Calculation Method for Self-Vibration Characteristics of Composite Beams with Corrugated Steel Webs and Related Equipment

Abstract

The present application relates to the technical field of composite beams with corrugated steel webs, and discloses a calculation method for the self-vibration characteristics of composite beams with corrugated steel webs and related equipment. The method includes: constructing a mechanical model for describing a composite beam with corrugated steel webs based on the Zig-zag theory; constructing a self-vibration state equation according to the mechanical model, solving the self-vibration state equation, and obtaining a state vector equation for the dynamic problem of the composite beam with corrugated steel webs; introducing the internal force displacement continuity condition and constructing a continuity condition equation; constructing a self-vibration frequency equation for the continuous beam with corrugated steel webs according to the continuity condition equation, solving the self-vibration frequency equation, and obtaining the self-vibration frequency of the continuous beam with corrugated steel webs; determining the proportional relationship of each component in the column vector of undetermined parameters according to the self-vibration frequency and solving the state vector equation for the dynamic problem of the composite beam with corrugated steel webs to determine the vibration modes of the displacement and internal force corresponding to the self-vibration frequency. The embodiments of the present application can accurately determine the self-vibration characteristics of the composite beam with corrugated steel webs.

Description

Technical Field

[0001] The present application relates to the technical field of composite girders with corrugated steel webs, in particular to a method for calculating the self-vibration characteristics of a composite girder with corrugated steel webs and related equipment. Background Art

[0002] The composite girder with corrugated steel webs is a new type of bridge structure, which consists of concrete top and bottom plates, external prestressing tendons and corrugated steel webs. Using corrugated steel plates to replace traditional concrete webs can reduce the self-weight of the structure and improve the shear resistance and durability of the structure.

[0003] How to accurately obtain the self-vibration characteristics of a composite girder with corrugated steel webs is the main task in the preliminary design of a composite girder with corrugated steel webs. However, due to the differences between the materials of the corrugated steel webs and the concrete top and bottom plates, the in-plane deformation of the composite girder with corrugated steel webs does not conform to the plane section assumption. In addition, the current calculation of the vibration characteristics of the composite girder with corrugated steel webs is either based on the classical beam theory or on the improved Timoshenko beam theory, and it is difficult to accurately describe the differences in the deformation slopes of each layer in the plane of the composite girder with corrugated steel webs. Summary of the Invention

[0004] The purpose of the present application is to provide a method for calculating the self-vibration characteristics of a composite girder with corrugated steel webs and related equipment, which can provide an important basis for the design of a composite girder with corrugated steel webs.

[0005] The embodiments of the present application provide a method for calculating the self-vibration characteristics of a composite girder with corrugated steel webs, including:

[0006] Construct a mechanical model for describing a composite girder with corrugated steel webs based on the Zig-zag theory;

[0007] According to the mechanical model, construct the self-vibration state equation of the composite girder with corrugated steel webs, solve the self-vibration state equation, and obtain the state vector equation of the dynamic problem of the composite girder with corrugated steel webs; the state vector equation of the dynamic problem of the composite girder with corrugated steel webs describes the relationship between the self-vibration state vector and the product of the undetermined parameter column vector composed of the general solution of the self-vibration state equation and a coefficient matrix;

[0008] Introduce the internal force displacement continuity condition, and construct the continuity condition equation of the composite girder with corrugated steel webs;

[0009] According to the continuity condition equation, construct the self-vibration frequency equation of the continuous corrugated steel web girder, solve the self-vibration frequency equation, and obtain the self-vibration frequency of the continuous corrugated steel web girder;

[0010] According to the self-vibration frequency of the continuous corrugated steel web girder, determine the proportional relationship of each component in the undetermined parameter column vector;

[0011] According to the proportional relationship of each component in the column vector of the to-be-determined parameters, solve the state vector equation of the dynamic problem of the corrugated steel web composite beam to determine the vibration modes of the displacement and internal force corresponding to the natural vibration frequency of the continuous corrugated steel web beam.

[0012] In some embodiments, based on the Zig-zag theory, constructing a mechanical model for describing the corrugated steel web composite beam includes:

[0013] Based on the principle of material equivalence, the corrugated steel web composite beam is equivalent to a sandwich beam model with orthotropic layers.

[0014] On the sandwich beam model, introduce the Zig-zag displacement assumption of continuous interlayers and the assumption of transverse shear stress distribution in the cross-section of layered parabolas to obtain the mechanical model of the corrugated steel web composite beam.

[0015] In some embodiments, according to the mechanical model, constructing the natural vibration state equation of the corrugated steel web composite beam, and using a numerically stable solution method to solve the natural vibration state equation to obtain the state vector equation of the dynamic problem of the corrugated steel web composite beam, including:

[0016] Based on Hamilton's principle, construct the dynamic equilibrium equation of the corrugated steel web composite beam under the action of conservative forces.

[0017] Based on the energy principle, solve the dynamic equilibrium equation to obtain the equilibrium equation of the corrugated steel web composite beam.

[0018] According to the combined state information in the mechanical model, construct the natural vibration state vector; the combined state information includes the deflection, rotation angle and Zig-zag displacement of the corrugated steel web composite beam, and the sectional shear force, bending moment and additional bending moment caused by the Zig-zag displacement that are energy dual.

[0019] Based on the state space method, rewrite the equilibrium equation as the state equation of the corrugated steel web composite beam in the free state, and substitute the natural vibration state vector into the state equation to obtain the natural vibration state equation.

[0020] Solve the natural vibration state equation to obtain a column vector of to-be-determined parameters and a corresponding coefficient matrix composed of the general solution of the natural vibration state equation, and use the column vector of to-be-determined parameters and the coefficient matrix to construct the state vector equation of the dynamic problem of the corrugated steel web composite beam.

[0021] In some embodiments, introducing the continuity condition of internal force and displacement, constructing the continuity condition equation of the corrugated steel web composite beam, including:

[0022] According to the internal force displacement continuity condition, the continuity condition equation of the beam span and the continuity condition equation of the support of the corrugated steel web composite beam are constructed, and the corresponding end boundary conditions are determined;

[0023] The continuity condition equation of the beam span, the continuity condition equation of the support and the end boundary condition are combined to obtain the continuity condition equation of the corrugated steel web composite beam.

[0024] In some embodiments, constructing the natural frequency equation of the corrugated steel web continuous beam according to the continuity condition equation, solving the natural frequency equation, and obtaining the natural frequency of the corrugated steel web continuous beam includes:

[0025] Solving the continuity condition equation to obtain the natural frequency equation;

[0026] The natural frequency of the continuous beam with corrugated steel web is obtained by solving the natural frequency equation at several modal orders using a dichotomy method.

[0027] In some embodiments, determining the proportional relationship of each component in the undetermined parameter column vector according to the natural frequency of the corrugated steel web continuous beam includes:

[0028] The natural frequency of the corrugated steel web continuous beam is substituted into the natural frequency equation, and the non-zero solution of the undetermined parameter column vector is calculated to determine the proportional relationship between the components in the undetermined parameter column vector.

[0029] In some embodiments, solving the state vector equation of the dynamic problem of the corrugated steel web composite beam according to the proportional relationship of each component in the undetermined parameter column vector to determine the vibration mode of the displacement and internal force corresponding to the natural frequency of the corrugated steel web continuous beam includes:

[0030] According to the proportional relationship between the components in the undetermined parameter column vector, the undetermined parameter column vector that conforms to the proportional relationship of the components is determined and substituted into the state vector equation of the dynamic problem of the corrugated steel web composite beam to obtain the natural vibration state vector of the vibration mode corresponding to the displacement and internal force of the natural vibration frequency of the corrugated steel web continuous beam.

[0031] The embodiment of the present application also provides a device for calculating the natural vibration characteristics of a corrugated steel web composite beam, comprising:

[0032] The first module is used to construct a mechanical model describing the composite beam with corrugated steel webs based on the Zig-zag theory;

[0033] A second module, configured to construct an equation of the natural vibration state of the corrugated steel web composite beam according to the mechanical model, and solve the equation of the natural vibration state by using a numerically stable solution method to obtain a state vector equation of the dynamic problem of the corrugated steel web composite beam; the state vector equation of the dynamic problem of the corrugated steel web composite beam describes the relationship between the natural vibration state vector and the product of a column vector of undetermined parameters composed of the general solution of the equation of the natural vibration state and a coefficient matrix;

[0034] A third module, configured to introduce the internal force displacement continuity condition and construct a continuity condition equation of the corrugated steel web composite beam;

[0035] A fourth module, configured to construct an equation of the natural vibration frequency of the continuous corrugated steel web beam according to the continuity condition equation, and solve the equation of the natural vibration frequency to obtain the natural vibration frequency of the continuous corrugated steel web beam;

[0036] A fifth module, configured to determine the proportional relationship of each component in the column vector of undetermined parameters according to the natural vibration frequency of the continuous corrugated steel web beam;

[0037] A sixth module, configured to solve the state vector equation of the dynamic problem of the corrugated steel web composite beam according to the proportional relationship of each component in the column vector of undetermined parameters, so as to determine the vibration modes of the displacement and internal force corresponding to the natural vibration frequency of the continuous corrugated steel web beam.

[0038] An embodiment of the present application further provides an electronic device, which includes a memory and a processor, where the memory stores a computer program, and when the processor executes the computer program, the above method for calculating the natural vibration characteristics of the corrugated steel web composite beam is implemented.

[0039] An embodiment of the present application further provides a computer-readable storage medium, which stores a computer program, and when the computer program is executed by a processor, the above method for calculating the natural vibration characteristics of the corrugated steel web composite beam is implemented.

[0040] Advantages of this application: Based on the Zig-zag theory, a mechanical model of the composite beam with corrugated steel webs is constructed. According to the mechanical model, the free vibration state equation is constructed, and a numerical stable solution method is used to solve the free vibration state equation to obtain the state vector equation of the dynamic problem of the composite beam with corrugated steel webs. The internal force-displacement continuity condition is introduced, and the continuity condition equation is constructed. According to the continuity condition equation, the free vibration frequency equation of the continuous beam with corrugated steel webs is constructed, and the free vibration frequency equation is solved to obtain the free vibration frequency of the continuous beam with corrugated steel webs. Based on the free vibration frequency, the proportional relationship of each component in the column vector of undetermined parameters is determined and the state vector equation of the dynamic problem of the composite beam with corrugated steel webs is solved to determine the vibration mode of the displacement and internal force corresponding to the free vibration frequency. The embodiment of this application determines the free vibration characteristics of the composite beam with corrugated steel webs based on the Zig-zag theory and the state space method, breaks through the plane section assumption in the classical beam theory, can improve the calculation accuracy and efficiency of the free vibration characteristics of the composite beam with corrugated steel webs, can handle complex variable cross-section forms and different constraint types in practical engineering, meets the requirements of the initial design stage of the composite beam with corrugated steel webs, and provides an important basis for the structural design of the composite beam with corrugated steel webs. Description of the Drawings

[0041] Figure 1 It is a flowchart of the method for calculating the free vibration characteristics of the composite beam with corrugated steel webs provided by the embodiment of this application.

[0042] Figure 2 It is a flowchart of the specific method of step S102 provided by the embodiment of this application.

[0043] Figure 3 It is a schematic structural diagram of the device for calculating the free vibration characteristics of the composite beam with corrugated steel webs provided by the embodiment of this application.

[0044] Figure 4 It is a schematic hardware structure diagram of the electronic device provided by the embodiment of this application.

[0045] Figure 5 It is a schematic diagram of the beam segment division of the composite beam with corrugated steel webs provided by the embodiment of this application.

[0046] Figure 6 It is a schematic diagram of the Zig-zag displacement field and the transverse shear stress distribution of the composite beam with corrugated steel webs provided by the embodiment of this application.

[0047] Figure 7 It is a schematic diagram of the first three vibration modes of the composite beam with corrugated steel webs provided by the embodiment of this application. Detailed Embodiments

[0048] In order to make the purpose, technical solutions and advantages of this application clearer, the following further elaborates on this application in conjunction with the drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain this application and are not used to limit this application.

[0049] It should be noted that although the functional modules are divided in the schematic diagram of the device and the logical sequence is shown in the flowchart, in some cases, the steps shown can be executed in a different module division from that in the device or a different order from that in the flowchart. Terms such as "first" and "second" in the specification, claims and drawings are used to distinguish similar objects, rather than to describe a specific order or sequence.

[0050] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by those skilled in the technical field to which this application belongs. The terms used herein are only for the purpose of describing the embodiments of this application and are not intended to limit this application.

[0051] Figure 1 is a flowchart of a method for calculating the self-vibration characteristics of a composite beam with corrugated steel webs provided by an embodiment of this application. In some embodiments, referring to Figure 1 , this method may specifically include steps S101 to S106.

[0052] Step S101: Based on the Zig-zag theory, construct a mechanical model describing the composite beam with corrugated steel webs.

[0053] In some embodiments, step S101 specifically is to equivalent the composite beam with corrugated steel webs to a sandwich beam model with orthotropic layers, establish a space rectangular coordinate system along the span direction, beam width direction and beam height direction of the composite beam, determine the heights of the top plate, web and bottom plate of the composite beam with corrugated steel webs, the widths of the top plate and bottom plate and the thickness of the web, and determine the equivalent elastic modulus, shear modulus, density, deflection, rotation angle, zig-zag displacement of each layer of the composite beam with corrugated steel webs and the internal forces dual to the energy of the composite beam with corrugated steel webs, so as to establish a mechanical model of the composite beam with corrugated steel webs.

[0054] Step S102: According to the mechanical model, construct the self-vibration state equation of the composite beam with corrugated steel webs, solve the self-vibration state equation, and obtain the state vector equation of the dynamic problem of the composite beam with corrugated steel webs.

[0055] Among them, the state vector equation of the dynamic problem of the composite beam with corrugated steel webs describes the relationship between the self-vibration state vector and the product of the column vector of undetermined parameters composed of the general solution of the self-vibration state equation and a coefficient matrix.

[0056] In some embodiments, step S102 is specifically to establish the free vibration state equation of the composite beam with corrugated steel webs based on Hamilton's principle without considering the action of external loads. To avoid the numerical instability problem in the traditional transfer matrix method, a numerically stable solution method is used to solve the free vibration state equation. Through matrix transformation, the column vector of undetermined parameters in the general solution of the free vibration state equation is used as the basic unknown to replace the state variables in the transfer matrix method, and the relationship between the corresponding state vectors and the column vector of undetermined coefficients within the beam segment of the composite beam with corrugated steel webs is described by the coefficient matrix, thereby obtaining the state vector equation for the dynamic problem of the composite beam with corrugated steel webs.

[0057] Step S103: Introduce the internal force-displacement continuity conditions to construct the continuity condition equation of the composite beam with corrugated steel webs.

[0058] In some embodiments, step S103 is specifically to, without loss of generality, follow the displacement and internal force continuity conditions between beam segments to establish the continuity condition equation between the beam segments of the composite beam with corrugated steel webs. Considering the continuity of internal forces and displacements at the positions of the intermediate supports between the adjacent beam segments on the left and right, and the vertical displacement being constrained, there are 6×(n - 1) continuity condition equalities at the n - 1 hinged supports. Considering the boundary conditions at the positions of the end supports, the common boundary conditions in engineering include simply supported, fixed, and free. Taking the simply supported boundary as an example, the corresponding vertical displacement w, bending moment M, and internal force P at the position of the end support are all 0, and 6 boundary condition equalities can be obtained for the two end supports, thereby obtaining the continuity condition equation of the composite beam with corrugated steel webs.

[0059] Step S104: Based on the continuity condition equation, construct the natural frequency equation of the continuous beam with corrugated steel webs, and solve the natural frequency equation to obtain the natural frequency of the continuous beam with corrugated steel webs.

[0060] In some embodiments, step S104 is specifically to, based on the continuity condition equation, make the column vector of undetermined parameters a non-zero solution to construct the natural frequency equation of the continuous beam with corrugated steel webs, and use the bisection method to solve the natural frequency equation to solve the solutions of the natural frequency equation at several modal orders, thereby obtaining the natural frequency of the continuous beam with corrugated steel webs.

[0061] Step S105: Determine the proportional relationship of each component in the column vector of undetermined parameters based on the natural frequency of the continuous beam with corrugated steel webs.

[0062] In some embodiments, step S105 is specifically to substitute the natural frequency of the continuous beam with corrugated steel webs into the continuity condition equation to determine the proportional relationship of each component in the column vector of undetermined parameters.

[0063] Step S106: According to the proportional relationship of each component in the vector of undetermined parameters, solve the state vector equation of the dynamic problem of the composite beam with corrugated steel webs to determine the vibration modes of the displacement and internal force corresponding to the natural vibration frequency of the continuous corrugated steel web beam.

[0064] In some embodiments, step S106 is specifically to solve the state vector equation of the dynamic problem of the composite beam with corrugated steel webs after determining the proportional relationship of each component in the vector of undetermined parameters, so as to determine the natural vibration state vector that satisfies the proportional relationship condition of each component in the vector of undetermined parameters, and determine the vibration modes of the displacement and internal force corresponding to the natural vibration frequency of the continuous corrugated steel web beam according to the components of the natural vibration state vector.

[0065] In a specific embodiment, step S101 specifically includes: based on the principle of material equivalence, equivalent the composite beam with corrugated steel webs to a sandwich beam model with orthotropic layers; on the sandwich beam model, introduce the continuous Zig-zag displacement assumption between layers and the assumption of transverse shear stress distribution in the section with a layered parabola to obtain the mechanical model of the composite beam with corrugated steel webs. The structure of the composite beam with corrugated steel webs is as Figure 5 shown.

[0066] In specific implementation, according to the principle of material equivalence, the composite beam with corrugated steel webs can be equivalent to a sandwich beam model with orthotropic layers with a width of b0, ρ k , E k , μ k and G k are respectively the equivalent density, elastic modulus, k = 1, 2, 3, Poisson's ratio and shear modulus of the kth layer, and their equivalent formulas are:

[0067] ,

[0068] ,

[0069] ,

[0070] ,

[0071] ,

[0072] ,

[0073] ,

[0074] where, n csw is the number of corrugated steel webs in the beam, α w , b w , c w , h w , t w and θw They are respectively the straight plate width, the inclined plate projection width, the inclined plate width, the corrugation height, the plate thickness, and the corrugation angle of the corrugated steel web. ρ, E, and μ respectively represent density, elastic modulus, and Poisson's ratio. The subscripts c and s respectively correspond to concrete and steel. b1, h1, b3, and h3 respectively represent the top plate width, the top plate thickness, the bottom plate width, and the bottom plate thickness. h2 is the height of the corrugated steel web, t is the web thickness, and i is a positive integer (i = 1, 2, 3).

[0075] Introduce the Zig-zag displacement s to describe the discontinuity of the in-plane displacement slopes of each layer of the corrugated steel web composite beam between layers. At the same time, assume that the shear stress on the cross-section is distributed in a stratified parabolic shape, and the mechanical model of the corrugated steel web composite beam is obtained. As shown in Figure 6, the Zig-zag theoretical displacement field and the in-plane transverse shear stress are:

[0076] ,

[0077] ,

[0078] Among them, is the displacement component of the k-th layer of the corrugated steel web composite beam in the x direction, is the displacement component of the k-th layer of the corrugated steel web composite beam in the z direction, u(x) is the displacement in the x direction at z = 0, is the cross-section rotation angle, s is the Zig-zag displacement amplitude, is the dimensionless coordinate of the k-th layer of the corrugated steel web composite beam, , w(x) is the vertical deflection of the corrugated steel web composite beam, is the in-plane transverse shear stress of the k-th layer of the corrugated steel web composite beam in the x direction, τ k-1 and τ k are respectively the in-plane transverse shear stresses on the upper and lower surfaces of the k-th layer of the corrugated steel web composite beam, τ0 and τ3 are respectively the in-plane transverse shear stresses on the upper and lower surfaces of the corrugated steel web composite beam, Q k is the in-plane transverse shear stress of the k-th layer of the corrugated steel web composite beam.

[0079] Figure 2 is the flowchart of the specific method of step S102 provided by the embodiment of the present application. In some embodiments, refer to Figure 2 , the method includes but is not limited to steps S201 to S205.

[0080] Step S201, based on Hamilton's principle, construct the dynamic equilibrium equation of the corrugated steel web composite beam under the action of conservative forces.

[0081] Step S202: Based on the energy principle, solve the dynamic equilibrium equation to obtain the equilibrium equation of the composite beam with corrugated steel webs.

[0082] Step S203: Construct the natural vibration state vector according to the combination state information in the mechanical model.

[0083] Among them, the combination state information includes the deflection, rotation angle and zig-zag displacement of the composite beam with corrugated steel webs, and the sectional shear force, bending moment and additional bending moment caused by the zig-zag displacement of the energy dual.

[0084] Step S204: Based on the state space method, rewrite the equilibrium equation as the state equation of the composite beam with corrugated steel webs in the free state, and substitute the natural vibration state vector into the state equation to obtain the natural vibration state equation.

[0085] Step S205: Solve the natural vibration state equation to obtain the column vector of undetermined parameters composed of the general solution of the natural vibration state equation and the corresponding coefficient matrix, and construct the state vector equation of the dynamic problem of the composite beam with corrugated steel webs by using the column vector of undetermined parameters and the coefficient matrix.

[0086] In specific implementation, based on Hamilton's principle, the dynamic equilibrium equation of the composite beam with corrugated steel webs under the action of conservative forces is:

[0087] ,

[0088] ,

[0089] ,

[0090] Among them, L = ∏ d -∏ R is the Lagrangian function, ∏ d is the system kinetic energy of the composite beam with corrugated steel webs, ∏ R is the functional of the generalized potential energy. A dot above a letter represents the derivative with respect to time. is the rotation angle of the k-th layer of the composite beam with corrugated steel webs. is the mass per unit length of the cross-section of the entire composite beam with corrugated steel webs. is the moment of inertia of the entire cross-section of the composite beam with corrugated steel webs. t0 and t1 are the time nodes respectively, h k is the thickness of the k-th layer, I k is the moment of inertia of the k-th layer cross-section.

[0091] According to the energy principle, by solving the dynamic equilibrium equation of the composite beam with corrugated steel webs, the equilibrium equation of the composite beam with corrugated steel webs can be obtained as follows:

[0092] ,

[0093] ,

[0094] ,

[0095] ,

[0096] ,

[0097] Among them, the superscript symbol represents the derivative with respect to the coordinate x, is the second derivative of the rotation angle with respect to time t, is the second derivative of the zig-zag displacement s with respect to time t, I is the moment of inertia of the cross-section, A is the cross-sectional area, and the corresponding boundary conditions are:

[0098] ,

[0099] ,

[0100] ,

[0101] ,

[0102] Among them, Q, M, N, and P are the shear stress, bending moment, axial force, and additional bending moment of the entire cross-section of the composite beam with corrugated steel webs, respectively. represents the variation, l is the bridge span, and the relationship between its internal force and displacement is:

[0103] ,

[0104] ,

[0105] ,

[0106] The relationship between its interlayer shear stress and the cross-section is:

[0107] ,

[0108] Among them, τ0 and τ3 are the in-plane transverse shear stresses on the upper and lower surfaces of the composite beam with corrugated steel webs, respectively. Therefore, τ0 = τ3 = 0.

[0109] Considering that the composite beam with corrugated steel web is not subjected to external axial force, that is, N=0, the axial force N and axial displacement μ are eliminated. Based on the state space method, the deflection w, rotation angle φ and zig-zag displacement s of the composite beam with corrugated steel web, and the additional bending moment P caused by the section shear force Q, bending moment M and zig-zag displacement of its energy duality are taken as state vectors. The equilibrium equation of the composite beam with corrugated steel web is combined, and the equilibrium equation of the composite beam with corrugated steel web is controlled in a matrix form. The state equation of the composite beam with corrugated steel web in the free state is obtained:

[0110] ,

[0111] In the state equation of the corrugated steel web composite beam in the free state,

[0112] ,

[0113] ,

[0114] ,

[0115] ,

[0116] ,

[0117] ,

[0118] ,

[0119] ,

[0120] Among them, H ij is the element corresponding to the i-th row and j-th column in the matrix H, A i , B j and C2 are constants related to geometric dimensions and material parameters, i∈[1,3], j∈[1,4], A i , B j The calculation formulas for C and C2 are:

[0121] ,

[0122] ,

[0123] ,

[0124] ,

[0125] ,

[0126] ,

[0127] ,

[0128] ,

[0129] For free vibration, the natural vibration state vector can be constructed as:

[0130] ,

[0131] Where v is the natural oscillation state vector, , , , and are the vibration mode functions of the deflection, rotation, zig-zag displacement, shear stress, bending moment and additional bending moment of the composite beam with corrugated steel webs, ω is the circular frequency of free vibration of the composite beam with corrugated steel webs, e is a natural constant, and T represents the transpose of the matrix.

[0132] Substituting the constructed natural vibration state vector into the state equation of the corrugated steel web composite beam in the free state, the natural vibration state equation is obtained:

[0133] ,

[0134] In the self-oscillating state equation,

[0135] ,

[0136] Considering numerical stability, the natural vibration state equation is solved by numerical stability method, and the state vector equation of the dynamic problem of the corrugated steel web composite beam is obtained:

[0137] ,

[0138] in, is the natural vibration state vector of the beam segment in the i-th span and the j-th segment, is the column vector of unknown parameters, for The corresponding coefficient matrix.

[0139] In a specific embodiment, step S103 specifically includes: constructing the continuity condition equation of the beam span and the continuity condition equation of the support of the corrugated steel web composite beam based on the internal force displacement continuity condition, and determining the corresponding end boundary conditions; combining the continuity condition equation of the beam span, the continuity condition equation of the support and the end boundary conditions to obtain the continuity condition equation of the corrugated steel web composite beam.

[0140] According to the jth beam segment and j+1th beam segment adjacent to the i-th beam span at the node Continuity conditions for internal forces and displacements at It can be obtained that:

[0141] ,

[0142] where \(i = 1, 2, 3, \cdots, n\) and \(j = 1, 2, 3, \cdots, m\). i -1.

[0143] The continuity condition equations of the beam span can be obtained within the entire beam span:

[0144] ,

[0145] where \(D\) is determined by the beam segment coefficient matrix, \(c\) is the undetermined column vector, and in the formula,

[0146] ,

[0147] ,

[0148] ,

[0149] For an \(n\)-span continuous beam, there are \(n - 1\) intermediate supports. The internal force and displacement continuity conditions of the adjacent beam segments on the left and right sides of the \(i\)-th intermediate support at the support node are:

[0150] ,

[0151] ,

[0152] ,

[0153] ,

[0154] ,

[0155] The above formula can be expressed in matrix form, that is, the continuity condition equation of the support, as:

[0156] ,

[0157] In the formula,

[0158] ,

[0159] ,

[0160] Taking the simply supported ends of a composite beam with corrugated steel webs as an example, the corresponding boundary conditions are: , , , , , , written in matrix form, i.e. the end boundary condition, is:

[0161] ,

[0162] In the formula,

[0163] ,

[0164] in, and The constant matrix that determines the continuity conditions for the intermediate supports, To determine the constant matrix of boundary conditions, For the nth span The coefficient matrix corresponding to the beam segment is: For the nth span The coordinates of the right end node of the beam segment.

[0165] Without loss of generality, other boundary conditions can always be written in the form of the above formula. Finally, the continuity condition equation of the beam span, the continuity condition equation of the support and the end boundary conditions are combined and written in matrix form to obtain the continuity condition equation of the corrugated steel web composite beam:

[0166] ,

[0167] Among them, K is Order square array.

[0168] In a specific embodiment, step S104 specifically includes: solving the continuity condition equation to obtain the natural frequency equation; using the dichotomy method to solve the natural frequency equation at several modal orders to obtain the natural frequency of the corrugated steel web continuous beam.

[0169] The continuity condition equation of the composite beam with corrugated steel web is a homogeneous linear algebraic equation system. The condition for its non-zero solution is that the determinant K of the coefficient matrix must be 0, that is:

[0170] ,

[0171] The above formula is the natural frequency equation of the composite beam with corrugated steel webs. It is a transcendental equation about ω and has infinite solutions. The binary method is used to solve it. After solving the solutions under several modal orders, the natural frequency of the continuous beam with corrugated steel webs is obtained. As shown in Table 1, the errors of the natural frequencies of the first three orders of the composite beam with corrugated steel webs obtained by the method of this application and the finite element calculation method are all within 5%, and the consistency is good. In addition, Figure 7 As shown, the calculation results of the first three modes of the corrugated steel web composite beam are given. The analytical method of the present invention is in good agreement with the ABAQUS numerical calculation results.

[0172]

[0173] Table 1

[0174] In a specific embodiment, step S105 specifically includes: substituting the natural vibration frequency of the corrugated steel web continuous beam into the natural vibration frequency equation, calculating the non-zero solution of the column vector of undetermined parameters, so as to determine the proportional relationship of each component in the column vector of undetermined parameters.

[0175] In a specific embodiment, step S106 specifically includes: according to the proportional relationship of each component in the column vector of undetermined parameters, determining the column vector of undetermined parameters that conforms to the proportional relationship of this component and substituting it into the state vector equation of the dynamic problem of the corrugated steel web composite beam, so as to obtain the natural vibration state vector that characterizes the vibration mode of the displacement and internal force corresponding to the natural vibration frequency of the corrugated steel web continuous beam.

[0176] Please refer to Figure 3 , the embodiment of the present application also provides a calculation device for the natural vibration characteristics of a corrugated steel web composite beam, which can implement the above calculation method for the natural vibration characteristics of a corrugated steel web composite beam. The device includes:

[0177] The first module 301 is used to construct a mechanical model describing the corrugated steel web composite beam based on the Zig-zag theory;

[0178] The second module 302 is used to construct the natural vibration state equation of the corrugated steel web composite beam according to the mechanical model, solve the natural vibration state equation, and obtain the state vector equation of the dynamic problem of the corrugated steel web composite beam; the state vector equation of the dynamic problem of the corrugated steel web composite beam describes the relationship between the natural vibration state vector and the product of the column vector of undetermined parameters composed of the general solution of the natural vibration state equation and a coefficient matrix;

[0179] The third module 303 is used to introduce the internal force displacement continuity condition and construct the continuity condition equation of the corrugated steel web composite beam;

[0180] The fourth module 304 is used to construct the natural vibration frequency equation of the corrugated steel web continuous beam according to the continuity condition equation, solve the natural vibration frequency equation, and obtain the natural vibration frequency of the corrugated steel web continuous beam;

[0181] The fifth module 305 is used to determine the proportional relationship of each component in the column vector of undetermined parameters according to the natural vibration frequency of the corrugated steel web continuous beam;

[0182] The sixth module 306 is used to solve the state vector equation of the dynamic problem of the corrugated steel web composite beam according to the proportional relationship of each component in the column vector of undetermined parameters, so as to determine the vibration mode of the displacement and internal force corresponding to the natural vibration frequency of the corrugated steel web continuous beam.

[0183] The specific implementation of the calculation device for the self-vibration characteristics of the composite beam with corrugated steel webs is basically the same as the specific embodiment of the calculation method for the self-vibration characteristics of the composite beam with corrugated steel webs described above, and will not be elaborated here.

[0184] Figure 4 is a block diagram of an electronic device shown in accordance with an exemplary embodiment.

[0185] Next, refer to Figure 4 to describe the electronic device 400 according to this embodiment of the present disclosure. Figure 4 The electronic device 400 shown is merely an example and should not impose any limitation on the functions and usage scope of the embodiments of the present disclosure.

[0186] As Figure 4 shown, the electronic device 400 is presented in the form of a general-purpose computing device. The components of the electronic device 400 may include, but are not limited to: at least one processing unit 410, at least one storage unit 420, a bus 430 connecting different system components (including the storage unit 420 and the processing unit 410), a display unit 440, etc.

[0187] Among them, the storage unit stores program code, and the program code can be executed by the processing unit 410, so that the processing unit 410 executes the steps according to various exemplary embodiments of the present disclosure described in the part of the calculation method for the self-vibration characteristics of the composite beam with corrugated steel webs in this specification.

[0188] The storage unit 420 may include a readable medium in the form of a volatile storage unit, such as a random access storage unit (RAM) 4201 and / or a cache storage unit 4202, and may further include a read-only storage unit (ROM) 4203.

[0189] The storage unit 420 may also include a program / utility 4204 having a set (at least one) of program modules 4205. Such program modules 4205 include, but are not limited to: an operating system, one or more application programs, other program modules, and program data. Each or some combination of these examples may include the implementation of a network environment.

[0190] The bus 430 may represent one or more of several types of bus structures, including a storage unit bus or a storage unit controller, a peripheral bus, a graphics acceleration port, a processing unit, or a local bus using any bus structure in a variety of bus structures.

[0191] The electronic device 400 can also communicate with one or more external devices 400' (such as a keyboard, a pointing device, a Bluetooth device, etc.), and can also communicate with one or more devices that enable a user to interact with the electronic device 400, and / or communicate with any device that enables the electronic device 400 to communicate with one or more other computing devices (such as a router, a modem, etc.). Such communication can be carried out through the input / output (I / O) interface 450. Moreover, the electronic device 400 can also communicate with one or more networks (such as a local area network (LAN), a wide area network (WAN), and / or a public network, such as the Internet) through the network adapter 460. The network adapter 460 can communicate with other modules of the electronic device 400 through the bus 430. It should be understood that although not shown in the figure, other hardware and / or software modules can be used in combination with the electronic device 400, including but not limited to: microcode, device drivers, redundant processing units, external disk drive arrays, RAID systems, tape drives, and data backup storage systems, etc.

[0192] An embodiment of the present application also provides a computer-readable storage medium. The computer-readable storage medium stores a computer program, and when the computer program is executed by a processor, the above-mentioned calculation method for the self-vibration characteristics of the corrugated steel web composite beam is implemented.

[0193] The calculation method for the self-vibration characteristics of the corrugated steel web composite beam and related devices provided by the embodiment of the present application constructs a mechanical model describing the corrugated steel web composite beam based on the Zig-zag theory, constructs a self-vibration state equation according to the mechanical model, uses a numerically stable solution method to solve the self-vibration state equation to obtain the state vector equation of the dynamic problem of the corrugated steel web composite beam, introduces the internal force displacement continuity condition, constructs a continuity condition equation, constructs the self-vibration frequency equation of the continuous corrugated steel web beam according to the continuity condition equation, solves the self-vibration frequency equation to obtain the self-vibration frequency of the continuous corrugated steel web beam, determines the proportional relationship of each component in the undetermined parameter column vector according to the self-vibration frequency and solves the state vector equation of the dynamic problem of the corrugated steel web composite beam to determine the vibration mode of the displacement and internal force corresponding to the self-vibration frequency. The embodiment of the present application determines the self-vibration characteristics of the corrugated steel web composite beam based on the Zig-zag theory and the state space method, breaks through the plane section assumption in the classical beam theory, can improve the calculation accuracy and efficiency of the self-vibration characteristics of the corrugated steel web composite beam, can cope with complex variable cross-section forms and different constraint types in practical engineering, meets the requirements of the initial design stage of the corrugated steel web composite beam, and provides an important basis for the structural design of the corrugated steel web composite beam.

[0194] Those skilled in the art can easily understand from the description of the above embodiments that the exemplary embodiments described herein can be implemented by software or by a combination of software and necessary hardware. Therefore, the technical solutions according to the embodiments of the present disclosure can be embodied in the form of a software product, which can be stored in a non-volatile storage medium (such as a CD-ROM, a USB flash drive, a mobile hard disk, etc.) or on a network, including several instructions to enable a computing device (such as a personal computer, a server, or a network device, etc.) to execute the above methods according to the embodiments of the present disclosure.

[0195] The program product can adopt any combination of one or more readable media. The readable media can be a readable signal medium or a readable storage medium. The readable storage medium can be, for example, but not limited to, an electrical, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, or device, or any combination of the above. More specific examples (a non-exhaustive list) of the readable storage medium include: an electrical connection with one or more wires, a portable disk, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or flash memory), an optical fiber, a portable compact disk read-only memory (CD-ROM), an optical storage device, a magnetic storage device, or any suitable combination of the above.

[0196] The computer-readable storage medium may include a data signal propagated in a baseband or as part of a carrier wave, which carries the readable program code. Such a propagated data signal can take various forms, including but not limited to electromagnetic signals, optical signals, or any suitable combination of the above. The readable storage medium can also be any readable medium other than the readable storage medium, which can send, propagate, or transmit a program for use by or in connection with an instruction execution system, apparatus, or device. The program code contained on the readable storage medium can be transmitted by any appropriate medium, including but not limited to wireless, wired, optical cable, RF, etc., or any suitable combination of the above.

[0197] Those skilled in the art can understand that the above-mentioned modules can be distributed in the device according to the description of the embodiments, or can be correspondingly changed and distributed in one or more devices that are different from the present embodiment. The modules of the above embodiments can be combined into one module, or further split into multiple sub-modules.

[0198] The above specifically shows and describes the exemplary embodiments of the present disclosure. It should be understood that the present disclosure is not limited to the detailed structures, settings, or implementation methods described herein; on the contrary, the present disclosure is intended to cover various modifications and equivalent settings included within the spirit and scope of the appended claims.

Claims

1. A calculation method for the self-vibration characteristics of a composite beam with corrugated steel webs, characterized in that including: Based on the Zig-zag theory, construct a mechanical model to describe the composite beam with corrugated steel webs; According to the mechanical model, construct the free vibration state equation of the composite beam with corrugated steel webs, solve the free vibration state equation, and obtain the state vector equation of the dynamic problem of the composite beam with corrugated steel webs; the state vector equation of the dynamic problem of the composite beam with corrugated steel webs describes the relationship between the free vibration state vector and the product of the column vector of undetermined parameters composed of the general solution of the free vibration state equation and a coefficient matrix; Introduce the internal force-displacement continuity condition, and construct the continuity condition equation of the composite beam with corrugated steel webs; According to the continuity condition equation, construct the free vibration frequency equation of the continuous beam with corrugated steel webs, solve the free vibration frequency equation, and obtain the free vibration frequency of the continuous beam with corrugated steel webs; According to the free vibration frequency of the continuous beam with corrugated steel webs, determine the proportional relationship of each component in the column vector of undetermined parameters; According to the proportional relationship of each component in the column vector of undetermined parameters, solve the state vector equation of the dynamic problem of the composite beam with corrugated steel webs to determine the vibration modes of the displacement and internal force corresponding to the free vibration frequency of the continuous beam with corrugated steel webs; The step of constructing the free vibration state equation of the composite beam with corrugated steel webs according to the mechanical model, solving the free vibration state equation, and obtaining the state vector equation of the dynamic problem of the composite beam with corrugated steel webs includes: Based on Hamilton's principle, construct the dynamic equilibrium equation of the composite beam with corrugated steel webs under conservative forces; Based on the energy principle, solve the dynamic equilibrium equation to obtain the equilibrium equation of the composite beam with corrugated steel webs; According to the combined state information in the mechanical model, construct the free vibration state vector; the combined state information includes the deflection, rotation angle and zig-zag displacement of the composite beam with corrugated steel webs, and the sectional shear force, bending moment and additional bending moment caused by the zig-zag displacement that are energy dual; Based on the state space method, rewrite the equilibrium equation as the state equation of the composite beam with corrugated steel webs in the free state, substitute the free vibration state vector into the state equation, and obtain the free vibration state equation; Solve the free vibration state equation to obtain the column vector of undetermined parameters composed of the general solution of the free vibration state equation and the corresponding coefficient matrix, and use the column vector of undetermined parameters and the coefficient matrix to construct the state vector equation of the dynamic problem of the composite beam with corrugated steel webs.

2. The calculation method for the natural vibration characteristics of the composite beam with corrugated steel webs according to claim 1, characterized in that The step of constructing a mechanical model to describe the composite beam with corrugated steel webs based on the Zig-zag theory includes: Based on the principle of material equivalence, equivalent the composite beam with corrugated steel webs to a sandwich beam model with orthotropic layers; On the sandwich beam model, introduce the Zig-zag displacement assumption of continuous interlayers and the assumption of transverse shear stress distribution in the section with a stratified parabola to obtain the mechanical model of the composite beam with corrugated steel webs.

3. The calculation method for the self-vibration characteristics of the composite beam with corrugated steel webs according to claim 1, characterized in that, The step of introducing the internal force-displacement continuity condition and constructing the continuity condition equation of the composite beam with corrugated steel webs includes: According to the internal force-displacement continuity condition, construct the continuity condition equations of the beam span and the supports of the composite beam with corrugated steel webs, and determine the corresponding end boundary conditions; The continuity condition equation of the beam span, the continuity condition equation of the support and the end boundary condition are combined to obtain the continuity condition equation of the corrugated steel web composite beam.

4. The calculation method for the natural vibration characteristics of the composite beam with corrugated steel webs according to claim 1, characterized in that The method of constructing the natural frequency equation of the corrugated steel web continuous beam according to the continuity condition equation and solving the natural frequency equation to obtain the natural frequency of the corrugated steel web continuous beam includes: Solving the continuity condition equation to obtain the natural frequency equation; The natural frequency of the continuous beam with corrugated steel web is obtained by solving the natural frequency equation at several modal orders using a dichotomy method.

5. The calculation method for the self-vibration characteristics of the composite beam with corrugated steel webs according to claim 1, wherein, Determining the proportional relationship of each component in the undetermined parameter column vector according to the natural frequency of the corrugated steel web continuous beam includes: The natural frequency of the corrugated steel web continuous beam is substituted into the natural frequency equation, and the non-zero solution of the undetermined parameter column vector is calculated to determine the proportional relationship between the components in the undetermined parameter column vector.

6. The calculation method for the natural vibration characteristics of the composite beam with corrugated steel webs according to claim 1, characterized in that, The method of solving the state vector equation of the dynamic problem of the corrugated steel web composite beam according to the proportional relationship of each component in the undetermined parameter column vector to determine the vibration mode of the displacement and internal force corresponding to the natural frequency of the corrugated steel web continuous beam comprises: According to the proportional relationship between the components in the undetermined parameter column vector, the undetermined parameter column vector that conforms to the proportional relationship of the components is determined and substituted into the state vector equation of the dynamic problem of the corrugated steel web composite beam to obtain the natural vibration state vector of the vibration mode corresponding to the displacement and internal force of the natural vibration frequency of the corrugated steel web continuous beam.

7. A calculation device for the self-vibration characteristics of a composite beam with corrugated steel webs, characterized in that, include: The first module is used to construct a mechanical model describing the composite beam with corrugated steel webs based on the Zig-zag theory; The second module is used to construct the self-oscillation state equation of the corrugated steel web composite beam according to the mechanical model, solve the self-oscillation state equation, and obtain the state vector equation of the dynamic problem of the corrugated steel web composite beam; the state vector equation of the dynamic problem of the corrugated steel web composite beam describes the relationship between the self-oscillation state vector and the product of the column vector of undetermined parameters formed by the general solution of the self-oscillation state equation and a coefficient matrix; The third module is used to introduce the internal force displacement continuity condition and construct the continuity condition equation of the corrugated steel web composite beam; The fourth module is used to construct the natural frequency equation of the corrugated steel web continuous beam according to the continuity condition equation, solve the natural frequency equation, and obtain the natural frequency of the corrugated steel web continuous beam; The fifth module is used to determine the proportional relationship of each component in the undetermined parameter column vector according to the natural frequency of the corrugated steel web continuous beam; The sixth module is used to solve the state vector equation of the dynamic problem of the corrugated steel web composite beam according to the proportional relationship of each component in the undetermined parameter column vector, so as to determine the vibration mode of the displacement and internal force corresponding to the natural frequency of the corrugated steel web continuous beam; The method of constructing the natural vibration state equation of the corrugated steel web composite beam based on the mechanical model and solving the natural vibration state equation to obtain the state vector equation of the dynamic problem of the corrugated steel web composite beam includes: Based on Hamilton's principle, construct the dynamic equilibrium equation of the composite beam with corrugated steel webs under the action of conservative forces; Based on the energy principle, solve the dynamic equilibrium equation to obtain the equilibrium equation of the composite beam with corrugated steel webs; Construct the natural vibration state vector according to the combined state information in the mechanical model; the combined state information includes the deflection, rotation angle and zig-zag displacement of the composite beam with corrugated steel webs, and the sectional shear force, bending moment and additional bending moment caused by the zig-zag displacement that are energy dual; Based on the state space method, rewrite the equilibrium equation as the state equation of the composite beam with corrugated steel webs in the free state, and substitute the natural vibration state vector into the state equation to obtain the natural vibration state equation; Solve the natural vibration state equation to obtain the column vector of undetermined parameters composed of the general solution of the natural vibration state equation and the corresponding coefficient matrix, and use the column vector of undetermined parameters and the coefficient matrix to construct the state vector equation of the dynamic problem of the composite beam with corrugated steel webs.

8. An electronic device, characterized in that, The electronic device includes a memory and a processor. The memory stores a computer program. When the processor executes the computer program, it implements the method for calculating the natural vibration characteristics of the composite beam with corrugated steel webs according to any one of claims 1 to 6.

9. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by the processor, it implements the method for calculating the natural vibration characteristics of the composite beam with corrugated steel webs according to any one of claims 1 to 6.

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