Combined beam dynamic characteristic analysis method considering shear slippage and vertical lifting

Abstract

The invention discloses a combined beam dynamic characteristic analysis method considering shear slip and vertical lifting, and belongs to the field of bridge dynamics and civil engineering structural analys.The method comprises the steps that a combined beam motion control equation and natural boundary conditions are established according to material characteristics of a combined beam; deforming the motion control equation of the composite beam to obtain a deformed motion control equation; substituting the input frequency into the deformed motion control equation, and solving a coefficient matrix eigenvalue of the motion control equation; obtaining a general solution of a motion control equation; constructing a matrix vector product expression form of the force and the displacement boundary to obtain a dynamic stiffness matrix; substituting the input frequency into the dynamic stiffness matrix to obtain a dynamic stiffness matrix corresponding to the input frequency in the current round; and according to the determinant of the dynamic stiffness matrix corresponding to the input frequency in the current round, judging whether the input frequency is the frequency corresponding to the composite beam. The problems that shearing slippage and vertical tilting are not considered in a current method, and efficiency and precision cannot be considered at the same time are solved.

Description

Technical Field

[0001] The present invention belongs to the fields of bridge dynamics and civil engineering structure analysis, and particularly relates to a method for analyzing the dynamic characteristics of composite beams considering shear slip and vertical uplift. Background Art

[0002] Composite beams are widely used in bridge structures, and their common form is steel-concrete composite beams, where the steel beam and the concrete slab are combined through shear connectors. Under the action of external dynamic loads, their dynamic characteristics directly affect the safety and service life of the bridge. However, traditional analysis methods for composite beams usually ignore the effects of shear slip and vertical uplift, resulting in inaccurate calculation results. Such simplification may not be able to truly reflect the behavior of composite beams under dynamic loads.

[0003] The existing analysis methods mainly focus on the following deficiencies:

[0004] 1. Neglect of shear slip effect: The performance degradation of shear connectors under dynamic loads will significantly change the modal parameters of composite beams. Existing analysis methods often simplify it to a fully connected state and cannot reflect the actual working conditions.

[0005] 2. Insufficient study of the influence of vertical uplift effect: Due to the deformation differences between the concrete slab and the steel beam, the vertical uplift phenomenon may cause significant modal changes, and it is difficult for existing analysis models to accurately simulate.

[0006] 3. Complexity and efficiency issues: Traditional analysis requires constructing a refined finite element model or solving through high-complexity numerical methods, resulting in low efficiency and being difficult to meet the actual engineering requirements. Summary of the Invention

[0007] In view of the above deficiencies in the prior art, a method for analyzing the dynamic characteristics of composite beams considering shear slip and vertical uplift provided by the present invention solves the problems that the current method does not consider shear slip and vertical uplift and cannot balance efficiency and accuracy.

[0008] To achieve the above invention objective, the technical solution adopted by the present invention is: A method for analyzing the dynamic characteristics of composite beams considering shear slip and vertical uplift, comprising:

[0009] Obtaining the material properties of the composite beam;

[0010] According to the material properties of the composite beam, establishing the motion control equation and natural boundary conditions of the composite beam;

[0011] Using the method of separation of variables to deform the motion control equation of the composite beam to obtain the deformed motion control equation;

[0012] Preset the self-vibration frequency increment and calculate the input frequency based on the self-vibration frequency increment; substitute the input frequency into the transformed motion control equation and solve for the eigenvalues of the coefficient matrix of the motion control equation;

[0013] According to the number of real roots and imaginary roots of the eigenvalues of the coefficient matrix of the motion control equation and the natural boundary conditions, obtain the general solution of the motion control equation;

[0014] Combined with the general solution of the motion control equation, construct the matrix-vector product expression form of the force and displacement boundaries to obtain the dynamic stiffness matrix;

[0015] Substitute the input frequency into the dynamic stiffness matrix to obtain the dynamic stiffness matrix corresponding to the input frequency in the current round;

[0016] Solve the determinant of the dynamic stiffness matrix corresponding to the input frequency in the current round. According to the determinant of the dynamic stiffness matrix corresponding to the input frequency in the current round, adjust the self-vibration frequency increment, and based on whether the determinant of the dynamic stiffness matrix corresponding to the input frequency in the current round converges. If so, the input frequency in the current round is the corresponding frequency of the composite beam, and the dynamic characteristic analysis of the composite beam considering shear slip and vertical uplift is completed. Otherwise, return to calculate the input frequency of the next round.

[0017] Furthermore, the material properties include elastic modulus, shear modulus, shear correction coefficient, material density, moment of inertia, cross-sectional area, distance from the centroid to the contact surface, calculated span of the composite beam, shear slip stiffness value, and vertical uplift stiffness value; the vertical uplift stiffness value ≥ 1000 × shear slip stiffness value.

[0018] Furthermore, based on the material properties of the composite beam, establish the motion control equation and natural boundary conditions of the composite beam, specifically:

[0019] According to the material properties of the composite beam, construct the relative slip relationship of the composite beam interface considering the shear deformation and vertical uplift of the composite beam:

[0020] u 12 = u 2 - u 1 + h 1 θ 1 + h 2 θ 2

[0021] where u 12 is the relative sliding displacement of the composite beam interface; u 2 is the sliding displacement of the lower structure of the composite beam; u 1 is the sliding displacement of the upper structure of the composite beam; h 1 is the distance from the centroid of the upper structure to the contact surface of the two materials; θ 1 is the shear slip rotation angle of the upper structure; h2 is the distance from the centroid of the lower - layer structure to the contact surface of the two materials; θ 2 is the shear - slip rotation angle of the lower - layer structure;

[0022] According to the relative - slip relationship of the composite - beam interface, the dynamic - response problem of the composite beam is expressed by the variational method and Hamilton's principle:

[0023]

[0024] where δ is the variation; t 1 is the starting time of shear - slip; t 2 is the starting time of shear - slip; T is the kinetic energy of shear - slip; U is the strain energy of shear - slip; U s is the potential energy of shear - slip; d is the differential symbol; L is the calculated span of the composite beam; f is the material - position identifier, being 1 for the upper - layer material and 2 for the lower - layer material; ρ f is the material density, which is the upper - layer material density when f = 1 and the lower - layer material density when f = 2; F f is the cross - sectional area of the material, which is the upper - layer material cross - sectional area when f = 1 and the lower - layer material cross - sectional area when f = 2; w f is the vertical displacement, which is the upper - layer vertical displacement when f = 1 and the lower - layer vertical displacement when f = 2; E f is the elastic modulus of the material, which is the upper - layer material elastic modulus when f = 1 and the lower - layer material elastic modulus when f = 2; u f,x is u f the derivative value of u with respect to x; u f is the sliding displacement, which is the sliding displacement of the upper - layer structure of the composite beam when f = 1 and the sliding displacement of the lower - layer structure of the composite beam when f = 2; I f is the flexural moment of inertia, which is the upper - layer structure flexural moment of inertia when f = 1 and the lower - layer structure flexural moment of inertia when f = 2; θ f,x is θ f the derivative value of θ with respect to x; θ f is the shear - slip rotation angle, which is the upper - layer structure shear - slip rotation angle when f = 1 and the lower - layer structure shear - slip rotation angle when f = 2; k f is the material shear - correction coefficient, which is the upper - layer material shear - correction coefficient when f = 1 and the lower - layer material shear - correction coefficient when f = 2; G f is the structural shear modulus, which is the upper - layer structure shear modulus when f = 1 and the lower - layer structure shear modulus when f = 2; w f.x is w f the derivative value of w with respect to x; K 1 is the shear stiffness of the composite beam; w 12 is the vertical - displacement difference between the upper and lower layer structures; K 2is the vertical lifting stiffness of the composite beam; u 12 is the relative sliding displacement of the composite beam interface;

[0025] According to the dynamic response problem of the composite beam, establish the motion control equation and natural boundary conditions of the composite beam:

[0026]

[0027] where, m 1 is the mass of the upper layer material; m 2 is the mass of the lower layer material; N is the overall shear force of the composite beam; N 1 is the shear force of the upper structure; N 2 is the shear force of the lower structure; M 1 is the bending moment of the upper structure; M 2 is the bending moment of the lower structure; Q 1 is the axial force of the upper structure; Q 2 is the axial force of the lower structure; EF is an intermediate variable.

[0028] Furthermore, the separation of variables method is used to deform the motion control equation of the composite beam to obtain the deformed motion control equation, specifically:

[0029] Separate the motion control equation in time and space to obtain the deformed motion control equation:

[0030] {w 1 w 2 u 12 θ 1 θ 2}={W 1 (x) W 2 (x) U 12 (x) Θ 1 (x) Θ 2 (x)}sin(ωt+ψ)

[0031]

[0032] where, w 1 is the vertical displacement of the upper layer; w 2 is the vertical displacement of the lower layer; u 12 is the relative sliding displacement of the composite beam interface; θ 1 is the shear slip rotation angle of the upper structure; θ 2 is the shear slip rotation angle of the lower structure; W 1 (x) is the vertical displacement mode function of the upper layer material; W 2 (x) is the vertical displacement mode function of the lower layer material; U 12 (x) is the relative sliding displacement mode function of the interface; Θ 1(x) is the shear slip rotation mode function of the upper structure; Θ 2 (x) is the shear slip rotation mode function of the lower structure; ω is the frequency; t is the time; ψ is the initial phase; EF is the intermediate variable; λ is the eigenvalue of the coefficient matrix; K 2 is the vertical lifting stiffness of the composite beam; L is the calculated span of the composite beam; h 1 is the distance from the centroid of the upper structure to the contact surface of the two materials; h 2 is the distance from the centroid of the lower structure to the contact surface of the two materials; k 1 is the shear correction coefficient of the upper material; G 1 is the shear modulus of the upper structure; E 1 is the elastic modulus of the upper material; k 2 is the shear correction coefficient of the lower material; G 2 is the shear modulus of the lower structure; E 2 is the elastic modulus of the lower material; I 1 is the flexural moment of inertia of the upper structure; F 1 is the cross-sectional area of the upper material; F 2 is the cross-sectional area of the lower material; I 2 is the flexural moment of inertia of the lower structure; K 1 is the shear stiffness of the composite beam; m 1 is the mass of the upper material; m 2 is the mass of the lower material; Z 1 、Z 2 、Z 3 、Z 4 and Z 5 are all undetermined coefficients after separating variables.

[0033] Furthermore, the expression of the input frequency is:

[0034] ω j+1 =ω j +Δω

[0035] ω 1 =0

[0036] where, ω j+1 is the input frequency of the (j + 1)-th iteration; ω j is the input frequency of the j-th iteration; Δω is the natural frequency increment; ω 1 is the initial frequency.

[0037] Furthermore, the general solution expression of the motion control equation is:

[0038]

[0039] where, W 1 (ξ) is the vertical displacement corresponding to the upper material at position ξ; The value ranges from 0 to 1; x is the coordinate along the beam length; A 2i-1 、A 2i 、A 9 and A 10 are all general solution coefficients of the vertical displacement of the upper layer material; λ i is the eigenvalue; λ 5 is the 5th eigenvalue; W 2 (ξ) is the corresponding vertical displacement of the lower layer material at position ξ; B 2i-1 、B 2i 、B 9 and B 10 are all general solution coefficients of the vertical displacement of the lower layer material; U 12 (ξ) is the relative slip displacement at the interface at position ξ; C 2i-1 、C 2i 、C 9 and C 10 are all general solution coefficients of the relative slip displacement at the interface; Θ 1 (ξ) is the shear slip angle of the upper layer structure at position ξ; D 2i-1 、D 2i 、D 9 and D 10 are all general solution coefficients of the shear slip angle of the upper layer structure; Θ 2 (ξ) is the shear slip angle of the lower layer structure at position ξ; E 2i-1 、E 2i 、E 9 and E 10 are all general solution coefficients of the shear slip angle of the lower layer structure; γ i is the relationship coefficient between the vertical displacement of the lower layer material and the vertical displacement of the upper layer material; η i is the relationship coefficient between the relative slip displacement at the interface and the vertical displacement of the upper layer material; κ i is the relationship coefficient between the shear slip angle of the upper layer structure and the vertical displacement of the upper layer material; μ i is the relationship coefficient between the shear slip angle of the lower layer structure and the vertical displacement of the upper layer material; and are all equation reference symbols; EF is an intermediate variable; K 2 is the vertical lifting stiffness of the composite beam; L is the calculated span of the composite beam; h 1 is the distance from the centroid of the upper layer structure to the contact surface of the two materials; h 2 is the distance from the centroid of the lower layer structure to the contact surface of the two materials; k 1 is the shear correction coefficient of the upper layer material; G 1 is the shear modulus of the upper layer structure; F 1 is the cross-sectional area of the upper layer material; E 1 is the elastic modulus of the upper layer material; I1 is the flexural moment of inertia of the upper structure; k 2 is the shear correction factor of the lower material; G 2 is the shear modulus of the lower structure; F 2 is the cross-sectional area of the lower material; E 2 is the elastic modulus of the lower material; I 2 is the flexural moment of inertia of the lower structure; m 1 is the mass of the upper material; K 1 is the shear stiffness of the composite beam; ω is the frequency.

[0040] Furthermore, by combining the general solution of the motion control equation and constructing the matrix-vector product expression form of the force and displacement boundaries, the dynamic stiffness matrix is obtained, specifically:

[0041] According to the general solution of the motion control equation, the nodal displacement vector u e is represented by the constant vector a to obtain the matrix-vector product expression form of the displacement boundary:

[0042] u e = N e a

[0043] u e = {W 1 (0)W 2 (0)U 12 (0)θ 1 (0)θ 2 (0)W 1 (1)W 2 (1)U 12 (1)θ 1 (1)θ 2 (1)}

[0044] a = {A 1 A 2 A 3 A 4 A 5 A 6 A 7 A 8 A 9 A 10}

[0045]

[0046] S = sinλ 5

[0047] C = cosλ 5

[0048] where, N e is the displacement boundary matrix; W 1$(0)$ is the vertical displacement corresponding to the upper layer material at the position $\xi = 0$; $W$ 2 $(0)$ is the vertical displacement corresponding to the lower layer material at the position $\xi = 0$; $U$ 12 $(0)$ is the relative sliding displacement of the interface at the position $\xi = 0$; $\theta$ 1 $(0)$ is the shear slip angle of the upper layer structure at the position $\xi = 0$; $\theta$ 2 $(0)$ is the shear slip angle of the lower layer structure at the position $\xi = 0$; $W$ 1 $(1)$ is the vertical displacement corresponding to the upper layer material at the position $\xi = 1$; $W$ 2 $(1)$ is the vertical displacement corresponding to the lower layer material at the position $\xi = 1$; $U$ 12 $(1)$ is the relative sliding displacement of the interface at the position $\xi = 1$; $\theta$ 1 $(1)$ is the shear slip angle of the upper layer structure at the position $\xi = 1$; $\theta$ 2 $(1)$ is the shear slip angle of the lower layer structure at the position $\xi = 1$; $A$ 1 、$A$ 2 、$A$ 3 、$A$ 4 、$A$ 5 、$A$ 6 、$A$ 7 、$A$ 8 、$A$ 9 and $A$ 10 are all general solution coefficients of the vertical displacement of the upper layer material; $\lambda$ 1 is the first eigenvalue of the coefficient matrix; $\lambda$ 2 is the second eigenvalue of the coefficient matrix; $\lambda$ 3 is the third eigenvalue of the coefficient matrix; $\lambda$ 4 is the fourth eigenvalue of the coefficient matrix; $\gamma$ 1 、$\gamma$ 2 、$\gamma$ 3 、$\gamma$ 4 and $\gamma$ 5 are all relationship coefficients between the vertical displacement of the lower layer material and the vertical displacement of the upper layer material; $\eta$ 1 、$\eta$ 2 、$\eta$ 3 、$\eta$ 4 and $\eta$ 5 are all relationship coefficients between the relative sliding displacement of the interface and the vertical displacement of the upper layer material; $\kappa$ 1 、$\kappa$ 2 、$\kappa$ 3 、$\kappa$ 4 and $\kappa$ 5 are all relationship coefficients between the shear slip angle of the upper layer structure and the vertical displacement of the upper layer material; $\mu$ 1 、$\mu$ 2 、$\mu$ 3 、$\mu$ 4 and $\mu$ 5They are all the relationship coefficients between the shear slip angle of the lower structure and the vertical displacement of the upper material; both S and C are the symbolic notations of the equations; λ 5 is the 5th eigenvalue of the coefficient matrix;

[0049] Express the nodal force vector p e in terms of the constant vector a, and obtain the matrix-vector product expression of the force:

[0050] p e = M e a

[0051] p e = {Q 1 (0)Q 2 (0)N(0)M 1 (0)M 2 (0)Q 1 (1)Q 2 (1)N(1)M 1 (1)M 2 (1)}

[0052]

[0053] Η i = k 1 G 1 F 1 (λ i - κ i )

[0054] Λ i = k 2 G 2 F 2 (γ i λ i - μ i )

[0055] Γ i = EFλ i (η i - h 1 κ i - h 2 μ i )

[0056] Ω i = E 1 I 1 λ i κ i

[0057] Ψ i = E 2 I 2 λ i μ i

[0058] Among them, M e is the force boundary matrix; Q 1 (0) is the axial force corresponding to the upper layer material at the position ξ = 0; Q 2 (0) is the axial force corresponding to the lower layer material at the position ξ = 0; N(0) is the shear force corresponding to the composite structure at the position ξ = 0; M 1 (0) is the bending moment corresponding to the upper layer material at the position ξ = 0; M 2 (0) is the bending moment corresponding to the lower layer material at the position ξ = 0; Q 1 (1) is the axial force corresponding to the upper layer material at the position ξ = 1; Q 2 (1) is the axial force corresponding to the lower layer material at the position ξ = 1; N(1) is the shear force corresponding to the composite structure at the position ξ = 1; M 1 (1) is the bending moment corresponding to the upper layer material at the position ξ = 1; M 2 (1) is the bending moment corresponding to the lower layer material at the position ξ = 1; Η i , Λ i , Γ i , Ω i and Ψ i are all equation reference symbols; k 1 is the shear correction coefficient of the upper layer material; G 1 is the shear modulus of the upper layer structure; F 1 is the cross-sectional area of the upper layer material; γ i is the relationship coefficient between the vertical displacement of the lower layer material and the vertical displacement of the upper layer material; κ i is the relationship coefficient between the shear slip angle of the upper layer structure and the vertical displacement of the upper layer material; k 2 is the shear correction coefficient of the lower layer material; G 2 is the shear modulus of the lower layer structure; F 2 is the cross-sectional area of the lower layer material; μ i is the relationship coefficient between the shear slip angle of the lower layer structure and the vertical displacement of the upper layer material; λ i is the eigenvalue; η i is the relationship coefficient between the relative slip displacement of the interface and the vertical displacement of the upper layer material; EF is an intermediate variable; h 1 is the distance from the centroid of the upper layer structure to the contact surface of the two materials; h 2 is the distance from the centroid of the lower layer structure to the contact surface of the two materials; E 1 is the elastic modulus of the upper layer material; I 1 is the flexural moment of inertia of the upper layer structure; E 2 is the elastic modulus of the lower layer material; I 2 is the flexural moment of inertia of the lower layer structure;

[0059] Based on the matrix-vector product expression form of force and the matrix-vector product expression form of displacement boundary, the dynamic stiffness matrix K is obtained. e :

[0060]

[0061] Further, the natural vibration frequency increment is adjusted according to the determinant of the dynamic stiffness matrix corresponding to the input frequency in the current round, specifically: If then set Δω = -Δω / 2; otherwise, the natural vibration frequency increment remains unchanged; where is the determinant of the dynamic stiffness matrix corresponding to the input frequency in the current round; is the determinant of the dynamic stiffness matrix corresponding to the input frequency in the previous round; Δω is the natural vibration frequency increment.

[0062] Further, it is determined whether the determinant of the dynamic stiffness matrix corresponding to the input frequency in the current round converges, specifically: When it is considered convergent, otherwise, it is non-convergent; where is the determinant of the dynamic stiffness matrix corresponding to the input frequency in the current round; ε is the convergence threshold.

[0063] The beneficial effects of the present invention are as follows: This analysis method fully considers the shear deformation and vertical lifting effect of the composite structure, and can accurately capture the actual working state of the composite structure. Compared with the traditional analysis method, it does not need to construct a cumbersome finite element model or rely on highly complex numerical solutions, thus significantly improving the work efficiency. This method not only meets the dual requirements of accuracy and efficiency in engineering practice, but also better adapts to the application requirements of actual engineering projects. BRIEF DESCRIPTION OF THE DRAWINGS

[0064] Figure 1 is the method flow chart of the present invention.

[0065] Figure 2 is a schematic diagram of a composite beam considering shear slip and vertical lifting in the present invention.

[0066] Figure 3 is a schematic diagram of the relative slip relationship of the interface of the composite beam in the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0067] The following describes the specific embodiments of the present invention to facilitate those skilled in the art to understand the present invention, but it should be clear that the present invention is not limited to the scope of the specific embodiments. For those of ordinary skill in the art, as long as various changes are within the spirit and scope of the present invention defined and determined by the appended claims, these changes are obvious, and all inventions made using the concept of the present invention are within the scope of protection.

[0068] AsFigure 1 As shown, in one embodiment of the present invention, a method for analyzing dynamic characteristics of a composite beam considering shear slip and vertical lifting includes:

[0069] Obtain material properties of composite beams;

[0070] According to the material properties of composite beams, the motion control equations and natural boundary conditions of composite beams are established;

[0071] The variable separation method is used to deform the composite beam motion control equation to obtain the deformed motion control equation.

[0072] Preset the natural frequency increment, and calculate the input frequency based on the natural frequency increment; substitute the input frequency into the deformed motion control equation, and solve the eigenvalue of the coefficient matrix of the motion control equation;

[0073] According to the number of real roots and imaginary roots of the eigenvalues ​​of the coefficient matrix of the motion control equation and the natural boundary conditions, the general solution of the motion control equation is obtained;

[0074] Combined with the general solution of the motion control equation, the matrix-vector product expression of the force and displacement boundary is constructed to obtain the dynamic stiffness matrix;

[0075] Substitute the input frequency into the dynamic stiffness matrix to obtain the dynamic stiffness matrix corresponding to the input frequency in the current round;

[0076] Solve the determinant of the dynamic stiffness matrix corresponding to the input frequency in the current round, adjust the natural frequency increment according to the determinant of the dynamic stiffness matrix corresponding to the input frequency in the current round, and check whether the determinant of the dynamic stiffness matrix corresponding to the input frequency in the current round converges. If so, the input frequency of the current round is the corresponding frequency of the composite beam, and the dynamic characteristics analysis of the composite beam considering shear slip and vertical lifting is completed. Otherwise, return to calculate the input frequency of the next round.

[0077] In this embodiment, the overall process of the solution is:

[0078] S1: Construct a calculation model for considering shear deformation and vertical lifting of composite beams (such as Figure 1 , Figure 2 As shown), the relative slip relationship of the composite beam interface is obtained;

[0079] S2: Using the variational method and Hamilton's principle, based on the established model and interface slip relationship, the composite beam motion control equation and natural boundary conditions are further established;

[0080] S3: Using the separation of variables method, the composite beam motion control equation is deformed and the eigenvalue of the motion control equation coefficient matrix is ​​solved;

[0081] S4: According to the number of eigenvalues of the coefficient matrix of the motion control equation, give the general solution expression of the motion control equation, and substitute the general solution expression into the deformed motion control equation of the composite beam to obtain the relationship between the unknowns in the general solution expression.

[0082] S5: According to the force and displacement boundary conditions at the beam ends, combined with the analytical solution of the motion differential equation, construct the matrix-vector product expression form of the force and displacement boundaries. Then obtain the relationship matrix between the force and displacement vectors, which is the dynamic stiffness matrix.

[0083] S6: By writing a MATLAB program, assign values to the characteristic frequencies starting from "0". When the absolute value of the determinant of the dynamic stiffness matrix corresponding to the characteristic frequency is less than 0.1, this frequency is the corresponding frequency of the composite beam.

[0084] The material properties include elastic modulus, shear modulus, shear correction coefficient, material density, moment of inertia, cross-sectional area, distance from the centroid to the contact surface, calculated span of the composite beam, shear slip stiffness value, and vertical lifting stiffness value; the vertical lifting stiffness value ≥ 1000 × the shear slip stiffness value.

[0085] As Figure 3 shown in the schematic diagram of the relative slip relationship of the composite beam interface, based on the material properties of the composite beam, establish the motion control equation and natural boundary conditions of the composite beam, specifically:

[0086] According to the material properties of the composite beam, construct the relative slip relationship formula of the composite beam interface considering the shear deformation and vertical lifting of the composite beam:

[0087] u 12 = u 2 - u 1 + h 1 θ 1 + h 2 θ 2

[0088] where, u 12 is the relative sliding displacement of the composite beam interface; u 2 is the sliding displacement of the lower structure of the composite beam; u 1 is the sliding displacement of the upper structure of the composite beam; h 1 is the distance from the centroid of the upper structure to the contact surface of the two materials; θ 1 is the shear slip angle of the upper structure; h 2 is the distance from the centroid of the lower structure to the contact surface of the two materials; θ 2 is the shear slip angle of the lower structure;

[0089] According to the relative slip relationship formula of the composite beam interface, represent the dynamic response problem of the composite beam through the variational method and Hamilton's principle:

[0090]

[0091] Among them, δ is the variation; t 1 is the starting time of shear slip; t 2 is the starting time of shear slip; T is the kinetic energy of shear slip; U is the strain energy of shear slip; U s is the potential energy of shear slip; d is the differential symbol; L is the calculated span of the composite beam; f is the material position identifier, being 1 for the upper layer material and 2 for the lower layer material; ρ f is the material density, being the upper layer material density when f = 1 and the lower layer material density when f = 2; F f is the cross-sectional area of the material, being the cross-sectional area of the upper layer material when f = 1 and the cross-sectional area of the lower layer material when f = 2; w f is the vertical displacement, being the upper layer vertical displacement when f = 1 and the lower layer vertical displacement when f = 2; E f is the elastic modulus of the material, being the elastic modulus of the upper layer material when f = 1 and the elastic modulus of the lower layer material when f = 2; u f,x is u f the derivative value of u with respect to x; u f is the sliding displacement, being the sliding displacement of the upper structure of the composite beam when f = 1 and the sliding displacement of the lower structure of the composite beam when f = 2; I f is the flexural moment of inertia, being the flexural moment of inertia of the upper structure when f = 1 and the flexural moment of inertia of the lower structure when f = 2; θ f,x is θ f the derivative value of θ with respect to x; θ f is the shear slip rotation angle, being the shear slip rotation angle of the upper structure when f = 1 and the shear slip rotation angle of the lower structure when f = 2; k f is the material shear correction coefficient, being the upper layer material shear correction coefficient when f = 1 and the lower layer material shear correction coefficient when f = 2; G f is the structural shear modulus, being the upper layer structural shear modulus when f = 1 and the lower layer structural shear modulus when f = 2; w f.x is w f the derivative value of w with respect to x; K 1 is the shear stiffness of the composite beam; w 12 is the vertical displacement difference between the upper and lower structures; K 2 is the vertical lifting stiffness of the composite beam; u 12 is the relative sliding displacement of the composite beam interface;

[0092] According to the dynamic response problem of the composite beam, establish the motion control equation and natural boundary conditions of the composite beam:

[0093]

[0094] Among them, m 1 is the mass of the upper layer material; m 2 is the mass of the lower layer material; N is the overall shear force of the composite beam; N 1 is the shear force of the upper structure; N 2 is the shear force of the lower structure; M 1 is the bending moment of the upper structure; M 2 is the bending moment of the lower structure; Q 1 is the axial force of the upper structure; Q 2 is the axial force of the lower structure; EF is an intermediate variable.

[0095] The separation of variables method is adopted to deform the motion control equation of the composite beam to obtain the deformed motion control equation, specifically:

[0096] Separate the motion control equation in time and space to obtain the deformed motion control equation:

[0097] {w 1 w 2 u 12 θ 1 θ 2} = {W 1 (x) W 2 (x) U 12 (x) Θ 1 (x) Θ 2 (x)}sin(ωt + ψ)

[0098]

[0099] Among them, w 1 is the vertical displacement of the upper layer; w 2 is the vertical displacement of the lower layer; u 12 is the relative sliding displacement of the composite beam interface; θ 1 is the shear slip rotation angle of the upper structure; θ 2 is the shear slip rotation angle of the lower structure; W 1 (x) is the vertical displacement mode function of the upper layer material; W 2 (x) is the vertical displacement mode function of the lower layer material; U 12 (x) is the relative sliding displacement mode function of the interface; Θ 1 (x) is the shear slip rotation angle mode function of the upper structure; Θ 2 (x) is the shear slip rotation angle mode function of the lower structure; ω is the frequency; t is the time; ψ is the initial phase; EF is an intermediate variable; λ is the eigenvalue of the coefficient matrix; K 2 is the vertical lifting stiffness of the composite beam; L is the calculated span of the composite beam; h 1 is the distance from the centroid of the upper structure to the contact surface of the two materials; h 2is the distance from the centroid of the lower - layer structure to the contact surface of the two materials; k 1 is the shear correction coefficient of the upper - layer material; G 1 is the shear modulus of the upper - layer structure; E 1 is the elastic modulus of the upper - layer material; k 2 is the shear correction coefficient of the lower - layer material; G 2 is the shear modulus of the lower - layer structure; E 2 is the elastic modulus of the lower - layer material; I 1 is the flexural moment of inertia of the upper - layer structure; F 1 is the cross - sectional area of the upper - layer material; F 2 is the cross - sectional area of the lower - layer material; I 2 is the flexural moment of inertia of the lower - layer structure; K 1 is the shear stiffness of the composite beam; m 1 is the mass of the upper - layer material; m 2 is the mass of the lower - layer material; Z 1 、Z 2 、Z 3 、Z 4 and Z 5 are all undetermined coefficients after separating variables.

[0100] The expression of the input frequency is:

[0101] ω j+1 =ω j +Δω

[0102] ω 1 =0

[0103] where, ω j+1 is the input frequency of the (j + 1)-th iteration; ω j is the input frequency of the j - th iteration; Δω is the increment of the natural vibration frequency; ω 1 is the initial frequency.

[0104] The general solution expression of the motion control equation is:

[0105]

[0106] where, W 1 (ξ) is the vertical displacement corresponding to the upper - layer material at the position ξ; takes values from 0 to 1; x is the coordinate along the beam length direction; A 2i-1 、A 2i 、A 9 and A 10 are all general solution coefficients of the vertical displacement of the upper - layer material; λ i is the eigenvalue; λ 5 is the 5th eigenvalue; W 2 (ξ) is the vertical displacement corresponding to the lower - layer material at the position ξ; B2i-1 , B 2i , B 9 and B 10 are all general solution coefficients of the vertical displacement of the lower layer material; U 12 (ξ) is the relative slip displacement of the interface at position ξ; C 2i-1 , C 2i , C 9 and C 10 are all general solution coefficients of the relative slip displacement of the interface; Θ 1 (ξ) is the shear slip rotation angle of the upper structure at position ξ; D 2i-1 , D 2i , D 9 and D 10 are all general solution coefficients of the shear slip rotation angle of the upper structure; Θ 2 (ξ) is the shear slip rotation angle of the lower structure at position ξ; E 2i-1 , E 2i , E 9 and E 10 are all general solution coefficients of the shear slip rotation angle of the lower structure; γ i is the relationship coefficient between the vertical displacement of the lower layer material and the vertical displacement of the upper layer material; η i is the relationship coefficient between the relative slip displacement of the interface and the vertical displacement of the upper layer material; κ i is the relationship coefficient between the shear slip rotation angle of the upper structure and the vertical displacement of the upper layer material; μ i is the relationship coefficient between the shear slip rotation angle of the lower structure and the vertical displacement of the upper layer material; and are all equation reference symbols; EF is an intermediate variable; K 2 is the vertical lifting stiffness of the composite beam; L is the calculated span of the composite beam; h 1 is the distance from the centroid of the upper structure to the contact surface of the two materials; h 2 is the distance from the centroid of the lower structure to the contact surface of the two materials; k 1 is the shear correction coefficient of the upper layer material; G 1 is the shear modulus of the upper structure; F 1 is the cross-sectional area of the upper layer material; E 1 is the elastic modulus of the upper layer material; I 1 is the flexural moment of inertia of the upper structure; k 2 is the shear correction coefficient of the lower layer material; G 2 is the shear modulus of the lower structure; F 2 is the cross-sectional area of the lower layer material; E 2 is the elastic modulus of the lower layer material; I 2 is the flexural moment of inertia of the lower structure; m 1 is the mass of the upper layer material; K 1is the shear stiffness of the composite beam; ω is the frequency.

[0107] Combined with the general solution of the motion control equation, the matrix-vector product expression of the force and displacement boundaries is constructed to obtain the dynamic stiffness matrix, specifically:

[0108] According to the general solution of the motion control equation, the nodal displacement vector u e is represented by the constant vector a to obtain the matrix-vector product expression of the displacement boundary:

[0109] u e = N e a

[0110] u e = {W 1 (0) W 2 (0) U 12 (0) θ 1 (0) θ 2 (0) W 1 (1) W 2 (1) U 12 (1) θ 1 (1) θ 2 (1)}

[0111] a = {A 1 A 2 A 3 A 4 A 5 A 6 A 7 A 8 A 9 A 10}

[0112]

[0113] S = sinλ 5

[0114] C = cosλ 5

[0115] Among them, N e is the displacement boundary matrix; W 1 (0) is the vertical displacement corresponding to the upper layer material at the position ξ = 0; W 2 (0) is the vertical displacement corresponding to the lower layer material at the position ξ = 0; U 12 (0) is the interfacial relative sliding displacement at the position ξ = 0; θ 1 (0) is the shear slip angle of the upper layer structure at the position ξ = 0; θ 2 (0) is the shear slip angle of the lower layer structure at the position ξ = 0; W1 $(1)$ is the vertical displacement corresponding to the upper layer material at the position $\xi = 1$; $W$ 2 $(1)$ is the vertical displacement corresponding to the lower layer material at the position $\xi = 1$; $U$ 12 $(1)$ is the relative sliding displacement of the interface at the position $\xi = 1$; $\theta$ 1 $(1)$ is the shear slip angle of the upper layer structure at the position $\xi = 1$; $\theta$ 2 $(1)$ is the shear slip angle of the lower layer structure at the position $\xi = 1$; $A$ 1 、$A$ 2 、$A$ 3 、$A$ 4 、$A$ 5 、$A$ 6 、$A$ 7 、$A$ 8 、$A$ 9 and $A$ 10 are all general solution coefficients of the vertical displacement of the upper layer material; $\lambda$ 1 is the first eigenvalue of the coefficient matrix; $\lambda$ 2 is the second eigenvalue of the coefficient matrix; $\lambda$ 3 is the third eigenvalue of the coefficient matrix; $\lambda$ 4 is the fourth eigenvalue of the coefficient matrix; $\gamma$ 1 、$\gamma$ 2 、$\gamma$ 3 、$\gamma$ 4 and $\gamma$ 5 are all relationship coefficients between the vertical displacement of the lower layer material and the vertical displacement of the upper layer material; $\eta$ 1 、$\eta$ 2 、$\eta$ 3 、$\eta$ 4 and $\eta$ 5 are all relationship coefficients between the relative sliding displacement of the interface and the vertical displacement of the upper layer material; $\kappa$ 1 、$\kappa$ 2 、$\kappa$ 3 、$\kappa$ 4 and $\kappa$ 5 are all relationship coefficients between the shear slip angle of the upper layer structure and the vertical displacement of the upper layer material; $\mu$ 1 、$\mu$ 2 、$\mu$ 3 、$\mu$ 4 and $\mu$ 5 are all relationship coefficients between the shear slip angle of the lower layer structure and the vertical displacement of the upper layer material; $S$ and $C$ are both equation reference symbols; $\lambda$ 5 is the fifth eigenvalue of the coefficient matrix;

[0116] Express the nodal force vector $p$ e in terms of the constant vector $a$, and obtain the matrix - vector product expression of the force:

[0117] $p$e = M e a

[0118] p e = {Q 1 (0)Q 2 (0)N(0)M 1 (0)M 2 (0)Q 1 (1)Q 2 (1)N(1)M 1 (1)M 2 (1)}

[0119]

[0120] Η i = k 1 G 1 F 1 (λ i - κ i )

[0121] Λ i = k 2 G 2 F 2 (γ i λ i - μ i )

[0122] Γ i = EFλ i (η i - h 1 κ i - h 2 μ i )

[0123] Ω i = E 1 I 1 λ i κ i

[0124] Ψ i = E 2 I 2 λ i μ i

[0125] Among them, M e is the force boundary matrix; Q 1 (0) is the axial force corresponding to the upper layer material at the position ξ = 0; Q 2 (0) is the axial force corresponding to the lower layer material at the position ξ = 0; N(0) is the shear force corresponding to the composite structure at the position ξ = 0; M 1 (0) is the bending moment corresponding to the upper layer material at the position ξ = 0; M2 (0) is the bending moment corresponding to the lower layer material at the position ξ = 0; Q 1 (1) is the axial force corresponding to the upper layer material at the position ξ = 1; Q 2 (1) is the axial force corresponding to the lower layer material at the position ξ = 1; N(1) is the shear force corresponding to the composite structure at the position ξ = 1; M 1 (1) is the bending moment corresponding to the upper layer material at the position ξ = 1; M 2 (1) is the bending moment corresponding to the lower layer material at the position ξ = 1; Η i 、Λ i 、Γ i 、Ω i and Ψ i are all equation reference symbols; k 1 is the shear correction coefficient of the upper layer material; G 1 is the shear modulus of the upper layer structure; F 1 is the cross-sectional area of the upper layer material; γ i is the relationship coefficient between the vertical displacement of the lower layer material and the vertical displacement of the upper layer material; κ i is the relationship coefficient between the shear slip angle of the upper layer structure and the vertical displacement of the upper layer material; k 2 is the shear correction coefficient of the lower layer material; G 2 is the shear modulus of the lower layer structure; F 2 is the cross-sectional area of the lower layer material; μ i is the relationship coefficient between the shear slip angle of the lower layer structure and the vertical displacement of the upper layer material; λ i is the eigenvalue; η i is the relationship coefficient between the relative slip displacement of the interface and the vertical displacement of the upper layer material; EF is an intermediate variable; h 1 is the distance from the centroid of the upper layer structure to the contact surface of the two materials; h 2 is the distance from the centroid of the lower layer structure to the contact surface of the two materials; E 1 is the elastic modulus of the upper layer material; I 1 is the flexural moment of inertia of the upper layer structure; E 2 is the elastic modulus of the lower layer material; I 2 is the flexural moment of inertia of the lower layer structure;

[0126] According to the matrix-vector product expression form of force and the matrix-vector product expression form of displacement boundary, the dynamic stiffness matrix K e :

[0127]

[0128] The self-vibration frequency increment is adjusted according to the determinant of the dynamic stiffness matrix corresponding to the input frequency in the current round, specifically: if then set Δω = -Δω / 2; otherwise, the self-vibration frequency increment remains unchanged; where, is the determinant of the dynamic stiffness matrix corresponding to the input frequency in the current round; is the determinant of the dynamic stiffness matrix corresponding to the input frequency in the previous round; Δω is the natural vibration frequency increment.

[0129] According to whether the determinant of the dynamic stiffness matrix corresponding to the input frequency in the current round converges, specifically: when it is considered convergent, otherwise, it is non - convergent; where is the determinant of the dynamic stiffness matrix corresponding to the input frequency in the current round; ε is the convergence threshold.

[0130] In this embodiment, by comprehensively considering the shear - slip effect and the vertical uplifting effect between the steel beam and the concrete slab, the composite beam considering the vertical uplifting stiffness is as Figure 2 shown, which can provide more accurate modal analysis during the bridge design and operation stages. Through a self - written MATLAB program, users only need to input material and geometric parameters to efficiently obtain frequency information. The present invention not only improves the accuracy of the calculation results, but also significantly improves the calculation efficiency, and is applicable to the dynamic performance analysis, design optimization, and health monitoring of various types of composite beam bridges.

Claims

1. A method for analyzing the dynamic characteristics of a composite beam considering shear slip and vertical uplift, characterized in that: include: Obtain material properties of composite beams; According to the material properties of composite beams, the motion control equations and natural boundary conditions of composite beams are established; The variable separation method is used to deform the composite beam motion control equation to obtain the deformed motion control equation. Preset the natural frequency increment, and calculate the input frequency based on the natural frequency increment; substitute the input frequency into the deformed motion control equation, and solve the eigenvalue of the coefficient matrix of the motion control equation; According to the number of real roots and imaginary roots of the eigenvalues ​​of the coefficient matrix of the motion control equation and the natural boundary conditions, the general solution of the motion control equation is obtained; Combined with the general solution of the motion control equation, the matrix-vector product expression of the force and displacement boundary is constructed to obtain the dynamic stiffness matrix; Substitute the input frequency into the dynamic stiffness matrix to obtain the dynamic stiffness matrix corresponding to the input frequency in the current round; Solve the determinant of the dynamic stiffness matrix corresponding to the input frequency in the current round, adjust the natural frequency increment according to the determinant of the dynamic stiffness matrix corresponding to the input frequency in the current round, and check whether the determinant of the dynamic stiffness matrix corresponding to the input frequency in the current round converges. If so, the input frequency of the current round is the corresponding frequency of the composite beam, and the dynamic characteristics analysis of the composite beam considering shear slip and vertical lifting is completed. Otherwise, return to calculate the input frequency of the next round.

2. The method for analyzing dynamic characteristics of composite beams considering shear slip and vertical lifting according to claim 1 is characterized in that: The material properties include elastic modulus, shear modulus, shear correction coefficient, material density, moment of inertia, cross-sectional area, distance from center of mass to contact surface, calculated span of composite beam, shear slip stiffness value and vertical uplift stiffness value; the vertical uplift stiffness value is ≥1000×shear slip stiffness value.

3. The method for analyzing dynamic characteristics of composite beams considering shear slip and vertical lifting according to claim 1 is characterized in that: According to the material properties of the composite beam, the composite beam motion control equation and the natural boundary conditions are established, which are specifically: According to the material properties of the composite beam, the relative slip relationship of the composite beam interface considering the shear deformation and vertical lifting of the composite beam is constructed: u 12 =u2-u1+h1θ1+h2θ2 Among them, u 12 is the relative sliding displacement of the composite beam interface; u2 is the sliding displacement of the lower structure of the composite beam; u1 is the sliding displacement of the upper structure of the composite beam; h1 is the distance from the centroid of the upper structure to the contact surface of the two materials; θ1 is the shear slip angle of the upper structure; h2 is the distance from the centroid of the lower structure to the contact surface of the two materials; θ2 is the shear slip angle of the lower structure; According to the relative slip relationship of the composite beam interface, the dynamic response problem of the composite beam is expressed by the variational method and Hamilton principle: Where, δ is the variation; t1 is the start time of shear slip; t2 is the start time of shear slip; T is the kinetic energy of shear slip; U is the strain energy of shear slip; U s is the potential energy of shear slip; d is the differential symbol; L is the calculated span of the composite beam; f is the material position identifier, 1 indicates the upper material, and 2 indicates the lower material; ρ f is the material density. When f=1, it is the density of the upper material. When f=2, it is the density of the lower material. f is the cross-sectional area of ​​the material. When f=1, it is the cross-sectional area of ​​the upper material. When f=2, it is the cross-sectional area of ​​the lower material. f is the vertical displacement. When f=1, it is the vertical displacement of the upper layer. When f=2, it is the vertical displacement of the lower layer. E f is the elastic modulus of the material. When f=1, it is the elastic modulus of the upper material. When f=2, it is the elastic modulus of the lower material. f,x for u f The derivative of x; u f is the sliding displacement. When f=1, it is the sliding displacement of the composite beam upper structure. When f=2, it is the sliding displacement of the composite beam lower structure. f is the bending inertia moment. When f=1, it is the bending inertia moment of the upper structure. When f=2, it is the bending inertia moment of the lower structure. f,x is θ f The derivative of x; θ f is the shear slip angle. When f=1, it is the shear slip angle of the upper structure. When f=2, it is the shear slip angle of the lower structure. f is the material shear correction coefficient. When f=1, it is the shear correction coefficient of the upper material. When f=2, it is the shear correction coefficient of the lower material. G f is the structural shear modulus. When f = 1, it is the shear modulus of the upper structure. When f = 2, it is the shear modulus of the lower structure. f.x w f The derivative value with respect to x; K1 is the shear stiffness of the composite beam; w 12 is the vertical displacement difference between the upper and lower structures; K2 is the vertical lifting stiffness of the composite beam; u 12 is the relative sliding displacement of the composite beam interface; According to the dynamic response problem of composite beams, the composite beam motion control equations and natural boundary conditions are established: δu1: δu2: sth1: sth2: δw1: δw2: Among them, m1 is the mass of the upper material; m2 is the mass of the lower material; N is the overall shear force of the composite beam; N1 is the shear force of the upper structure; N2 is the shear force of the lower structure; M1 is the bending moment of the upper structure; M2 is the bending moment of the lower structure; Q1 is the axial force of the upper structure; Q2 is the axial force of the lower structure; EF is an intermediate variable.

4. The method for analyzing dynamic characteristics of composite beams considering shear slip and vertical lifting according to claim 1 is characterized in that: The variable separation method is used to deform the composite beam motion control equation to obtain the deformed motion control equation, which is specifically: Separating the motion control equations in time and space, we get the deformed motion control equations: {w1 w2 u 12 θ1 θ2}={W1(x) W2(x) U 12 (x) Θ1(x) Θ2(x)}sin(ωt+ψ) Where, w1 is the vertical displacement of the upper layer; w2 is the vertical displacement of the lower layer; u 12 is the relative sliding displacement of the composite beam interface; θ1 is the shear slip angle of the upper structure; θ2 is the shear slip angle of the lower structure; W1(x) is the vertical displacement formation function of the upper material; W2(x) is the vertical displacement formation function of the lower material; U 12 (x) is the interface relative sliding displacement formation function; Θ1(x) is the upper structure shear slip angle formation function; Θ2(x) is the lower structure shear slip angle formation function; ω is the frequency; t is the time; ψ is the initial phase; EF is the intermediate variable; λ is the coefficient matrix eigenvalue; K2 is the vertical uplift stiffness of the composite beam; L is the calculated span of the composite beam; h1 is the distance from the upper structure centroid to the contact surface of the two materials; h2 is the distance from the lower structure centroid to the contact surface of the two materials; k1 is the shear correction coefficient of the upper material ; G1 is the shear modulus of the upper structure; E1 is the elastic modulus of the upper material; k2 is the shear correction coefficient of the lower material; G2 is the shear modulus of the lower structure; E2 is the elastic modulus of the lower material; I1 is the bending moment of inertia of the upper structure; F1 is the cross-sectional area of ​​the upper material; F2 is the cross-sectional area of ​​the lower material; I2 is the bending moment of inertia of the lower structure; K1 is the shear stiffness of the composite beam; m1 is the mass of the upper material; m2 is the mass of the lower material; Z1, Z2, Z3, Z4 and Z5 are all unknown coefficients after separation of variables.

5. The method for analyzing dynamic characteristics of composite beams considering shear slip and vertical lifting according to claim 1 is characterized in that: The expression of the input frequency is: oh j+1 =ω j +See ω1=0 Among them, ω j+1 is the input frequency of the j+1th iteration; ω j is the input frequency of the jth iteration; Δω is the increment of the natural frequency; ω1 is the initial frequency.

6. The method for analyzing dynamic characteristics of composite beams considering shear slip and vertical lifting according to claim 1 is characterized in that: The general solution of the motion control equation is expressed as: Where W1(ξ) is the vertical displacement corresponding to the upper material at position ξ; The value is 0-1; x is the coordinate along the length of the beam; A 2i-1 , A 2i , A9 and A 10 are the general solution coefficients of the vertical displacement of the upper material; i is the eigenvalue; λ5 is the fifth eigenvalue; W2(ξ) is the vertical displacement corresponding to the lower material at position ξ; B 2i-1 , B 2i , B9 and B 10 are the general solution coefficients of the vertical displacement of the lower material; U 12 (ξ) is the relative sliding displacement of the interface at position ξ; C 2i-1 , C 2i , C9 and C 10 are the general solution coefficients of the relative sliding displacement of the interface; Θ1(ξ) is the shear slip angle of the superstructure at position ξ; D 2i-1 , D 2i , D9 and D 10 are the general solution coefficients of the shear slip angle of the upper structure; Θ2(ξ) is the shear slip angle of the lower structure at position ξ; E 2i-1 、E 2i , E9 and E 10 are the general solution coefficients of the shear slip angle of the lower structure; γ i is the relationship coefficient between the vertical displacement of the lower material and the vertical displacement of the upper material; η i is the relationship coefficient between the relative sliding displacement of the interface and the vertical displacement of the upper material; κ i is the relationship coefficient between the shear slip angle of the upper structure and the vertical displacement of the upper material; μ i is the relationship coefficient between the shear slip angle of the lower structure and the vertical displacement of the upper material; and All are symbols for equations; EF is the intermediate variable; K2 is the vertical uplift stiffness of the composite beam; L is the calculated span of the composite beam; h1 is the distance from the centroid of the upper structure to the contact surface of the two materials; h2 is the distance from the centroid of the lower structure to the contact surface of the two materials; k1 is the shear correction coefficient of the upper material; G1 is the shear modulus of the upper structure; F1 is the cross-sectional area of ​​the upper material; E1 is the elastic modulus of the upper material; I1 is the bending moment of inertia of the upper structure; k2 is the shear correction coefficient of the lower material; G2 is the shear modulus of the lower structure; F2 is the cross-sectional area of ​​the lower material; E2 is the elastic modulus of the lower material; I2 is the bending moment of inertia of the lower structure; m1 is the mass of the upper material; K1 is the shear stiffness of the composite beam; ω is the frequency.

7. The method for analyzing dynamic characteristics of composite beams considering shear slip and vertical lifting according to claim 1 is characterized in that: The general solution of the combined motion control equation is used to construct the matrix-vector product expression of the force and displacement boundary to obtain the dynamic stiffness matrix, which is specifically: According to the general solution of the motion control equation, the node displacement vector u e Using the constant vector a, we get the matrix-vector product expression of the displacement boundary: u e =N e a u e ={W1(0)W2(0)U 12 (0)θ1(0)θ2(0)W1(1)W2(1)U 12 (1)θ1(1)θ2(1)} <h2 style=";text-align:left;direction:ltr">a={A1 A2 A3 A4 A5 A6 A7 A8 A9 A<h2 style=";text-align:left;direction:ltr"> 10 <h2 style=";text-align:left;direction:ltr">} S=sinλ5 C=cosλ5 Among them, N e is the displacement boundary matrix; W1(0) is the vertical displacement corresponding to the upper material at position ξ = 0; W2(0) is the vertical displacement corresponding to the lower material at position ξ = 0; U 12 (0) is the relative sliding displacement of the interface at position ξ = 0; θ1(0) is the shear slip angle of the upper structure at position ξ = 0; θ2(0) is the shear slip angle of the lower structure at position ξ = 0; W1(1) is the vertical displacement corresponding to the upper material at position ξ = 1; W2(1) is the vertical displacement corresponding to the lower material at position ξ = 1; U 12 (1) is the relative sliding displacement of the interface at position ξ = 1; θ1(1) is the shear slip angle of the upper structure at position ξ = 1; θ2(1) is the shear slip angle of the lower structure at position ξ = 1; A1, A2, A3, A4, A5, A6, A7, A8, A9 and A 10 are all general solution coefficients of the vertical displacement of the upper material; λ1 is the first eigenvalue of the coefficient matrix; λ2 is the second eigenvalue of the coefficient matrix; λ3 is the third eigenvalue of the coefficient matrix; λ4 is the fourth eigenvalue of the coefficient matrix; γ1, γ2, γ3, γ4 and γ5 are all relationship coefficients between the vertical displacement of the lower material and the vertical displacement of the upper material; η1, η2, η3, η4 and η5 are all relationship coefficients between the relative sliding displacement of the interface and the vertical displacement of the upper material; κ1, κ2, κ3, κ4 and κ5 are all relationship coefficients between the shear slip angle of the upper structure and the vertical displacement of the upper material; μ1, μ2, μ3, μ4 and μ5 are all relationship coefficients between the shear slip angle of the lower structure and the vertical displacement of the upper material; S and C are equation designators; λ5 is the fifth eigenvalue of the coefficient matrix; The node force vector p e Using a constant vector a, we get the matrix-vector product form of the force: p e =M e a p e ={Q1(0)Q2(0)N(0)M1(0)M2(0)Q1(1)Q2(1)N(1)M1(1)M2(1)} OR i =k1G1F1(λ i -k i ) L i =k2G2F2(γ i l i -m i ) C i =EFλ i (or i -h1k i -h2m i ) Oh i =E1I1λ i k i P i =E2I2λ i m i Among them, M e is the force boundary matrix; Q1(0) is the axial force corresponding to the upper material at position ξ=0; Q2(0) is the axial force corresponding to the lower material at position ξ=0; N(0) is the shear force corresponding to the combined structure at position ξ=0; M1(0) is the bending moment corresponding to the upper material at position ξ=0; M2(0) is the bending moment corresponding to the lower material at position ξ=0; Q1(1) is the axial force corresponding to the upper material at position ξ=1; Q2(1) is the axial force corresponding to the lower material at position ξ=1; N(1) is the shear force corresponding to the combined structure at position ξ=1; M1(1) is the bending moment corresponding to the upper material at position ξ=1; M2(1) is the bending moment corresponding to the lower material at position ξ=1; H i ,Λ i , Γ i ,Ω i and i All are symbols for equations; k1 is the shear correction coefficient of the upper material; G1 is the shear modulus of the upper structure; F1 is the cross-sectional area of ​​the upper material; γ i is the relationship coefficient between the vertical displacement of the lower material and the vertical displacement of the upper material; κ i is the relationship coefficient between the shear slip angle of the upper structure and the vertical displacement of the upper material; k2 is the shear correction coefficient of the lower material; G2 is the shear modulus of the lower structure; F2 is the cross-sectional area of ​​the lower material; μ i is the relationship coefficient between the shear slip angle of the lower structure and the vertical displacement of the upper material; i is the characteristic value; η i is the coefficient of relationship between the relative sliding displacement of the interface and the vertical displacement of the upper material; EF is the intermediate variable; h1 is the distance from the mass center of the upper structure to the contact surface of the two materials; h2 is the distance from the mass center of the lower structure to the contact surface of the two materials; E1 is the elastic modulus of the upper material; I1 is the bending moment of inertia of the upper structure; E2 is the elastic modulus of the lower material; I2 is the bending moment of inertia of the lower structure; According to the matrix-vector product expression of force and the matrix-vector product expression of displacement boundary, the dynamic stiffness matrix K is obtained. e :

8. The method for analyzing dynamic characteristics of composite beams considering shear slip and vertical lifting according to claim 1 is characterized in that: The natural frequency increment adjustment is performed according to the determinant of the dynamic stiffness matrix corresponding to the input frequency in the current round, specifically: Then set Δω=-Δω / 2; otherwise, the natural frequency increment remains unchanged; where, is the determinant of the dynamic stiffness matrix corresponding to the input frequency in the current round; is the determinant of the dynamic stiffness matrix corresponding to the input frequency in the previous round; Δω is the increment of the natural frequency.

9. The method for analyzing dynamic characteristics of composite beams considering shear slip and vertical lifting according to claim 1, characterized in that: According to whether the determinant of the dynamic stiffness matrix corresponding to the input frequency in the current round converges, specifically: It is convergent, otherwise, it is not convergent; among them, is the determinant of the dynamic stiffness matrix corresponding to the input frequency in the current round; ε is the convergence threshold.

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