force.

5. The method for calculating the deformation and internal force of an anchor rod frame beam based on the transfer matrix method according to claim 3, characterized in that In S2, the expressions of the rotation angle, bending moment, and shear force of any section on the i-th section of the longitudinal beam are respectively: Among them, θ i (x) represents the rotation angle of the i-th longitudinal beam under the action of the anchoring force; M i (x) represents the bending moment of the i-th longitudinal beam under the action of the anchoring force; Q i (x) represents the shear force of the i-th longitudinal beam under the action of the anchoring force.

6. The calculation method for the deformation and internal force of an anchor rod frame beam based on the transfer matrix method according to claim 5, characterized in that S3 includes the following steps: S31. The deflection, rotation angle, bending moment, and shear force of any section on the i-th section of the longitudinal beam are expressed in matrix form as follows: Among them, [A i represents the coefficient matrix before the undetermined constants C 4i-3 , C 4i-2 , C 4i-1 , C 4i ; represents the influence matrix of the anchoring force P i on the deflection, rotation angle, bending moment and shear force of the i-th longitudinal beam; S32. Take the left endpoint x on the i-th longitudinal beam i and the right endpoint x i+1 , and substitute them into the matrix in S31 respectively. Since C 4i-3 , C 4i-2 , C 4i-1 , and C 4i are the same in the same beam segment, the transfer relationship between the left and right ends of the i-th longitudinal beam is as follows: Among them, [T i = [A i [A i+1 -1 , T i is an intermediate parameter; [A i+1 -1 represents the inverse matrix of the coefficient matrix before C i+1 , C 4i-3 , C 4i-2 , C 4i-1 , C 4i at the right endpoint x respectively represent the influence matrices of the anchoring force on the longitudinal beam deflection, rotation angle, bending moment and shear force at the left endpoint x i , right endpoint x i+1 .​​ S33. Since the frame beam is rigidly connected at the beam section connection, the deflection, rotation angle, bending moment, and shear force at the end of the i-th beam are equal to the deflection, rotation angle, bending moment, and shear force at the starting point of the (i + 1)-th beam; therefore, substitute the starting point and end point relationship of each beam into the matrix in S32, and then substitute the starting point matrix of the next beam into the end point matrix of the previous beam one by one, so as to realize the transfer from the starting point of the first beam to the end point of the N-th beam.

7. The method for calculating the deformation and internal force of an anchor rod frame beam based on the transfer matrix method according to claim 6, characterized in that The transfer relationship from the starting point of the first beam to the end point of the N-th beam in S33 is: Among them, according to the Heaviside function existing in the matrix When x < x p At this time, the function value is always 0. Therefore, at the starting point, At the starting point of the first beam segment, the boundary conditions are M1(x1) = 0 and Q1(x1) = 0; at the end of the Nth beam segment, the boundary conditions are M N+1 (x N+1 ) = 0 and Q N+1 (x N+1 ) = 0; thus, we obtain: Among them, is the matrix element, is inverse matrix; is the matrix element, and the numerical values of w1(x1) and θ1(x1) are obtained by solving; Substitute the values of w1(x1) and θ1(x1) into the transfer relationship from the starting point of the first beam segment to the ending point of the Nth beam segment, and then the values of w N+1 (x N+1 ) and θ N+1 (x N+1 ) can be obtained, that is, the deflection, rotation angle, bending moment, and shear force solutions at any position in the Nth beam segment are obtained.

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