A time-domain calculation method for dynamic response of a suspended tunnel under multiple load coupling

Abstract

This invention relates to a time-domain calculation method for the dynamic response of suspended tunnels under multi-load coupling, belonging to the field of dynamic response analysis of suspended tunnels. Addressing the problems of poor versatility and difficulty in handling multi-load coupling and structural variations caused by inconsistent governing equations in traditional methods, this invention proposes establishing a unified equation of motion for the tunnel body. This equation is decomposed into a system of generalized coordinate ordinary differential equations using the modal superposition method. An implicit Newmark-β method is employed for solving the equations, and a sub-loop iteration mechanism is embedded to achieve dynamic coupling between loads and tunnel motion. By independently modeling anchor cable constraint reactions and using a modular environmental load interface, the method is compatible with complex terrain and various load coupling effects. This invention improves the versatility and computational efficiency of dynamic response analysis of suspended tunnels, solves the accuracy and stability challenges under multi-load interaction, and provides a reliable basis for engineering design.

Description

Technical Field

[0001] This invention belongs to the field of dynamic response analysis of suspended tunnels, and relates to a time-domain calculation method for the dynamic response of suspended tunnels under multiple load coupling. Background Technology

[0002] Suspended tunnels, as one of the most promising new transportation structures of the 21st century, mainly consist of a tube, anchor cables, foundations, and revetment sections. They are characterized by large spanning capacity, strong deep-water adaptability, and excellent environmental and economic benefits. However, due to the complexity of marine and river environments, no actual projects have been built globally. The key challenge lies in the fact that the suspended tunnel tube must withstand the coupled effects of multiple loads, including anchor cable constraints, wave flow, vehicle traffic, and external impacts. Furthermore, there is a dynamic interaction between the loads and the tube's movement, resulting in an extremely complex dynamic response mechanism. Existing research is insufficient to fully analyze the multi-load coupling effects.

[0003] Traditional methods for modeling the dynamic response of suspended tunnels are typically derived based on specific anchor cable types, revetment boundaries, and load types. While these methods can be optimized for specific scenarios, they result in a lack of a unified expression for the governing equations. Once the structural form or load conditions change, the governing equations must be reconstructed, significantly limiting the versatility of the calculation method. Furthermore, existing technologies struggle to achieve modular programming and cannot flexibly handle variables such as seabed topography and anchor cable layout, hindering efficient analysis of the dynamic response of suspended tunnels in complex marine environments. Therefore, there is an urgent need for a calculation method with a unified mathematical framework, high versatility, and compatibility with multiple load couplings to overcome the technical bottlenecks in the engineering application of suspended tunnels. Summary of the Invention

[0004] In view of this, the purpose of this invention is to provide a time-domain calculation method for the dynamic response of suspended tunnels under multiple load coupling, which solves the problem of calculating the dynamic response of suspended tunnels in complex marine environments.

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

[0006] A time-domain calculation method for the dynamic response of a suspended tunnel under multiple load coupling includes the following steps:

[0007] Step 1: Establish the governing equations for the pipe's motion. Based on Hamilton's principle, derive a system of partial differential equations containing horizontal displacement u(x,t), vertical displacement w(x,t), and torsional displacement θ(x,t). These equations include the distributed load F. qu (x,t),F qw (x,t),F qθ (x,t) and concentrated load F pui (t), F pwi (t), F pθi(t) acting term;

[0008] Step 2: Determine the environmental loads acting on the pipe body, including the cable anchor restraint reaction force, wave load, vehicle load, and impact load. The cable anchor restraint reaction force is calculated by coordinate transformation to determine the installation point position based on the current motion state of the pipe body, and is updated based on the cable anchor stiffness and deformation;

[0009] Step 3: Decompose the pipe body displacement into a linear combination of modal functions by the modal superposition method to obtain the generalized coordinates U n (t), W n (t), Θ n (t) ordinary differential equations;

[0010] Step 4: Numerically solve the generalized coordinate equation using the implicit Newmark-β method. The environmental loads and the pipe body motion state are updated by sub-cycle iteration within each time step until the generalized displacement residual meets the error requirement.

[0011] Furthermore, the calculation method of the cable anchor restraint reaction force in Step 2 includes:

[0012] Calculate the new position P′ of the cable anchor installation point through rigid body coordinate transformation according to the displacements u, w, θ of the centroid of the pipe body cross-section t ;

[0013] Calculate the linear elastic restraint reaction force according to the difference between the current length l′ and the initial length l of the cable anchor where K is the axial stiffness of the cable anchor, P a is the coordinate of the seabed anchor point;

[0014] When l′ < l, the restraint reaction force F m is set to zero to simulate the non-compressible state of the cable anchor.

[0015] Furthermore, the selection of the modal functions in Step 3 includes:

[0016] When the bank end of the pipe body is a simply supported boundary, the modal function is Eigenvalue

[0017] When the bank end of the pipe body is a fixed boundary, the horizontal and vertical modal functions are determined by the beam fixed-end vibration mode equation, and the torsional modal function is Eigenvalue

[0018] Furthermore, the parameter settings of the implicit Newmark-β method in Step 4 are γ = 0.5, β = 1 / 6, and the acceleration under-relaxation technique is used in the calculation of the generalized acceleration, with the relaxation coefficient α = 0.85.

[0019] Furthermore, the sub-cycle iteration in Step 4 includes:

[0020] Within each time step, update the environment state based on the current generalized coordinates, velocity, and acceleration;

[0021] Calculate the updated environmental loads and substitute them into the generalized coordinate equations to solve for the acceleration at the next moment;

[0022] The generalized displacement and velocity are iteratively corrected using the Newmark-β prediction-correction scheme until the displacement residual between adjacent iterations is less than a set threshold.

[0023] Furthermore, the environmental load includes a harmonic uniformly distributed load with a phase difference of π / 4 between its horizontal and vertical components, an amplitude of 80 kN / m, and a period of 12.69 s.

[0024] Furthermore, the modal superposition method selects the first 15 modes for calculation.

[0025] Furthermore, in step one, the pipe is regarded as a homogeneous Euler-Bernoulli beam with uniform cross-section, axial deformation is ignored, and the constraint form of the revetment end is fixed simply supported or consolidated.

[0026] Furthermore, the torque generated by the anchor cable constraint reaction force on the pipe body in step two is calculated by the vector cross product from the centroid to the installation point.

[0027] Furthermore, the modal damping coefficient c in the generalized coordinate equation un c wn and c θn Determined based on anchor cable stiffness and pipe damping model.

[0028] The beneficial effects of this invention are as follows:

[0029] (1) By constructing a unified form of pipe motion control equation and generalized coordinate equation, this invention realizes modular processing of various anchor cable forms, revetment boundaries and load types, which significantly improves the universality and adaptability of the calculation method and can efficiently meet the dynamic response analysis needs of multiple load coupling effects in complex marine environments.

[0030] (2) An independent modeling method for calculating anchor cable constraint reaction force is adopted. Each anchor cable is treated as an independent object and its geometric and physical properties are dynamically updated. This effectively avoids nonlinear errors caused by large deformation of the pipe body. At the same time, it is compatible with the fine simulation of complex seabed topography and improves the accuracy of constraint reaction force calculation.

[0031] (3) By combining the implicit Newmark-β method with the sub-loop iteration technique, the environmental load and the pipe motion state are fully coupled within the time step, which solves the numerical stability problem under the interaction of multiple loads and ensures the accuracy and reliability of dynamic response calculation.

[0032] (4) Through the modular design of the environmental load interface, it supports the flexible expansion of complex load forms such as waves, impacts, and fluid-structure interaction, providing an efficient tool for the study of the dynamic behavior of suspended tunnels in real marine environments and accelerating the feasibility verification of engineering applications.

[0033] (5) Based on the modal superposition method and orthogonal basis function decomposition, the partial differential equations are transformed into a low-dimensional ordinary differential equation system, which greatly reduces the computational complexity and takes into account both computational efficiency and engineering accuracy requirements, providing a reliable theoretical basis for the design optimization and safety assessment of suspended tunnels.

[0034] 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

[0035] 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:

[0036] Figure 1 This is a schematic diagram of a typical suspended tunnel structure and the coordinate system of this method;

[0037] Figure 2 A simplified diagram for calculating anchor cable constraint reaction force;

[0038] Figure 3 A schematic diagram of the dynamic response algorithm for the pipe under multiple load coupling.

[0039] Figure 4 A schematic diagram of the ABAQUS finite element model;

[0040] Figure 5 This is a time history curve of the mid-span displacement of the pipe under uniformly distributed harmonic load.

[0041] Figure 6 The diagram shows the steady-state peak displacement of the pipe under uniformly distributed harmonic load.

[0042] Figure 7 This is a schematic diagram of impact load application;

[0043] Figure 8 This is a time history curve of the mid-span displacement of the pipe under impact load. 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 1 This is a schematic diagram of a typical suspended tunnel structure, with the coordinate system shown in the figure. The origin of the world coordinate system is located on the still water surface, the length direction of the tube is the X-axis, and the Z-axis is perpendicular to the water surface and upwards. A right-handed coordinate system is used, and the cross-section at the left revetment end of the tube is coplanar with the YZ plane. The displacement components of the tube at each position are represented by u(x,t), w(x,t), and θ(x,t), respectively, in the horizontal, vertical, and torsional directions around the centroidal axis of the cross-section. u and w are taken as positive in the positive directions of the Y and Z axes, respectively, and θ is taken as positive in the counterclockwise rotation shown in the figure.

[0048] This calculation method is based on the following two fundamental assumptions:

[0049] (1) The tube is regarded as a homogeneous Euler-Bernoulli beam with uniform cross-section, and the beam only undergoes elastic deformation. The revetment end has a fixed constraint form. The tube cross-section only considers the degrees of freedom in the horizontal, vertical and torsional directions.

[0050] (2) Each anchor cable object has independent geometric and physical properties, and the anchor cable constraint force is applied to the pipe body in the form of concentrated load, but the axial load component of the pipe body is ignored.

[0051] A time-domain calculation method for the dynamic response of a suspended tunnel under multiple load coupling includes the following steps:

[0052] Step 1: Establish the control equations for the pipe motion under multiple load coupling.

[0053] First, based on the fundamental assumptions, the following control equations for the tube's motion are derived using Hamilton's principle:

[0054]

[0055] In the above formula, M0 represents the mass per unit length of the tube; J0 represents the moment of inertia per unit length of the tube; E and G represent the elastic modulus and shear modulus of the tube, respectively; I u I w and I p Let c represent the horizontal moment of inertia, the vertical moment of inertia, and the polar moment of inertia of the centroid of the tube cross-section, respectively; u c w and c θ These represent the damping coefficients in the horizontal, vertical, and torsional directions of the pipe, respectively; F qu F qw and F qθ These represent the total distributed loads on the pipe body in the horizontal, vertical, and torsional directions, respectively; F pui F pwi and F pθi These represent the concentrated loads or concentrated moments in the horizontal, vertical, and torsional directions at the i-th point of application on the pipe, respectively, i = 0, 1, 2, ..., I, where I is the total number of concentrated loads on the pipe; x i Let be the X-coordinate of the i-th point of force application; δ(·) is the Dirac function.

[0056] Step 2: Determine the environmental loads to be considered.

[0057] A key feature of this invention is that all loads acting on the tube are considered as environmental effects. Each environment needs to update its own environmental state based on the current motion state of the tube, thereby calculating the loads acting on the tube. This process is modularly encapsulated according to different load types, and all modules have a unified return interface, i.e., distributed loads or concentrated loads acting on the tube. Users can provide different load modules according to their needs, thereby realizing the multi-load effects of complex marine environments on the suspended tunnel tube.

[0058] The anchor cable restraint load is an essential type of environmental load, and different mathematical models of anchor cables have different calculation methods for the restraint reaction force. The present invention uses a linear elastic discrete anchor cable model. As Figure 2 shown, the anchor cable and the pipe cross-section are located in the same Y-Z plane. The axial stiffness of the anchor cable is K, and the original length (initial length) is l. In the world coordinate system of the Y-Z plane, the anchoring point of the anchor cable on the seabed is P a , the installation point at the initial moment is P t , the centroid of the pipe cross-section is P c , the distance between the installation point and the centroid of the cross-section is r. After a period of time, the position of the installation point of the anchor cable is P′ t , the position of the centroid of the cross-section is P′ c , the length of the anchor cable is l′, and the restraint reaction force at this time is F m . According to the current motion state of the pipe, the displacements u, w, and θ in three directions of the cross-section where the anchoring point is located are calculated. According to the coordinate transformation formula of rigid body motion, the position of the installation point at the current moment is calculated:

[0059]

[0060] Then, the restraint reaction force F of the anchor cable is calculated from Equation (3) m :

[0061]

[0062] After obtaining the restraint reaction force, the torque of the restraint reaction force on the centroid of the cross-section can be calculated by cross-multiplying the displacement vector from the centroid of the cross-section to the action point and the restraint reaction force. Additionally, if the compression state of the anchor cable is not allowed, when l′ < l, the restraint reaction force of the anchor cable can be set to 0

[0063] Step 3: Derive the generalized coordinate equation by the modal superposition method

[0064] The present invention uses the modal superposition method to solve the pipe control equation obtained in the previous section. The displacement function to be solved is written in the form of a superposition of a set of complete and orthogonal basis functions:

[0065]

[0066] In the above formula, X un (x), X wn (x) and X θn (x) respectively represent the nth-order modal functions of the horizontal, vertical, and torsional motion equations of the pipe. U n (t), W n (t) and Θ n (t) respectively represent the generalized coordinate functions corresponding to the respective nth-order modes. Substituting the modal functions into Equation (1) and using the orthogonality of the modal functions, a system of ordinary differential equations satisfied by the generalized coordinate functions can be obtained:

[0067]

[0068] In the above formula, L is the length of the tube; c un c wn and c θn λ represents the modal damping coefficients of the tube in the horizontal, vertical, and torsional directions, respectively, for the nth mode. These coefficients can be determined based on existing damping models and the constraint stiffness of the anchor cables; un , λ wn and λ θn These represent the eigenvalues ​​corresponding to the nth-order modal functions in the three directions, respectively; ||X un (x)||,||X wn (x)|| and ||X θn (x)|| denotes the mode function on the interval [0,L]. 2 The square of the norm.

[0069] Different boundary conditions for pipe revetments result in different modal functions and corresponding eigenvalues. This invention provides the modal functions and corresponding eigenvalues ​​under two different revetment boundary conditions:

[0070] 1. Simply supported boundary conditions at both ends

[0071]

[0072] 2. Consolidation boundary conditions at both ends

[0073]

[0074] In the formula, the eigenvalue λ n The sequence of positive roots of the nonlinear equation (18) is determined (arranged in ascending order).

[0075] cos(λ n L)cosh(λ n L)=0 (8)

[0076] When λ n When the value of L exceeds a certain upper limit, λ should be used to avoid "large number" numerical errors. n The asymptotic function that replaces the original modal function when L approaches infinity is given by equation (9):

[0077]

[0078] Step 4: Solve the generalized coordinate equations using the Newmark-β method.

[0079] In this study, the Newmark-β numerical motion solution method is used to solve equation (5). This method has second-order computational accuracy and good convergence and stability. The iterative format is shown in equation (10):

[0080]

[0081] In the above formula, Φ, and Let represent the generalized coordinate vector, generalized velocity vector, and generalized acceleration vector, respectively. The subscripts “n” and “o” represent “next moment” and “previous moment”, respectively. The superscript “[i]” represents the i-th sub-loop iteration step within the time step. Δt represents the time step size. γ and β are algorithm parameters. This invention suggests taking values ​​of γ = 0.5 and β = 1 / 6, that is, assuming that the generalized acceleration changes linearly within each time step. Since some environmental loads are related to the acceleration of the pipe (e.g., wave loads), in order to reduce the influence of the additional mass of the pipe on the calculation and further improve the stability of the fluid-structure interaction calculation, the acceleration sub-relaxation technique will be applied to the solution of the motion equation. That is, whenever the generalized acceleration at the current moment is calculated by Equation (5), its result is always weighted and averaged with the generalized acceleration at the previous moment as the final generalized acceleration, as shown in Equation (11):

[0082]

[0083] In the above formula The generalized relaxation acceleration is α, and the relaxation coefficient is α = 0.85. This invention suggests that the value be α = 0.85.

[0084] Figure 3 The calculation process under multiple load coupling is presented. This paper treats all loads acting on the pipe as environmental effects, and the location, direction, and magnitude of these effects are related to the current state of the environment, such as the anchor cable inclination angle, the shape of the free water surface, the surface velocity of the pipe, and the position and attitude of the vehicle. Therefore, the time step iteration is divided into the following parts:

[0085] Step 1: Update the environmental state based on the generalized coordinates, generalized velocity, and generalized acceleration of the tube at time t and the previous time.

[0086] Step 2: Calculate the environmental load at time t based on the updated environmental state, and return it in the form of concentrated load or distributed load.

[0087] Step 3: Calculate the generalized acceleration at time t using the generalized coordinate equation formula (5).

[0088] Step 4: Predict the generalized coordinates and generalized velocity at time t+Δt using the Newmark-β scheme.

[0089] Step 5: Update the environmental state at time t+Δt and calculate the environmental load and generalized acceleration.

[0090] Step 6: Correct the generalized coordinates and generalized velocity of the tube at time t+Δt according to the Newmark-β format.

[0091] Step 7: Return to step 5 and perform sub-loop iterations within the time step until the generalized displacement residuals of two adjacent sub-iteration steps are less than the specified error limit. Then, end the sub-loop and start the next time step iteration.

[0092] Based on the above calculation steps, given the initial generalized coordinates and generalized velocity of the tube, the time-domain calculation of the dynamic response of the suspended tunnel under multiple load coupling can be completed. It is generally assumed that at the initial moment, the tube is stationary in its original position, so the generalized coordinates and generalized velocity are both 0. After solving for the generalized coordinates of the tube at each moment, the dynamic response of the tube is obtained by equation (4). In addition, this invention suggests taking the first 15 modes for calculation to ensure calculation accuracy.

[0093] This method establishes the general motion control equations for a suspended tunnel under multi-load coupling based on classical dynamics theory. Then, the motion equations are transformed into a system of ordinary differential equations satisfied by generalized coordinates using the modal superposition method. Finally, the equations are numerically solved using the Newmark-β motion integral. To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and examples.

[0094] Example 1

[0095] To verify the accuracy of the proposed method, the results calculated by 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 given. All parameters of the model are determined according to the basic structural parameters of the suspended tunnel in Table 1.

[0096] Table 1

[0097]

[0098] In this embodiment, a harmonic uniformly distributed load is applied to the pipe in both the horizontal and vertical directions. The load values ​​in both directions are 80 kN / m, the load variation period is 12.69 s, and the phase difference between the horizontal and vertical loads is π / 4 to simulate the effect of linear wave load. Figure 5 The paper presents the time histories of mid-span pipe interface displacement under two conditions: simply supported revetment and consolidated revetment. Figure 6The peak displacement distribution curve of the pipe under stable vibration state is given. It is easy to see from the calculation results that the calculation method proposed in this invention agrees well with the calculation results of ABAQUS, indicating that the calculation method proposed in this invention is accurate and reliable, and that the calculation efficiency of this method is significantly higher than that of the ABAQUS finite element approximation solution.

[0099] Example 2

[0100] To further verify the accuracy of the proposed method, the structural parameters of the tube in this example remain the same as in Table 1. However, the load is changed to an impact load applied at the mid-span section of the tube to verify the free vibration characteristics of the structure. The magnitude of the impact load is 2E+5kN, and the direction of action is shown in the loading curve. Figure 7 As shown. Figure 8 The time histories of mid-span tube interface displacement are presented under two conditions: simply supported and consolidated support. The calculation results show that the proposed method is in good agreement with the ABAQUS calculations. The mid-span torsional displacement calculated by the proposed method is slightly smaller than that calculated by ABAQUS, and it decays more rapidly due to damping. This further demonstrates the accuracy and reliability of the proposed method.

[0101] 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 time-domain calculation method for the dynamic response of a suspended tunnel under multi-load coupling, characterized in that: Includes the following steps: Step 1: Establish the control equations for the pipe's motion, and derive the equations including horizontal displacement based on Hamilton's principle. u ( x , t Vertical displacement w ( x , t and torsional displacement θ ( x , t A system of partial differential equations, including distributed loads. , , and concentrated loads , , The function term; Step 2: Determine the environmental loads acting on the pipe, including anchor cable constraint reaction force, wave load, vehicle load and impact load. The anchor cable constraint reaction force is calculated based on the current motion state of the pipe through coordinate transformation to determine the installation point position and is updated based on the anchor cable stiffness and deformation. Step 3: Decompose the pipe displacement into a linear combination of modal functions using the modal superposition method to obtain the generalized coordinates. , , A system of ordinary differential equations; Step 4: The implicit Newmark-β method is used to numerically solve the generalized coordinate equation. In each time step, the environmental load and the motion state of the pipe are updated iteratively through a sub-loop until the generalized displacement residual meets the error requirements. The calculation method for the anchor cable constraint reaction force in step two includes: Based on the displacement of the centroid of the pipe cross section u , w , θ The new position of the anchor cable installation point is calculated through rigid body coordinate transformation. P′ t ; Based on the current length of the anchor cable l′ With initial length l The difference is used to calculate the linear elastic constraint reaction force. ,in K For the axial stiffness of the anchor cable, The coordinates of the seabed anchorage point; when l′ < l At that time, constraint reaction force F m Set to zero to simulate the uncompressible state of the anchor cable; The selection of mode functions in step three includes: When the revetment end of the pipe body is a simply supported boundary, the modal function is: eigenvalues ; When the revetment end of the pipe is a consolidated boundary, the horizontal and vertical modal functions are determined by the beam's fixed-support vibration mode equations, and the torsional modal function is... eigenvalues .

2. The method for calculating the dynamic response of a suspended tunnel under load coupling as described in claim 1, characterized in that: The parameters of the implicit Newmark-β method in step four are set as follows: γ =0.5、 β =1 / 6, Generalized acceleration calculations employ sub-relaxation techniques, relaxation coefficient α =0.

85.

3. The method for calculating the dynamic response of a suspended tunnel under load coupling as described in claim 1, characterized in that: The sub-loop iteration in step four includes: Within each time step, update the environment state based on the current generalized coordinates, velocity, and acceleration; Calculate the updated environmental loads and substitute them into the generalized coordinate equations to solve for the acceleration at the next moment; The generalized displacement and velocity are iteratively corrected using the Newmark-β prediction-correction scheme until the displacement residual between adjacent iterations is less than a set threshold.

4. The method for calculating the dynamic response of a suspended tunnel under load coupling as described in claim 1, characterized in that: The environmental load includes a harmonic uniformly distributed load, the phase difference between its horizontal and vertical components being [missing information]. π / 4, amplitude of 80kN / m, period of 12.69s.

5. The method for calculating the dynamic response of a suspended tunnel under load coupling as described in claim 1, characterized in that: The modal superposition method selects the first 15 modes for calculation.

6. The method for calculating the dynamic response of a suspended tunnel under load coupling as described in claim 1, characterized in that: In step one, the pipe is regarded as a homogeneous Euler-Bernoulli beam with uniform cross-section, axial deformation is ignored, and the restraint form of the revetment end is fixed simply supported or consolidated.

7. The method for calculating the dynamic response of a suspended tunnel under load coupling as described in claim 1, characterized in that: In step two, the torque generated by the anchor cable constraint reaction force on the pipe body is calculated by the vector cross product from the centroid to the installation point.

8. The method for calculating the dynamic response of a suspended tunnel under load coupling as described in claim 1, characterized in that: Modal damping coefficients in the generalized coordinate equation c un , c wn and c θn Determined based on anchor cable stiffness and pipe damping model.

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