Analysis method for single symmetrical thin-walled curved girder bridge under moving simple harmonic load

Abstract

The application discloses a kind of mobile simple harmonic load under single symmetrical thin-walled curved girder bridge analysis method, it is related to thin-walled curved girder evaluation field.Vehicle is modeled as simple harmonic mobile load, and curved girder is constructed as Bernoulli-Euler beam;Based on the finite element formula of asymmetric section curved girder, the relationship between energy principle and force and beam displacement is applied, and the motion equation of curved girder is constructed;The motion equation of curved girder is converted from time domain to s domain by Laplace transformation, and the analytical solution of the motion equation of curved girder in s domain is solved;The analytical solution in s domain is applied to obtain the time domain analytical solution by Laplace inverse transformation, and the curved girder bridge analysis is completed.The application provides strong technical support for calculating the dynamic response of single symmetrical section curved girder under mobile simple harmonic load, and also greatly reduces the difficulty of calculation in the design process of curved girder.Through the analysis of analytical solution, it also provides theoretical support for exploring the in-plane vibration and out-of-plane vibration coupling mechanism of curved girder.

Description

Technical Field

[0001] This invention relates to the field of thin-walled curved beam evaluation, and more specifically to a method for analyzing a single-symmetric thin-walled curved beam bridge under moving harmonic loads. Background Technology

[0002] Thin-walled curved beams are widely used in curved railways, subways, highway bridges, and other engineering projects. In practical applications, their cross-sections include three types: biaxially symmetric, uniaxially symmetric, and asymmetric. Due to the different cross-sectional shapes, curved beams exhibit complex mechanical behaviors. Therefore, it is necessary to study the dynamic response of thin-walled curved beams with different cross-sectional shapes under moving loads.

[0003] First, because the center of gravity of a curved beam coincides with its shear center, many researchers have paid close attention to the vibration problem of curved beams with biaxially symmetric sections. This characteristic means that the in-plane vibration of a curved beam with a biaxially symmetric section is independent of its out-of-plane vibration. For example: Vlasov established the equilibrium equations for the radial, vertical, and torsional motions of curved beams under static loads; Yang focused on the vibration problem of rectangular cross-section horizontal curved beams under a series of moving loads, using the Galerkin method and considering only the first-order vibration mode to derive simplified analytical solutions describing the in-plane and out-of-plane dynamic responses of the beam; Wu and Chiang considered shear deformation and rotational inertia effects to study the dynamic response of circular curved Timoshenko beams triggered by moving loads; Lee conducted out-of-plane free vibration analysis on curved beams with variable curvature, including the effects of rotational inertia, torsional inertia, and shear deformation; Nikkhoo proposed a numerical solution for the dynamic analysis of semi-circular curved beams under moving loads; Rosstam studied the vibration suppression problem of typical uniform circular curved beams under eccentric moving loads; Li analyzed the transient dynamic behavior of curved beams under three-dimensional moving loads based on the Galerkin method and modal superposition method; Poojary compared the in-plane radial vibration characteristics of intact and cracked circular curved beams under moving loads.

[0004] Unlike biaxially symmetric sections, uniaxially symmetric curved beams induce coupling effects of in-plane and out-of-plane vibrations. Currently, a considerable number of numerical studies based on the finite element method have considered the coupling effects of four-dimensional vibrations in beams. For example, Morris and Ho established the stiffness matrix of curved beam elements using only equilibrium conditions; Chai proposed a general matrix formula for the displacement and frequency of spatial elements in thin-walled curved beams, applicable to all linear and linearized structural problems without approximating curvature or warping; Heins and Sahin obtained the natural frequencies using the finite difference method and compared them with measured results from several curved box girders; Lu studied the shear lag effect of thin-walled curved box girders under vehicle loads using the three-dimensional finite element method and verified the model's correctness through experimental comparison; Marcello used the Hamiltonian structural analysis method to numerically analyze the shear lag, non-uniform torsion, and distortion effects of curved box girder bridges. Compared to the widespread application of numerical solutions, analytical solutions for uniaxially symmetric sections remain insufficient. Most existing analytical studies neglect the in-plane / out-of-plane vibration coupling effect. For example, Luo conducted analytical and numerical studies on the free and forced vibrations of Euler-Bernoulli curved beams and extended the derived semi-analytical solution of beam vibration characteristics to the train-track spatial interaction; Khaloo considered the mass inertia effect of a single-symmetric cross-section railway curved bridge and derived its in-plane and out-of-plane vibration semi-analytical solutions, but neglected the coupling effect caused by the offset between the beam's center of gravity and shear center; Cai derived a closed-loop solution for the dynamic response of a curved beam under moving mass based on the differential transformation method. However, none of the above methods fully capture the unique dynamic characteristics of a single-axisymmetric curved beam under the in-plane / out-of-plane coupling effect.

[0005] For asymmetric cross-sections, the vibration coupling effect of beams is more pronounced, and their vibration behavior is more complex than that of symmetric cross-sections. Lebeck and Knowlton established a finite element loop model for asymmetric cross-sections, considering the coupling effects of in-plane and out-of-plane responses. Kawakami et al. derived approximate solutions for the in-plane and out-of-plane free vibration responses of curved beams with arbitrary shapes and variable cross-sections. Kim applied technical calculation programs to the free vibration and spatial stability analysis of thin-walled curved beams with asymmetric cross-sections. Zhou established a transfer matrix model for asymmetric non-uniform bending beams. However, the above studies still focus on numerical solutions to investigate the mechanical behavior of bending beams.

[0006] Previous theoretical studies on the dynamic vibration of curved beams have mainly focused on biaxially symmetric sections, while research on the coupling effects of in-plane and out-of-plane vibrations caused by single-axisymmetric sections is relatively limited. Furthermore, harmonic moving loads are a fundamental dynamic load frequently encountered in engineering applications. Therefore, studying the dynamic vibration of single-axisymmetric curved beams under simple harmonic moving loads has significant practical implications. Summary of the Invention

[0007] In view of this, the present invention provides an analysis method for a single-symmetric thin-walled curved beam bridge under moving harmonic loads, and reveals the vibration characteristics of a single-symmetric cross-section curved beam by comparing it with a double-axis symmetric curved beam.

[0008] To achieve the above objectives, the present invention adopts the following technical solution:

[0009] An analysis method for a single-symmetric thin-walled curved beam bridge under moving harmonic loads includes the following steps:

[0010] The vehicle is modeled as a simple harmonic moving load, and the curved beam is constructed as a Bernoulli-Euler beam;

[0011] Based on the finite element formula for asymmetric cross-section curved beams, and applying the energy principle and the relationship between force and beam displacement, the motion equations of the curved beams are constructed.

[0012] The equations of motion of the curved beam are transformed from the time domain to the s domain by using the Laplace transform, and the analytical solution of the equations of motion of the curved beam in the s domain is solved.

[0013] The analytical solution in the s-domain is obtained by applying the inverse Laplace transform to the analytical solution in the time domain, thus completing the analysis of the curved beam bridge.

[0014] Optionally, verifying the time-domain analytical solution specifically includes: generating comparative data using the finite element method, and establishing the global equations of motion for the thin-walled curved beam under moving loads as follows:

[0015]

[0016] Where [M], [C], and [K] represent the mass matrix, damping matrix, and stiffness matrix, respectively; {f b This indicates the displacement of the beam and the moving loads acting on it. and{u b} are the acceleration vector, velocity vector, and displacement vector of the beam, respectively; after the global equation of motion is established, the Newmark-β method is used for time-domain incremental analysis.

[0017] Optionally, during the establishment of the global motion equations, based on the energy principle, the integral expressions of the element mass matrix and stiffness matrix are derived, the element mass matrix and stiffness matrix of the curved beam are calculated, and integrated into the global mass matrix and stiffness matrix; among them, the global damping matrix is ​​calculated using the Rayleigh damping assumption, and the moving load is treated as an equivalent nodal load updated in real time.

[0018] Optionally, when establishing the motion equations for the curved beam, the energy principle and the relationship between force and beam displacement are applied, and the offset between the shear center and the center of mass is considered. The governing motion equations for a single-axis symmetric curved beam under simple harmonic moving loads are expressed as follows, covering axial vibration, radial vibration, vertical vibration, and torsional vibration:

[0019]

[0020]

[0021] Where E represents the elastic modulus, I y I represents the moment of inertia along the y-axis. z I represents the moment of inertia along the z-axis. ω Represents the warping constant, I r Represents the torsional moment of inertia, a z Represents the offset of point A from point O along the y-axis, R represents the radius, A represents the cross-sectional area, e represents the eccentricity, and u represents the offset. x (x,t), u y (x,t) and u z (x,t) represent the axial, radial, and vertical displacements, respectively; θ(x,t) represents the torsional angle about the shear center S; δ represents the Dirac function; J represents the torsional moment of inertia; G represents the shear modulus; and u x ,u y ,u z Represents three rotational displacements, r is the polar radius of inertia, and x represents the X-axis.

[0022] Optional, the expression for a simple harmonic moving load is as follows:

[0023]

[0024] F z (x,t)=m v g sin(ωt);

[0025] Where, m v F represents the vehicle's mass; g represents gravitational acceleration. y (x,t) and F z (x,t) represent the radial external load and the vertical external load, respectively.

[0026] Optionally, the lateral and torsional displacements of the thin-walled beam are constrained by the end diaphragms, and the vertical displacement is constrained by the supports. The end sections are allowed to warp freely. Based on this, the boundary conditions and initial conditions for the curved beam are specified as follows:

[0027] u x (0,t)=u x (K,t)=u x (0,t)=u x (L,t)=0;

[0028] u y (0,t)=u y (L,t)=u” y (0,t)=u”y (L,t)=0;

[0029] u z (0,t)=u z (L,t)=u” z (0,t)=u” z (L,t)=0;

[0030] θ(0,t)=θ(L,t)=θ”(0,t)=θ”(L,t)=0;

[0031] Where u x (0,t),u x (L,t) represents the displacement boundary conditions of the curved beam at the left end (x=0) and the right end (x=L), u x (0,t) is u x "(L,t) represents the bending moment boundary conditions of the curved beam at the left end (x=0) and the right end (x=L) in the axial direction. Similarly, y represents the radial direction, z represents the vertical direction, and θ represents the torsional direction.

[0032] The initial conditions are as follows:

[0033] u z (x,0)=0,

[0034] θ(x,0)=0,

[0035] u x (x,0)=0,

[0036] u y (x,0)=0,

[0037] In the formula Let represent the derivatives with respect to time t;

[0038] Based on the boundary conditions of a single-symmetric bending beam, the displacements of the beam in the four directions can be expressed as:

[0039]

[0040] Among them, X n (t), Y n (t), Z n (t) and θ n (t) represents the nth modal coordinates of the axial, radial, vertical, and torsional responses, respectively. Using the orthogonality constraint of trigonometric functions, integrating x from 0 to L and multiplying by sin(jπx / L), the modal motion equations of the thin-walled beam in the axial, radial, vertical, and torsional directions are derived as follows:

[0041]

[0042]

[0043] By order The homogeneous part is written as:

[0044]

[0045] Or it can be expressed in matrix form as:

[0046]

[0047] Among them W n V n ,P n and Q n These represent the vibration amplitudes under axial, radial, vertical, and torsional external loads, respectively.

[0048] For the eigenvalue problem to have non-zero solutions, the determinant of its matrix must be equal to 0, i.e.:

[0049] The equation is simplified to a fourth-order polynomial equation with a zero determinant of the coefficient matrix. Using MATLAB's `solve` function, the natural frequencies of the curved beam are calculated, yielding four real roots. Where ω n1 ,ω n2 ,ω n3 ,ω n4 The four-directional coupled frequencies of a single-axis symmetric cross-section curved beam are associated with the coupled vibration directions of vertical, radial, torsional, and axial directions, respectively.

[0050] Optionally, the equations of motion of the curved beam can be transformed from the time domain to the s-domain using Laplace transform, and the analytical solution of the equations of motion of the curved beam in the s-domain can be solved. The specific calculation process is as follows:

[0051]

[0052]

[0053] in, r1 = Ω n +ω,r2=Ω n -ω; r1 and r2 are the left and right shift frequencies, respectively;

[0054] s is a complex variable, Ω n The velocity parameter representing the vehicle speed v, based on the Laplace transform, is transformed from the time domain to the s domain, yielding:

[0055]

[0056]

[0057] Where, ω n1 ,ω n2 ,ω n3 ,ω n4 The coupled four-directional frequencies of a single-axis symmetric cross-section curved beam are associated with the coupled vibration directions in the vertical, radial, torsional, and axial directions, respectively. z Represents the offset of point A from point O along the y-axis, e represents the eccentricity, L represents the length, and u represents the distance between the two points. x (x,t), u y (x,t) and u z (x,t) represent the axial displacement, radial displacement, and vertical displacement, respectively, and θ(x,t) represents the torsional angle about the shear center S. and These represent the uncoupled and coupled frequencies in the axial, radial, vertical, and torsional directions of the curved beam, respectively, where the subscript x indicates the uncoupled frequency and the subscript xx indicates the coupled frequency. and The m represents the vertical and torsional coupling frequency of the curved beam. L The moment of inertia is the rotational inertia in the torsional direction, and m is the mass of the thin-walled curved beam per unit length.

[0058] As can be seen from the above technical solution, compared with the prior art, this invention discloses an analysis method for a single-symmetric thin-walled curved beam bridge under moving harmonic loads, providing strong technical support for calculating the dynamic response of a single-symmetric cross-section curved beam under moving harmonic loads, and greatly reducing the calculation difficulty in the design process of curved beams. The analysis of the analytical solution also provides theoretical support for exploring the coupling mechanism of in-plane and out-of-plane vibrations of curved beams. Attached Figure Description

[0059] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0060] Figure 1 A schematic diagram of a thin-walled bending beam with a single symmetrical cross-section under harmonic moving load provided by the present invention;

[0061] Figure 2 A cross-sectional view of a thin-walled curved beam provided for this invention;

[0062] Figure 3 A schematic diagram of a curved beam element with a length of Le provided by the present invention;

[0063] Figure 4a This refers to the mid-span displacement response of the radially thin-walled curved beam of the present invention;

[0064] Figure 4b This refers to the mid-span displacement response of the vertical thin-walled curved beam of the present invention;

[0065] Figure 4c This refers to the mid-span displacement response of the torsional thin-walled curved beam of the present invention. Detailed Implementation

[0066] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0067] This invention discloses an analysis method for a single-symmetric thin-walled curved beam bridge under moving harmonic loads, comprising the following steps:

[0068] The vehicle is modeled as a simple harmonic moving load, and the curved beam is constructed as a Bernoulli-Euler beam;

[0069] Based on the finite element formula for asymmetric cross-section curved beams, and applying the energy principle and the relationship between force and beam displacement, the motion equations of the curved beams are constructed.

[0070] The equations of motion of the curved beam are transformed from the time domain to the s domain by using the Laplace transform, and the analytical solution of the equations of motion of the curved beam in the s domain is solved.

[0071] The analytical solution in the s-domain is obtained by applying the inverse Laplace transform to the analytical solution in the time domain, thus completing the analysis of the curved beam bridge.

[0072] Specifically, in this simplified embodiment, the vehicle is modeled as a harmonic moving load. Since the depth of the curved beam is much smaller than its equivalent longitudinal span, the beam can be considered a Bernoulli-Euler beam, and the effects of its shear deformation and rotational inertia are ignored in this embodiment. Therefore, Figure 1 A simply supported thin-walled curved beam with a single axisymmetric cross-section is drawn, which is subjected to a simple harmonic load F moving at a velocity v. z sin(ωt) and F y sin(ωt) acts, and its cross-section is as follows Figure 2 As shown in the figure. In the figure, R, L, and Φ represent the horizontal radius of curvature, length, and central angle of the curved beam, respectively. Figure 2As shown, this single-axis symmetric section has the characteristic that its shear center and centroid do not coincide, where S represents the shear center and C represents the centroid. A right-handed coordinate system is used, where the x-axis represents the centroidal axis of the beam; the y-axis and z-axis represent the vertical axis and the horizontal axis, respectively. To simplify the problem, considering that the harmonic load acting on the curved beam has transient characteristics, the damping of the curved beam is assumed to be zero.

[0073] Based on the finite element method for asymmetric curved beams, the energy principle and the relationship between force and beam displacement are applied when establishing the equations of motion for the curved beam, and the offset between the shear center and the center of mass is considered. Accordingly, the governing equations of motion for a single-axis symmetric curved beam under harmonic moving loads can be expressed as follows, which covers axial vibration, radial vibration, vertical vibration, and torsional vibration:

[0074]

[0075]

[0076] Meanwhile, the simple harmonic moving load is expressed as follows:

[0077]

[0078] F z (x,t)=m v g sin(ωt) (6)

[0079] In the formula, u x (x,t), u y (x,t) and u z (x,t) represent axial displacement, radial displacement, and vertical displacement, respectively; θ(x,t) represents the torsional angle about the shear center S. δ represents the Dirac function. The other parameters in equations (1)-(4) are defined as follows: (E,G) = elastic modulus and shear modulus; (I x ,I z ,I y = Moments of inertia of the cross sections about the x-axis, z-axis, and y-axis; I ω = Warping constant; J = Torsional moment of inertia; m = Mass per unit length; a z = The offset of the shear center S from the centroid C along the z-axis; r = Polar radius of gyration; For moving loads: F z ,F y = Load amplitude; ω = External load frequency; v = Moving speed; m v = Vehicle mass; g = Gravitational acceleration; e = Eccentricity from the beam centerline.

[0080] Equations (1)-(4) clearly show that, due to the characteristics of a single-axisymmetric section, the in-plane and out-of-plane vibration behaviors of a curved beam are inherently coupled. In contrast, for a curved beam with a double-axisymmetric section, its in-plane vibration is independent of its out-of-plane vibration, which indicates that a single-axisymmetric curved beam has unique dynamic characteristics.

[0081] In practice, the lateral and torsional displacements of thin-walled beams are constrained by the end diaphragms, while the vertical displacement is constrained by the supports, but the end sections can warp freely. Based on this, the boundary and initial conditions for this curved beam are specified as follows:

[0082] u x (0,t)=u x (L,t)=u x (0,t)=u x (L,t)=0 (7)

[0083] u y (0,t)=u y (L,t)=u” y (0,t)=u” y (L,t)=0 (8)

[0084] u z (0,t)=u z (L,t)=u” z (0,t)=u” z (L,t)=0 (9)

[0085] θ(0,t)=θ(L,t)=θ”(0,t)=θ”(L,t)=0 (10)

[0086] The initial conditions are as follows:

[0087] u z (x,0)=0,

[0088] θ(x,0)=0,

[0089] u x (x,0)=0,

[0090] u y (x,0)=0,

[0091] In the formula, (′) and (·) represent the derivatives with respect to distance x and time t, respectively.

[0092] Based on the boundary conditions of a single-symmetric bending beam, the displacements of the beam in the four directions can be expressed as:

[0093]

[0094] Among them, X n (t), Y n (t), Z n (t) and θ n (t) represents the nth modal coordinates of the axial, radial, vertical, and torsional responses, respectively. Using the orthogonality constraint of trigonometric functions, integrating x from 0 to L, multiplying by sin(jπx / L), and substituting equations (15)-(18) into equations (1)-(3), the modal motion equations of the thin-walled beam can be derived as follows:

[0095]

[0096] Among them, parameters and and The relevant coefficients are listed in Appendix A for ease of derivation.

[0097] By order The homogeneous parts in equations (19)-(22) can be written as:

[0098]

[0099] Or it can be expressed in matrix form as:

[0100]

[0101] For the eigenvalue problem described in equation (27) to have a non-zero solution, the determinant of its matrix should be equal to 0, that is:

[0102]

[0103] Equation (28) can be simplified to a fourth-order polynomial equation (it is necessary to reduce the polynomial to a fourth-order polynomial equation). (Consider it as a global variable). By setting the determinant of the coefficient matrix in equation (28) to zero and solving it using MATLAB's solve function, the natural frequency of the curved beam can be calculated. This naturally yields four real roots. Where ω n1 ,ω n2 ,ω n3 ,ω n4 The four-directional coupled frequencies of a single-axis symmetric cross-section curved beam are associated with the coupled vibration directions of vertical, radial, torsional, and axial directions, respectively.

[0104] The coupled equations (19)-(22) are difficult to solve directly in the time domain, so the Laplace transform method is used to process the equations (which must satisfy zero initial conditions, i.e., equations (11)-(14)). Specifically:

[0105]

[0106]

[0107] in

[0108]

[0109] Here, s is a complex variable, and Ω represents the velocity parameter of vehicle speed v. Based on the Laplace transform, the coupled equations (19)-(22) can be transformed from the time domain to the s domain, and we can obtain:

[0110]

[0111]

[0112] By applying the inverse Laplace transform, the s-domain solutions in equations (34)-(37) can be transformed to the time domain. However, the derivation process of the closed-loop solution is quite complex, and this embodiment cannot fully present its expression. The relevant derivation details can be implemented in the Maple computing platform. Based on the above method, this embodiment presents for the first time the analytical solution of the in-plane-out-of-plane coupled response of a horizontal curved beam with a single axisymmetric cross-section.

[0113] To verify the validity of the above analytical solution, this embodiment uses the finite element method (FEM) to generate comparative data. Figure 3 It shows a length of L e A curved beam element considering in-plane and out-of-plane vibration coupling effects is described. This beam element has seven degrees of freedom at each of its two end nodes i and j, specifically including three translational displacements: u x ,u y ,u z Three rotational displacements: θ x ,θ y ,θ z And a warping deformation: χ

[0114] The coefficient matrix H of the displacement function can be found in existing literature. This matrix comprehensively considers both in-plane and out-of-plane vibrations, and its expression is as follows:

[0115]

[0116] In the formula, 0 represents the zero matrix; H inq (q=1,2,3) and H outp (p = 1, 2, 3, 4) represent the corresponding coefficient matrices for the seven degrees of freedom, as detailed in Appendix B. The shape function N of the displacement is defined as follows:

[0117]

[0118] In the formula, ζ∈[0,L e [] represents the local axial coordinates of the beam. Based on the energy principle, the integral expressions for the element mass matrix and stiffness matrix are derived, detailed in Appendix B. Then, the element mass matrix and stiffness matrix of the curved beam are calculated and integrated into the global mass matrix and stiffness matrix. The global damping matrix is ​​calculated using the Rayleigh damping assumption. Moving loads are treated as equivalent nodal loads updated in real time. Finally, the global equations of motion for the thin-walled curved beam under moving loads are established as follows:

[0119]

[0120] In the formula, [M], [C], and [K] represent the mass matrix, damping matrix, and stiffness matrix, respectively; {f b} represents the displacement of the beam and the moving load acting on the beam; and{u b Let $\mathbf{x}$ represent the acceleration vector, velocity vector, and displacement vector of the beam, respectively. After establishing the global equations of motion, the Newmark-β method is used for incremental time-domain analysis. Parameters $β = 0.25$ and $γ = 0.5$ are selected to ensure unconditional stability, the time step is set to 0.0001 s, and the initial conditions are zero.

[0121] The material parameters of the thin-walled curved beam are detailed in Table 1. The amplitudes of the vertical and radial harmonic moving loads are set to 20 kN and 2 kN, respectively. The load moves eccentrically along the beam at a constant speed of 10 m / s, and the external excitation frequency is 3 Hz. To ensure the accuracy of the numerical results, the first 15 vibration modes are included in the calculation.

[0122] Table 1. Material properties of thin-walled curved beams

[0123]

[0124] Based on the beam parameters mentioned above, Table 2 presents the analytical solutions for the first three natural frequencies of the beam structure. As shown in the table, the analytical solutions agree well with the numerical solutions. Furthermore, Figures 4a-4c The paper presents a comparison between analytical and numerical solutions for the mid-span displacement response of a thin-walled curved beam. It is evident that the analytical results for beam displacement remain consistent with the numerical solutions within a reasonable range. These two comparative studies fully verify the accuracy and reliability of the proposed analytical solution in calculating the natural frequencies and dynamic responses of a single-axisymmetric thin-walled curved beam under moving loads.

[0125] Table 2 Natural frequencies of the beam (Hz)

[0126]

[0127]

[0128] The parameters in equations (19)-(22) are defined as follows:

[0129]

[0130] in

[0131]

[0132] m L =mr 2 (A10)

[0133] In the formula: and These represent the uncoupled and coupled frequencies in the axial (x), radial (y), vertical (z), and torsional (L) directions of the curved beam, respectively, where the subscript x indicates the uncoupled frequency and the subscript xx indicates the coupled frequency.

[0134] Wherein, the mass m of the curve beam element in equation (38) is... e and stiffness k e It can be represented as:

[0135]

[0136] In the formula, N represents the shape function matrix, N x N y N z and These represent the shape functions for the axial, radial, vertical, and torsional directions of the curved beam, respectively.

[0137]

[0138] H in2 =[1 cos(ζ / R) sin(ζ / R) (ζ / R)*cos(ζ / R) (ζ / R)*sin(ζ / R) 0] (B4)

[0139] H out1 =[1 ζ / R cosh(α*ζ / R) sinh(α*ζ / R) cos(ζ / R) sin(ζ / R) (ζ / R)*cos(ζ / R)] (B5)

[0140]

[0141] in γ=2β 2 / (α 2 +β 2 +1),FF=1-2I z / (AR 2 ), H inq (q=1,2,3) and Houtp (p=1,2,3,4) represent the corresponding coefficient matrices for the seven degrees of freedom.

[0142] The various embodiments in this specification are described in a progressive manner, with each embodiment focusing on its differences from other embodiments. Similar or identical parts between embodiments can be referred to interchangeably. For the apparatus disclosed in the embodiments, since they correspond to the methods disclosed in the embodiments, the description is relatively simple; relevant parts can be referred to the method section.

[0143] The above description of the disclosed embodiments enables those skilled in the art to make or use the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined in these embodiments may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the invention is not to be limited to the embodiments shown herein, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A method for analyzing a single-symmetric thin-walled curved beam bridge under moving harmonic loads, characterized in that, Includes the following steps: The vehicle is modeled as a simple harmonic moving load, and the curved beam is constructed as a Bernoulli-Euler beam; Based on the finite element formula for asymmetric cross-section curved beams, and applying the energy principle and the relationship between force and beam displacement, the motion equations of the curved beams are constructed. The equations of motion for the curved beam are transformed from the time domain to the time domain using the Laplace transform. s In the domain, solve the equations of motion for a curved beam. s Analytical solutions for the domain; right s The analytical solution in the domain is applied to the inverse Laplace transform to obtain the analytical solution in the time domain, thus completing the analysis of the curved beam bridge; The equations of motion for the curved beam are transformed from the time domain to the time domain using the Laplace transform. s In the domain, solve the equations of motion for a curved beam. s The analytical solution for the domain is calculated in the following process: ； ； ； in, ; and These are the frequencies for left shift and right shift, respectively. s is a complex variable, The velocity parameter representing the vehicle speed v, based on the Laplace transform, is transformed from the time domain to the s domain, yielding: ； ； ； ； in, The coupled four-directional frequencies of a single-axis symmetric cross-section curved beam are associated with the coupled vibration directions in the vertical, radial, torsional, and axial directions, respectively. a z This represents the relationship between points A and O. y Offset in the axial direction, e Represents the eccentricity. L Represents length, x ( x , t ), y ( x , t )and z ( x , t These represent axial displacement, radial displacement, and vertical displacement, respectively. ( x , t ) indicates cutting around the center S The angle of twist, , , , , , , , These represent the uncoupled and coupled frequencies in the axial, radial, vertical, and torsional directions of the curved beam, respectively, where the subscript x indicates the uncoupled frequency and the subscript xx indicates the coupled frequency. and Indicates the vertical and torsional coupling frequencies of the curved beam. The moment of inertia is the rotational inertia in the direction of reversal. The mass of a thin-walled curved beam is expressed as a unit of linear meter.

2. The method for analyzing a single-symmetric thin-walled curved beam bridge under moving harmonic load according to claim 1, characterized in that, Verification of the time-domain analytical solution specifically includes: generating comparative data using the finite element method, and establishing the global equations of motion for the thin-walled curved beam under moving loads as follows: ； in, , and These represent the mass matrix, damping matrix, and stiffness matrix, respectively. This indicates the displacement of the beam and the moving loads acting on it. , and These are the acceleration vector, velocity vector, and displacement vector of the beam, respectively; after establishing the global equations of motion, Newmark- The method is used to perform time-domain incremental analysis.

3. The method for analyzing a single-symmetric thin-walled curved beam bridge under moving harmonic load according to claim 2, characterized in that, In the process of establishing the global motion equations, based on the energy principle, the integral expressions of the element mass matrix and stiffness matrix are derived, the element mass matrix and stiffness matrix of the curved beam are calculated, and integrated into the global mass matrix and stiffness matrix; among them, the global damping matrix is ​​calculated using the Rayleigh damping assumption, and the moving load is treated as an equivalent nodal load updated in real time.

4. The method for analyzing a single-symmetric thin-walled curved beam bridge under moving harmonic load as described in claim 1, characterized in that, When establishing the equations of motion for a curved beam, the energy principle and the relationship between force and beam displacement are applied, and the offset between the shear center and the center of mass is considered. The governing equations of motion for a single-axis symmetric curved beam under simple harmonic moving loads are expressed as follows, covering axial vibration, radial vibration, vertical vibration, and torsional vibration: ； ； ； ； in, E Represents the elastic modulus ，I y represent y moment of inertia of the shaft I z represent z moment of inertia of the shaft I ω Represents the warping constant. I r Represents the torsional moment of inertia. a z This represents the relationship between points A and O. y Offset in the axial direction, R Represents radius, A Represents the cross-sectional area. e Represents the eccentricity. x ( x , t ), y ( x , t )and z ( x , t These represent axial displacement, radial displacement, and vertical displacement, respectively. ( x , t ) indicates cutting around the center S The angle of twist, This represents the Dirac function. J Represents the torsional moment of inertia. G Represents shear modulus, , , Represents three rotational displacements, r is the polar radius of inertia, and x represents the X-axis.

5. The method for analyzing a single-symmetric thin-walled curved beam bridge under moving harmonic load according to claim 1, characterized in that, The expression for a simple harmonic moving load is as follows: ； ； in, G represents the vehicle's mass; g represents the acceleration due to gravity. and These represent radial external loads and vertical external loads, respectively.

6. The method for analyzing a single-symmetric thin-walled curved beam bridge under moving harmonic load according to claim 1, characterized in that, The lateral and torsional displacements of the thin-walled beam are constrained by the end diaphragms, while the vertical displacement is constrained by the supports. The end sections are allowed to warp freely. Based on this, the boundary and initial conditions for the curved beam are defined as follows: ； ； ； ； in , This represents the axial displacement boundary conditions of the curved beam at the left and right ends. for The y-axis represents the bending moment boundary conditions at the left and right ends of the curved beam; similarly, y represents the radial direction and z represents the vertical direction. This is a twisting direction; The initial conditions are as follows: ； ； ； ； In the formula , , , Let represent the derivatives with respect to time t; Based on the boundary conditions of a single-symmetric bending beam, the displacements of the beam in the four directions can be expressed as: ； ； ； ； in, ( ), ( ), ( )and ( The first three components of the axial response, radial response, vertical response, and torsional response are respectively the first three components of the axial response, radial response, vertical response, and torsional response. n Modal coordinates; using the orthogonality constraint of trigonometric functions, from 0 to... L right x Integral, multiplied by sin( j / The modal motion equations of the thin-walled beam in the axial, radial, vertical, and torsional directions are derived as follows: ； ； ； ； By order The homogeneous part is written as: ； ； ； ； Or it can be expressed in matrix form as: ； in , , and These represent the vibration amplitudes under axial, radial, vertical, and torsional external loads, respectively. For the eigenvalue problem to have non-zero solutions, the determinant of its matrix must be equal to 0, i.e.: This simplifies to a fourth-order polynomial equation where the determinant of the coefficient matrix is ​​zero, and is further simplified using MATLAB. solve Solve the function to calculate the natural frequency of the curved beam; thus, obtain four real roots. ,in The four-directional coupled frequencies of a single-axis symmetric cross-section curved beam are associated with the coupled vibration directions of vertical, radial, torsional, and axial directions, respectively.

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