A dynamic analysis method for composite beams considering shear slip and vertical uplift

Abstract

The present invention discloses a method for analyzing the dynamic characteristics of a composite beam taking into account shear slip and vertical uplift, belonging to the field of bridge dynamics and civil engineering structure analysis. The method comprises establishing a composite beam motion control equation and natural boundary conditions based on the material properties of the composite beam; deforming the composite beam motion control equation to obtain a deformed motion control equation; substituting the input frequency into the deformed motion control equation to solve the eigenvalues ​​of the coefficient matrix of the motion control equation; obtaining a general solution to the motion control equation; constructing a matrix-vector product expression of the force and 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 judging whether the input frequency is a frequency corresponding to the composite beam based on the determinant of the dynamic stiffness matrix corresponding to the input frequency in the current round. The present invention solves the problems that current methods do not consider shear slip and vertical uplift and cannot strike a balance between efficiency and accuracy.

Description

Technical Field

[0001] The invention belongs to the field of bridge dynamics and civil engineering structure analysis, and in particular relates to a method for analyzing the dynamic characteristics of a composite beam taking shear slip and vertical uplift into consideration. Background Art

[0002] Composite beams are widely used in bridge structures. A common form is the steel-concrete composite beam, which combines a steel beam with a concrete slab via shear connectors. Their dynamic properties under external dynamic loads directly impact the safety and service life of the bridge. However, traditional composite beam analysis methods often ignore the effects of shear slip and vertical uplift, resulting in inaccurate calculations. This simplification may not accurately reflect the behavior of composite beams under dynamic loads.

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

[0004] 1. Ignoring the shear slip effect: The performance degradation of shear connectors under dynamic loads will significantly change the modal parameters of the composite beam. Existing analysis methods are often simplified to full connection, which cannot reflect the actual working conditions.

[0005] 2. The impact of vertical uplift has not been fully studied: Due to the deformation differences between concrete slabs and steel beams, vertical uplift may cause significant modal changes, which are difficult to accurately simulate with existing analytical models.

[0006] 3. Complexity and efficiency issues: Traditional analysis requires the construction of sophisticated finite element models or the use of highly complex numerical methods, which leads to low efficiency and makes it difficult to adapt to actual engineering needs. Summary of the Invention

[0007] In response to the above-mentioned deficiencies in the prior art, the present invention provides a method for analyzing the dynamic characteristics of a composite beam taking into account shear slip and vertical uplift, which solves the problem that the current method does not take shear slip and vertical uplift into account and cannot strike a balance between efficiency and accuracy.

[0008] In order to achieve the above-mentioned object, the present invention adopts a technical solution: a method for analyzing the dynamic characteristics of a composite beam considering shear slip and vertical uplift, comprising:

[0009] Obtain material properties of composite beams;

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

[0011] The variable separation method is used to deform the composite beam motion control equation to obtain the deformed motion control equation.

[0012] 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;

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

[0014] Combining 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;

[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, and adjust the natural frequency increment according to the determinant of the dynamic stiffness matrix corresponding to the input frequency in the current round. 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 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.

[0017] Furthermore, 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.

[0018] Furthermore, according to the material properties of the composite beam, the composite beam motion control equation and natural boundary conditions are established, specifically:

[0019] 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:

[0020] u 12 =u2-u1+h1θ1+h2θ2

[0021] Among them, u 12 is the relative sliding displacement of the composite beam interface; u2 is the sliding displacement of the composite beam lower structure; u1 is the sliding displacement of the composite beam upper structure; h1 is the distance from the mass center 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 mass center of the lower structure to the contact surface of the two materials; θ2 is the shear slip angle of the lower 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 principle:

[0023]

[0024] Where, δ is the variation; t1 is the starting time of shear slip; t2 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, 1 indicates the upper layer material, and 2 indicates the lower layer 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. I 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;

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

[0026]

[0027] 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.

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

[0029] Separating the motion control equations in time and space, we get the deformed motion control equations:

[0030] {w1 w2 u 12 θ1 θ2}={W1(x) W2(x) U 12 (x) Θ1(x) Θ2(x)}sin(ωt+ψ)

[0031]

[0032] Among them, 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 shear slip angle formation function of the upper structure; Θ2(x) is the shear slip angle formation 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; 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 center of mass of the upper structure to the contact surface of the two materials; h2 is the distance from the center of mass 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; 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.

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

[0034] ω j+1 =ω j +Δω

[0035] ω1=0

[0036] 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 natural frequency increment; ω1 is the initial frequency.

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

[0038]

[0039] Where W1(ξ) is the vertical displacement of 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; λ i is the eigenvalue; λ5 is the fifth eigenvalue; W2(ξ) is the vertical displacement of 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 layer material and the vertical displacement of the upper layer material; η i is the coefficient of relationship between the relative sliding displacement of the interface and the vertical displacement of the upper material; κ i is the coefficient of relationship between the shear slip angle of the upper structure and the vertical displacement of the upper material; μ i is the coefficient of relationship between the shear slip angle of the lower structure and the vertical displacement of the upper material; and All are symbols used in equations; EF is an 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 center of mass of the upper structure to the contact surface of the two materials; h2 is the distance from the center of mass 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.

[0040] Furthermore, the general solution of the combined motion control equation is used to construct a matrix-vector product expression of the force and displacement boundary to obtain the dynamic stiffness matrix, which is specifically:

[0041] According to the general solution of the motion control equation, the node displacement vector u e Using the constant vector a, we can get the matrix-vector product expression of the displacement boundary:

[0042] u e =N e a

[0043] u e ={W1(0)W2(0)U 12 (0)θ1(0)θ2(0)W1(1)W2(1)U 12 (1)θ1(1)θ2(1)}

[0044] a={A1 A2 A3 A4 A5 A6 A7 A8 A9 A 10}

[0045]

[0046] S=sinλ5

[0047] C=cosλ5

[0048] Among them, N e is the displacement boundary matrix; W1(0) is the vertical displacement of the upper material at position ξ=0; W2(0) is the vertical displacement of 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 both equation designators; λ5 is the fifth eigenvalue of the coefficient matrix;

[0049] The nodal force vector p e Using a constant vector a, we get the matrix-vector product form of the force:

[0050] p e =M e a

[0051] p e ={Q1(0)Q2(0)N(0)M1(0)M2(0)Q1(1)Q2(1)N(1)M1(1)M2(1)}

[0052]

[0053] Η i =k1G1F1(λ i -κ i )

[0054] Λ i =k2G2F2(γ i λ i -μ i )

[0055] Γ i =EFλ i (η i -h1κ i -h2μ i )

[0056] Ω i =E1I1λ i κ i

[0057] Ψ i =E2I2λ i μ i

[0058] 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; Η i , Λ i , Γ i ,Ω i and Ψ i are all 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 layer material and the vertical displacement of the upper layer material; κ i is the coefficient of the relationship 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 coefficient of relationship 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 center of mass of the upper structure to the contact surface of the two materials; h2 is the distance from the center of mass 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;

[0059] 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 :

[0060]

[0061] Furthermore, 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: if 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.

[0062] Furthermore, according to whether the determinant of the dynamic stiffness matrix corresponding to the input frequency in the current round converges, specifically: when It is convergent, otherwise, it is not convergent; 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 this invention are: this analysis method fully considers the shear deformation and vertical uplift effects of composite structures, accurately capturing the actual operating conditions of composite structures. Compared to traditional analysis methods, it eliminates the need for complex finite element modeling or reliance on highly complex numerical solutions, significantly improving 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 Flow chart of the method of the present invention.

[0065] Figure 2 Schematic diagram of the composite beam considering shear slip and vertical uplift in the present invention.

[0066] Figure 3 Schematic diagram of the relative slip relationship of the composite beam interface in the present invention. DETAILED DESCRIPTION

[0067] The specific embodiments of the present invention are described below to facilitate understanding of the present invention by those skilled in the art. However, it should be clear that the present invention is not limited to the scope of the specific embodiments. For those skilled in the art, as long as various changes are within the spirit and scope of the present invention as defined and determined by the appended claims, these changes are obvious, and all inventions and creations utilizing the concepts of the present invention are protected.

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

[0069] Obtain material properties of composite beams;

[0070] According to the material properties of the composite beam, the composite beam motion control equation and natural boundary conditions 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 and imaginary roots 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] Combining 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, and adjust the natural frequency increment according to the determinant of the dynamic stiffness matrix corresponding to the input frequency in the current round. 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 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 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 calculus of variations and Hamilton's principle, based on the established model and the interface slip relationship, further establish the composite beam motion control equations and natural boundary conditions;

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

[0081] S4: According to the number of eigenvalues ​​of the coefficient matrix of the motion control equation, the general solution expression of the motion control equation is given. The general solution expression is substituted into the motion control equation of the deformed composite beam to obtain the relationship between the various unknown quantities in the general solution expression.

[0082] S5: Based on the force and displacement boundary conditions at the beam end and the analytical solution to the differential equation of motion, construct a matrix-vector product representation of the force and displacement boundary. This then yields the relationship matrix between the force and displacement vectors, which is the dynamic stiffness matrix.

[0083] S6: Through the written MATLAB program, the characteristic frequency is assigned 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, the 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 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.

[0085] like Figure 3 The figure shows the relative slip relationship of the composite beam interface. Based on the material properties of the composite beam, the composite beam motion control equation and natural boundary conditions are established, specifically:

[0086] 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:

[0087] u 12 =u2-u1+h1θ1+h2θ2

[0088] Among them, u 12 is the relative sliding displacement of the composite beam interface; u2 is the sliding displacement of the composite beam lower structure; u1 is the sliding displacement of the composite beam upper structure; h1 is the distance from the mass center 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 mass center of the lower structure to the contact surface of the two materials; θ2 is the shear slip angle of the lower structure;

[0089] 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:

[0090]

[0091] Where, δ is the variation; t1 is the starting time of shear slip; t2 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, 1 indicates the upper layer material, and 2 indicates the lower layer 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. Ef 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. I 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;

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

[0093]

[0094] 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.

[0095] The variable separation method is used to deform the composite beam motion control equation to obtain the deformed motion control equation, which is specifically:

[0096] Separating the motion control equations in time and space, we get the deformed motion control equations:

[0097] {w1 w2 u 12 θ1 θ2}={W1(x) W2(x) U 12 (x) Θ1(x) Θ2(x)}sin(ωt+ψ)

[0098]

[0099] Among them, 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 shear slip angle formation function of the upper structure; Θ2(x) is the shear slip angle formation 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; 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 center of mass of the upper structure to the contact surface of the two materials; h2 is the distance from the center of mass 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; 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.

[0100] The expression of the input frequency is:

[0101] ω j+1 =ω j +Δω

[0102] ω1=0

[0103] 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 natural frequency increment; ω1 is the initial frequency.

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

[0105]

[0106] Where W1(ξ) is the vertical displacement of 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; λ iis the eigenvalue; λ5 is the fifth eigenvalue; W2(ξ) is the vertical displacement of 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 layer material and the vertical displacement of the upper layer material; η i is the coefficient of relationship between the relative sliding displacement of the interface and the vertical displacement of the upper material; κ i is the coefficient of relationship between the shear slip angle of the upper structure and the vertical displacement of the upper material; μ i is the coefficient of relationship between the shear slip angle of the lower structure and the vertical displacement of the upper material; and All are symbols used in equations; EF is an 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 center of mass of the upper structure to the contact surface of the two materials; h2 is the distance from the center of mass 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.

[0107] 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:

[0108] According to the general solution of the motion control equation, the node displacement vector u e Using the constant vector a, we can get the matrix-vector product expression of the displacement boundary:

[0109] u e =Ne a

[0110] u e ={W1(0) W2(0) U 12 (0) θ1(0) θ2(0) W1(1) W2(1) U 12 (1) θ1(1) θ2(1)}

[0111] a={A1 A2 A3 A4 A5 A6 A7 A8 A9 A 10}

[0112]

[0113] S=sinλ5

[0114] C=cosλ5

[0115] Among them, N e is the displacement boundary matrix; W1(0) is the vertical displacement of the upper material at position ξ=0; W2(0) is the vertical displacement of 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 both equation designators; λ5 is the fifth eigenvalue of the coefficient matrix;

[0116] The nodal force vector p eUsing a constant vector a, we get the matrix-vector product form of the force:

[0117] p e =M e a

[0118] p e ={Q1(0)Q2(0)N(0)M1(0)M2(0)Q1(1)Q2(1)N(1)M1(1)M2(1)}

[0119]

[0120] Η i =k1G1F1(λ i -κ i )

[0121] Λ i =k2G2F2(γ i λ i -μ i )

[0122] Γ i =EFλ i (η i -h1κ i -h2μ i )

[0123] Ω i =E1I1λ i κ i

[0124] Ψ i =E2I2λ i μ i

[0125] 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; Η i , Λ i , Γ i ,Ω i and Ψ iare all 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 layer material and the vertical displacement of the upper layer material; κ i is the coefficient of the relationship 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 coefficient of relationship 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 center of mass of the upper structure to the contact surface of the two materials; h2 is the distance from the center of mass 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;

[0126] 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 :

[0127]

[0128] 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.

[0129] 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; 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, the method comprehensively considers the shear slip effect between the steel beam and the concrete slab and the vertical lifting effect, and considers the composite beam with vertical lifting stiffness. Figure 2As shown, this method can provide more accurate modal analysis during bridge design and operation. By using a custom MATLAB program, users only need to input material and geometric parameters to efficiently obtain frequency information. This method not only improves the accuracy of calculation results but also significantly enhances computational efficiency, making it suitable for 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 composite beams considering shear slip and vertical uplift, characterized in that: include: Obtain material properties of composite beams; According to the material properties of the composite beam, the composite beam motion control equation and natural boundary conditions 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; the expression of the general solution of the motion control equation is: in, For upper material in position The corresponding vertical displacement at , the value is 0-1; x Coordinates along the length of the beam; 、 、 and are the general solution coefficients of the vertical displacement of the upper material; is the eigenvalue; is the fifth eigenvalue; For the lower material in position The corresponding vertical displacement at 、 、 and are the general solution coefficients of the vertical displacement of the lower material; For the location The relative sliding displacement of the interface at 、 、 and are the general solution coefficients of the relative sliding displacement of the interface; For the location The shear slip angle of the superstructure at ; 、 、 and are the general solution coefficients of the shear slip angle of the superstructure; For the location The shear slip angle of the underlying structure at ; 、 、 and are the general solution coefficients of the shear slip angle of the lower structure; is the relationship coefficient between the vertical displacement of the lower material and the vertical displacement of the upper material; is the relationship coefficient between the relative sliding displacement of the interface and the vertical displacement of the upper material; is the coefficient of relationship between the shear slip angle of the upper structure and the vertical displacement of the upper material; is the coefficient of relationship between the shear slip angle of the lower structure and the vertical displacement of the upper material; 、 、 、 、 、 、 、 、 、 、 、 、 、 、 、 and All are symbols for equations; is an intermediate variable; is the vertical lifting stiffness of the composite beam; Calculate spans for composite beams; is the distance from the center of mass of the superstructure to the contact surface of the two materials; is the distance from the center of mass of the lower structure to the contact surface of the two materials; is the shear correction factor of the upper material; is the shear modulus of the superstructure; is the cross-sectional area of ​​the upper material; is the elastic modulus of the upper material; is the bending moment of inertia of the superstructure; is the shear correction factor of the underlying material; is the shear modulus of the underlying structure; is the cross-sectional area of ​​the underlying material; is the elastic modulus of the underlying material; is the bending moment of inertia of the lower structure; For the quality of the upper material; is the shear stiffness of the composite beam; is the frequency; Combining 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, and adjust the natural frequency increment according to the determinant of the dynamic stiffness matrix corresponding to the input frequency in the current round. 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 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 dynamic characteristics analysis method of composite beams considering shear slip and vertical uplift 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 1000× shear slip stiffness value.

3. The dynamic characteristics analysis method of composite beams considering shear slip and vertical uplift according to claim 1 is characterized in that: According to the material properties of the composite beam, the composite beam motion control equation and natural boundary conditions are established, 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: in, is the relative sliding displacement of the composite beam interface; is the sliding displacement of the composite beam substructure; is the sliding displacement of the composite beam superstructure; is the distance from the center of mass of the superstructure to the contact surface of the two materials; is the shear slip angle of the superstructure; is the distance from the center of mass of the lower structure to the contact surface of the two materials; 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: in, is a variation; is the shear slip start time; is the shear slip start time; is the kinetic energy of shear slip; is the strain energy of shear slip; is the potential energy of shear slip; is the differential symbol; Calculate spans for composite beams; The material position identifier is 1, indicating the upper material, and 2, indicating the lower material; is the material density, when =1, is the density of the upper material, when =2, it is the density of the lower material; is the cross-sectional area of ​​the material, when =1, is the cross-sectional area of ​​the upper material, when =2, it is the cross-sectional area of ​​the lower material; is the vertical displacement, when =1, it is the vertical displacement of the upper layer. =2, it is the vertical displacement of the lower layer; is the elastic modulus of the material, when =1, is the elastic modulus of the upper material, when =2, it is the elastic modulus of the lower material; for right The derivative of ; is the sliding displacement, when =1, is the sliding displacement of the composite beam superstructure. =2, it is the sliding displacement of the composite beam lower structure; is the bending moment of inertia, when =1, is the bending inertia moment of the superstructure, when =2, it is the bending inertia moment of the lower structure; for right The derivative of ; is the shear slip angle, when =1, is the shear slip angle of the superstructure. =2, it is the shear slip angle of the lower structure; is the material shear correction factor, when =1, is the shear correction coefficient of the upper material. =2, it is the shear correction coefficient of the lower material; is the structural shear modulus, when =1, is the shear modulus of the superstructure, when =2, it is the shear modulus of the lower structure; for right The derivative of ; is the shear stiffness of the composite beam; is the vertical displacement difference between the upper and lower structures; is the vertical lifting stiffness of the composite beam; is the relative sliding displacement of the composite beam interface; According to the dynamic response problem of the composite beam, the composite beam motion control equation and natural boundary conditions are established: in, For the quality of the upper material; The quality of the underlying material; is the overall shear force of the composite beam; is the shear force of the superstructure; is the shear force of the lower structure; is the superstructure bending moment; is the lower structure bending moment; is the superstructure axial force; is the axial force of the lower structure; is an intermediate variable.

4. The dynamic characteristics analysis method of composite beams considering shear slip and vertical uplift 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: in, is the vertical displacement of the upper layer; is the vertical displacement of the lower layer; is the relative sliding displacement of the composite beam interface; is the shear slip angle of the superstructure; is the shear slip angle of the lower structure; is the vertical displacement formation function of the upper material; is the vertical displacement formation function of the lower material; is the interface relative sliding displacement formation function; is the shear slip rotation angle formation function of the superstructure; is the shear slip rotation angle formation function of the lower structure; is the frequency; t For time; is the initial phase; is an intermediate variable; is the coefficient matrix eigenvalue; is the vertical lifting stiffness of the composite beam; Calculate spans for composite beams; is the distance from the center of mass of the superstructure to the contact surface of the two materials; is the distance from the center of mass of the lower structure to the contact surface of the two materials; is the shear correction factor of the upper material; is the shear modulus of the superstructure; is the elastic modulus of the upper material; is the shear correction factor of the underlying material; is the shear modulus of the underlying structure; is the elastic modulus of the underlying material; is the bending moment of inertia of the superstructure; is the cross-sectional area of ​​the upper material; is the cross-sectional area of ​​the underlying material; is the bending moment of inertia of the lower structure; is the shear stiffness of the composite beam; For the quality of the upper material; The quality of the underlying material; 、 、 、 and All are unknown coefficients after separation of variables.

5. The dynamic characteristics analysis method of composite beams considering shear slip and vertical uplift according to claim 1 is characterized in that: The expression of the input frequency is: in, For the The input frequency of the iteration; For the The input frequency of the iteration; is the natural frequency increment; is the initial frequency.

6. The dynamic characteristics analysis method of composite beams considering shear slip and vertical uplift 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 Using a constant vector Indicates that the matrix-vector product expression of the displacement boundary is obtained: in, is the displacement boundary matrix; For upper material in position =0 corresponding vertical displacement; For the lower material in position =0 corresponding vertical displacement; For the location =0 interface relative sliding displacement; For the superstructure in position =0 shear slip angle; For the lower structure in position =0 shear slip angle; For the upper material in position =1 corresponding vertical displacement; For the lower material in position =1 corresponding vertical displacement; For the location =1 interface relative sliding displacement; For the superstructure in position =1 shear slip angle; For the lower structure in position =1 shear slip angle; 、 、 、 、 、 、 、 、 and are the general solution coefficients of the vertical displacement of the upper material; is the first eigenvalue of the coefficient matrix; is the second eigenvalue of the coefficient matrix; is the third eigenvalue of the coefficient matrix; is the 4th eigenvalue of the coefficient matrix; 、 、 、 and are the relationship coefficients between the vertical displacement of the lower layer material and the vertical displacement of the upper layer material; 、 、 、 and are the relationship coefficients between the relative sliding displacement of the interface and the vertical displacement of the upper material; 、 、 、 and are the coefficients of relationship between the shear slip angle of the upper structure and the vertical displacement of the upper material; 、 、 、 and are the coefficients of relationship between the shear slip angle of the lower structure and the vertical displacement of the upper material; and All are symbols for equations; is the fifth eigenvalue of the coefficient matrix; The node force vector Using a constant vector Expressed as, the matrix-vector product expression of the force is obtained: in, is the force boundary matrix; For upper material in position =0 corresponding axial force; For the lower material in position =0 corresponding axial force; For the combined structure in position =0 corresponding shear force; For upper material in position =0 corresponding bending moment; For the lower material in position =0 corresponding bending moment; For upper material in position =1 corresponding axial force; For the lower material in position =1 corresponding axial force; For the combined structure in position =1 corresponding to the shear force; For upper material in position =1 corresponding bending moment; For the lower material in position =1 corresponding bending moment; 、 、 、 and All are symbols for equations; is the shear correction factor of the upper material; is the shear modulus of the superstructure; is the cross-sectional area of ​​the upper material; is the relationship coefficient between the vertical displacement of the lower material and the vertical displacement of the upper material; is the coefficient of relationship between the shear slip angle of the upper structure and the vertical displacement of the upper material; is the shear correction factor of the underlying material; is the shear modulus of the underlying structure; is the cross-sectional area of ​​the underlying material; is the coefficient of relationship between the shear slip angle of the lower structure and the vertical displacement of the upper material; is the characteristic value; is the relationship coefficient between the relative sliding displacement of the interface and the vertical displacement of the upper material; is an intermediate variable; is the distance from the center of mass of the superstructure to the contact surface of the two materials; is the distance from the center of mass of the lower structure to the contact surface of the two materials; is the elastic modulus of the upper material; is the bending moment of inertia of the superstructure; is the elastic modulus of the underlying material; 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 is ​​obtained : 。 7. The method for analyzing dynamic characteristics of composite beams considering shear slip and vertical uplift 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 ; 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; is the natural frequency increment.

8. The dynamic characteristics analysis method of composite beams considering shear slip and vertical uplift according to claim 1 is 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; is the convergence threshold.

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