Analysis method for key local vibration of fluid-structure interaction bridge based on substructure method

Abstract

The application discloses an analysis method for key local vibration of fluid-solid coupling bridges based on a substructure method, and relates to the technical field of bridge vibration analysis, and comprises the following steps: dividing a response key area in a bridge structure into substructures, performing static condensation, and obtaining a condensed substructure model; establishing a whole bridge-wave coupling model, and adopting a dynamic explicit solving strategy based on a coupling Euler-Lagrange method; independently solving the substructures by using a dynamic implicit method; coupling the substructures to the whole model by an interface coordination condition, and jointly calculating the whole explicit model and the local implicit model. The analysis method adopts an explicit method to process a strong nonlinear fluid-solid coupling process at a whole level, and establishes a fine substructure model through Guyan static condensation at a local area, so that the analysis precision is effectively improved.

Description

Technical Field

[0001] This invention relates to the field of bridge vibration analysis technology, and in particular to an analysis method based on the substructure method for critical local vibrations of fluid-structure coupled bridges. Background Technology

[0002] The dynamic response analysis of long-span bridges under wave loads is a nonlinear dynamic problem involving complex fluid-structure interaction. The displacement response characteristics of the mid-span region are crucial for assessing structural safety and service performance. Traditional numerical simulation methods based on single algorithms often struggle to achieve an ideal balance between overall response and detailed local analysis, particularly in capturing the complex dynamic behavior of critical regions. Summary of the Invention

[0003] This invention provides an analysis method for critical local vibrations of fluid-structure interaction bridges based on the substructure method. By integrating the substructure method with a dynamic explicit-implicit joint solution strategy, it achieves high-precision analysis of the dynamic response of critical local areas of the bridge.

[0004] To solve the above-mentioned technical problems, the technical solution proposed by this invention is as follows:

[0005] An analysis method for critical local vibrations of fluid-structure interaction bridges based on the substructure method includes the following steps:

[0006] The critical response region in the bridge structure is divided into substructures, and static condensation is performed to obtain the condensed substructure model.

[0007] A holistic bridge-wave coupled model was established, and a dynamic explicit solution strategy based on the coupled Eulerian-Lagrange method was adopted.

[0008] The substructures are solved independently using a dynamic implicit method;

[0009] By coordinating interface conditions, substructures are coupled into the overall model, enabling joint computation of the overall explicit model and the local implicit model.

[0010] A further improvement to the above technical solution is as follows:

[0011] Preferably, the degrees of freedom of the sub-nodes in the substructure are divided into two categories: boundary degrees of freedom and internal degrees of freedom. The boundary degrees of freedom are the degrees of freedom of nodes connected to the main structure, and the internal degrees of freedom are the degrees of freedom of nodes inside the substructure that are not directly connected to the main structure.

[0012] Preferably, the static condensation is performed using the Guyan static condensation method, which condenses the internal degrees of freedom of the substructure to the boundary degrees of freedom. The equation of the condensed substructure is as follows:

[0013]

[0014] in, For the boundary degrees of freedom, The velocity vector represents the boundary degrees of freedom. Let be the acceleration vector of the boundary degrees of freedom. , , The mass matrix, damping matrix, and stiffness matrix after condensation are given. This represents the equivalent external force vector acting on the boundary of the substructure after condensation.

[0015] Preferably, the implicit dynamic method uses the Newmark-β method for time integration.

[0016] Preferably, the interface coordination condition includes interface displacement coordination and force balance condition. The interface displacement coordination condition is that the displacement of the substructure boundary node is consistent with the displacement of the corresponding interface node of the overall model. The force balance condition is that the reaction force of the substructure boundary and the interface force applied by the overall model satisfy the action-reaction relationship.

[0017] Preferably, the joint computation involves data exchange and synchronization between the overall explicit model and the local implicit substructure at each time step, and execution of force balance and displacement coordination conditions at the coupling interface.

[0018] Preferably, the Newton-Raphson iterative method and the Newmark-β method are combined to solve the nonlinear dynamic equations.

[0019] Preferably, the combination process of the Newton-Raphson iterative method and the Newmark-β method is as follows:

[0020] First, an initialization phase is performed. At each time step, the predicted values ​​of displacement, velocity, and acceleration are initialized as the initial guesses for the Newton-Raphson iterative method.

[0021] Within each time step, the solution is iteratively obtained using the Newton-Raphson iterative method until convergence.

[0022] For the implicit solution of the substructure, after the Newton-Raphson iterative method converges, the Newmark-β method is used to update the displacement, velocity and acceleration.

[0023] Back-substitute the acceleration of the internal nodes of the substructure and update the external forces;

[0024] If the residual of the nonlinear equation system is less than the set tolerance, it is considered to have converged and proceeds to the next time step; otherwise, it returns to iteration until convergence.

[0025] Preferably, the wave load of the bridge is calculated based on Stokes wave theory, including linear and nonlinear terms.

[0026] The substructure-based analysis method for critical local vibrations of fluid-structure interaction bridges provided by this invention has the following advantages compared with existing technologies:

[0027] (1) The substructure method of the present invention is used to analyze the key local vibration of fluid-structure coupled bridges. It proposes an efficient joint solution framework. By deeply integrating the substructure method with the CEL framework, high-precision analysis of the key area in the middle of the span is achieved. The framework adopts an explicit method to handle the strong nonlinear fluid-structure coupled process at the overall level, and establishes a fine substructure model through Guyan static condensation in the local area, which effectively improves the analysis accuracy.

[0028] (2) The substructure method of the present invention is used to analyze the key local vibration of fluid-structure coupled bridges. It adopts a strategy that combines the Newton-Raphson iterative method with the Newmark-β implicit integral, which effectively handles the nonlinear problem of the system and ensures the numerical stability and convergence efficiency of the solution process. Attached Figure Description

[0029] Figure 1 This is a numerical verification diagram of wave load for the present invention.

[0030] Figure 2 This is a schematic diagram of the coupling between the substructure and the main structure of the present invention.

[0031] Figure 3 This is a schematic diagram of the finite element model and structural dimensions of the bridge of the present invention.

[0032] Figure 4 This is a coarse mesh diagram of the main structure of the substructure model used for experimental verification of this invention.

[0033] Figure 5 This is a fine mesh diagram of the substructure model used for experimental verification of the present invention.

[0034] Figure 6 The wave-bridge fluid-structure interaction overall finite element model is used for experimental verification of this invention.

[0035] Figure 7 Comparison of mid-span lateral displacement-time curves when the wave velocity is 1 m / s during the experimental verification of this invention.

[0036] Figure 8 Comparison of mid-span lateral displacement-time curves during experimental verification of this invention at a wave velocity of 3 m / s.

[0037] Figure 9Comparison of mid-span lateral displacement-time curves during experimental verification of this invention at a wave velocity of 6 m / s.

[0038] Figure 10 Comparison of mid-span lateral acceleration-time curves for wave velocity of 1 m / s during experimental verification of this invention.

[0039] Figure 11 Comparison of mid-span lateral acceleration-time curves for wave velocity of 3 m / s during experimental verification of this invention.

[0040] Figure 12 Comparison of mid-span lateral acceleration-time curves for a wave velocity of 6 m / s during experimental verification of this invention.

[0041] Figure 13 (a) is a scatter plot of the simulated and measured values ​​of the mid-span lateral displacement of the coarse grid model when the wave speed is 1 m / s during the experimental verification of this invention.

[0042] Figure 13 (b) is a scatter plot of the simulated and measured values ​​of the mid-span lateral displacement of the fine mesh model when the wave speed is 1 m / s during the experimental verification of this invention.

[0043] Figure 13 (c) is a scatter plot of the simulated and measured values ​​of the mid-span lateral displacement of the substructure model when the wave speed is 1 m / s during the experimental verification of this invention.

[0044] Figure 14 (a) is a scatter plot of the simulated and measured values ​​of the mid-span lateral displacement of the coarse mesh model when the wave speed is 3 m / s during the experimental verification of this invention.

[0045] Figure 14 (b) is a scatter plot of the simulated and measured values ​​of the mid-span lateral displacement of the fine mesh model when the wave speed is 3 m / s during the experimental verification of this invention.

[0046] Figure 14 (c) is a scatter plot of the simulated and measured values ​​of the mid-span lateral displacement of the substructure model when the wave speed is 3 m / s during the experimental verification of this invention.

[0047] Figure 15 (a) is a scatter plot of the simulated and measured values ​​of the mid-span lateral displacement of the coarse mesh model when the wave speed is 6 m / s during the experimental verification of this invention.

[0048] Figure 15 (b) is a scatter plot of the simulated and measured values ​​of the mid-span lateral displacement of the fine mesh model when the wave speed is 6 m / s during the experimental verification of this invention.

[0049] Figure 15(c) is a scatter plot of the simulated and measured values ​​of the mid-span lateral displacement of the substructure model when the wave speed is 6 m / s during the experimental verification of this invention.

[0050] Figure 16 (a) is a radar chart of the robustness evaluation index of each model when the wave speed is 1 m / s during the experimental verification of this invention.

[0051] Figure 16 (b) is a radar chart of the robustness evaluation index of each model when the wave speed is 3m / s during the experimental verification of the present invention.

[0052] Figure 16 (c) is a radar chart of the robustness evaluation index of each model when the wave speed is 6m / s during the experimental verification of this invention. Detailed Implementation

[0053] The following provides a detailed description of specific embodiments of the present invention. It should be understood that the specific embodiments described herein are for illustrative and explanatory purposes only and are not intended to limit the scope of the invention.

[0054] The substructure method, as an effective numerical calculation strategy, provides a new approach to improving local calculation accuracy by decoupling and analyzing response-sensitive regions in a structural system. Meanwhile, the coupled Eulerian-Lagrange method exhibits unique advantages in handling fluid-structure interaction problems with large deformations. This invention's vibration detection method deeply integrates these two methods, establishing a completely new computational framework. The vibration detection method proposes a dynamic explicit-implicit joint solution strategy, achieving a significant breakthrough at the algorithmic level. In the overall analysis, explicit integration based on the CEL method is used to handle the strong nonlinear interaction between waves and the structure, while for the critical mid-span region, a refined substructure model is established through Guyan static condensation and solved independently using an implicit method. The two types of algorithms are organically integrated through strict interface coordination conditions, fully leveraging the advantages of explicit algorithms in handling complex fluid dynamics while retaining the high accuracy of implicit algorithms in calculating structural dynamic responses.

[0055] The Coupled Eulerian-Lagrangian (CEL) method, as an advanced numerical computation framework, aims to efficiently solve strongly nonlinear problems involving large deformation and material flow in fluid-structure interaction (FSI) systems.

[0056] In the Lagrange description, the computational mesh is consolidated with the material and moves synchronously, allowing for precise tracking of the structural boundary deformation process. Its governing equations are:

[0057] (1)

[0058] In the formula: The structural mass matrix; Here is the structural damping matrix; Here is the structural stiffness matrix; Let be the displacement vector of the structure; External forces acting on the structure.

[0059] In the Euler description, the computational mesh remains fixed in space, while fluid material flows between mesh cells, effectively avoiding mesh distortion caused by large material deformation. Fluid motion is governed by the Navier-Stokes equations:

[0060] (2)

[0061] In the formula: For fluid velocity field, it represents the velocity of the fluid at each point in space; For time; Density; For fluid pressure; For gravity; This refers to kinematic viscosity.

[0062] The two physical domains are coupled using an Eulerian-Lagrange contact algorithm, and kinematic coordination and dynamic equilibrium conditions must be met at the interface. The material distribution is tracked in a fixed mesh by introducing the Euler volume fraction (EVF, with values ​​ranging from 0 to 1).

[0063] The CEL method typically employs explicit time integration schemes (such as the central difference method) to solve highly nonlinear coupled systems. The numerical stability of this scheme is dependent on the critical time step (Δt). crit The constraint is calculated using the following formula:

[0064] (3)

[0065] (4)

[0066] In the formula, The element length is given, and the dilatation wave velocity is given. , For Young's modulus, Let ν be the density and ν be the Poisson's ratio of the material.

[0067] The present invention provides an analysis method for critical local vibrations of fluid-structure interaction bridges based on the substructure method, which specifically includes the following steps:

[0068] Step S1, wave numerical simulation.

[0069] S1-1, Derivation of Wave Load

[0070] For wave forces acting on small-scale piles, their effect on the structure mainly consists of viscous effects and added mass. The wave force on a single pile can be decomposed into drag force and inertial force. The expression is as follows:

[0071] (5)

[0072] In the formula, The density of water; and These are the drag force coefficient and the inertia force coefficient, respectively; S is the projected area of ​​the structure per unit height perpendicular to the direction of wave travel; Volume per unit height of the column; and These represent the velocity and acceleration of the water particles, respectively.

[0073] Taking a circular cross-section beam as an example, its two ends are rigidly fixed and it is subjected to flow velocity. A uniform flow field with a velocity of 1 m / s acts on the bridge pier. The pier's outer diameter is D = 0.03 m, length l = 10 m, elastic modulus E = 210 GPa, and Poisson's ratio v = 0.3. (The coordinate system is defined as: x is horizontal, y is vertical, and y = 0 at the water surface). The drag coefficient is used in the calculation. =0.7.

[0074] Based on theoretical calculations, the line load (load per unit length) for:

[0075] (6)

[0076] In the formula, This represents the drag force per unit length.

[0077] The reaction force of the main support is:

[0078] (7)

[0079] The maximum disturbance at the midpoint of the beam is:

[0080] (8)

[0081] In the formula, This indicates the vertical displacement of the beam. Indicates the total load. This indicates the bending stiffness of the beam.

[0082] Simulated by ABAQUS (finite element analysis software) and Consistent with the theoretical value. Figure 1 The deflection curves from numerical simulation and theoretical derivation were compared (x in the figure is the horizontal coordinate along the bridge axis, and z is the coordinate perpendicular to the water surface). The results show that the two are in excellent agreement, which verifies the accuracy of the flow field definition and hydrodynamic calculation in the software.

[0083] S1-2, Wave Force Derivation

[0084] According to the basic principles of fluid-structure interaction, wave force is generated by the pressure of the fluid on the surface of the structure, and can be obtained by integrating the pressure on the surface. Assuming the superstructure of the bridge (e.g., the bridge deck) coincides with the surface where the waves act, the expression for the wave force when fluid forces act on the surface of the structure is:

[0085] (9)

[0086] In the formula, For time Wave force represents the force exerted by the fluid on the structure; Let be the surface area of ​​the structure in contact with the fluid, assumed to be a region on the xy plane; For position and time Wave pressure; The formula, which calculates the total wave force by integrating the pressure on the structural surface, is applicable to wave force calculations for large-scale structures such as bridge superstructures. denoted as a small surface area element; x is the coordinate vector of a point on the surface.

[0087] Wave pressure It can be calculated using either linear or nonlinear wave theory, depending on the complexity of the waves. In linear wave theory, the expression for wave pressure is:

[0088] (10)

[0089] In the formula: The density of water; It is the acceleration due to gravity; This represents the wavefront displacement.

[0090] Stokes wave theory, as a nonlinear theory describing surface waves, is suitable for characterizing non-steep waves. It considers wave nonlinear effects, allowing for a more accurate description of the dynamic characteristics of high-altitude waves. When considering large or nonlinear waves, the nonlinear effects in wave force calculations need to be corrected; in this case, the wave pressure expression includes nonlinear terms.

[0091] (11)

[0092] In the formula: The nonlinear coefficients are based on Stokes' second-order wave theory and are determined according to the actual wave conditions. The wave displacement term is to consider the second-order nonlinear effect.

[0093] S1-3, Derivation of Wave Displacement

[0094] Stokes waves, as a type of finite amplitude wave theory, have the following characteristics: the wave centerline has superelevation relative to the still water surface; in addition to periodic horizontal and vertical components, the motion of water particles also exhibits static horizontal movement along the wave direction. Wave surface displacement. The linear part is usually a simple harmonic sine wave, and the expression for the first-order wave displacement is:

[0095] (12)

[0096] In the formula: The amplitude of the wave (half the wave height); For wave number, Wavelength; Angular frequency, For periodicity; and These represent the horizontal position and the time, respectively. This is a linear term for wave displacement, describing the basic vibration of waves.

[0097] The second-order wave displacement formula in Stokes theory introduces nonlinear effects, especially as wave amplitude increases. The second-order wave displacement term... The expression is:

[0098] (13)

[0099] Combining the first-order (linear) and second-order (nonlinear) wave displacement terms, the total wavefront displacement in Stokes wave theory The expression is:

[0100] (14)

[0101] In the formula: This is the linear part, describing the fundamental wave; This is the nonlinear part, representing the secondary effect of the wave due to the increase in wave amplitude, mainly manifested in the wave crests becoming sharper.

[0102] Step S2: Establish the coupled system based on the substructure method.

[0103] S2-1, Substructure Division and Degree of Freedom Condensation

[0104] First, the mid-span region, which has the most critical response in the bridge structure, is defined as a substructure, and its sub-node degrees of freedom are divided into two categories:

[0105] (1) Boundary degrees of freedom Degrees of freedom of nodes connected to the main structure.

[0106] (2) Internal degrees of freedom Degrees of freedom of nodes within a substructure that are not directly connected to the main structure.

[0107] Based on this, the stiffness matrix of the substructure The following blocks can be used:

[0108] (15)

[0109] In the formula, This represents the stiffness submatrix between boundary degrees of freedom. This represents the coupling stiffness submatrix between the boundary degrees of freedom and the interior degrees of freedom. This represents the coupling stiffness submatrix between the internal degrees of freedom and the boundary degrees of freedom. This represents the stiffness submatrix between internal degrees of freedom.

[0110] Assuming that the inertial force within the substructure is much smaller than its elastic restoring force in the dynamic response (this assumption is particularly reasonable in low-frequency vibrations), Guyan static condensation can be used to reduce the internal degrees of freedom. Condensed into boundary degrees of freedom through static relations superior:

[0111] (16)

[0112] Among them, the static condensation transformation matrix is:

[0113] (17)

[0114] In the formula: For the internal stiffness submatrix of the substructure; For the internal boundary stiffness submatrix and damping matrix of the substructure.

[0115] S2-2, Substructure equation after condensation

[0116] Wave-bridge coupled models typically include bridge vibration equations and wave vibration equations, which interact through contact forces. The coupled dynamic equations are expressed as follows:

[0117] (18)

[0118] In the formula: This is the mass matrix of the bridge, representing the mass distribution of each part of the bridge. Here is the damping matrix of the bridge, representing the damping effect of the bridge structure on vibration; This is the stiffness matrix of the bridge, representing the bridge structure's resistance to deformation. Let be the displacement vector of the bridge. Let V be the velocity vector of the bridge. Let be the acceleration vector of the bridge; The wave force is derived from the wave pressure effect of the CEL method.

[0119] The substructure is divided into:

[0120] (19)

[0121] in, These are the boundary degrees of freedom, i.e., the displacements of the boundary nodes. For internal degrees of freedom, i.e., internal node displacements, and for N, the number of substructures.

[0122] Using Guyan static polycondensation:

[0123] (20)

[0124] The inertial forces at the pseudo-internal nodes are negligible. This equation holds true only when the inertial forces at the internal nodes are much smaller than the elastic forces (applicable to low-frequency vibrations); otherwise, dynamic shrinkage (such as the Craig-Bampton method) is required. For the internal stiffness submatrix of the substructure, These are the stiffness submatrices of the substructure's interior and boundary, respectively.

[0125] The substructure equation after condensation is:

[0126] (twenty one)

[0127] in, The velocity vector represents the boundary degrees of freedom. Let be the acceleration vector of the boundary degrees of freedom. , , The mass matrix, damping matrix, and stiffness matrix after condensation are given. This represents the equivalent external force vector acting on the boundary of the substructure after condensation.

[0128] S2-3, System Coupling Assembly

[0129] like Figure 2As shown, the substructures are embedded into the main structure: In the final global model, the main structure (the part of the bridge excluding the mid-span) is modeled using a relatively coarse mesh. Then, the aforementioned condensed substructures are "assembled" to the corresponding nodes of the main structure through their boundary nodes, constructing a hybrid model. In this model, the main structure is responsible for rapidly calculating the overall response, while the refined mid-span response is calculated with high precision by solving the equations of the condensed substructures, ultimately achieving a balance between computational efficiency and local accuracy.

[0130] Step S3: Application of explicit and implicit power.

[0131] S3-1, Explicit solution of the dynamics of the overall model

[0132] For fluid-structure interaction systems involving wave generation, propagation, breaking, and their interaction with the overall structure, a dynamic explicit method based on the CEL framework is used for solution. The solution equations are in the form of:

[0133] (twenty two)

[0134] The superscript “exp” indicates explicit analysis, corresponding to the explicit time integration scheme of the CEL method; These are the explicit mass matrix, damping matrix, and stiffness matrix of the overall model, respectively. These are the explicit displacement, velocity, and acceleration vectors of the overall model, respectively. The explicit wave force experienced by the overall model.

[0135] S3-2, Implicit Dynamic Solution of Substructure Model

[0136] For the substructure representing the critical region across the middle and which has undergone condensation, its dynamic behavior is governed by the dimension-reduced equations, which are solved independently using a dynamic implicit method:

[0137] (twenty three)

[0138] The superscript "imp" indicates implicit analysis, corresponding to the static condensation and implicit solution logic of the substructure part. These are the implicit mass matrix, damping matrix, and stiffness matrix of the substructure, respectively. The implicit external forces acting on the substructure.

[0139] Implicit methods (such as the Newmark-β method) have unconditional stability, allow for time steps much larger than those of explicit methods, and offer higher numerical accuracy in solving structural dynamic responses. They also have controllable numerical damping, effectively filter out high-frequency numerical noise, and obtain smoother and more accurate displacement and stress time history results.

[0140] S3-3, Force-Displacement Coupling Condition

[0141] The key to achieving joint solution lies in realizing data exchange and synchronization between the global explicit model and the local implicit substructure at each time step, which is achieved by forcibly enforcing force balance and displacement compatibility conditions on the coupling interface:

[0142] The displacements of the substructure boundary nodes must be consistent with the displacements of the corresponding coupling interface nodes in the overall model, expressed as:

[0143] (twenty four)

[0144] In the formula: For implicit displacements of the substructure boundary nodes; For the explicit displacement of the interface nodes coupled to the overall model; "" indicates the coupling interface between the substructure and the main beam.

[0145] The forces applied to the boundaries of the substructure by the overall model and the boundary reactions calculated from the substructure must satisfy the law of action and reaction forces, expressed as follows:

[0146] (25)

[0147] In the formula: The implicit reaction force at the boundary of the substructure; Explicit forces are applied to the coupling interface of the overall model to ensure that the implicit analysis results of the substructure are compatible with the overall explicit model at the interface.

[0148] Step S4: The Newton-Raphson method combined with the Newmark β method

[0149] The Newton-Raphson iterative method is a classic iterative method for solving systems of nonlinear algebraic equations. Its core idea is to linearize the current nonlinear equation through Taylor expansion, gradually approximating the solution. For systems of nonlinear equations:

[0150] (26)

[0151] in, It is related to displacement The relevant nonlinear force vector. The basic iterative formula of the Newton-Raphson iterative method is:

[0152] (27)

[0153] in: This is the displacement vector for the current iteration; This is the displacement vector for the next iteration; It is a system of nonlinear equations; Let be the Jacobian matrix of the k-th iteration, and represent the system stiffness matrix.

[0154] The formula for the residual force vector is:

[0155] (28)

[0156] in: It is the residual vector; For the acceleration at the next time step; For the velocity of the next time step, It is a nonlinear internal force vector; It is an external force vector (including random traffic loads, etc.). This is the quality matrix; Here is the damping matrix.

[0157] In each iteration, the Newton-Raphson iterative method updates the displacement by calculating the tangent stiffness (Jacobian matrix) of the current displacement vector, thereby gradually approximating the solution of the nonlinear equation.

[0158] The displacement correction formula is:

[0159] (29)

[0160] in, This is the displacement correction amount for the (k+1)th iteration; Let be the Jacobian matrix for the k-th iteration; Let be the residual vector of the k-th iteration.

[0161] The formula for the Jacobian matrix is:

[0162] (30)

[0163] in, The effective stiffness matrix for the k-th iteration is shown in the following equation:

[0164] (31)

[0165] In the formula, Indicates displacement The nonlinear internal force vector under the following conditions This represents the displacement at the next time step.

[0166] The Newmark-β method is a commonly used implicit time integration method in structural dynamics problems. It is suitable for dynamic response analysis of long-span bridges and other structures, can handle nonlinear problems, and exhibits excellent computational efficiency and stability. Its basic equations are the time-discrete form of the structural dynamics equations:

[0167] (32)

[0168] in: This is the quality matrix; Here is the damping matrix; Here, u is the stiffness matrix; u is the displacement vector. It is the velocity vector; It is the acceleration vector; This refers to the external load vector (including wave force, traffic load, etc.).

[0169] The core of the Newmark-β method is to obtain the structural response at each time step by integrating acceleration, velocity, and displacement over time, assuming that at time... Given the acceleration, velocity, and displacement of the structure, solve for the time interval. The response.

[0170] The solution process combining the Newton-Raphson iterative method and the Newmark-β method is as follows:

[0171] S4-1, Initialization phase, setting the time step. Total calculation time Number of times per time step as follows:

[0172] (33)

[0173] Define the initial displacement of the structure ,speed and acceleration :

[0174] (34)

[0175] In the formula, initial represents the quantity at the initial time (t=0).

[0176] Initialize the mass matrix of the structure Stiffness matrix Damping matrix Define nonlinear force and external loads .

[0177] For each time step, calculate the nonlinear force at the current displacement and velocity. Calculate the stiffness matrix of the system (including linear and nonlinear components):

[0178] (35)

[0179] in, It includes the linear stiffness matrix and the additional stiffness caused by nonlinear factors. This represents the total internal force vector of the system.

[0180] At each time step, initialize the predicted values ​​for displacement, velocity, and acceleration (as initial guesses for the Newton-Raphson iteration):

[0181] (36)

[0182] (37)

[0183] (38)

[0184] In the formula: the superscript "*" indicates the predicted value. This represents the predicted displacement at the next time step. This represents the displacement vector at the current time step. This represents the velocity vector at the current time step. This represents the acceleration vector at the current time step. This represents the predicted displacement at the next time step. This represents the predicted value of the velocity at the next time step.

[0185] S4-2, within each time step, is solved iteratively using the Newton-Raphson iterative method until convergence. The process for each iteration is as follows:

[0186] Calculate the current displacement External loads below and nonlinear forces Then calculate the total external force:

[0187] (39)

[0188] Calculate the current displacement The corresponding current stiffness matrix It typically consists of two parts: a linear stiffness matrix and a nonlinear stiffness matrix.

[0189] Update the nonlinear force using the current displacement and predicted velocity and acceleration. and stiffness matrix .

[0190] Solve the Newton-Raphson linearized equations and iteratively update the displacements using the following formula. :

[0191] (40)

[0192] Calculate the updated displacement ,until If it is small enough, it means that it has converged.

[0193] S4-3, the implicit solution for the substructure is as follows:

[0194] (41)

[0195] Once the Newton-Raphson iterative method converges, the Newmark-β method can be used to update the displacement, velocity, and acceleration. Assuming the current displacement, velocity, and acceleration are known, the following formulas are used to update the displacement, velocity, and acceleration at the next time step:

[0196] The displacement is updated as follows:

[0197] (42)

[0198] in, For the displacement at the next time step; This represents the displacement at the current time step. The velocity at the current time step; The acceleration at the current time step; For the acceleration at the next time step; For time step; This is a parameter for the Newmark method, typically set to 0.25 to ensure numerical stability.

[0199] (43)

[0200] In this context, the superscript (i) represents the i-th substructure, and the subscript b explicitly represents the boundary node of the substructure, thus reflecting that the formula is used for the displacement update of the boundary node in the implicit solution of the substructure.

[0201] The speed update is as follows:

[0202] (44)

[0203] In the formula, The velocity for the next time step; This is another parameter of the Newmark method, usually set to 0.5 to ensure numerical stability.

[0204] (45)

[0205] In the formula, Indicates the first Substructure at time step The boundary node velocity vector, Indicates the first Substructure at time step The boundary node velocity vector, Indicates the first Substructure at time step The boundary node acceleration vector, Indicates the first Substructure at time step The boundary node acceleration vector.

[0206] This clarifies that the formula is applicable to the implicit velocity update of substructure boundary nodes.

[0207] Acceleration updates are as follows:

[0208] (46)

[0209] S4-4, the acceleration of internal nodes of the substructure needs to be back-substituted:

[0210] (47)

[0211] Wherein, the superscript (i) represents the i-th substructure, indicating that the formula is used to calculate the internal node acceleration in the implicit solution of the substructure; Indicates the first The acceleration vector of each node inside the substructure Indicates the first The inverse matrix of the internal stiffness matrix of each substructure Indicates the action in the External force vectors at the internal nodes of each substructure Indicates the first The coupling stiffness submatrix between the interior and boundary of each substructure Indicates the first The acceleration vector of each substructure boundary node.

[0212] The external force is updated as follows:

[0213] (48)

[0214] In the formula, Indicates at time step The system's total external force vector, Indicates time The external load vector, Indicates displacement The nonlinear internal force vector.

[0215] If the residuals of the nonlinear equation system If the error is less than the set tolerance, the computation is considered to have converged, and the process proceeds to the next time step. Otherwise, the process returns to iterate until convergence.

[0216] Step S5: Perform Abaqus numerical modeling for the key local vibrations of long-span bridges.

[0217] S5-1, taking a single-tower cable-stayed bridge as the engineering background, a three-dimensional numerical model of the entire bridge was established based on ABAQUS. Figure 3 The bridge is a single-tower, double-cable-stayed, steel-concrete composite beam structure with a span arrangement of 33 + 102 + 183 = 318m and a deck width of 44.5m. The substructure has four piers, with N03 as the main pier and N01, N02, and N04 as auxiliary piers; each pier consists of a concrete pier body, a round-ended abutment, and 2.0m diameter bored piles. The main girder adopts a hybrid box girder system: the main span is a steel box girder, and the side spans are prestressed concrete box girders to balance the dead load on both sides of the bridge tower. The bridge tower is an arched structure with a total height of 109.5m. The tower columns have rectangular sections, with a 5m thick box section at the top. The total width at the bottom of the bridge is 60.3m. Material parameters are detailed in Table 1.

[0218] Table 1 Material parameters of bridge components

[0219]

[0220] S5-2, Mesh Generation

[0221] To accurately capture the dynamic characteristics of the mid-span region of the bridge (the critical region with the most significant displacement response) under wave load, the mid-span region of the main girder is divided using a fine grid, while other regions are divided using a coarse grid.

[0222] S5-2-1, Substructure Model

[0223] To achieve a balance between computational efficiency and simulation accuracy, this invention innovatively introduces the substructure method. The core of this method lies in decoupling the response-sensitive mid-span region from the global model, establishing independent, refined substructures, such as... Figure 4 and Figure 5 The Guyan static condensation technique is used to condense the internal degrees of freedom of the substructure to the boundary nodes, generating a condensation matrix that retains complete mechanical properties. A coarser mesh is used in the main structure model to control the computational scale, and the condensed substructure is coupled to the main structure through boundary nodes (e.g., ...). Figure 6 As shown in the figure, this effectively improves computational efficiency while ensuring high-precision simulation of key areas.

[0224] Experimental verification

[0225] To verify the accuracy of the substructure coupling model in predicting the vibration response of key bridge components under wave loads, field measurements were conducted for this experimental verification. High-precision strain gauges were directly installed on the bridge deck in the mid-span region. A synchronous data acquisition system was used to record the dynamic strain time histories of the bridge under wave velocities of 1 m / s², 3 m / s², and 6 m / s². This measured data will serve as a benchmark to verify the mid-span displacement and acceleration response obtained from numerical simulations, thereby quantitatively evaluating the accuracy and reliability of the model in wave-bridge coupled vibration analysis.

[0226] (1) Comparison of displacement-time curves

[0227] Figures 7 to 9 The results show a comparison of the lateral displacement-time curves at the mid-span of the bridge under three wave velocities: 1 m / s, 3 m / s, and 6 m / s. The curve shapes clearly show that the simulation results of the coarse model differ significantly from the actual situation, while the response curves of the fine model and the substructure model are closer to the baseline, with the substructure model showing the best agreement.

[0228] Table 2 further quantifies the accuracy of different models using maximum error data. Under 1 m / s wave action, the substructure model's error is 0.56 mm, which is approximately 26.3% and 18.8% lower than the coarse model (0.76 mm) and the fine model (0.69 mm), respectively. As the wave speed increases to 3 m / s, the substructure model's error (0.67 mm) decreases by approximately 49.2% and 22.1% compared to the coarse model (1.32 mm) and the fine model (0.86 mm), respectively. Under the extreme condition of 6 m / s, the substructure model's error (0.97 mm) shows a more significant advantage, reducing the error by approximately 64.1% and 40.9% compared to the coarse model (2.70 mm) and the fine model (1.64 mm), respectively.

[0229] Table 2 Maximum error of mid-span displacement (mm)

[0230]

[0231] The above results show that, although the overall error of each model increases with the increase of wave velocity, the substructure model always exhibits the highest numerical accuracy. Its error is reduced by an average of about 46.5% compared with the coarse model and by an average of about 27.3% compared with the fine model, which fully demonstrates that the model has better reliability and applicability in simulating wave-bridge dynamic response.

[0232] (2) Comparison of acceleration-time curves

[0233] Figures 10 to 12The comparison of lateral acceleration-time curves at the mid-span of the bridge under three wave velocities: 1 m / s, 3 m / s, and 6 m / s is presented. The figures show significant differences in the acceleration responses of different models under various wave excitations. Taking the wave velocity of 3 m / s as an example, the response curve of the substructure model is closest to the actual measured curve in terms of phase and overall trend, especially showing good consistency in the timing of peak occurrence and waveform oscillation characteristics. This indicates that this model can more effectively capture the dynamic coupling effect of the structure under wave load. In contrast, although the response curves of the coarse-grid model and the fine-grid model are similar in trend, their overall amplitudes are significantly lower, reflecting a potential deficiency in simulating the dynamic amplification effect of waves on the structure.

[0234] Table 3 further lists the maximum mid-span acceleration (unit: m / s²) for each model under different wave velocities. Under a wave speed of 1 m / s, the actual measured acceleration is 0.137 m / s², with the coarse-grid model and fine-grid model measuring only 0.076 m / s² and 0.065 m / s², respectively, which are approximately 44.5% and 52.6% lower than the actual values. The substructure model, however, measures 0.328 m / s², exceeding the actual value by approximately 139.4%. As the wave speed increases to 3 m / s, the actual acceleration rises to 0.184 m / s², with the coarse-grid model and fine-grid model measuring 0.122 m / s² and 0.156 m / s², respectively, still approximately 33.7% and 15.2% lower than the actual values. The substructure model measures 0.292 m / s², exceeding the actual value by approximately 58.7%. Under the action of a 6 m / s wave, the actual acceleration is 0.175 m / s², while the coarse mesh model and fine mesh model are 0.085 m / s² and 0.126 m / s², respectively, which are about 51.4% and 28.0% lower than the actual values. The substructure model is 0.281 m / s², which is still about 60.6% higher than the actual value.

[0235] Table 3 Maximum Mid-Span Acceleration (m / s²) -2 )

[0236]

[0237] (3) Robustness comparison

[0238] To evaluate the simulated stability of different modeling methods under wave action, Figures 13 to 15 The correlation coefficient scatter plots are shown between the simulated mid-span lateral displacement of the bridge and the actual measured values ​​under three working conditions: wave speeds of 1 m / s, 3 m / s, and 6 m / s, using the coarse-grid model, fine-grid model, and substructure model. By calculating the coefficient of determination (R²) for each set of data, the degree of consistency between each model and the actual results can be quantified.

[0239] At a wave speed of 1 m / s, the R² of the coarse-grid model is 0.72, the fine-grid model increases to 0.85, and the substructure model reaches 0.94, with correlations improving by approximately 30.6% and 10.6% compared to the coarse and fine models, respectively. When the wave speed increases to 3 m / s, the R² of the coarse-grid model decreases to 0.65, the fine-grid model is 0.82, and the substructure model remains at 0.93, with correlations improving by approximately 43.1% and 13.4% compared to the coarse and fine models, respectively. At a higher wave speed of 6 m / s, the R² of the coarse-grid model further decreases to 0.58, the fine-grid model is 0.76, and the substructure model is 0.91, with correlations improving by approximately 56.9% and 19.7% compared to the former two, respectively.

[0240] From the distribution pattern of the scatter plot, the data points of the substructure model are most densely distributed near the diagonal at all wave velocities, especially at 3 m / s, indicating that it has excellent fitting accuracy and robustness under different load conditions. In contrast, the data point distribution of the coarse-grid model diverges significantly with increasing wave velocity, indicating that its simulation stability decreases significantly with increasing load; although the fine-grid model has a certain accuracy under low wave conditions, it still has significant deviations at high wave velocities.

[0241] In summary, the substructure model exhibits the highest correlation coefficient and strongest stability under all wave conditions. The consistency between its simulation results and actual values ​​is significantly better than that of the coarse-grid and fine-grid models, especially under high wave velocities. This demonstrates that the method has higher reliability and applicability in wave-bridge dynamic response simulation.

[0242] A multi-index comprehensive evaluation method was adopted to compare and analyze the robustness of coarse-grid, fine-grid, and substructure models in lateral displacement simulation at wave velocities of 1 m / s, 3 m / s, and 6 m / s. Evaluation indices included root mean square error (RMSE), mean absolute error (MAE), symmetric mean absolute percentage error (SMAPE), coefficient of determination (R²), and median absolute error (MedAE). Model performance was comprehensively evaluated from three dimensions: error magnitude, goodness of fit, and anti-interference capability. Figure 16 As shown.

[0243] In terms of error control, the substructure model performed best under all operating conditions. At 1 m / s, its RMSE (0.18) was 40.0% lower than the coarse-grid model (0.30); at 3 m / s, the RMSE (0.30) was 28.6% lower than the coarse-grid model (0.42); and at 6 m / s, the RMSE (0.49) was 34.7% lower than the coarse-grid model (0.75). The MAE and MedAE indices also showed significant advantages under all operating conditions, indicating that the method has good numerical stability. Regarding anti-interference capability, the substructure model had the lowest SMAPE and MedAE indices. At 1 m / s, the SMAPE (0.62) was 29.5% lower than that of the coarse mesh model (0.88); at 3 m / s, the SMAPE (0.54) was 23.9% lower than that of the coarse mesh model (0.71); and at 6 m / s, the SMAPE (0.58) was 23.7% lower than that of the coarse mesh model (0.76). The MedAE index also showed a significant improvement trend.

[0244] Comprehensive radar chart analysis shows that the substructure model maintains optimal and stable performance across all evaluation dimensions, with full polygonal contours and minimal fluctuations with load variations. This indicates that the proposed method exhibits excellent robustness under a wide range of wave excitations and is suitable for accurate simulation of bridge dynamic responses in complex marine environments.

[0245] This invention aims to improve the simulation accuracy of key local vibration responses of long-span bridges under wave action. It develops a dynamic explicit-implicit joint solution strategy based on the substructure method and verifies the superiority of the method by comparing numerical simulations with measured data.

[0246] Regarding the accuracy of displacement field simulation, comparative analysis of calculated cloud maps shows that the overall displacement field of the substructure model is highly consistent with that of the local fine model, with the maximum displacement difference not exceeding 0.81%, which is significantly better than the traditional coarse mesh model.

[0247] It exhibits excellent accuracy in key local response simulations. Under different wave speeds (1 m / s, 3 m / s, 6 m / s), the average error of the substructure model in mid-span displacement simulation is reduced by about 46.5% compared with the traditional coarse mesh model and by about 27.3% compared with the global fine mesh model.

[0248] In summary, both the coarse-grid model and the fine-grid model systematically underestimated the acceleration response of the structure under actual wave excitation across the entire wave velocity range, failing to fully reflect the actual acceleration response of the structure. While the substructure model generally had higher response amplitudes, it was closer to the actual dynamic behavior in terms of trend, showing its potential in capturing the dynamic characteristics of the structure. However, further calibration is still needed to improve its numerical accuracy.

[0249] The above embodiments are merely preferred examples of the present invention and are not intended to limit the present invention in any way. Although the present invention has been disclosed above with reference to preferred embodiments, it is not intended to limit the present invention. Therefore, any simple modifications, equivalent changes, and alterations made to the above embodiments based on the technical essence of the present invention without departing from the scope of the present invention should fall within the protection scope of the present invention.

Claims

1. A method for analyzing critical local vibrations of fluid-structure coupled bridges based on the substructure method, characterized in that, Includes the following steps: In the bridge structure, the mid-span region is the critical response region. The critical response region is divided into substructures, and static condensation is performed to obtain the condensed substructure model. A holistic bridge-wave coupled model was established, and a dynamic explicit solution strategy based on the coupled Eulerian-Lagrange method was adopted. The substructures are solved independently using a dynamic implicit method; By coordinating interface conditions, substructures are coupled into the overall model, enabling joint computation of the overall explicit model and the local implicit model.

2. The analysis method for critical local vibrations of fluid-structure coupled bridges based on the substructure method according to claim 1, characterized in that, The degrees of freedom of the sub-nodes in the substructure are divided into two categories: boundary degrees of freedom and internal degrees of freedom. The boundary degrees of freedom are the degrees of freedom of nodes connected to the main structure, and the internal degrees of freedom are the degrees of freedom of nodes inside the substructure that are not directly connected to the main structure.

3. The analysis method for critical local vibrations of fluid-structure coupled bridges based on the substructure method according to claim 2, characterized in that, The static condensation method employs the Guyan static condensation technique to condense the internal degrees of freedom of the substructure to the boundary degrees of freedom. The equations of the condensed substructure are as follows: ； in, For the boundary degrees of freedom, The velocity vector represents the boundary degrees of freedom. Let be the acceleration vector of the boundary degrees of freedom. , , The mass matrix, damping matrix, and stiffness matrix after condensation are given. This represents the equivalent external force vector acting on the boundary of the substructure after condensation.

4. The analysis method for critical local vibrations of fluid-structure coupled bridges based on the substructure method according to claim 3, characterized in that, The dynamic implicit method uses the Newmark-β method for time integration.

5. The analysis method for critical local vibrations of fluid-structure coupled bridges based on the substructure method according to claim 4, characterized in that, The interface coordination conditions include interface displacement coordination and force balance conditions. The interface displacement coordination condition is that the displacement of the substructure boundary node is consistent with the displacement of the corresponding interface node of the overall model. The force balance condition is that the reaction force of the substructure boundary and the interface force applied by the overall model satisfy the action-reaction relationship.

6. The analysis method for critical local vibrations of fluid-structure coupled bridges based on the substructure method according to claim 5, characterized in that, The joint computation involves data exchange and synchronization between the overall explicit model and the local implicit substructures at each time step, and execution of force balance and displacement coordination conditions at the coupling interface.

7. The analysis method for critical local vibrations of fluid-structure coupled bridges based on the substructure method according to claim 6, characterized in that, The nonlinear dynamic equations are solved by combining the Newton-Raphson iterative method with the Newmark-β method.

8. The analysis method for critical local vibrations of fluid-structure coupled bridges based on the substructure method according to claim 7, characterized in that, The combined process of the Newton-Raphson iterative method and the Newmark-β method is as follows: First, an initialization phase is performed. At each time step, the predicted values ​​of displacement, velocity, and acceleration are initialized as the initial guesses for the Newton-Raphson iterative method. Within each time step, the solution is iteratively obtained using the Newton-Raphson iterative method until convergence. For the implicit solution of the substructure, after the Newton-Raphson iterative method converges, the Newmark-β method is used to update the displacement, velocity and acceleration. Back-substitute the acceleration of the internal nodes of the substructure and update the external forces; If the residual of the nonlinear equation system is less than the set tolerance, it is considered to have converged and proceeds to the next time step; otherwise, it returns to iteration until convergence.

9. The analysis method for critical local vibrations of fluid-structure coupled bridges based on the substructure method according to claim 1, characterized in that, The wave load on the bridge is calculated based on Stokes wave theory, including linear and nonlinear terms.

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