A method for determining spatial dynamic response of a four-legged elastic support bridge

Abstract

The application relates to the technical field of bridge engineering and provides a method for determining the spatial dynamic response of a four-corner elastic support bridge, which comprises the following steps: introducing a virtual rigid arm, converting a complex four-corner elastic support bridge system into a simplified mechanical model with clear physical meaning, and obtaining a parameter-adjustable analytical simplified theoretical model; based on the parameter-adjustable analytical simplified theoretical model, solving bridge vertical and torsional dynamic response equations by using Duhamel integral, and outputting dynamic response analytical solutions. The application converts the traditional bridge dynamic response problem depending on high-dimensional finite element simulation into an analytical expression which can be obtained by using Duhamel integral, greatly improving the solution efficiency; the application can be used for bridge design, evaluation and health monitoring: the dynamic response analytical solution of the simplified theoretical model can be embedded into a bridge structure response analysis tool chain, realizing bridge performance prediction and diagnosis under the working conditions of extreme load action, support degradation and the like, and having good engineering popularization value.

Description

Technical Field

[0001] This invention relates to the field of bridge engineering technology and is applied to the field of bridge design technology. Background Technology

[0002] In the field of bridge engineering, traditional methods for dynamic response analysis are mainly based on simplified two-dimensional simply supported beam models and finite element numerical simulation techniques. Early studies typically simplified bridges with elastic supports at four corners into planar beam elements, using Euler-Bernoulli beam theory or Timoshenko beam theory to establish the equations of motion, and solving for the dynamic response using modal superposition or direct integration methods. While these methods are computationally simple, they have significant theoretical shortcomings: their two-dimensional simply supported modeling assumptions completely ignore the spatial coupling effect between supports, especially when the bridge is subjected to eccentric vehicle loads, failing to accurately reflect the interaction between torsional vibration and vertical vibration, leading to a systematic deviation between the calculated results and the actual dynamic response.

[0003] With the development of computer technology, the three-dimensional finite element method has gradually become the mainstream tool for bridge dynamic analysis. Researchers use commercial software such as ANSYS and ABAQUS to build detailed finite element models including components such as the bridge deck and bearings, and use subspace iteration or Lanczos methods for modal analysis. Although these methods improve computational accuracy to some extent, their modeling process is complex, requiring the division of a large number of elements to ensure computational accuracy, often resulting in extremely high computational costs when dealing with long-span bridges. More importantly, the finite element method is essentially a numerical approximation method, making it difficult to establish explicit theoretical relationships between loads and structural parameters and bridge response, resulting in a lack of clear mechanical guiding principles in the bridge bearing design process. Summary of the Invention

[0004] To address the aforementioned technical problems, this invention proposes a method for determining the spatial dynamic response of a four-corner elastically supported bridge. Through an innovative simplified mechanical and theoretical model, the complex spatial support system is simplified into an analytically solvable vibration model, significantly improving the efficiency of response analysis of bridges under moving loads (such as vehicles) and providing theoretical support for the optimized design of bridge supports.

[0005] To achieve the above objectives, the technical solution adopted by the present invention is as follows:

[0006] A method for determining the spatial dynamic response of a four-corner elastically supported bridge includes the following steps:

[0007] Step 1: Introduce a virtual rigid arm to transform the complex four-corner elastic support bridge system into a simplified mechanical model with clear physical meaning, and obtain an analytical simplified theoretical model with adjustable parameters;

[0008] Step 2: Based on the parameter-adjustable analytical simplified theoretical model from Step 1, solve the vertical and torsional dynamic response equations of the bridge using Duhamel integrals, and output the analytical solution of the dynamic response.

[0009] This application has the following beneficial effects:

[0010] By introducing a virtual rigid arm, the geometric coupling relationship of a complex spatially supported bridge system is explicitly transformed into an equivalent torsional stiffness term with mechanical significance, clearly revealing the coupling mechanism of the supported bridge.

[0011] Dynamic response can be solved analytically with a significant reduction in dimensionality: Based on the mode combination assumption (vibration is the sum of bending sinusoidal modes and rigid body displacement modes), the dynamic response problem of bridges, which traditionally relies on high-dimensional finite element simulation, is transformed into an analytical expression that can be obtained through Duhamel integration, which greatly improves the solution efficiency.

[0012] This invention can be used for bridge design, evaluation and health monitoring: the analytical solution of the dynamic response of the simplified theoretical model can be embedded in the bridge structural response analysis toolchain to realize the prediction and diagnosis of bridge performance under extreme loads, support degradation and other conditions, and has good engineering promotion value. Attached Figure Description

[0013] Figure 1 A simplified mechanical model of a bridge with elastic supports at four corners.

[0014] Figure 2 A simplified theoretical model for a bridge with elastic supports at four corners.

[0015] Figure 3 This is a schematic diagram of the torsional deformation conditions of an elastic support on the same side.

[0016] Figure 4 This is a schematic diagram of a vehicle traveling on a four-corner elastic support bridge.

[0017] Figure 5 The theoretical values ​​of the simplified theoretical model in the embodiments of the present invention are compared with the finite element mode shapes.

[0018] Figure 6 This paper compares the theoretical values ​​and finite element simulation values ​​of the vertical and torsional fundamental frequencies in the simplified theoretical model of the present invention for embodiments.

[0019] Figure 7 This invention relates to the mid-span response of a four-corner elastically supported bridge caused by a moving vehicle load, as described in an embodiment of the invention.

[0020] Figure 8 Different eccentricities e and support stiffness ratios are used in embodiments of the present invention. κ u The bridge mid-span response. Detailed Implementation

[0021] The technical solutions provided in this application will be further described below with reference to specific embodiments and accompanying drawings. The advantages and features of this application will become clearer from the following description.

[0022] Step 1: Introduce a virtual rigid arm to transform the complex four-corner elastic support bridge system into a simplified mechanical model with clear physical meaning, and obtain an analytical simplified theoretical model with adjustable parameters;

[0023] Step 1.1: Identify the physical composition of the bridge structure, reveal the spatial coupling mechanism, and establish a simplified mechanical model of the four-corner elastic support bridge system;

[0024] In practical bridge engineering, the superstructure of common multi-span bridges is often connected to the substructure via elastic supports arranged at four corner points, forming a four-corner elastic support bridge system. This four-corner elastic support bridge system has the following mechanical characteristics:

[0025] Characteristic 1. Each support possesses vertical elastic stiffness and can be considered as a single-degree-of-freedom linear spring, exhibiting stiffness. k ;

[0026] Characteristic 2. The supports are not collinear, meaning there is a lateral distance between the supports and the bridge's neutral axis. b ;

[0027] Characteristic 3. When the vehicle load deviates from the neutral axis, it will cause inconsistent settlement of different supports, which in turn will cause the bridge to twist around the longitudinal axis;

[0028] Characteristic 4. While the bridge undergoes torsional deformation, it is also undergoing vertical bending deformation. The reaction force of the support is further fed back to the bridge deck, which includes both the suppression of vertical displacement and the constraint of torsional angle, so that torsion and vertical deformation affect each other and form a vertical-torsion spatial coupling mechanism.

[0029] To simplify the spatial coupling mechanism in characteristic 4 into a structural unit model with clear mechanical meaning, this invention proposes the concept of a virtual rigid arm. The four-corner elastically supported bridge system is simplified as a simplified mechanical model composed of three interdependent parts, such as... Figure 1 As shown, the three interdependent parts are:

[0030] Main beam (primary load-bearing beam): A three-dimensional spatial beam that satisfies the Euler-Bernoulli beam theory, primarily concerned with the vertical displacement of the bridge. Angle of twist of the bridge about its longitudinal axis ;

[0031] End-distributed elastic support: adjacent intervals are 2 b Linear elastic support;

[0032] A virtual rigid arm that acts as a lever for force transmission.

[0033] Therefore, the connection method of the four-corner elastically supported bridge system can be equivalently represented as follows: one end is connected to the supporting foundation through elastic supports; the other end is connected to the torsional center of the main beam through a virtual rigid arm. Through the above steps, the originally complex four-corner elastically supported bridge system is simplified into a simplified mechanical model with two main degrees of freedom.

[0034] Step 1.2: Explicitly demonstrate the support effect and construct a simplified analytical theoretical model with adjustable parameters.

[0035] Step 1.2.1: Abstract and reduce the dimensionality of the simplified mechanical model in Step 1.1 to form an analytically simplified theoretical model, such as... Figure 2 As shown.

[0036] like Figure 3 As shown, under torsional deformation conditions, the vertical displacement of each elastic support... Through geometric coupling, directly with the torsional angle of the bridge about its longitudinal axis Related: Therefore, the resulting elastic support force Constrained by the structural relationship of elastic supports: , k The vertical elastic stiffness of a single elastic support is given; then, converting the elastic support force into a torsional constraint moment on the main beam, the equivalent torsional stiffness is... τ for:

[0037] (1)

[0038] Thus, by simplifying the elastic support-virtual rigid arm action in the simplified mechanical model of step 1.1 above into a low-dimensional torsional constraint, an analytical simplified theoretical model is obtained, as shown in formula (1). Specifically, while retaining the inherent vertical bending stiffness and torsional stiffness of the main beam structure, the spatial geometric coupling effect introduced by the four-corner elastically supported bridge system is innovatively equivalent to a set of concentrated torsional stiffness terms. Through this approach, the system's degrees of freedom and solution dimensionality are significantly reduced, enabling the dynamic response problem, which originally relied on high-dimensional finite element analysis, to be efficiently modeled and calculated within a low-dimensional analytical framework.

[0039] Step 1.2.2: Based on the analytical theoretical model in Step 1.2.1, further establish the explicit relationship between support stiffness and bridge response, clarify the physical meaning related to support stiffness, and form a simplified theoretical model with adjustable parameters.

[0040] Specifically, the vibration is approximated as the sum of rigid body displacement modes and bending sinusoidal modes, the expressions of which are as follows:

[0041] (2)

[0042] (3)

[0043] These are the vertical mode function and the torsional mode function, respectively. These are vertical rigid body displacement and torsional rigid body displacement, respectively. L Let x be the length of the bridge and x be the X-axis coordinate.

[0044] The vertical and torsional vibrations of the bridge are considered to comprehensively describe its dynamic response. (Bridge vertical displacement) and the angle of twist of the bridge about its longitudinal axis The expression is as follows:

[0045] (4)

[0046] (5)

[0047] In the formula, These are the generalized coordinates of the vertical and torsional vibration shapes, respectively. t represents time.

[0048] The boundary condition formulas for both ends of the main beam are:

[0049] (6)

[0050] (7)

[0051] In the formula, E and G These are the elastic modulus and shear modulus of the main beam, respectively. I z Main beam z Moment of inertia of the axis, J It is the torsional constant of the main beam.

[0052] Substituting formulas (1), (4), and (5) into formulas (6) and (7) yields the following results: and will Depend on To indicate:

[0053] (8)

[0054] In a physical sense, this ratio represents the stiffness ratio between the main beam and the elastic support. This ratio incorporates the relative contribution of the support rigid body displacement to the bridge's response and can be adjusted. To control the bridge response. The above is about... The solution and the combination of vibration modes also provide a basis for solving the dynamic response of bridges.

[0055] Step 2: Based on the parameter-adjustable analytical simplified theoretical model from Step 1, solve the vertical and torsional dynamic response equations of the bridge using Duhamel integrals, output the analytical solution of the dynamic response, and apply it to bridge response assessment and design.

[0056] Based on modal shape function (i.e., vertical mode) Torsional mode function The displacement response obtained (i.e., the vertical displacement of the bridge) The angle of twist of the bridge about its longitudinal axis By combining the partial differential equations of the interaction between the vehicle and bridge systems, the vibration response of the bridge under eccentric vehicle load is analytically solved.

[0057] Specifically, for Figure 4 The vehicle-axle system shown has the bridge modeled as having a length of L ,lie in xz A planar, horizontal spatial beam supported by four elastic supports; the elastic supports are symmetrically arranged at both ends of the main beam, with an adjacent width of 2. b The elastic stiffness is k Meanwhile, in the analysis, vehicles traveling on the bridge are modeled as a single-degree-of-freedom spring-damped mass system, eccentrically positioned along the centerline of the horizontal spatial beam. e Traveling along the route. Subject to a speed of... v The vertical and torsional dynamic response equations of the main beam affected by the moving vehicle are as follows:

[0058] (9)

[0059] (10)

[0060] in m The mass per unit length of the main beam, I x Main beam x The moment of inertia of the shaft due to mass. For the Dirac function, the contact force in the formula The expression is as follows:

[0061] (11)

[0062] In the formula, k v For vehicle suspension system stiffness, vehicle suspension system damping coefficient c v , m v For vehicle quality, qv Let represent the vertical displacement of the vehicle; (·) represents the first derivative with respect to time t.

[0063] Substitute the vertical and torsional displacement expressions, i.e., formulas (4) and (5), into the vertical and torsional dynamic response equations, i.e., formulas (9) and (10), and multiply both sides of the equations by the vertical and torsional modes, respectively. and Then x From 0 to L Integral, assuming vehicle mass m v Mass much smaller than that per unit length of the main beam m And by simplifying the calculation, we can obtain:

[0064] (12)

[0065] (13)

[0066] In the formula, (··) represents the second derivative with respect to time t. Using the Duhamel integral and substituting the zero initial conditions (initial displacement and velocity are zero) into formulas (12) and (13), the vertical generalized coordinate at time t can be obtained. q u and torque generalized coordinates q θ :

[0067] (14)

[0068] (15)

[0069] in:

[0070] (16)

[0071] Where Ω is the speed-related driving frequency. These are the velocity parameters for the vertical and torsional modes, respectively.

[0072] Multiply the above formulas (14) and (15) by the corresponding generalized coordinates respectively. The expressions for the vertical displacement and torsional angular displacement at time t and position x are as follows:

[0073] (17)

[0074] (18)

[0075] Based on the assumption of a rigid section in the plane, the response of the main beam in the XZ plane at time t is obtained. The expression is as follows:

[0076] (19)

[0077] z is the Z-axis coordinate value, which also represents the coordinates of a point within the width of the bridge. Using formula (19), the vertical displacement of any position on the bridge at any time can be calculated in real time without relying on finite element simulation, given the vehicle speed, mass, and travel path.

[0078] The meanings of the parameters in this invention are shown in Table 1.

[0079] Table 1 Parameter Annotation Table

[0080]

[0081] Verification of Examples

[0082] To evaluate the rationality of the analytically simplified theoretical model of the four-corner elastically supported bridge in this invention, a finite element model of the four-corner elastically supported bridge was established in this embodiment. The characteristics of the bridge and vehicles used are listed in Table 2. In this finite element model, the main beam is simulated as a six-degree-of-freedom spatial beam, including three axial displacements and three rotations around the axis; the virtual rigid arm is modeled as a rigid beam. Through natural vibration analysis of the finite element model of the four-corner elastically supported bridge, the vertical and torsional modes and their natural frequencies of the bridge were obtained.

[0083] Table 2 Calculation parameters for four-corner elastic-supported bridges and vehicles

[0084]

[0085] Figure 5 It shows different stiffness ratios ( A comparison of the vertical and torsional modes obtained by the method of the present invention with the modal results of finite element simulation shows that the vertical and torsional modes obtained by finite element simulation are in good agreement with the vertical and torsional modes obtained by the method of the present invention.

[0086] Based on the simplified theoretical model of this invention, the vertical and torsional fundamental frequencies of the four-corner elastically supported bridge can be estimated using the Rayleigh energy method as follows:

[0087] (18)

[0088] (19)

[0089] in, ω u0 , ω θ0 These are the vertical and torsional fundamental frequencies of the beam in the simply supported case.

[0090] (20)

[0091] from Figure 6 It can be seen that, based on the simplified theoretical model calculation of this invention, different The vertical and torsional fundamental frequencies are almost identical to the finite element simulation results. This result demonstrates that the simplified theoretical model proposed in this invention can accurately describe the vibration characteristics of bridges under four-corner elastic support conditions.

[0092] The analytical solution of the dynamic response proposed in step 2 of this invention is verified. Rubber bearings are selected for the bridge supports according to national standards, and the support stiffness of the bearings is assumed to be... k 3.47×10 9 N / m. Assume a vehicle crosses the bridge at a speed of 5 m / s, 4 m off the bridge's centerline. The vertical and torsional displacement responses of the bridge calculated using the simplified theoretical model based on the method of this invention are compared with those calculated using finite element simulation. Figure 7 As shown, ignoring minor differences, the analytical solution and the finite element simulation achieve good agreement, verifying the correctness of the analytical solution obtained from the simplified theoretical model. The analytical solution establishes an analytical mapping relationship between vehicle load, eccentricity path, support stiffness ratio, and structural response, opening up the entire path from "load input" to "response output." Specifically, as... Figure 8 As shown, the eccentricity of different vehicles e and support stiffness ratio The bridge response under different influences is different. Compared with the traditional FEM multi-round model update, analytical solutions can significantly reduce computational costs and are suitable for rapid screening of multiple parameters in the early design stage.

[0093] The above description is merely a description of preferred embodiments of this application and is not intended to limit the scope of this application in any way. Any changes or modifications made by those skilled in the art based on the above-disclosed technical content should be considered as equivalent and valid embodiments and fall within the scope of protection of the technical solution of this application.

Claims

1. A method for determining the spatial dynamic response of a four-corner elastically supported bridge, characterized in that, Includes the following steps: Step 1: Introduce a virtual rigid arm to transform the complex four-corner elastic support bridge system into a simplified mechanical model with clear physical meaning, and obtain an analytical simplified theoretical model with adjustable parameters; Step 2: Based on the parameter-adjustable analytical simplified theoretical model from Step 1, solve the vertical and torsional dynamic response equations of the bridge using Duhamel integrals, and output the analytical solution of the dynamic response. Step 1 includes: Step 1.1 Identify the physical composition of the bridge structure, reveal the spatial coupling mechanism, and establish a simplified mechanical model of the four-corner elastic support bridge system; Step 1.2 Explicitize the support effect and construct an analytically simplified theoretical model with adjustable parameters; Step 1.1: The four-corner elastic support bridge system is connected to the substructure pier system through elastic supports arranged at the four corner points, and is characterized as follows: Each elastic support has vertical elastic stiffness and is considered as a single-degree-of-freedom linear spring with stiffness k; the elastic supports are arranged non-collinearly, that is, there is a lateral distance b between the support and the neutral axis of the bridge; when the vehicle load deviates from the neutral axis, the bridge twists around the longitudinal axis; while the bridge undergoes torsional deformation, it also undergoes vertical bending deformation. The reaction force of the elastic support is fed back to the bridge deck, which includes both the suppression of vertical displacement and the constraint of torsional angle, so that torsion and vertical deformation affect each other and form a vertical-torsion spatial coupling mechanism. The four-corner elastically supported bridge system can be simplified and equivalently represented as a simplified mechanical model consisting of three interdependent parts: Main girder: A three-dimensional spatial beam that satisfies the Euler-Bernoulli beam theory, focusing on the vertical displacement of the bridge. Angle of twist of the bridge about its longitudinal axis ; End-distributed elastic support: linear elastic support with an adjacent interval of 2b; A virtual rigid arm that acts as a lever for force transmission; The connection method of the four-corner elastic support bridge system can be equivalently represented as follows: one end is connected to the support foundation through elastic support; the other end is connected to the torsional center of the main beam through a virtual rigid arm; in this way, the originally complex four-corner elastic support bridge system is simplified into a simplified mechanical model with two main degrees of freedom. Step 1.2: Step 1.2.1: Abstract and reduce the dimensionality of the simplified mechanical model in Step 1.1 to form an analytically simplified theoretical model; Under torsional deformation conditions, the vertical displacement of each elastic support Through geometric coupling, directly with the torsional angle of the bridge about its longitudinal axis Related: Therefore, the resulting elastic support force Constrained by the structural relationship of elastic supports: Let k be the vertical elastic stiffness of a single elastic support; then, converting the elastic support force into a torsional constraint moment on the main beam, the equivalent torsional stiffness τ is: (1) Thus, by simplifying the elastic support-virtual rigid arm action in the simplified mechanical model of step 1.1 above into a low-dimensional torsional constraint, an analytical simplified theoretical model is obtained. Step 1.2.2: Based on the analytical theoretical model in Step 1.2.1, further establish the explicit relationship between support stiffness and bridge response, clarify the physical meaning related to support stiffness, and form a simplified theoretical model with adjustable parameters, specifically as follows; The vibration is approximated as the sum of rigid body displacement modes and bending sinusoidal modes, the expressions of which are as follows: (2) (3) These are the vertical mode function and the torsional mode function, respectively. These represent the vertical rigid body displacement and the torsional rigid body displacement, respectively; L is the bridge length; and x is the X-axis coordinate value. Considering both vertical and torsional vibrations of the bridge to comprehensively describe its dynamic response; bridge vertical displacement and the angle of twist of the bridge about its longitudinal axis The expression is as follows: (4) (5) In the formula, These are the generalized coordinates of the vertical and torsional vibration shapes, respectively. t represents time; The boundary condition formulas for both ends of the main beam are: (6) (7) In the formula, E and G are the elastic modulus and shear modulus of the main beam, respectively, Iz is the moment of inertia of the main beam about the z-axis, and J is the torsional constant of the main beam. Substituting formulas (1), (4), and (5) into formulas (6) and (7) yields the results. and will Depend on To indicate: (8) In a physical sense, this ratio represents the stiffness ratio between the main beam and the elastic support. This ratio incorporates the relative contribution of the support rigid body displacement to the bridge's response. This can be adjusted... To control the bridge response; Step 2: Based on the displacement response obtained from the modal shape function, and combined with the partial differential equations of the vehicle-bridge system interaction, the vibration response of the bridge under eccentric vehicle load is analytically solved; the modal shape function is the vertical mode. and torsional mode function The displacement response refers to the vertical displacement of the bridge. and the angle of twist of the bridge about its longitudinal axis Specifically: In the vehicle-bridge system, the bridge is modeled as a horizontal spatial beam of length L, located in the xz plane, and supported by four elastic supports. These elastic supports are symmetrically arranged at both ends of the main beam, with an adjacent width of 2b and an elastic stiffness of k. Simultaneously, in the analysis, the vehicle traveling on the bridge is modeled as a single-degree-of-freedom spring-damped mass system, traveling along a path eccentrically e from the centerline of the horizontal spatial beam. The vertical and torsional dynamic response equations of the main beam affected by the moving vehicle at velocity v are as follows: (9) (10) Where m is the mass per unit length of the main beam, and Ix is the moment of inertia of the main beam about the x-axis. For the Dirac function, the contact force in the formula The expression is as follows: (11) In the formula, kv is the stiffness of the vehicle suspension system, cv is the damping coefficient of the vehicle suspension system, mv is the vehicle mass, qv is the vertical displacement of the vehicle; (·) represents the first derivative with respect to time t. Substitute the vertical and torsional displacement expressions, i.e., formulas (4) and (5), into the vertical and torsional dynamic response equations, i.e., formulas (9) and (10), and multiply both sides of the equations by the vertical and torsional modes, respectively. and Then, x is integrated from 0 to L, assuming the vehicle mass mv is much smaller than the mass m per unit length of the main beam, and the calculation is simplified to obtain: (12) (13) In the formula, (··) represents the second derivative with respect to time t; Using the Duhamel integral and substituting the initial conditions of zero displacement and zero velocity into formulas (12) and (13), the vertical generalized coordinate qu and the torque generalized coordinate qθ at time t are obtained: (14) (15) in: (16) Where Ω is the speed-related driving frequency. These are the velocity parameters for the vertical and torsional modes, respectively. These are the vertical frequency and the torsional frequency, respectively. Multiply the above formulas (14) and (15) by the corresponding generalized coordinates respectively. The expressions for the vertical displacement and torsional angular displacement at time t and position x are as follows: (17) (18) Based on the assumption of a rigid section in the plane, the response of the main beam in the XZ plane at time t is obtained. The expression is as follows: (19) z is the Z-axis coordinate value, which also represents the coordinates of a point within the width of the bridge; Using formula (19), given the vehicle speed, mass, and travel path, the vertical displacement of any position on the bridge at any time can be calculated.

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