A method and device for identifying the dynamic structural system of a box girder bridge considering the shear lag effect

Abstract

The present invention belongs to the technical field related to health monitoring, and discloses a method and device for identifying the dynamic structural system of a box girder bridge considering the shear lag effect. The steps are as follows: (1) Based on the one-dimensional beam element, the longitudinal displacement difference of the wing plate is introduced as the generalized degree of freedom of shear lag, and then the stiffness matrix and mass matrix of the box girder element considering the shear lag effect are constructed; (2) The structural undamped free vibration state equation of the box girder element is constructed, and then the boundary condition constraints are introduced to construct the dynamic observability state equation; the modal analysis of the beam element is carried out, and the pseudo-observations of the longitudinal displacement difference of the nodes are calculated based on the obtained modal parameters; (3) The modal parameters and the pseudo-observations of the longitudinal displacement difference of the nodes are used as input quantities and substituted into the dynamic observability state equation, and then the dynamic observability state equation is solved by using the null space matrix of the coefficient matrix of the observability state equation to obtain the axial stiffness and flexural stiffness of the box girder. The present invention improves the identification accuracy.

Description

Technical Field

[0001] The present invention belongs to the technical field related to health monitoring, and more specifically, relates to a method and device for identifying the dynamic structural system of a box girder bridge considering the shear lag effect. Background Technique

[0002] When conducting structural system identification, the influence of shear deformation is often easily overlooked. Based on the classical beam theory, Timoshenko first introduced shear deformation into the beam model and defined a shear coefficient to describe the distribution change of shear stress on the cross-section. In subsequent studies, different scholars respectively deduced the calculation formula of the shear coefficient for different cross-sections in static and dynamic analyses and compared the influence of the shear effect.

[0003] However, for wide flange box girders, the shear lag effect will be more significant, and the current specifications cannot fully consider its influence on the structural mechanical properties. Therefore, in this case, it is obviously insufficient to only use the shear coefficient to consider the effect of shear deformation. Summary of the Invention

[0004] In view of the above defects or improvement requirements of the prior art, the present invention provides a method and device for identifying the dynamic structural system of a box girder bridge considering the shear lag effect, aiming to solve the problem of low identification accuracy of the existing method.

[0005] To achieve the above object, according to one aspect of the present invention, a method for identifying the dynamic structural system of a box girder bridge considering the shear lag effect is provided. The method includes the following steps:

[0006] Step 1, introduce the longitudinal displacement difference of the wing plate as the shear lag generalized degree of freedom on the basis of the one-dimensional beam element, and then construct the stiffness matrix and mass matrix of the box girder element considering the shear lag effect, which are respectively:

[0007]

[0008] In the formula, ρ represents the density of the box girder element; T represents the transpose of the matrix, without special physical meaning; I s is the moment of inertia of the upper and lower wing plates about the centroidal axis x; E represents the elastic modulus; G is the shear modulus; [N v is the deflection displacement mode shape function of the beam segment; is the longitudinal displacement difference mode shape function of the beam segment; b is half of the width of the top plate; z is the ordinate along the bridge span;

[0009] Step 2: Based on the element stiffness and mass matrices considering the shear lag effect, construct the structural undamped free vibration state equation of the box girder element. Then, introduce boundary condition constraints and separate the known and unknown quantities in the structural undamped free vibration state equation of the box girder element to construct the dynamic observability state equation. At the same time, perform modal analysis on the beam element based on the structural dynamic responses collected under dynamic excitation or environmental excitation, and calculate the pseudo-observations of the longitudinal displacement difference at the nodes based on the obtained modal parameters.

[0010] Step 3: Substitute the modal parameters and the pseudo-observations of the longitudinal displacement difference at the nodes into the dynamic observability state equation. Then, use the null space matrix of the coefficient matrix of the observability state equation to determine the solution of the dynamic observability state equation, and thus obtain the axial stiffness and flexural stiffness of the box girder.

[0011] Furthermore, the calculation formula for the pseudo-observations of the longitudinal displacement difference at the nodes is:

[0012]

[0013] where is the pseudo-observation of the longitudinal displacement difference at the i-th node; θ i is the measured value of the rotational displacement at the i-th node; is the shear lag displacement coefficient at the i-th node.

[0014] Furthermore, Step 1 includes the following sub-steps:

[0015] S1.1: Introduce a longitudinal displacement function of the flange, i.e., the shear lag longitudinal warping displacement function, to describe the bending displacement of the flange of the box girder. The expression of the longitudinal displacement function of the flange is: where u(x, z, t) is the longitudinal dynamic displacement function of the flange; is the longitudinal dynamic displacement difference function of the flange; v(z, t) is the vertical dynamic deflection of the flange; z is the longitudinal coordinate along the bridge span; b is half of the width of the top plate; x is the transverse coordinate of the cross-section; h i is the distance from the centroid of the cross-section to the middle of the upper and lower flanges;

[0016] S1.2: On the degrees of freedom of the one-dimensional beam element, introduce the shear lag longitudinal warping displacement difference function as a displacement component into the node displacement vector of the beam element. Then, based on the formula derivation of the node displacement vector of the beam element, obtain the dynamic deflection displacement function and the dynamic longitudinal displacement difference function of each node of the beam segment.

[0017] S1.3: Calculate the energy of each part of the box girder cross-section based on the dynamic deflection displacement function and the dynamic longitudinal displacement difference function of each node of the beam segment.

[0018] S1.4. According to Hamilton's principle and the energies of various parts of the box girder section, the total potential energy and total kinetic energy of the box girder element considering the shear lag effect are obtained, and then the undamped free vibration equation of the box girder element is obtained. The undamped free vibration equation is matrix-transformed to obtain the stiffness and mass matrices of the box girder element considering the shear lag effect.

[0019] Furthermore, when the beam segment element vibrates, it is assumed that each node performs simple harmonic motion. Therefore, the dynamic deflection displacement function of each node of the beam segment and the dynamic longitudinal displacement difference function of the flange plate can be expressed as:

[0020] v(z, t) = v(z)sin(ωt + ζ) = [N v {δ} e sin(ωt + ζ) (1 - 5)

[0021]

[0022] where {δ} e is the column matrix of element node displacements; ω is the natural vibration frequency of the box girder; ζ is the initial phase angle during the vibration of the box girder.

[0023] Furthermore, the total kinetic energy of the beam segment element is calculated by taking the sum of the kinetic energy of the vertical deflection of the beam and the kinetic energy of the longitudinal displacement, that is, the calculation formula is:

[0024]

[0025] where ρ is the density of the box girder element.

[0026] Furthermore, the construction of the dynamic observability state equation includes the following sub-steps:

[0027] S2.1. First, establish the undamped free vibration state equation of the simply supported box girder element considering the shear lag effect:

[0028] K l {φ l} = λ l ·M l {φ l} (2 - 1)

[0029] where K l is the overall stiffness matrix of the simply supported beam considering the shear lag effect, {φ l} is the nodal vibration mode displacement, λ l is the structural vibration mode frequency, and M l is the overall mass matrix considering the shear lag effect;

[0030] Subsequently, boundary condition constraints are introduced, and the stiffness and mass matrices of the box girder element are processed by the "row and column deletion" method to obtain:

[0031]

[0032] Among them, N N is the number of nodes, and N B is the number of boundary conditions;

[0033] S2.2. Separate and transform the matrix in the structural undamped free vibration state equation, and perform centralized coupling on the target structural system parameters of the obtained structural undamped free vibration state equation to form a new parameter vector, so that the overall stiffness matrix and mass matrix originally considering the shear lag effect become matrices containing only constant coefficients. Then, couple the structural system parameters in the stiffness matrix and mass matrix with the modal displacement quantity to form a new observed variable D l , and then obtain the dynamic observability state equation.

[0034] Furthermore, the dynamic observability state equation is:

[0035]

[0036] In this equation, B l z l = D l is the dynamic observability state equation of the box girder considering the shear lag effect, and are submatrices of the stiffness matrix, and their sizes are determined by and ; and are submatrices of the mass matrix, and their sizes are determined by and ; z l contains all unknown variables, B l is the coefficient matrix corresponding to the unknown variables, and D l is a constant vector.

[0037] Furthermore, use the null space matrix to determine whether the dynamic observability equation has a solution:

[0038] [V] T {D} = 0

[0039] In the formula, [V] is the null space matrix of the coefficient matrix [B] of the observability state equation; {D} is a completely known vector; solve the null space matrix V of the coefficient matrix [B]. If [V] does not exist, that is, when [V] is zero-dimensional, there is a unique solution for all variables in the equation; when [V] exists and is not a zero matrix, the variables corresponding to its all-zero rows have a unique solution, and the corresponding unique particular solution variables are obtained through B\D, and then the axial stiffness and flexural stiffness are obtained.

[0040] The present invention also provides a dynamic structural system identification system for a box girder bridge considering shear lag effect. The system includes a memory and a processor. The memory stores a computer program, and when the processor executes the computer program, it executes the method for identifying the dynamic structural system of the box girder bridge considering shear lag effect as described above.

[0041] The present invention also provides a computer-readable storage medium. The computer-readable storage medium stores machine-executable instructions. When the machine-executable instructions are called and executed by a processor, the machine-executable instructions cause the processor to implement the method for identifying the dynamic structural system of the box girder bridge considering shear lag effect as described above.

[0042] Generally speaking, compared with the prior art by the above technical solutions conceived by the present invention, the method and device for identifying the dynamic structural system of the box girder bridge considering shear lag effect provided by the present invention mainly have the following beneficial effects:

[0043] 1. The present invention considers the shear lag effect to ensure the effectiveness of the identification result. However, the traditional method ignores the influence of the shear lag effect on the structural physical mechanism, which is generally applicable to narrow beam bridges. However, due to the more obvious shear lag effect of wide flange thin-walled box girder bridges, the influence of the shear lag effect on the cross-section deformation and stress distribution in the transverse direction cannot be ignored. Therefore, this method has higher identification accuracy compared with the traditional method.

[0044] 2. On the basis of one-dimensional beam elements, the longitudinal displacement difference of the wing plate is introduced as the generalized degree of freedom of shear lag to consider the shear lag effect of the wide flange thin-walled box girder structure, and its calculation process is relatively simple.

[0045] 3. The rotational displacement generated by one-dimensional beam elements includes the rotation corresponding to the classical beam theory and the generalized rotation corresponding to the shear lag effect. However, the traditional method can only determine the sum of the two, that is, the total rotational displacement of the beam element nodes. This method respectively obtains the rotation corresponding to the classical beam theory and the generalized rotation corresponding to the shear lag effect based on the analysis of the shear lag effect of the box girder. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 is a schematic diagram of the principle of a method for identifying the dynamic structural system of a box girder bridge considering shear lag effect provided by the present invention;

[0047] Figure 2 is a schematic diagram of the displacement degrees of freedom of a beam element considering shear lag effect;

[0048] Figure 3 is a schematic diagram of the boundary conditions of a continuous beam;

[0049] Figure 4 is a schematic diagram of the segmentation of a beam element;

[0050] Figure 5 It is a schematic diagram of the first-order vertical bending vibration mode;

[0051] Figure 6 It is a schematic diagram of the identification error of the flexural rigidity EI of the beam element. Specific implementation manner

[0052] In order to make the purpose, technical solutions and advantages of the present invention clearer, the present invention will be further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.

[0053] The present invention provides a method for identifying the dynamic structural system of a box girder bridge considering the shear lag effect. The method applies the dynamic finite element method considering the shear lag effect, the dynamic observability technology, and the box girder analysis theory of the box girder structure. Compared with the traditional stiffness identification method that does not consider the shear lag effect, it has higher accuracy and can be more reliably applied to the dynamic analysis of engineering structures.

[0054] Please refer to Figures 1 to 6 , the method mainly includes the following steps:

[0055] Step 1, on the basis of a one-dimensional beam element, introduce the longitudinal displacement difference of the flange as the generalized degree of freedom of shear lag, and then construct the stiffness matrix and mass matrix of the box girder element considering the shear lag effect, which are respectively:

[0056]

[0057] In the formula, ρ represents the density of the box girder element; T represents the transpose of the matrix, without special physical meaning; I s is the moment of inertia of the upper and lower flanges about the centroidal axis x; E represents the elastic modulus; G is the shear modulus; [N v is the deflection displacement mode shape function of the beam segment; is the longitudinal displacement difference mode shape function of the beam segment; b is half of the width of the top plate; z is the longitudinal coordinate along the bridge span.

[0058] Step 1 includes the following sub-steps:

[0059] S1.1, introduce a longitudinal displacement function of the flange, that is, the longitudinal warping displacement function of shear lag, to describe the bending displacement of the box girder flange; the expression of the longitudinal displacement function of the flange is:

[0060] Specifically, assuming that the box girder section is symmetrically deflected, at this time, analyze the flange on the right side of the symmetry axis of the box girder, and its longitudinal displacement expression is:

[0061]

[0062] In the formula, \(u(x, z, t)\) is the longitudinal dynamic displacement function of the flange; is the longitudinal dynamic displacement difference function of the flange plate; \(v(z, t)\) is the vertical dynamic deflection of the flange; \(z\) is the longitudinal coordinate along the bridge span; \(b\) is half of the width of the top plate; \(x\) is the transverse coordinate of the cross-section; \(h\) i is the distance from the centroid of the cross-section to the middle of the upper and lower flanges.

[0063] S1.2. On the degrees of freedom of the one-dimensional beam element, the shear lag longitudinal warping displacement difference function is introduced as a displacement component into the nodal displacement vector of the beam element, and then the dynamic deflection displacement function and the dynamic longitudinal displacement difference function of each node of the beam segment are derived based on the formula of the nodal displacement vector of the beam element.

[0064] On the degrees of freedom of the one-dimensional beam element (deflection and rotation angle), the longitudinal displacement difference of the wing plate is introduced as the shear lag generalized degree of freedom to consider the coupling effect of the shear lag effect and the bending deformation of the box girder structure. The dynamic deflection displacement function and the dynamic longitudinal displacement difference function of each node of the beam segment are derived based on the formula of the nodal displacement vector of the beam element.

[0065] Assuming that the longitudinal displacement of the beam element is not considered, the nodal displacement vector of the box girder element can be expressed as:

[0066]

[0067] In the formula, \(v\) i , \(v\) j are the vertical deflections at both ends of the beam element respectively; \(\theta\) i , \(\theta\) j are the rotation angles at both ends of the beam element respectively; are the longitudinal displacement differences at both ends of the beam element respectively.

[0068] Ignoring the axial deformation, the beam element displacement and the shear lag displacement can be expressed as:

[0069] \(v = [N\) v {\(\delta\)} (1 - 3)

[0070]

[0071] In the formula, \([N\) v is the deflection displacement mode shape function of the beam segment; is the longitudinal displacement difference mode shape function of the beam segment.

[0072] When the beam segment element vibrates, it is assumed that each node makes a simple harmonic motion. Therefore, the dynamic deflection displacement function of each node of the beam segment and the dynamic longitudinal displacement difference function of the flange plate can be expressed as:

[0073] \(v(z, t) = v(z)\sin(\omega t+\zeta)=[N\)v {δ} e sin(wt + ζ) (1-5)

[0074]

[0075] Where {δ} e is the displacement array of the unit node; w is the natural vibration frequency of the box girder; ζ is the initial phase angle during the vibration of the box girder.

[0076] S1.3. Calculate the energy of each part of the box girder cross-section based on the dynamic deflection displacement function and the dynamic longitudinal displacement difference function of each node of the beam segment.

[0077] Total potential energy of box girder vibration: For the strain potential energy of the box girder, the longitudinal strain energy and the tangential strain energy need to be considered separately for the upper and lower flange plates. Assuming that the web deformation satisfies the plane section assumption, only the vertical longitudinal strain energy is considered.

[0078] Among them, the expression of the strain potential energy of the web is:

[0079]

[0080] The expression of the strain potential energy of the lower flange plate is:

[0081]

[0082] Where ε b is the longitudinal strain of the lower flange plate; γ b is the tangential strain of the lower flange plate; I w is the moment of inertia of the web about the centroidal axis x; E is the elastic modulus; G is the shear modulus;

[0083] The formula for the strain potential energy of the upper flange plate is:

[0084]

[0085] Where ε u is the longitudinal strain of the upper flange plate; γ u is the tangential strain of the upper flange plate;

[0086] Total kinetic energy of the box girder element: The total kinetic energy of the beam segment element is calculated by taking the sum of the kinetic energy of the vertical deflection of the beam and the kinetic energy of the longitudinal displacement, that is, the calculation formula is:

[0087]

[0088] Where ρ is the density of the box girder element.

[0089] Since it is assumed that the web deformation conforms to the classical beam theory, the plane section assumption is still adopted, and only the kinetic energy of the classical beam displacement is considered. The expression of its longitudinal displacement is:

[0090]

[0091] The kinetic energy of the web is obtained as:

[0092]

[0093] In Equation (1-12), the "dot" represents the derivative with respect to time t.

[0094] Substituting Equations (1-5) to (1-6) into Equation (1-10), the expression of the kinetic energy of the lower flange can be obtained, that is:

[0095]

[0096] Similarly, according to the calculation method of the kinetic energy of the lower flange, the expression of the kinetic energy of the upper flange can be obtained:

[0097]

[0098] S1.4. According to Hamilton's principle and the energies of each part of the box girder section, the total potential energy and total kinetic energy of the box girder element considering the shear lag effect are obtained. Furthermore, the undamped free vibration equation of the box girder element is obtained, and the undamped free vibration equation is matrix-transformed to obtain the stiffness and mass matrices of the box girder element considering the shear lag effect.

[0099] The total potential energy of the box girder element considering the shear lag effect can be expressed as:

[0100]

[0101] In the formula, the "prime" represents the differentiation with respect to the longitudinal dz of the beam element. Substituting Equations (1-5) and (1-6) into (1-7) and converting it into matrix form, it is expressed as:

[0102]

[0103] In the formula, I s is the moment of inertia of the upper and lower flanges about the centroidal axis x, which can be calculated as: I s = I b + I u ; I is the moment of inertia of the box girder cross-section about the centroidal axis x, which can be calculated as: I = I s + I w ; I w is the moment of inertia of the web about the centroidal axis x.

[0104] The total kinetic energy can be expressed as:

[0105]

[0106] In the formula, T u is the kinetic energy of the web; T b is the kinetic energy of the lower flange; T w is the kinetic energy of the upper flange; ρ is the density of the box girder element.

[0107] Select a vibration period for integration, and the difference between the kinetic energy and the potential energy can be calculated as follows:

[0108]

[0109]

[0110] According to Hamilton's principle, perform variational on the above formula (1-18), that is:

[0111]

[0112] Transform formula (1-19) and eliminate π to obtain the following form:

[0113]

[0114] Formula (1-20) is the undamped free vibration equation of the beam element, and it is expressed as a matrix equation as:

[0115]

[0116] w is the natural vibration frequency, {δ} e is the beam-end vibration mode displacement vector corresponding to it. Substitute formula (1-21) into (1-20) to obtain the stiffness and mass matrices of the box girder element considering the shear lag effect as:

[0117]

[0118] Step 2: Based on the element stiffness and mass matrices considering the shear lag effect, construct the structural undamped free vibration state equation of the box girder element, and then introduce boundary condition constraints, and separate the known and unknown quantities in the structural undamped free vibration state equation of the box girder element to construct the dynamic observability state equation; at the same time, perform modal analysis on the beam element based on the structural dynamic responses collected under dynamic excitation or environmental excitation, and calculate the pseudo-observations of the longitudinal displacement difference at the nodes based on the obtained modal parameters.

[0119] The construction of the dynamic observability state equation includes the following sub-steps:

[0120] S2.1, First, establish the structural undamped free vibration state equation of the simply supported box girder element considering the shear lag effect:

[0121] Kl {φ l} = λ l ·M l {φ l} (2-1)

[0122] where K l is the overall stiffness matrix of the simply supported beam considering shear lag effect, {φ l} is the nodal mode displacement, λ l is the structural mode frequency, and M l is the overall mass matrix considering shear lag effect. Here, the simply supported beam considers a single-element system. If the structure consists of multiple elements, the element stiffness and mass matrices need to be assembled into the overall stiffness and mass matrices according to the nodal positions in sequence.

[0123] Subsequently, boundary condition constraints are introduced, and the stiffness and mass matrices of the box girder element are processed by the "row and column deletion" method to obtain:

[0124]

[0125] where N N is the number of nodes, and N B is the number of boundary conditions.

[0126] S2.2. Separate and transform the matrices in the structural undamped free vibration state equation, and centrally couple the target structural system parameters of the obtained structural undamped free vibration state equation to form a new parameter vector, so that the original overall stiffness matrix and mass matrix considering shear lag effect become matrices containing only constant coefficients. Then, couple the structural system parameters in the stiffness matrix and mass matrix with the mode displacement quantity to form a new observed variable D l , and then obtain the dynamic observability state equation.

[0127] Transform Equation (2-2) into the following form:

[0128]

[0129] In the formula, and are constant coefficient matrices containing only the element length l, is the coupling vector containing the stiffness matrix system parameters (EI s , EI, EA, GI, GI s ) and the mode displacement , is the coupling vector containing the mass matrix system parameters (ρI, ρI s , ρA), the mode frequency λ, and the mode displacement.

[0130] Subsequently, separate the known and unknown quantities in Equation (2-3), and move them to both sides of the equation respectively to obtain the dynamic observability state equation:

[0131]

[0132] This equation B l z l = D l is the dynamic observability state equation of the box girder considering the shear lag effect. and are submatrices of the stiffness matrix, and their sizes are determined by and respectively. and are submatrices of the mass matrix, and their sizes are determined by and respectively. The subscripts 0 and 1 represent unknown and known respectively.

[0133] At this time, z l contains all unknown variables, B l is the coefficient matrix corresponding to the unknown variables, and D l is the constant vector.

[0134] Obtaining the pseudo-observations of the longitudinal displacement difference of nodes: Based on the modal analysis of the structural dynamic responses (i.e., the rotation angles, deflections, and vibration frequencies of each observation point) collected under dynamic excitation or environmental excitation, and calculate the pseudo-observations of the longitudinal displacement difference of nodes based on the modal parameters (natural frequencies, damping ratios, modal vibration modes).

[0135] The translational and rotational displacements of the observation points can be measured by displacement sensors and inclinometers, and the vibration frequency can be calculated using acceleration sensors. However, the observations of the longitudinal displacement difference of the shear lag generalized degrees of freedom cannot be directly obtained by instruments. Combining the box girder analytical theory, D'Alembert's principle, and the superposition principle, and ignoring the errors generated in the actual measurement process, the calculation formula of the shear lag displacement function μ φ is derived, and the calculation formula of the pseudo-observations of the longitudinal displacement difference of nodes is proposed:

[0136]

[0137] In the formula, is the pseudo-observation of the longitudinal displacement difference of the i-th node; θ i is the measured value of the rotational displacement of the i-th node; is the shear lag displacement coefficient of the i-th node.

[0138] Derive the shear lag displacement coefficient successively through the analytical solution of the longitudinal displacement difference and the calculation of the node displacements during the structural vibration, and then use Equation (3-1) to calculate the pseudo-observations of the longitudinal displacement difference:

[0139] S2.11, Analytical solution of longitudinal displacement difference

[0140] According to the principle of minimum potential energy, when the structure is in equilibrium under the action of external loads, the variation of the total potential energy of the structure system is always zero. Then, the necessary and sufficient condition for the total potential energy to reach an extreme value is:

[0141]

[0142] where v is the vertical deflection, is the longitudinal displacement difference function.

[0143] Equation (3-2) is a differential equation considering the shear lag effect. According to the boundary constraint conditions of the longitudinal warping displacement function of shear lag, this equation is sorted out to obtain:

[0144]

[0145] where:

[0146] The general solution form of the non-homogeneous differential equation (3-3) is:

[0147]

[0148] In Equation (3-5), is the particular solution of the longitudinal displacement difference of shear lag, which is only related to the Q(z) situation at the nodes. The integral constants C1 and C2 in the general solution depend on the boundary conditions of the specific structure system. Equation (3-4) is deformed to obtain the following expression:

[0149]

[0150] In Equation (3-6), the first term M(z) on the right side of the equation is the bending moment corresponding to the classical beam theory when the beam bends, and the second term M f is the bending moment corresponding to the additional displacement of the shear lag effect. Therefore, when considering the positive shear lag effect, the effective stiffness of the box girder flange will be affected and reduced, which will lead to an increase in deflection and rotation angle. In this case, the cross-sectional deformation of the beam no longer satisfies the plane section assumption. From Equation (3-7), it can be seen that M f is related to the first derivative of the longitudinal displacement difference function u(z) of the box girder rib and is positively correlated with the bending stiffness EI s of the flange.

[0151] For a simply supported beam with a concentrated force acting at the mid-span, the bending moment and shear force along the beam segment can be calculated by the following equations:

[0152]

[0153] where, l is the length of the beam; P is the concentrated force; a and b are the distances from the load application points to the two ends of the beam respectively.

[0154] Substituting Eqs. (3-8) and (3-9) into Eq. (3-5), the general solution of the longitudinal displacement difference function u(z) of the box girder slab rib is:

[0155]

[0156] Introducing the boundary conditions, the longitudinal displacement difference function of the beam segment is obtained:

[0157]

[0158] After differentiating Eq. (3-12) and substituting it into Eq. (3-6), the curvature of the beam segment considering the shear lag effect is obtained:

[0159]

[0160] Substituting Eqs. (3-8) and (3-9) into Eq. (3-13) and integrating once, the deflection and rotation angle of the beam segment considering the shear lag effect are obtained.

[0161] v = v z + v f (3-14)

[0162] θ = θ z + θ f (3-15)

[0163] In the formula, v z is the deflection displacement corresponding to the classical beam theory; v f is the generalized deflection displacement corresponding to the shear lag effect; θ z is the rotation angle displacement corresponding to the classical beam theory; θ f is the generalized rotation angle displacement corresponding to the shear lag effect.

[0164] When 0 ≤ z ≤ a:

[0165]

[0166] When a ≤ z ≤ l:

[0167]

[0168] In the formula,

[0169] For a cantilever beam subjected to a concentrated load P, at this time, the bending moment and shear force along the beam segment are both piecewise functions:

[0170]

[0171]

[0172] The general solution of the longitudinal displacement difference function u(z) of the box girder slab rib is as follows:

[0173]

[0174] Introducing the boundary conditions, the longitudinal displacement difference function of the beam segment is obtained:

[0175]

[0176] Substitute the derivative of Equation (3-27) into Equation (3-6) to obtain the curvature of the beam segment considering the shear lag effect:

[0177]

[0178] Substitute Equations (3-24) and (3-25) into Equation (3-28), integrate once to obtain the rotation angle of the beam segment considering the shear lag effect, and integrate twice to obtain the deflection of the beam segment considering the shear lag effect.

[0179] When 0 ≤ z ≤ a:

[0180]

[0181] When a ≤ z ≤ l:

[0182]

[0183] v f = 0 (3-34)

[0184]

[0185] θ f = 0 (3-36)

[0186] S2.12: Shear lag displacement coefficient

[0187] First, the differential motion equilibrium equation of the beam element is established. Combining the D'Alembert principle and the structural main vibration mode function, the inertial force during the vibration of the beam element is solved, and the analytical solution expressions of the rotation displacement and longitudinal displacement difference along the beam length are calculated respectively based on the superposition principle. Considering undamped free vibration, the analytical solutions of the modal parameters of a homogeneous equal-section simply supported beam and a cantilever beam are used for example analysis.

[0188] (1) Solve the undamped free vibration function of the beam element

[0189] For a homogeneous equal-section straight beam with a flexural rigidity of EI, a density of ρ, and a cross-sectional area of A, when shear deformation and rotational inertia are not considered and the influence of axial deformation is assumed to be negligible, the free vibration equation of the beam can be written as:

[0190]

[0191] In the formula, v is the vertical deflection; t is the time.

[0192] For a damped free vibration system, it can be decomposed into an infinite number of degrees-of-freedom systems. That is, when the system vibrates according to a certain principal vibration mode, all the mass points on it perform corresponding simple harmonic motions. For the solution of Equation (3-37), using the method of separation of variables, it can be expressed as:

[0193] w(x, t) = Y(x)·T(t) (3-38)

[0194] The above formula shows that free vibration is a motion in which the amplitude changes with time according to T(t) and proceeds in the specified shape Y(x). Using a prime to denote the derivative with respect to x and a dot to denote the derivative with respect to t, substituting Equation (3-38) into (3-37) gives:

[0195]

[0196] Separating variables in the above formula gives the following equation:

[0197]

[0198] where a 4 is a constant. From (3-38), it can be seen that

[0199] Solving Equations (3-40) and (3-41) gives the forms of the solutions for T(t) and Y(x) as:

[0200] T(t) = B1cosωt + B2sinωt (3-42)

[0201] Y(x) = A1sin(ax) + A2cos(ax) + A3sinh(ax) + A4cosh(ax) (3-43)

[0202] For the obtained response function, six unknown parameters B1, B2, A1, A2, A3, and A4 need to be determined. Since the form of the solution of its synchronous vibration is the familiar abbreviated vibration solution, substituting two general initial conditions can obtain the unknown parameters B1 and B2; for the unknown parameters A1, A2, A3, and A4, they need to be determined according to the boundary conditions (displacement, slope, bending moment, or shear force) of different structural systems. For the convenience of calculation, when determining these constants, any one of the constants can be used to represent the other three constants, obtaining the expression of the frequency equation. The expression of this frequency equation can be used to solve the frequency parameter a.

[0203] For a simply supported beam, its boundary conditions are:

[0204] Y(0) = 0, M(0) = EIY''(0) = 0 (3 - 44)

[0205] Y(l) = 0, M(l) = EIY''(l) = 0 (3 - 45)

[0206] Substitute equations (3 - 44) and (3 - 45) into the main vibration mode function (3 - 43), we get A1 = A3 = A4 = 0, then the main vibration mode function is:

[0207]

[0208] For a cantilever beam, its boundary conditions are:

[0209] Y(0) = 0, Y'(0) = 0 (3 - 47)

[0210] M(l) = EIY''(l) = 0, V(l) = EIY'''(l) = 0 (3 - 48)

[0211] Substitute equations (3 - 47) and (3 - 48) into the main vibration mode function (3 - 43), we get A1 = -A3, A2 = -A4, then the main vibration mode function is:

[0212]

[0213] Among them, when i ≤ 3, after converting the circular frequency, the square of the characteristic root Natural frequency ω i is

[0214] (2) D'Alembert's principle

[0215] According to D'Alembert's principle, for any structural physical system, the sum of the virtual work values done by the internal inertial forces or the applied external forces is zero.

[0216] Interpret the motion equation of the structural system by Newton's second law: The rate of change of momentum of any mass element is equal to the force acting on it. That is, when the mass does not change with time, the force vector p(t) satisfies:

[0217]

[0218] At this time, the on the right side of the equation is called the inertial force that resists the mass acceleration. According to D'Alembert's principle, the inertial force can be considered as including many kinds of forces acting on the mass. Furthermore, the dynamic problems can be studied by the static method, that is, the method of static - dynamic equivalence.

[0219] For a free - vibrating homogeneous straight beam with a constant cross - section, the inertial force of any micro - element dx on the beam can be written as:

[0220]

[0221] Therefore, for a single-degree-of-freedom undamped free vibration system with mass \(m\), stiffness \(k\), and displacement \(y(t)\), the equilibrium state equation can be written as:

[0222]

[0223] Substitute (3 - 38) into (3 - 52), and cancel out \(T(t)\) on both sides of the equation, we can get:

[0224] ω 2 \(mY(x)+kY(x) = 0\) (3 - 53)

[0225] The expression of the inertial force at this time is only related to the position \(x\) on the beam:

[0226] \(P(x)=ω\) 2 \(mY(x)\)

[0227] (3) Calculation of shear lag displacement coefficient

[0228] For a simply supported beam under a concentrated force, according to Equation (3 - 54), the vibration modes of the beam can be regarded as the effects generated by countless concentrated forces with the same magnitude and opposite direction as the inertial force acting on the beam, and this distributed force changes along the beam length according to the specified shape \(Y(x)\).

[0229] When only considering the inertial force of the \(dx\) microelement, the longitudinal displacement difference at position \(z\) on the beam can be calculated according to Equation (3 - 12).

[0230] According to the superposition principle, when calculating the shear lag effect of a statically determinate structure, the shear lag effects under multiple loads can be equivalent to the sum of the shear lag effects of the corresponding statically determinate system under a single load and redundant forces. Therefore, when considering the inertial forces of all microelements on the beam, the longitudinal displacement difference on the beam can be calculated by integration:

[0231]

[0232] Similarly, according to the superposition principle, when considering the inertial forces of all microelements on the beam, the rotation angles at each position on the beam can be calculated as:

[0233]

[0234] According to Equation (3 - 27), similarly, the calculation formula for the longitudinal displacement difference of a cantilever beam can be obtained as:

[0235]

[0236] The rotation displacement considering the shear lag effect is:

[0237]

[0238] The shear lag displacement coefficient can be derived from Equations (3-55)-(3-58). The calculation formula is as follows:

[0239]

[0240] Step 3: Substitute the modal parameters and the pseudo-observations of the longitudinal displacement difference of the nodes into the dynamic observability state equation, and then use the null space matrix of the coefficient matrix of the dynamic observability state equation to determine the solution of the dynamic observability state equation, so as to obtain the axial stiffness and flexural stiffness of the box girder.

[0241] Specifically, use the null space matrix to determine whether the dynamic observability equation has a solution:

[0242] [V] T {D}=0

[0243] In the formula, [V] is the null space matrix of the coefficient matrix [B] of the dynamic observability state equation; {D} is a completely known vector.

[0244] Solve the null space matrix V of the coefficient matrix [B]. If [V] does not exist, that is, when [V] is zero-dimensional, there is a unique solution for all variables in the equation; when [V] exists and is not a zero matrix, the variables corresponding to all zero rows have a unique solution, and the corresponding unique particular solution variables are obtained by B\D. Substitute the particular solution, that is, the observability variable, as a known quantity into Step 2) for iterative calculation to obtain all target parameters including the axial and flexural stiffnesses of the structure.

[0245] The present invention also provides a dynamic structural system identification system for a box girder bridge considering the shear lag effect. The system includes a memory and a processor. The memory stores a computer program, and when the processor executes the computer program, it executes the dynamic structural system identification method for a box girder bridge considering the shear lag effect as described above.

[0246] The present invention also provides a computer-readable storage medium. The computer-readable storage medium stores machine-executable instructions. When the machine-executable instructions are called and executed by the processor, the machine-executable instructions cause the processor to implement the dynamic structural system identification method for a box girder bridge considering the shear lag effect as described above.

[0247] It is easy for those skilled in the art to understand that the above is only a preferred embodiment of the present invention, and it is not intended to limit the present invention. Any modifications, equivalent replacements, and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.

Claims

1. A method for identifying the dynamic structural system of a box girder bridge considering the shear lag effect, characterized in that, The method includes the following steps: Step 1, on the basis of one-dimensional beam elements, introduce the longitudinal displacement difference of the flange as the generalized degree of freedom of shear lag, and then construct the stiffness matrix and mass matrix of the box girder element considering the shear lag effect, which are respectively: Where, ρ represents the density of the box girder element; T represents the transpose of the matrix; I s is the moment of inertia of the upper and lower flange plates about the centroidal axis x; E represents the elastic modulus; G is the shear modulus; [N v is the deflection displacement mode shape function of the beam segment; is the longitudinal displacement difference mode shape function of the beam segment; b is half of the width of the top plate; z is the ordinate along the bridge span; Step 2, based on the element stiffness and mass matrices considering the shear lag effect, construct the structural undamped free vibration state equation of the box girder element, and then introduce boundary condition constraints, and separate the known and unknown quantities in the structural undamped free vibration state equation of the box girder element to construct the dynamic observability state equation; at the same time, based on the structural dynamic responses collected under dynamic excitation or environmental excitation, perform modal analysis on the beam element, and calculate the pseudo-observations of the longitudinal displacement difference of the nodes based on the obtained modal parameters; Step 3, substitute the modal parameters and the pseudo-observations of the longitudinal displacement difference of the nodes into the dynamic observability state equation, and then use the null space matrix of the coefficient matrix of the observability state equation to judge the solution of the dynamic observability state equation, and then obtain the axial stiffness and flexural stiffness of the box girder; The construction of the dynamic observability state equation includes the following sub-steps: S2.1, first, establish the structural undamped free vibration state equation of the simply supported box girder element considering the shear lag effect: K l {φ l} = λ l ·M l {φ l} (2-1) Among which, K l is the overall stiffness matrix considering shear lag effect of the simply supported beam, {φ l} is the nodal mode displacement, λ l is the structural mode frequency, and M l is the overall mass matrix considering shear lag effect; Subsequently, introduce boundary condition constraints, and process the stiffness and mass matrices of the box girder element by the "row and column deletion" method to obtain: Among them, N N is the number of nodes, and N B is the number of boundary conditions; S2.2, perform a separation transformation on the matrix in the structural undamped free vibration state equation, centrally couple the target structural system parameters of the obtained structural undamped free vibration state equation to form a new parameter vector, so that the overall stiffness matrix and mass matrix originally considering the shear lag effect become matrices containing only constant coefficients, and then couple the structural system parameters in the stiffness matrix and mass matrix with the modal displacement to form a new observation variable D l , and then obtain the dynamic observability state equation; The dynamic observability state equation is: Equation B l z l = D l is the dynamic observability state equation of the box girder considering the shear lag effect, and are submatrices of the stiffness matrix, and their sizes are determined by and ; and are submatrices of the mass matrix, and their sizes are determined by and ; z l contains all unknown variables, B l is the coefficient matrix corresponding to the unknown variables, and D l is a constant vector.

2. The method for identifying the dynamic structural system of a box girder bridge considering the shear lag effect according to claim 1, wherein: The calculation formula for the pseudo-observations of the longitudinal displacement difference of the nodes is: In the formula, is the pseudo-observation value of the longitudinal displacement difference of the i-th node; θ i is the measured value of the rotational displacement of the i-th node; is the shear lag displacement coefficient of the i-th node.

3. The method for identifying the dynamic structural system of a box girder bridge considering the shear lag effect according to claim 1, characterized in that: Step 1 includes the following sub-steps: S1.1, introduce a longitudinal displacement function of the flange plate, that is, the longitudinal shear lag warping displacement function, to describe the bending displacement of the box girder flange plate; the expression of the longitudinal displacement function of the flange plate is: In the formula, u(x, z, t) is the longitudinal dynamic displacement function of the flange; is the longitudinal dynamic displacement difference function of the flange plate; v(z, t) is the vertical dynamic deflection of the flange; z is the longitudinal coordinate along the bridge span; b is half of the width of the top plate; x is the transverse coordinate of the section; h i is the distance from the centroid of the section to the middle of the upper and lower flanges; S1.2, on the degrees of freedom of the one-dimensional beam element, introduce the shear lag longitudinal warping displacement difference function as a displacement component into the node displacement vector of the beam element, and then derive the dynamic deflection displacement function and dynamic longitudinal displacement difference function of each node of the beam segment based on the formula of the node displacement vector of the beam element; S1.3, calculate the energies of each part of the box girder section based on the dynamic deflection displacement function and dynamic longitudinal displacement difference function of each node of the beam segment; S1.4, according to Hamilton's principle and the energies of each part of the box girder section, obtain the total potential energy and total kinetic energy of the box girder element considering the shear lag effect, and then obtain the undamped free vibration equation of the box girder element, and perform matrix transformation on the undamped free vibration equation to obtain the stiffness and mass matrices of the box girder element considering the shear lag effect.

4. The method for identifying the dynamic structural system of a box girder bridge considering the shear lag effect according to claim 3, wherein: When the beam segment element vibrates, it is assumed that each node performs simple harmonic motion, so the dynamic deflection displacement function of each node of the beam segment and the dynamic longitudinal displacement difference function of the flange can be expressed as: v(z,t) = v(z)sin(ωt + ζ) = [N v {δ} e sin(ωt + ζ) (1-5) where {δ} e is the displacement array of element nodes; w is the natural vibration frequency of the box girder; ζ is the initial phase angle during the vibration of the box girder.

5. The method for identifying the dynamic structural system of a box girder bridge considering the shear lag effect according to claim 4, wherein: The total kinetic energy of the beam segment element is calculated by taking the sum of the kinetic energy of the vertical deflection of the beam and the kinetic energy of the longitudinal displacement, that is, the calculation formula is: In the formula, ρ is the density of the box girder element.

6. The method for identifying the dynamic structural system of a box girder bridge considering the shear lag effect according to any one of claims 1-5, characterized in that: Use the null space matrix to judge whether the dynamic observability equation has a solution: [V] T {D} = 0 In the formula, [V] is the null space matrix of the coefficient matrix [B] of the observability state equation; {D} is a completely known vector; solve the null space matrix V of the coefficient matrix [B], if [V] does not exist, that is, [V] is zero-dimensional, there is a unique solution for all variables in the equation; when [V] exists and is not a zero matrix, the variables corresponding to the all-zero rows of it have a unique solution, and the corresponding unique particular solution variables are obtained through B\D, and then the axial stiffness and flexural stiffness are obtained.

7. A dynamic structural system identification system for a box girder bridge considering shear lag effect, characterized in that: The system includes a memory and a processor. The memory stores a computer program. When the processor executes the computer program, it executes the method for identifying the dynamic structural system of a box girder bridge considering shear lag effect according to any one of claims 1-6.

8. A computer-readable storage medium, characterized in that: The computer-readable storage medium stores machine-executable instructions. When the machine-executable instructions are called and executed by a processor, the machine-executable instructions cause the processor to implement the method for identifying the dynamic structural system of a box girder bridge considering shear lag effect according to any one of claims 1-6.

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