A method for calculating dynamic response of a suspended tunnel in a wave environment in a frequency domain

Abstract

The present application relates to a kind of wave environment under suspended tunnel dynamic response frequency domain calculation method, belong to the field of tunnel structure.Based on classical dynamics theory, the elastic foundation beam simplification of pipe body with discrete anchor is carried out, the motion control equation of suspended tunnel is established using Hamilton principle, the wave force on suspended tunnel pipe body is calculated by potential flow theory and combined with boundary element numerical calculation method, finally the dynamic response of suspended tunnel is obtained by solving wave-solid coupling equation using modal superposition method.Based on potential flow theory, the accuracy and efficiency of the solution can be focused on simultaneously, it is suitable for large-scale structure, and the combination of elastic foundation beam model makes the calculation of wave-solid coupling not consume too much computing resources, comprehensive multi-parameter multi-condition analysis is carried out on the structure, the coupling effect of pipe body horizontal displacement and torsional displacement is considered, compared with the previous method, the calculation of the situation that pipe body appears larger horizontal displacement or torsional displacement will be more accurate.

Description

Technical Field

[0001] This invention belongs to the field of tunnel structures and relates to a frequency domain calculation method for the dynamic response of a suspended tunnel under wave conditions. Background Technology

[0002] Submerged Floating Tunnels (SFTs) mainly consist of a tube body, anchor cables, foundations, and revetment sections. They possess advantages such as large span capacity, adaptability to deep-water construction, all-weather operation, and good environmental and economic benefits, making them considered one of the most challenging and promising new transportation structures of the 21st century. However, there are currently no real-world examples of submerged tunnel engineering projects, and concerns remain regarding their safety and comfort in complex marine environments. Wave loads are the most significant environmental dynamic load experienced by submerged tunnels, and the complexity and difficulty of calculating the wave-structure interaction dynamic response of submerged tunnels have become key obstacles to the technological development of this technology. Solving this problem requires efficient and accurate calculation of the wave forces acting on the moving tube body, while simultaneously solving for the tube body's motion under the influence of these wave forces. Currently, the empirical formula Morison equation is commonly used for wave force calculation; however, the Morison equation has relatively low accuracy for large-scale structures like submerged tunnels. Using high-precision numerical calculation methods (such as Computational Fluid Dynamics (CFD)) requires a significant amount of computation time, and simultaneously solving for the overall dynamic response of the submerged tunnel structure would inevitably consume substantial computational resources.

[0003] Therefore, the above methods are insufficient to comprehensively and accurately analyze the dynamic response characteristics of suspended tunnels under wave action, making it difficult to select the structure of suspended tunnels and formulate relevant technical specifications. Thus, there is an urgent need for an efficient and accurate calculation and analysis method for the dynamic response of suspended tunnels under wave conditions. Summary of the Invention

[0004] In view of this, the purpose of this invention is to provide a frequency domain calculation method for the dynamic response of a suspended tunnel in a wave environment.

[0005] To achieve the above objectives, the present invention provides the following technical solution:

[0006] A method for frequency domain calculation of the dynamic response of a suspended tunnel under wave conditions, comprising the following steps:

[0007] S1: Solve for the velocity potential function using the boundary element method;

[0008] like Figure 1As shown, the fluid region is divided into three parts: Ω1, Ω2, and Ω3. Ω3 is the computational region, and Γ1 and Γ2 represent the left and right boundary surfaces of the computational region. Their horizontal distance from the surface of the underwater structure should not be less than twice the maximum dimension of the structure. The fluid velocity potential function... The superposition is divided into the following five parts:

[0009]

[0010] in and represents the incident potential, diffraction potential, and radiation potential components generated by the unit amplitude vibration of the tube in the horizontal, vertical, and torsional directions, respectively; x, y, and z are the coordinate components of the structure in the opposite, horizontal, and vertical directions of the central axis, respectively, with the origin at the still water surface and the structure's center of mass passing through the Z-axis; v u ,v w and v θ These represent the vibration amplitudes of the tube in the horizontal, vertical, and torsional directions, respectively; the incident potential is calculated according to the following formula based on linear wave theory:

[0011]

[0012] Where i is the imaginary unit, H is the wave height, ω is the incident wave angular frequency, k is the incident wave number, d is the water depth, and g is the gravitational acceleration. All satisfy the two-dimensional Laplace equation, and their boundary value problem is expressed as:

[0013]

[0014]

[0015] in, Represents the diffraction potential within the calculation region and radiation potential components One of them, V j The value of the normal vector of the corresponding velocity potential is given by formula (4), where n is the average surface normal vector of the structure, and r is the radius vector. It represents the average surface area.

[0016]

[0017] Where, r s The distance from the center of torsion of the structure to the surface of the object is represented by γ, and the angle between the line connecting a point on the surface of the object and the center of torsion and the positive direction of the y-axis is represented by n. y ,n z These are the projections of the unit normal vector n onto the y-axis and z-axis, respectively. The boundary element method or other numerical methods are used to solve the Laplace equation boundary value problem above.

[0018] S2: Calculate the wave force on the pipe body

[0019] The distribution of hydrodynamic pressure in the fluid is obtained from the velocity potential function based on the linearized Bernoulli equation. The wave force acting on the pipe is obtained by integrating the pressure along the pipe surface, and is expressed as:

[0020]

[0021] Where ρ is the density of the fluid, f u (x), f w (x), f θ (x) are the complex domain expressions of the horizontal wave force, vertical wave force, and wave torque experienced by the structure, respectively. The line integral in formula (5) can be directly calculated. For convenience, the above formula is denoted as:

[0022]

[0023] The constant coefficients are as follows:

[0024]

[0025] S3: Calculate the dynamic response of the tube in a steady state;

[0026] The formula for calculating the beam model and anchor cable constraint reaction force on an elastic foundation is as follows:

[0027]

[0028]

[0029] Among them, F eu ,F ew ,F eθ θ represents the constraint reaction forces of the anchor cable in the horizontal, vertical, and torsional directions, respectively; u, w, and θ represent the pipe displacements in the three directions; α and β represent the tilt angle and installation angle, respectively; E c Indicates the elastic modulus of the anchor cable; A c Let represent the cross-sectional area of ​​the anchor cable, s represent the spacing between the anchor cables, and l represent the length of the anchor cable. Based on the above formulas for calculating the anchor cable constraint reaction force and the wave force, the governing equations for the pipe are given by Hamilton's principle:

[0030]

[0031] Where: E is the elastic modulus of the tube, G is the shear modulus of the tube; m0 is the mass per unit length of the tube, c u c w c θThese are the damping coefficients of the tube in the horizontal, vertical, and torsional directions, respectively; I u I w I θ Let X, Y, and y be the moments of inertia of the tube about the X-axis, Y-axis, and pole, respectively. According to the modal superposition method, the steady vibration state of the tube is described as follows:

[0032]

[0033] Where L represents the total length of the tube, differentiating both sides of the above equation with respect to time t yields the velocity expression for the tube, and further, the velocity amplitude v at each point in the tube. u (x),v w (x),v θ The expression for (x) is then substituted into formula (10) along with formula (11) to obtain the generalized coordinates U of each modality of the tube. n W n ,Θ n The equations satisfied are:

[0034]

[0035] Solve the above set of equations to obtain the generalized coordinates of each mode, and obtain the dynamic response of the tube through formula (11).

[0036] The beneficial effects of this invention are: it can calculate the dynamic response of suspended tunnels with different cross-sectional shapes under wave loads very efficiently and relatively accurately, providing a reliable means for structural safety analysis and structural design parameter optimization required in the design and construction of suspended tunnels, and also providing a powerful tool for various numerical experiments carried out in the research of suspended tunnels.

[0037] Other advantages, objectives, and features of the invention will be set forth in part in the description which follows, and in part will be apparent to those skilled in the art from the following examination, or may be learned from practice of the invention. The objectives and other advantages of the invention can be realized and obtained through the following description. Attached Figure Description

[0038] To make the objectives, technical solutions, and advantages of the present invention clearer, the preferred embodiments of the present invention will be described in detail below with reference to the accompanying drawings, wherein:

[0039] Figure 1 This is a schematic diagram of a typical suspended tunnel structure and the coordinate system of this method;

[0040] Figure 2 This is a schematic diagram of the domain division for boundary element method calculation;

[0041] Figure 3A comparison chart showing the calculation results using different modal stacking numbers and different computational region half-widths; Figure 3 (a) The maximum horizontal displacement of each point in the tube calculated for different numbers of superimposed modes and different half-widths of the computational region; Figure 3 (b) shows the calculation results for different calculation region widths;

[0042] Figure 4 A schematic diagram of the ABAQUS finite element model;

[0043] Figure 5 This is a comparison chart of the calculation results of the present invention with those of ABAQUS analysis and previous calculation methods; Figure 5 (a) represents the maximum horizontal and maximum vertical displacements at various points in the pipe; Figure 5 (b) represents the torsional displacement at various points on the tube. Detailed Implementation

[0044] The following specific examples illustrate the implementation of the present invention. Those skilled in the art can easily understand other advantages and effects of the present invention from the content disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and various details in this specification can be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that the illustrations provided in the following embodiments are only schematic representations of the basic concept of the present invention. Unless otherwise specified, the following embodiments and features can be combined with each other.

[0045] The accompanying drawings are for illustrative purposes only and are schematic diagrams, not actual pictures. They should not be construed as limiting the invention. To better illustrate the embodiments of the invention, some parts in the drawings may be omitted, enlarged, or reduced, and do not represent the actual product dimensions. It is understandable to those skilled in the art that some well-known structures and their descriptions may be omitted in the drawings.

[0046] In the accompanying drawings of the embodiments of the present invention, the same or similar reference numerals correspond to the same or similar components. In the description of the present invention, it should be understood that if terms such as "upper," "lower," "left," "right," "front," and "rear" indicate the orientation or positional relationship based on the orientation or positional relationship shown in the drawings, they are only for the convenience of describing the present invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, the terms used to describe positional relationships in the drawings are only for illustrative purposes and should not be construed as limiting the present invention. For those skilled in the art, the specific meaning of the above terms can be understood according to the specific circumstances.

[0047] Figure 1This is a schematic diagram and coordinate system of a typical suspended tunnel structure. Based on two-dimensional linear potential flow theory and combined with the boundary element numerical method, this invention calculates the stable simple harmonic vibration state of the tunnel body under wave force, i.e., a frequency domain calculation method for the dynamic response of a suspended tunnel in a wave environment, including the following steps:

[0048] S1: Solve for the velocity potential function using the boundary element method;

[0049] like Figure 1 As shown, the fluid region is divided into three parts Ω1, Ω2, and Ω3, where Ω3 is the computational region. The horizontal distance between its two boundary surfaces Γ1 and Γ2 and the surface of the underwater structure should not be less than twice the maximum size of the structure. The fluid velocity potential function... The superposition is divided into the following five parts:

[0050]

[0051] in and represents the incident potential, diffraction potential, and radiation potential components generated by the unit amplitude vibration of the tube in the horizontal, vertical, and torsional directions, respectively; x, y, and z are the coordinate components of the structure in the opposite, horizontal, and vertical directions of the central axis, respectively, with the origin at the still water surface and the structure's center of mass passing through the Z-axis; v u ,v w and v θ These represent the vibration amplitudes of the tube in the horizontal, vertical, and torsional directions, respectively; the incident potential is calculated according to the following formula based on linear wave theory:

[0052]

[0053] Where i is the imaginary unit, H is the wave height, ω is the incident wave angular frequency, k is the incident wave number, d is the water depth, and g is the gravitational acceleration. All satisfy the two-dimensional Laplace equation, and their boundary value problem is expressed as:

[0054]

[0055]

[0056] in, Represents the diffraction potential within the calculation region and radiation potential components One of the velocity potential functions, where n is the average surface normal vector of the structure per unit area, and r is the radius vector. V represents the average surface area. j V represents d V u1 V w1 V θ1One of them, whose definition and the mean surface condition of four different velocity potentials are given by formula (4):

[0057]

[0058] Where, r s The distance from the center of torsion of the structure to the surface of the object is represented by γ, and the angle between the line connecting a point on the surface of the object and the center of torsion and the positive direction of the y-axis is represented by n. y ,n z These are the projections of the unit normal vector n onto the y-axis and z-axis, respectively. The boundary element method or other numerical methods are used to solve the Laplace equation boundary value problem above.

[0059] S2: Calculate the wave force on the pipe body

[0060] The distribution of hydrodynamic pressure in the fluid is obtained from the velocity potential function based on the linearized Bernoulli equation. The wave force acting on the pipe is obtained by integrating the pressure along the pipe surface, and is expressed as:

[0061]

[0062] Where ρ is the density of the fluid, f u (x), f w (x), f θ (x) are the complex domain expressions of the horizontal wave force, vertical wave force, and wave torque experienced by the structure, respectively. The line integral in formula (5) can be directly calculated. For convenience, the above formula is denoted as:

[0063]

[0064] The constant coefficients are as follows:

[0065]

[0066] S3: Calculate the dynamic response of the tube in a steady state;

[0067] The formula for calculating the beam model and anchor cable constraint reaction force on an elastic foundation is as follows:

[0068]

[0069]

[0070] Among them, F eu ,F ew ,F eθ θ represents the constraint reaction forces of the anchor cable in the horizontal, vertical, and torsional directions, respectively; u, w, and θ represent the pipe displacements in the three directions; α and β represent the tilt angle and installation angle, respectively; E c Indicates the elastic modulus of the anchor cable; A cLet represent the cross-sectional area of ​​the anchor cable, s represent the spacing between the anchor cables, and l represent the length of the anchor cable. Based on the above formulas for calculating the anchor cable constraint reaction force and the wave force, the governing equations for the pipe are given by Hamilton's principle:

[0071]

[0072] Where: E is the elastic modulus of the tube, G is the shear modulus of the tube; m0 is the mass per unit length of the tube, c u c w c θ These are the damping coefficients of the tube in the horizontal, vertical, and torsional directions, respectively; I u I w I θ Let X, Y, and y be the moments of inertia of the tube about the X-axis, Y-axis, and pole, respectively. According to the modal superposition method, the steady vibration state of the tube is described as follows:

[0073]

[0074] Where L represents the total length of the tube, differentiating both sides of the above equation with respect to time t yields the velocity expression for the tube, and further, the velocity amplitude v at each point in the tube. u (x),v w (x),v θ The expression for (x) is then substituted into formula (10) along with formula (11) to obtain the generalized coordinates U of each modality of the tube. n W n ,Θ n The equations satisfied are:

[0075]

[0076] Solve the above set of equations to obtain the generalized coordinates of each mode, and obtain the dynamic response of the tube through formula (11).

[0077] This invention discloses a frequency domain method for calculating the dynamic response of a suspended tunnel under wave loads. This method establishes the kinematic control equations of the suspended tunnel based on classical dynamics theory, calculates the wave forces on the tunnel body using potential flow theory combined with boundary element numerical calculation, and finally obtains the dynamic response of the suspended tunnel by solving the wave-structure interaction equations using the modal superposition method. Compared with previous methods for calculating the dynamic response of suspended tunnels, this method balances accuracy and efficiency, making it suitable for multi-parameter, multi-condition dynamic response analysis of suspended tunnels. To make the objectives, technical solutions, and advantages of this invention clearer, the following detailed description is provided with reference to the accompanying drawings and examples.

[0078] Example 1

[0079] To verify the convergence and computational efficiency of this calculation method, a dynamic response analysis of a typical suspended tunnel structure under wave load was performed using this method. The model and environmental parameters are shown in Table 1. First, the boundary element method was used to solve the partial differential equation boundary value problem shown in Equation 3. In this embodiment, different computational domain widths were selected for calculation to verify the influence of the computational domain width on the calculation results. Then, the integral coefficients in Equation 5 were calculated, and the generalized coordinates of each mode were solved according to Equation 11. Finally, the dynamic response of the tunnel body was obtained according to Equation 10. In this example, different numbers of modes were selected for superposition to find the minimum number of modes required to meet the accuracy requirements, thereby improving computational efficiency. Figure 3 A comparison chart showing the calculation results using different modal stacking numbers and different computational region half-widths; Figure 3 (a) presents the maximum horizontal displacement of each point on the pipe calculated with different numbers of modal superpositions and different computational domain half-widths. Figure 3 It is easy to see from (a) that the results of superimposing the first 9 modes are basically the same as those of superimposing the first 11 modes, indicating that when using this method, only the superposition of the first 9 modes is needed to ensure the calculation accuracy. Similarly, Figure 3 (b) shows the calculation results for different calculation region widths. The calculation results for a calculation region half-width of 40m and 48m are basically the same, indicating that the accuracy of numerical calculation can be guaranteed when the calculation region width is selected to be greater than 40m. The calculation results of this example show that the method has good convergence and can guarantee calculation efficiency.

[0080] Example 2

[0081] To further verify the accuracy of the proposed method, the results of the proposed calculation method were compared with those calculated by the commercial finite element analysis software ABAQUS. Figure 4 A schematic diagram of the finite element model is provided. All parameters of the model are determined according to the data in Table 1. The wave force calculated using potential flow theory is applied to the pipe body as a distributed load, and the dynamic response of the pipe body is calculated using the modal dynamics module of ABAQUS. Furthermore, to demonstrate the superiority of the novel anchor cable constraint reaction force calculation formula in the proposed method, the calculation results of the proposed method are compared with those obtained using the traditional constraint reaction force calculation formula. Figure 5 This is a comparison chart of the calculation results of the present invention with those of ABAQUS analysis and previous calculation methods. Figure 5 (a) The maximum horizontal and maximum vertical displacements at various points in the pipe are given. Figure 5(b) The torsional displacements at various points in the tube are given. The calculation results show that the proposed method is highly consistent with the results of the finite element analysis, both for the maximum translational displacement and the torsional displacement of the tube. In contrast, the results obtained by using the traditional anchor cable constraint reaction formula show a large deviation in the horizontal and torsional displacements. This result verifies the accuracy of the proposed method and also shows that the proposed method is more suitable for situations where the horizontal or torsional displacement of the tube is large, making it very suitable for the dynamic response calculation of suspended tunnels in wave environments.

[0082] Table 1 Model parameters and environmental parameters

[0083]

[0084] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and are not intended to limit it. Although the present invention has been described in detail with reference to preferred embodiments, those skilled in the art should understand that modifications or equivalent substitutions can be made to the technical solutions of the present invention without departing from the spirit and scope of the present invention, and all such modifications or substitutions should be covered within the scope of the claims of the present invention.

Claims

1. A method for frequency domain calculation of the dynamic response of a suspended tunnel under wave conditions, characterized in that: The method includes the following steps: S1: Solve for the velocity potential function using the boundary element method; The fluid region is divided into three parts: Ω1, Ω2, and Ω3. Ω3 is the computational region, and Γ1 and Γ2 represent the left and right boundary surfaces of the computational region, with their horizontal distances from the surface of the underwater structure not less than twice the maximum dimension of the structure. The fluid velocity potential function is... The superposition is divided into the following five parts: in and represents the incident potential, diffraction potential, and radiation potential components generated by the unit amplitude vibration of the tube in the horizontal, vertical, and torsional directions, respectively; x, y, and z are the coordinate components of the structure in the opposite, horizontal, and vertical directions of the central axis, respectively, with the origin at the still water surface and the structure's center of mass passing through the Z-axis; v u ,v w and v θ These represent the vibration amplitudes of the tube in the horizontal, vertical, and torsional directions, respectively; the incident potential is calculated according to the following formula based on linear wave theory: Where i is the imaginary unit, H is the wave height, ω is the incident wave angular frequency, k is the incident wave number, d is the water depth, and g is the gravitational acceleration. All satisfy the two-dimensional Laplace equation, and their boundary value problem is expressed as: in, Represents the diffraction potential within the calculation region and radiation potential components One of them, V j The value of the normal vector of the corresponding velocity potential is given by formula (4), where n is the average surface normal vector of the structure, and r is the radius vector. Indicates the average surface area; Where, r s The distance from the center of torsion of the structure to the surface of the object is represented by γ, and the angle between the line connecting a point on the surface of the object and the center of torsion and the positive direction of the y-axis is represented by n. y ,n z These are the projections of the unit normal vector n onto the y-axis and z-axis, respectively. The boundary element method or other numerical methods are used to solve the Laplace equation boundary value problem above. S2: Calculate the wave force on the pipe body The distribution of hydrodynamic pressure in the fluid is obtained from the velocity potential function based on the linearized Bernoulli equation. The wave force acting on the pipe is obtained by integrating the pressure along the pipe surface, and is expressed as: Where ρ is the density of the fluid, f u (x), f w (x), f θ (x) are the complex domain expressions of the horizontal wave force, vertical wave force, and wave torque experienced by the structure, respectively. The line integral in formula (5) is directly calculated. For convenience, the above formula is denoted as: The constant coefficients are as follows: S3: Calculate the dynamic response of the tube in a steady state; The formula for calculating the beam model and anchor cable constraint reaction force on an elastic foundation is as follows: Among them, F eu ,F ew ,F eθ θ represents the constraint reaction forces of the anchor cable in the horizontal, vertical, and torsional directions, respectively; u, w, and θ represent the pipe displacements in the three directions; α and β represent the tilt angle and installation angle, respectively; E c Indicates the elastic modulus of the anchor cable; A c Let represent the cross-sectional area of ​​the anchor cable, s represent the spacing between the anchor cables, and l represent the length of the anchor cable. Based on the above formulas for calculating the anchor cable constraint reaction force and the wave force, the governing equations for the pipe are given by Hamilton's principle: Where: E is the elastic modulus of the tube, G is the shear modulus of the tube; m0 is the mass per unit length of the tube, c u c w c θ These are the damping coefficients of the tube in the horizontal, vertical, and torsional directions, respectively; I u I w I θ Let X, Y, and y be the moments of inertia of the tube about the X-axis, Y-axis, and pole, respectively. According to the modal superposition method, the steady vibration state of the tube is described as follows: Where L represents the total length of the tube, differentiating both sides of the above equation with respect to time t yields the velocity expression for the tube, and further, the velocity amplitude v at each point in the tube. u (x),v w (x),v θ The expression for (x) is then substituted into formula (10) along with formula (11) to obtain the generalized coordinates U of each modality of the tube. n W n ,Θ n The equations satisfied are: Solve the above set of equations to obtain the generalized coordinates of each mode, and obtain the dynamic response of the tube through formula (11).

Images