An analytical method for internal forces and deformations of immersed tunnels under tsunami load.

Abstract

This invention discloses an analytical method for the internal forces and deformations of immersed tunnels under tsunami loads. The method includes simulating a tsunami using a first-order solitary wave, introducing fluid dynamics methods to analyze the tsunami force acting on the seabed, simplifying the tsunami force using an exponential form, and obtaining the seepage pressure acting on the immersed tunnel caused by the tsunami using Boit consolidation theory. A semi-infinite elastic foundation beam model is established at both ends to simulate the immersed tunnel, neglecting the deformation of the tunnel section cross-section, treating the tunnel section as an Euler beam, and the foundation as a Pasternak foundation, thus obtaining the beam deformation control equation for the foundation beam model under tsunami loads. The method introduces joint bending and shear coefficients to simulate the bending and shearing of the joints, obtaining the boundary conditions controlling the continuity of the immersed tunnel at the joints. Finally, parameters such as settlement, rotation, shear force, and bending moment of the tunnel sections and joints under tsunami loads are obtained. This invention provides a reference for understanding the dynamic response of the seawater-seabed-tunnel multi-media system under disaster conditions.

Description

Technical Field

[0001] This invention belongs to the field of underground engineering technology, specifically relating to an analysis method for the internal forces and deformation of immersed tunnels under tsunami load. Background Technology

[0002] The marine environment and conditions are extremely complex. Not only are the topography significantly different from that of land, but the complex nearshore engineering geological environment is also formed by seawater, soft seabed sedimentary layers, and water-saturated fractured rock masses. Furthermore, marine structures are often subjected to numerous natural disasters during their normal operating lifespan, such as waves, earthquakes, tsunamis, and seafloor movements. Therefore, considering the performance of immersed tunnels under tsunami loads is of practical significance in the research and analysis of cross-sea immersed tunnels. When subjected to tsunami loads, immersed tunnels often experience significant vertical deformation. However, the presence of flexible tunnel joints can often reduce the internal forces of the overall structure to a certain extent, thus buffering the damage to the tunnel sections.

[0003] Existing research on immersed tunnels mainly focuses on issues such as sand and soft soil foundation treatment, seismic resistance, waterproofing of tunnel joints, and various stress paths. However, research on the impact of ocean waves and tsunamis in marine areas is relatively limited, and a simple and effective method to reflect the performance of tunnel sections and joints under extreme natural disasters is lacking. In terms of theoretical solutions, most current methods do not consider the continuity conditions of immersed tunnels at joints.

[0004] Since the joint location is where the deformation and internal forces of immersed tunnels are most complex, a practical method for calculating the deformation and internal forces of tunnel sections and joints is urgently needed to prevent excessive deformation under tsunami loads that could lead to joint damage or leakage. Currently, domestic and international research on the performance of structures under wave action mainly focuses on two aspects: First, in terms of theoretical methods, the focus is primarily on the dynamic response of nearshore bridges and underground pipelines under non-extreme wave conditions such as linear waves and elliptical cosine waves. Furthermore, the theoretical calculations for buried pipelines mostly use Winkler beams, neglecting the continuity of the foundation. Second, in terms of numerical simulation and wave testing, the focus is mainly on laboratory wave generation and its response under load. There are relatively few theoretical calculation methods for immersed tunnels under tsunami loads, and their development is urgently needed. Summary of the Invention

[0005] To address the aforementioned technical problems, the present invention aims to provide a simple and accurate method for calculating the internal forces and deformations of immersed tunnel sections and joints under tsunami load wave forces. This method facilitates the allowance of appropriate margins for relevant mechanical parameters during the structural design of immersed tunnels, thereby further ensuring the structural safety under extreme natural disaster conditions.

[0006] To achieve the above objectives, the present invention provides the following solution:

[0007] A method for analyzing the internal forces and deformations of an immersed tunnel under tsunami load, characterized by the following steps:

[0008] Step 1: Using a first-order solitary wave, the hydrodynamic method is introduced to analyze the tsunami force acting on the seabed. The tsunami force is simplified in an exponential form, and the Boit consolidation theory is used to obtain the seepage pressure acting on the immersed tunnel caused by the tsunami.

[0009] Step 2: In the finite element software ABAQUS, two semi-infinite length elastic Euler foundation beam models are established to simulate the immersed tunnel, and bending and shear springs are established at the joint locations.

[0010] Step 3: In the analysis process, the deformation of the pipe section cross section is ignored; the pipe section is regarded as Euler beam and the foundation is regarded as Pasternak foundation. The tsunami seepage pressure in Step 1 is introduced to obtain the beam deformation control equation of the foundation beam model under the action of tsunami.

[0011] Step four: Introduce the joint bending resistance coefficient and shear resistance coefficient to simulate the bending and shearing of the joint, and obtain the boundary conditions controlling the continuity of the immersed tunnel at the joint; obtain the settlement, rotation, shear force and bending moment parameters of the immersed tunnel segments and joints under tsunami load.

[0012] Furthermore, in step one, the tsunami force is represented in exponential form, and the tsunami force along the tunnel direction is expressed as:

[0013]

[0014] In the formula, p| z＝h ρ represents the water pressure at the seabed caused by a tsunami, measured in Pa, where z = h represents the depth at the vertical axis; w The density of seawater is 10. 3 kg / m 3 g is the acceleration due to gravity, with units of m / s². 2 ; d is the distance from the free surface to the seabed, in meters; H is the height of the solitary wave, in meters; The unit is m -1 ; x represents the longitudinal coordinate of the tunnel, in meters; t is the propagation time of the tsunami wave; c is the propagation speed of the tsunami wave, in meters per second, which can be obtained from... Calculated.

[0015] Furthermore, the feature is that, in step one, the seepage pressure acting on the immersed tunnel caused by the tsunami is:

[0016]

[0017] In the formula, F(x,t) represents the seepage pressure acting on the immersed tunnel, in N; x is the longitudinal coordinate of the tunnel, in m; t is the tsunami wave propagation time; p is the pore water pressure, in Pa; b is the width of the immersed tunnel, in m; D is the height of the immersed tunnel, in m; ρ w The density of seawater is 10. 3 kg / m 3 g is the acceleration due to gravity, with units of m / s². 2 H is the solitary wave height, in meters (m); C1 and C2 are parameters obtained by solving the partial differential equation of pore water pressure at any point in the seabed soil layer based on the boundary conditions of the seabed; hD / 2 is the burial depth of the immersed tunnel, in meters (m); h is the distance from the tunnel's neutral axis to the seabed surface, in meters (m); c is the propagation speed of the tsunami wave, in meters per second (m / s), which can be obtained from... The calculated value is d, which is the distance between the free water surface and the seabed, in meters.

[0018] Furthermore, the aforementioned

[0019] in γ w The specific weight of water, expressed in kg / m³. 3 ;n s The porosity of the seabed soil; k s n is the permeability coefficient of the seabed soil, expressed in m / s. s K represents the porosity of the seabed soil. w The bulk modulus of water is 2.18 × 10⁻⁶. 9 N / m 2 G s ρ is the shear modulus of the soil; μ is the Poisson's ratio of the soil.

[0020] Furthermore, in step three, the governing equation for beam deformation of the foundation beam model under tsunami action can be expressed as:

[0021]

[0022] In the formula, w is the equation of the foundation beam deflection curve; EI is the flexural stiffness of the immersed tunnel, in N·m. 2 K is the compression coefficient of the elastic foundation beam, in Pa; G is the shear coefficient of the elastic foundation beam, in Pa; d represents the differential operation; F(x) represents the longitudinal seepage pressure acting on the immersed tunnel at the most dangerous moment, and x represents the longitudinal position coordinate of the tunnel, in meters.

[0023] Because the stiffness and bending and shear resistance of the joints in immersed tunnels are weaker than those of the tunnel sections, the joints become the weakest link in the tunnel system when a tsunami propagates longitudinally along the tunnel. Furthermore, the deformation at the joint is greatest when the maximum seepage pressure caused by the tsunami acts on it, making the tunnel system most vulnerable to damage. Therefore, by solving the time history expression for the seepage pressure caused by the tsunami, F(x,t), at the moment of maximum load at the joint location x=0, the time factor in the expression can be eliminated, yielding F(x) as follows.

[0024]

[0025] That is

[0026]

[0027] Furthermore, the method for calculating the compression coefficient of the elastic foundation beam is as follows:

[0028]

[0029] The shear coefficient G is calculated as follows:

[0030]

[0031] In the formula E s This is the elastic modulus of the foundation soil, expressed in N / m³. 2 μ is the Poisson's ratio of the soil; D is the height of the immersed tunnel in meters; EI is the flexural stiffness of the immersed tunnel in N·m. 2 ; 'a' represents the depth of influence, which is generally taken as a = 2.5D, in meters.

[0032] Furthermore, in step four, the continuity boundary condition at the joint can be expressed as:

[0033]

[0034] In the formula, x represents the longitudinal position coordinate of the tunnel, in meters; w is the equation of the foundation beam deflection curve; θ is the rotation angle of the immersed tunnel, in rad; M is the bending moment, in N·m; and Q is the shear force, in N.

[0035] δw represents the displacement difference between the two sides of the tunnel joint.

[0036] δθ represents the angle difference between the two sides of the tunnel joint.

[0037] In the formula k Q For shear stiffness; k M For bending stiffness.

[0038] Furthermore, the equation of the deflection curve is obtained by solving the boundary conditions at the joint:

[0039]

[0040] Angle equation:

[0041]

[0042] Bending moment equation:

[0043]

[0044] Shear force equation:

[0045]

[0046] The unknown parameters α and β are obtained by solving the fourth-order homogeneous ordinary differential equation. To obtain, that is Since it is assumed that the tunnel is infinitely long on both sides, that is, A1, A2, A7, and A8 are all zero; the calculation of A3-A6 is obtained by solving the transfer matrix;

[0047] In the formula, x represents the longitudinal coordinate of the tunnel, in meters; ρ w The density of seawater is 10. 3 kg / m 3 g is the acceleration due to gravity, with units of m / s². 2 H represents the isolated wave height in meters (m); w is the equation of the foundation beam deflection curve; EI is the flexural stiffness of the immersed tunnel in N·m. 2 K is the compression coefficient of the elastic foundation beam, in Pa; G is the shear coefficient of the elastic foundation beam, in Pa; b is the width of the immersed tunnel, in m; C1 and C2 are parameters obtained by solving the partial differential equation of pore water pressure at any point in the seabed soil layer based on the boundary conditions of the seabed; hD / 2 is the burial depth of the immersed tunnel, in m; D is the height of the immersed tunnel, in m. The unit is m -1 .

[0048] Furthermore, steps one, three, and four employ programming languages ​​to implement the input of tsunami force, the calculation of matrix transmission, and the output of vertical displacement, rotation angle, and internal force of immersed tunnel sections and joints.

[0049] Furthermore, for immersed tunnel sections and joints, within permissible limits, the smaller the settlement deformation and internal forces generated under tsunami load, the better; and for structural design, under the condition of ensuring structural safety and optimal efficiency, the most suitable bending and shear stiffness of the sections and joints should be selected.

[0050] Compared with existing technologies and calculation methods, the present invention has the following beneficial effects:

[0051] This invention provides a method for analyzing the performance of immersed tunnel segments and joints under tsunami loads. Building upon current analytical solutions for immersed tunnels, this method considers the impact of tsunamis on the tunnel. It uses a first-order solitary wave to simulate a tsunami, introduces fluid dynamics methods to analyze the tsunami force acting on the seabed, and simplifies the tsunami force using an exponential form. A semi-infinite elastic foundation beam model is established at both ends to simulate the immersed tunnel, neglecting the deformation of the segment cross-section during the analysis. The segment is considered an Euler beam, and the foundation is considered a Pasternak foundation, yielding the beam deformation control equations for the foundation beam model under tsunami loads. The method introduces bending and shear coefficients for the joints to simulate bending and shear stresses, obtaining the boundary conditions controlling the continuity of the immersed tunnel at the joints. This allows for the acquisition of parameters such as settlement, rotation, shear force, and bending moment of the immersed tunnel segments and joints under tsunami loads. These parameters provide comprehensive and reliable data for the structural design of immersed tunnels.

[0052] The method described in this invention can be programmed using MATLAB software to assign characteristic parameters of tsunamis, tunnels, and soil layers, enabling rapid evaluation of the performance of immersed tunnels under tsunami loads. Under the conditions of ensuring structural safety and optimal benefits, the most suitable bending and shear stiffness of pipe sections and joints can be selected, which has certain application prospects. Attached Figure Description

[0053] Figure 1 This is a schematic diagram of the response of an immersed tunnel caused by the transfer of tsunami load to the immersed tunnel, provided by the present invention.

[0054] Figure 2 These are comparison diagrams of surface pressure and seabed water pressure over the tunnel provided by this invention, as well as comparison diagrams of simplified model dynamic water pressure and theoretical dynamic water pressure results;

[0055] Figure 3 This is a schematic diagram of the Pasternak elastic foundation beam model provided by the present invention;

[0056] Figure 4 This is a schematic diagram of the stress analysis of the tunnel segment micro-element provided by the present invention;

[0057] Figure 5 This is the MATLAB programming flowchart provided by the present invention;

[0058] Figure 6 This is a comparison diagram of vertical deformation and shear force of immersed tunnels with and without joints under tsunami load, provided by the present invention.

[0059] Figure 7This is a comparison diagram of the influence of the shear stiffness of the immersed tunnel section on the internal forces of the tunnel under tsunami load, provided by the present invention.

[0060] Figure 8 This is a comparison chart of the theoretical calculation results of the influence of tsunami load intensity on tunnel deformation and internal forces provided by this invention. Detailed Implementation

[0061] The embodiments of the present invention will now be clearly and completely described with reference to the accompanying drawings. The scope of the present invention is not limited thereto.

[0062] Example

[0063] This embodiment provides a performance analysis method for immersed tunnel sections and joints under tsunami loads, the steps of which are as follows:

[0064] Step 1: Using a first-order solitary wave, the hydrodynamic method is introduced to analyze the tsunami force acting on the seabed. The tsunami force is simplified in an exponential form, and the Boit consolidation theory is used to obtain the seepage pressure caused by the tsunami acting on the immersed tunnel.

[0065] Step two: In the finite element software ABAQUS, create two semi-infinite-length elastic Euler foundation beam models to simulate the immersed tunnel, and establish bending and shear springs at the joint locations. For example... Figure 1 As shown, two springs (illustrated) are installed between the two pipe sections. The upper spring is an anti-bending spring, and the lower spring is an anti-shear spring.

[0066] Step 3: Ignore the deformation of the pipe section cross-section; in the analysis, the pipe section is regarded as an Euler beam and the foundation is regarded as a Pasternak foundation. Introduce the tsunami seepage pressure from Step 1 to obtain the beam deformation control equation of the foundation beam model under the action of tsunami.

[0067] Step four: Introduce the joint bending resistance coefficient and shear resistance coefficient to simulate the bending and shearing of the joint, and obtain the boundary conditions for the continuity of the immersed tunnel at the joint; obtain parameters such as settlement, rotation angle, shear force, and bending moment of the immersed tunnel segments and joints under tsunami load.

[0068] The specific steps are as follows:

[0069] See Figure 1 Because the joint of the immersed tunnel is the weakest point in the structure, when a tsunami reaches the joint, it can easily cause excessive deformation and damage to the structure. Since the tsunami waves propagate from left to right, the left tunnel segment experiences the force first, which is then transferred to the right tunnel segment through the joint. The tsunami force acting on the tunnel can be calculated using fluid mechanics and permeability mechanics. The specific calculation method is as follows:

[0070] The potential function of particles in tsunami water satisfies the Laplace equation and the Lagrange integral, that is:

[0071]

[0072]

[0073] In the formula, φ is the potential function; ρ is the Laplace operator; x represents the longitudinal coordinate of the immersed tunnel, i.e., x = 0 is the location of the tunnel joint, x < 0 is the tunnel segment before the joint, and x > 0 is the tunnel segment after the joint, in meters; z represents the vertical distance of any point in the water-soil-tunnel system from the neutral axis of the immersed tunnel; p is the pressure at any point in the fluid, in Pa; d is the distance from the free water surface to the seabed, in meters; ρ w The density of seawater is expressed in kg / m³. 3 g is the acceleration due to gravity, with units of m / s². 2 hD / 2 is the burial depth of the immersed tunnel, in meters; u and v are the velocities of the fluid particles in the x and z directions, respectively, in meters per second; t is the time it takes for the tsunami wave to propagate, in seconds.

[0074] Based on the satisfied boundary conditions, the wave surface equation of the first-order solitary wave is obtained as follows:

[0075]

[0076] In the formula, η is the wavefront equation; H is the isolated wave height, in meters (m). The unit is m -1 x represents the longitudinal coordinate of the tunnel, in meters; c represents the wave propagation speed, in meters per second; t represents the time it takes for the tsunami wave to travel, in seconds.

[0077] When tsunami force is expressed in exponential form, the tsunami force along the x-direction is:

[0078]

[0079] In the formula, p| z＝h ρ represents the water pressure at the bottom surface caused by the tsunami, in Pa; z = h represents the depth position at the vertical axis, in meters (m); w The density of seawater is 10. 3 kg / m 3 g is the acceleration due to gravity, with units of m / s². 2 ; d is the distance from the free surface to the seabed, in meters; H is the height of the solitary wave, in meters; The unit is m -1; x represents the longitudinal coordinate of the tunnel, in meters; t is the propagation time of the tsunami wave; c is the propagation speed of the tsunami wave, in meters per second, which can be obtained from... The calculated value is d, which is the distance between the free water surface and the seabed, in meters.

[0080] According to Boit's consolidation theory, if the soil skeleton is considered as the isolated body, based on the effective stress principle and neglecting volume forces, the governing equation of the soil skeleton is:

[0081]

[0082] In the formula u s v s Let G be the strain of the soil skeleton in the x and z directions, respectively; x and z represent the directions along the tunnel length and vertically upward; G is the shear modulus of the soil, and G... s =E s / (2+2μ), where E s λ is the elastic modulus of the soil, in Pa; μ is the Poisson's ratio of the soil; λ satisfies λ = μE s / [(1+μ)(1-2μ)];ε V γ represents volumetric strain; p represents pore water pressure at any point in the soil, in Pa; γ represents the unit weight of the soil, in kg / m³. 3 .

[0083] Considering the compressibility of the fluid in the pores, the governing equation for the continuity of water flow is:

[0084]

[0085] In the formula k s γ is the permeability coefficient of the seabed soil, in m / s; p is the pore water pressure, in Pa; w The specific weight of water, expressed in kg / m³. 3 ;n s K represents the porosity of the seabed soil. w The bulk modulus of water is 2.18 × 10⁻⁶. 9 N / m 2 ;ε v t represents volumetric strain; t represents time, in seconds.

[0086] Combining the above three equations and decoupling them, we can obtain:

[0087]

[0088] Based on the boundary conditions of the seabed, the pore water pressure at any point in the seabed soil layer can be calculated as follows:

[0089]

[0090] Where p(x,z,t) represents the pore water pressure at any time under any coordinate in the soil;

[0091] Let the height of the immersed tunnel be D and the width be b. Therefore, the seepage pressure acting on the immersed tunnel due to the tsunami is:

[0092]

[0093] In the formula, F(x,t) represents the seepage pressure acting on the immersed tunnel caused by the tsunami, x is the longitudinal coordinate of the immersed tunnel, t is the tsunami propagation time; p is the pore water pressure in Pa; b is the width of the immersed tunnel in m; D is the height of the immersed tunnel in m; ρ w The density of seawater is 10. 3 kg / m 3 g is the acceleration due to gravity, with units of m / s². 2 ; d is the distance from the free surface to the seabed, in meters; H is the height of the solitary wave, in meters; The unit is m -1 ; x represents the longitudinal coordinate of the tunnel, in meters; t is the propagation time of the tsunami wave; c is the propagation speed of the tsunami wave, in meters per second, which can be obtained from... The calculations show that C1 and C2 are parameters obtained by solving the partial differential equation of pore water pressure at any point in the seabed soil layer based on the boundary conditions of the seabed; hD / 2 is the burial depth of the immersed tunnel in meters; and h is the distance from the tunnel's neutral axis to the seabed surface in meters.

[0094] The following is a specific example for verifying tsunami load, assuming the wave height of the tsunami is 5m and the seawater density is 1025kg / m³. 3 The bulk modulus is 2.18 × 10⁻⁶. 9 N / m 2 The seabed soil layer is a homogeneous layer with a natural unit weight of 1.84 × 10⁻⁶. 4 N / m 3 The elastic modulus is 1.0 × 10⁻⁶. 6 N / m 2 The Poisson's ratio μ is 0.3, and the permeability coefficient of the soil in the seabed is 5 × 10⁻⁶. -5 m / s, porosity 0.66. See also Figure 2 The left side shows the water pressure on the seabed and the tunnel overlying surface. It can be seen that the overlying soil layer absorbs most of the tsunami energy. When the depth of the overlying soil layer is 2 meters, the seepage pressure transmitted to the tunnel is approximately one-third that at the seabed surface. This indicates that the overlying soil layer of the immersed tunnel provides good protection for the tunnel. (See also...) Figure 2The right side shows a comparison between the water pressure at the sea surface when using a one-dimensional solitary wave and expressing it exponentially and the theoretical water pressure. It can be seen that when using a first-order solitary wave and expressing it exponentially, the water pressure is slightly lower than that under theoretical conditions, but it can reflect the characteristics of tsunami loads relatively well, with an error of 6.5%, which is relatively small.

[0095] Once the tsunami-induced load is obtained, the deformation of the immersed tunnel can be solved theoretically. The tunnel is simplified as a Winker beam model, and the foundation adopts the Pasternak foundation. (See [link to relevant documentation]). Figure 3 The foundation is simplified to foundation springs and a shear layer; these two types of springs are used to simulate the effect of the foundation on the tunnel. See also Figure 4 When a tsunami load acts on an immersed tunnel, the stress state of its tunnel segment's infinitesimal element is shown in the figure, i.e.:

[0096] The governing equation for beam deformation under tsunami load on the foundation beam model can be expressed as:

[0097]

[0098] In the formula, w is the equation of the foundation beam deflection curve; EI is the flexural stiffness of the immersed tunnel, in N·m. 2 K is the compression coefficient of the elastic foundation beam, in Pa; G is the shear coefficient of the elastic foundation beam, in Pa; d represents the differential operation; F(x) represents the longitudinal seepage pressure acting on the immersed tunnel at the most dangerous moment, and x represents the longitudinal position coordinate of the tunnel, in meters.

[0099] Because the stiffness and bending and shear resistance of the joints in immersed tunnels are weaker than those of the tunnel sections, the joints become the weakest link in the tunnel system when a tsunami propagates longitudinally along the tunnel. Furthermore, the deformation at the joint is greatest when the maximum seepage pressure caused by the tsunami acts on it, making the tunnel system most vulnerable to damage. Therefore, by solving the time history expression for the seepage pressure caused by the tsunami, F(x,t), at the moment of maximum load at the joint location x=0, the time factor in the expression can be eliminated, yielding F(x) as follows.

[0100]

[0101] That is

[0102]

[0103] The compression coefficient and shear coefficient of the elastic foundation beam can be determined by equations (12) and (13), respectively:

[0104]

[0105]

[0106] In the formula E s This is the elastic modulus of the foundation soil, expressed in N / m³. 2 μ is the Poisson's ratio of the soil; D is the height of the immersed tunnel in meters; a is the depth of influence, generally taken as a = 2.5D in meters.

[0107] Furthermore, the continuity boundary condition at the joint in step four can be expressed as:

[0108]

[0109] In the formula, x is the position coordinate of the immersed tunnel along its length; w is the equation of the foundation beam deflection curve; θ is the rotation angle of the immersed tunnel, in rad; M is the bending moment, in N·m; and Q is the shear force, in N.

[0110] For a joint, its stress-deformation satisfies the following relationship:

[0111] δw represents the displacement difference between the two sides of the tunnel joint.

[0112] δθ represents the difference in rotation angle between the two sides of the tunnel joint.

[0113] In the formula k Q For shear stiffness; k M For bending stiffness.

[0114] The equation of the deflection curve is obtained by solving the problem as follows:

[0115]

[0116] The unknown parameters α and β are obtained by solving the fourth-order homogeneous ordinary differential equation. To obtain, that is Since it is assumed that the tunnel is infinitely long on both sides, the deflection on both sides is finite, that is, A1, A2, A7, and A8 are all zero; the calculation of A3-A6 is obtained by solving the transfer matrix;

[0117] In the formula, x represents the longitudinal coordinate of the tunnel, in meters; ρ w The density of seawater is 10. 3 kg / m 3 g is the acceleration due to gravity, with units of m / s². 2 H represents the isolated wave height in meters (m); w is the equation of the foundation beam deflection curve; EI is the flexural stiffness of the immersed tunnel in N·m. 2K is the compression coefficient of the elastic foundation beam, in Pa; G is the shear coefficient of the elastic foundation beam, in Pa; d represents differential operation; b is the width of the immersed tunnel, in meters; C1 and C2 are parameters obtained by solving the partial differential equation of pore water pressure at any point in the seabed soil layer based on the boundary conditions of the seabed; hD / 2 is the burial depth of the immersed tunnel, in meters; D is the height of the immersed tunnel, in meters. The unit is m -1 .

[0118] The transfer matrix for parameters A3-A6 is calculated as follows:

[0119]

[0120] In the formula, A3-A6 are demand coefficients, and EI is the flexural stiffness of the immersed tunnel, in N·m. 2 The parameters α and β are obtained by solving the fourth-order homogeneous ordinary differential equation. To obtain, that is k Q For shear stiffness; k M For bending stiffness; The unit is m -1 ; ρ w The density of seawater is 10. 3 kg / m 3 g is the acceleration due to gravity, with units of m / s². 2 H is the isolated wave height in meters; b is the tunnel width in meters; K is the compression coefficient of the elastic foundation beam in Pa; G is the shear coefficient of the elastic foundation beam in Pa.

[0121] Therefore, the differential relationships of deflection, rotation angle, bending moment, and shear force are listed below in the order described above:

[0122]

[0123]

[0124]

[0125]

[0126] In the formula, w is the equation of the deflection curve; θ is the equation of the rotation angle; M is the equation of the bending moment; and Q is the equation of the shear force.

[0127] See Figure 5 To realize the calculation of the above process, a programming language is used to implement the input of tsunami force, matrix transfer operation, and the output of vertical displacement, rotation angle and internal force of immersed tunnel sections and joints.

[0128] The following is an example of verifying the internal forces and deformation of an immersed tunnel. The immersed tunnel is 37.95m wide and 5.7m high, with a longitudinal bending stiffness of 1.464 × 10⁻⁶. 11 N·m 2 The compressibility coefficient of the foundation soil is 500, and the shear coefficient of the foundation soil is 1.9 × 10⁻⁶. 9 The tunnel is 2m deep; the still water depth is 5m; the tsunami wave height is 5m; the bending stiffness of the joint is 2.44×10⁻⁶. 8 N·m 2 The shear stiffness of the joint is 1.1 × 10⁻⁶. 9 N / m.

[0129] See Figure 6 When there are no tunnel joints, the settlement and shear force of the immersed tunnel are significantly less than when there are joints. With joints, the settlement is 6.8 mm and the shear force is Q = 4.38 × 10⁻⁶. 6 N; When there are no joints, the immersed tunnel experiences almost no settlement, and the shear force is only Q = 1.00 × 10⁻⁶. 5 N. This is because the integrity of the tunnel is better when there are no joints than when there are joints. The presence of joints reduces the stiffness of the tunnel. In a homogeneous soil layer, there is no uneven settlement of the soil at both ends of the joint under continuous stress conditions. Therefore, the overall bending stiffness of the tunnel is an important factor affecting the settlement and shear force of the pipe sections and joints under tsunami load conditions. This indicates that joints should not be set in locations with good soil conditions in immersed tunnels, as this will reduce the performance of the tunnel.

[0130] See Figure 7 When the shear stiffness of the immersed tunnel joint is changed, at 0.1k Q -10k Q Within the specified range, the shear force and bending moment at the tunnel joint location remain almost unchanged, which further confirms that in a homogeneous soil layer, the shear stiffness has minimal impact on internal forces and deformations, and the performance of the immersed tunnel is mainly determined by the bending stiffness of the joint.

[0131] See Figure 8 As the wave height of the tsunami increases, the settlement of the immersed tunnel also increases from 6mm to 9.5mm. This shows that when the tsunami is higher, the deformation and settlement are also greater. The joint can only withstand deformation within a limited range. When the deformation is too large, it is easy to cause the joint to break, which can lead to tunnel leakage or even structural damage.

[0132] The results of this invention can predict the deformation and internal forces of tunnel sections under tsunami loads, thus playing a preventive role in the protection and disaster prevention of tunnels under extreme natural disaster conditions.

[0133] Those skilled in the art can make various modifications and variations to this invention without departing from its spirit and scope. Therefore, if these modifications and variations fall within the scope of the claims and their equivalents, this invention is also intended to include these modifications and variations.

[0134] The above examples are for illustrative purposes only and are not intended to limit the scope of the invention. Any simple modifications to the invention are within the protection scope of the invention.

[0135] The above description is merely a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any modifications, equivalent substitutions, and improvements made by those skilled in the art within the scope of the technology disclosed in the present invention should be included within the scope of protection of the invention.

Claims

1. A method for analyzing the internal forces and deformations of an immersed tunnel under tsunami load, characterized in that, Includes the following steps: Step 1: Using a first-order solitary wave, the fluid dynamics method is introduced to analyze the tsunami force acting on the seabed. The tsunami force is simplified using an exponential form, and the Boit consolidation theory is used to obtain the seepage pressure acting on the immersed tunnel caused by the tsunami. In step one, the tsunami force is expressed in exponential form. Therefore, the tsunami force along the longitudinal direction of the tunnel can be expressed as: In the formula, The water pressure at the bottom surface caused by a tsunami, measured in units of... , z=h The vertical axis is represented as The depth position; ρ w The density of seawater is expressed in units of 1000 kJ / m³. ; g This is the acceleration due to gravity, measured in m / s². 2 ; The distance from the free surface of the water to the seabed, in units of ; The solitary wave height is expressed in units of 1. ; The unit is ; Represents the longitudinal position coordinates of the tunnel, in units of ; t This refers to the time it takes for the tsunami wave to propagate. c The speed of tsunami wave propagation, in units of 1000 m / s. , can be Calculated; In step one, the seepage pressure acting on the immersed tunnel caused by the tsunami is: In the formula, This represents the seepage pressure acting on the immersed tunnel, in units of... , x These are the longitudinal coordinates of the tunnel, in units of... , t This refers to the time it takes for the tsunami wave to propagate. Pore ​​water pressure, unit: ; b The width of the immersed tunnel is indicated by units of 1000 mm. ; D This indicates the height of the immersed tunnel, in meters (m). ρ w The density of seawater is expressed in units of 1000 kJ / m³. ; g This is the acceleration due to gravity, measured in m / s². 2 ; The solitary wave height is expressed in units of 1. ; C 1. C 2 represents the parameters obtained by solving the partial differential equation of pore water pressure at any point in the seabed soil layer based on the boundary conditions of the seabed. The depth of the immersed tunnel is expressed in units of 1. ; c The speed of tsunami wave propagation, in units of 1000 m / s. ,Depend on Calculations show that The distance from the free surface of the water to the seabed, in units of ; Step 2: In the finite element software ABAQUS, two semi-infinite length elastic Euler foundation beam models are established to simulate the immersed tunnel, and bending and shear springs are established at the joint locations. Step 3: In the analysis process, the deformation of the pipe section cross section is ignored; the pipe section is regarded as Euler beam and the foundation is regarded as Pasternak foundation. The tsunami seepage pressure in Step 1 is introduced to obtain the beam deformation control equation of the foundation beam model under the action of tsunami. Step four: Introduce the joint bending resistance coefficient and shear resistance coefficient to simulate the bending and shearing of the joint, and obtain the boundary conditions controlling the continuity of the immersed tunnel at the joint; obtain the settlement, rotation, shear force and bending moment parameters of the immersed tunnel segments and joints under tsunami load.

2. The method for analyzing the internal forces and deformations of an immersed tunnel under tsunami load as described in claim 1, characterized in that, The ; in , The specific gravity of water, measured in units of 1000 kJ / m². ; The porosity of the seabed soil; The permeability coefficient of the seabed soil is given in units of... ; Let be the bulk modulus of water, and its value is... ; G s The shear modulus of the soil; μ Poisson's ratio for soil.

3. The method for analyzing the internal forces and deformations of an immersed tunnel under tsunami load as described in claim 2, characterized in that, In step three, the beam deformation control equation of the foundation beam model under tsunami action can be expressed as: In the formula The equation for the deflection curve of the foundation beam; The bending stiffness of the immersed tunnel is expressed in units of 1. ; The compression coefficient of the elastic foundation beam is given by units of 1. ; The shear coefficient of the elastic foundation beam is given in units of 1. ; d Represents differentiation operations; F(x) This represents the longitudinal seepage pressure acting on the immersed tunnel at the most dangerous moment. x Represents the longitudinal position coordinates of the tunnel, in units of ; By solving the time history expression of the seepage pressure caused by the tsunami At the joint position At the moment of maximum load, we obtain as follows: That is 。 4. The method for analyzing the internal forces and deformations of an immersed tunnel under tsunami load as described in claim 3, characterized in that, The compressibility coefficient of the elastic foundation beam K The calculation method is as follows: The shear coefficient The calculation method is as follows: In the formula The elastic modulus of the foundation soil, in units of... ; Poisson's ratio for soil; The height of the immersed tunnel is given in units of 1. ; The bending stiffness of the immersed tunnel is expressed in units of 1. ; To indicate the depth of influence, the unit is... .

5. The method for analyzing the internal forces and deformations of an immersed tunnel under tsunami load as described in claim 1, characterized in that, In step four, the continuity boundary condition at the joint can be expressed as: In the formula, x These are the longitudinal coordinates of the tunnel, in units of... ; The equation for the deflection curve of the foundation beam; The angle of the immersed tunnel is expressed in units of 1. ; The bending moment is expressed in units of 1000 ppm. ; Shear force, unit: ; This indicates the displacement difference between the two sides of the tunnel joint. ; This indicates the angle difference between the two sides of the tunnel joint. ; In the formula For shear stiffness; For bending stiffness.

6. The method for analyzing the internal forces and deformations of an immersed tunnel under tsunami load as described in claim 5, characterized in that, The equation of the deflection curve is obtained by solving the boundary conditions at the joint: Angle equation: Bending moment equation: Shear force equation: Unknown parameters , Solving fourth-order homogeneous ordinary differential equations To obtain, that is , Assuming the tunnel is infinitely long on both sides, that is... , , , All are zero; The calculation is obtained by solving the transfer matrix; In the formula, x Represents the longitudinal position coordinates of the tunnel, in units of ; ρ w The density of seawater is expressed in units of 1000 kJ / m³. ; g This is the acceleration due to gravity, measured in m / s². 2 ; The solitary wave height is expressed in units of 1. ; The equation for the deflection curve of the foundation beam; The bending stiffness of the immersed tunnel is expressed in units of 1. ; The compression coefficient of the elastic foundation beam is given by units of 1. ; The shear coefficient of the elastic foundation beam is given in units of 1. ; b Width of the immersed tunnel; unit: ; C 1. C 2 represents the parameters obtained by solving the partial differential equation of pore water pressure at any point in the seabed soil layer based on the boundary conditions of the seabed. The depth of the immersed tunnel is expressed in units of 1. ; The height of the immersed tunnel is given in units of 1. ; The unit is .

7. The method for analyzing the internal forces and deformations of an immersed tunnel under tsunami load as described in claim 1, characterized in that: Steps one, three, and four employ programming languages ​​to input tsunami force, perform matrix transfer calculations, and output the vertical displacement, rotation angle, and internal forces of immersed tunnel sections and joints.

8. The method for analyzing the internal forces and deformations of an immersed tunnel under tsunami load as described in claim 1, characterized in that: For immersed tunnel sections and joints, within permissible limits, the smaller the settlement deformation and internal forces under tsunami load, the better; while for structural design, under the condition of ensuring structural safety and optimal efficiency, the most suitable bending and shear stiffness of the sections and joints should be selected.

Images