Immersed tunnel joint numerical simulation method considering foundation differential settlement effect

Abstract

The invention discloses an immersed tunnel joint numerical simulation method considering a foundation differential settlement effect, and belongs to the technical field of settlement simulation, the method comprises the following steps: constructing a joint rigidity weakening equivalent PD shell model based on a near field dynamics plate shell basic theory, and obtaining a motion equation under a global coordinate system through coordinate transformation; an ideal elastic-plastic soil spring is used for representing soil-tunnel interaction, and a flexible joint is simulated through a local weakening method; dispersing the immersed tunnel by a meshless method, applying a soil spring by volume force and applying differential settlement in a quasi-static state; and solving the equation by adopting a self-adaptive dynamic relaxation format, and outputting pipe joint longitudinal deformation, joint opening amount and cracking risk results. According to the method, the pipe joint and joint deformation of the immersed tunnel under continuous and discontinuous working conditions can be evaluated more accurately, and the possible cracking risk can be effectively predicted.

Description

Technical Field

[0001] This invention relates to the field of settlement simulation technology, and in particular to a numerical simulation method for immersed tunnel joints that takes into account the effects of differential settlement of the foundation. Background Technology

[0002] In recent years, with the continuous growth of transportation demand in riverside and coastal areas, underwater tunnels crossing rivers and seas have become crucial connecting hubs in the land transportation system, playing an indispensable role in the coordinated development of riverside and coastal cities, islands, and inland areas. Among numerous tunnel construction technologies, the immersed tube method, with its superior geological adaptability, shallow burial depth, excellent waterproof performance, flexible cross-section design, and efficient space utilization, is particularly suitable for tunnel construction in urban rivers and near-shore waters. Immersed tube tunnels typically consist of multiple prefabricated tube sections connected by specific joints, and the sensitivity of these joints to longitudinal settlement is extremely critical.

[0003] As differential settlement of the foundation occurs during the long-term operation of immersed tunnels, underwater structures will be affected to varying degrees, potentially leading to structural damage, leakage, and other problems. In severe cases, this could even jeopardize the normal operation and structural safety of the tunnel. Therefore, accurately predicting the impact of differential foundation settlement on the longitudinal deformation response of immersed tunnel joints, longitudinal deformation of tunnel sections, and potential cracking risks is crucial for the later maintenance and safety assurance of the tunnel.

[0004] Differential settlement of the foundation can induce additional axial forces in the joint area of ​​the immersed tunnel, leading to bending deformation of the tunnel structure. However, as immersed tunnels are large-volume concrete structures with large segment dimensions (e.g., the length, width, and height of each segment of the Hong Kong-Zhuhai-Macau immersed tunnel are 22.5 m, 37.95 m, and 11.4 m, respectively), multiple sets of full-scale on-site model tests are necessary for high-precision simulation of the joint's mechanical behavior. However, full-scale on-site tests are often limited by economic constraints and practical construction conditions due to their high cost and limited applicability.

[0005] To overcome this limitation, indoor model experiments using scaled-down models to study the longitudinal deformation and joint damage mechanical properties of tunnel sections have become an important research method. For example, Xiao et al. used a 1 / 10 scaled-down model test to study the mechanical properties of immersed tunnel joints with steel shear keys under axial load and horizontal bending, and obtained the range of stiffness ratio between the joint and tunnel components under service conditions. However, the large scale and high cost of the experiment limit its widespread application. In theoretical research, the traditional "two-stage analysis method" assumes that the structure does not interact with the soil. It first calculates the free displacement field of the soil caused by external loads, and then inputs the displacement field as a boundary condition into a beam-spring model to calculate longitudinal deformation and internal force distribution. However, this method fails to fully consider the nonlinear response and cracking risk of the joint.

[0006] In numerical simulation, researchers such as Peng Haikuo have found that plate and shell elements can effectively and accurately simulate the deformation of immersed tunnels under complex working conditions, but the analysis of concrete cracking and complex soil responses remains incomplete. While current numerical analysis methods can provide a certain degree of prediction of mechanical behavior, they suffer from insufficient numerical convergence, high model complexity, and low computational efficiency, severely limiting the comprehensive assessment of complex deformation and damage in immersed tunnels. Particularly in the study of reinforced concrete materials, although traditional finite element analysis simplifies the modeling of reinforcement distribution, it still cannot effectively capture the entire process of crack propagation and material strength damage in concrete. Summary of the Invention

[0007] The purpose of this invention is to provide a numerical simulation method for immersed tunnel joints that considers the effect of differential settlement of the foundation, so as to solve the problems of insufficient numerical convergence, high complexity of model conditions and low computational efficiency of existing methods, and to achieve a comprehensive assessment of the complex deformation and damage of immersed tunnels.

[0008] To achieve the above objectives, the following technical solution is adopted: A numerical simulation method for immersed tunnel joints considering differential foundation settlement, the method comprising: Based on the fundamental theory of near-field dynamics plate and shell, a PD shell model equivalent to weakening the stiffness of the immersed tunnel joint is constructed. The degrees of freedom of each mass point in the PD shell model are defined, and the motion equations of the mass points are established. The degrees of freedom of each mass point include three displacements and three rotations in the global coordinate system, and three displacements and three rotations in the local coordinate system. The PD shell model is subjected to coordinate transformation between the local coordinate system and the global coordinate system. The mechanical quantities in the local coordinate system are converted into mechanical quantities in the global coordinate system and superimposed to obtain the motion equation of the PD shell model in the global coordinate system. Soil-tunnel interaction is represented by a soil spring with ideal elastic-plastic properties; The flexible joint of the immersed tunnel was simulated, and the stiffness of the joint area was weakened by the local weakening method. The stiffness and local stiffness matrix of the joint were dynamically updated. The immersed tunnel is discretized using a meshless method, which discretizes the immersed tunnel into particles with uniform spacing. The horizontal integral is calculated by summing the centroids of all particles. Soil springs are applied to the material layer near the boundary in the form of volume forces to apply differential settlement of the strata in a quasi-static manner. An adaptive dynamic relaxation scheme is used to solve the motion equations of the PD shell model in the static or quasi-static case in the global coordinate system. Based on the solution results, the simulation results of longitudinal deformation, joint opening, and potential cracking risk of the immersed tunnel segment are output.

[0009] Furthermore, the equation of motion for the particle is expressed as: ;

[0010] in, ρ Indicates mass density, u and b These represent displacement and volume force vectors, respectively. δ The range of influence of a certain particle. H X It is a family of particles in the near field. t Representing a point of matter k Acting on a point of matter j Force density on X As a point mass, For point mass X acceleration vector, for X Near field range H X Any particle inside, The volume of each particle, f Let be a bivariate force function, representing a point mass. X 'Acting on a point mass' X Force vector on; In the PD shell model, under the local coordinate system, based on the Lagrange equations and using the principle of virtual work, the following equations of motion are derived: ;

[0011] in q i Indicates degrees of freedom. express q i Time derivative, L The Lagrangian function, representing the difference between kinetic energy and total potential energy, is expressed as: ;

[0012] in, The strain energy density per unit volume; It is a velocity vector. For volume forces, It is a displacement vector. For cross-sectional area, This is the quality matrix; Considering the interaction states between particles, the PD equation for the shell structure can be expressed using the Lagrange equation as follows: ;

[0013] in, Point k andj The interaction state between them; A vector representing force; V j Represents a point mass j Volume; Force vectors in the equations of motion The calculation formula is: ;

[0014] in, The strain energy density per unit volume; , and These are the particles orbiting each other in the local coordinate system. X , Y and Z The amount of rotation of the shaft, , and The particles are respectively in X , Y and Z In-plane displacement in three directions, Defined as: ;

[0015] in, This represents the energy release rate of the interaction between particles k and j; g c This represents the average critical energy release rate of the interaction between the corresponding material particles; ;

[0016] in, η Indicates the thickness of the shell; Indicates the distance between adjacent particles; and Represented as point masses k and j The micro-potential energy of the interaction between them; V (j) and V (k) They represent point masses respectively. j and k Volume; ;

[0017] in, , and These represent the micro-potential energy resulting from plane strain, shear deformation, and bending deformation, respectively. Average critical energy release rate corresponding to the interaction between material particlesg c The calculation formula is: ,in, G c The critical energy release rate of the immersed tunnel material. N c This represents the total number of interactions traversing a unit crack region. Local damage degree is introduced to characterize the degree of damage to the structure. The formula for calculating the local damage degree is as follows: ;

[0018] in, ( X , t () represents the local damage coefficient. N The total number of interacting particles within the field of view. X (j) and X (k) represent j and k Point mass.

[0019] Furthermore, a coordinate transformation is performed on the PD shell model between the local and global coordinate systems. The mechanical quantities in the local coordinate system are converted into mechanical quantities in the global coordinate system and then superimposed to obtain the motion equations of the PD shell model in the global coordinate system, including: Establish a local triorthogonal basis for each particle; Based on the local triorthogonal basis and the superposition rule determined by formula (4), the motion equations of the PD shell model in the global coordinate system are obtained.

[0020] Furthermore, a local triorthogonal basis is established for each particle, including: In the global coordinate system ( X , Y , Z Next, give the first k For each particle, define the unit vectors for the local tangent and normal directions, as follows: ;

[0021] in, For point mass k Local coordinate system X Towards the unit vector, For point mass k Local coordinate system Y Towards the unit vector, For point mass k Local coordinate system Z Towards the unit vector, , and Local X To the unit vector in the global X , Y and Z Projected length on the axis , and Local Y To the unit vector in the global X , Y and Z Projected length on the axis , and Local Z To the unit vector in the global X , Y and Z Projected length on the axis; Will , and The direction cosine matrix H is formed by stacking rows. k , is represented as: ;

[0022] According to the direction cosine matrix H k The coordinate relationship is expressed as: ;

[0023] in, For a particle in local coordinates k , For a point mass in global coordinates k ; Each particle has 6 degrees of freedom, and a 6×6 displacement transformation matrix T is defined. k for: ;

[0024] According to the displacement transformation matrix T k get ,on the contrary H k Orthogonal For displacement vector transformation in local coordinate system, This is a displacement vector transformation in the global coordinate system.

[0025] Furthermore, the superposition rule is expressed as: , , ; in, The force vector in the global coordinate system. Body strength in the global coordinate system; This is the mass matrix in the global coordinate system.

[0026] Furthermore, the relevant calculation formula for the soil spring is as follows: ;

[0027] in, Indicates the weight per unit soil volume; Indicates the tunnel depth; Indicates the width of the immersed tunnel; This represents the coefficient of lateral pressure on the soil when it is at rest. This represents the frictional resistance between the tunnel surface and the soil. Indicates the internal friction angle of the soil surrounding the tunnel; Indicates the horizontal bearing capacity coefficient; Indicates the vertical lift coefficient; Indicates the vertical bearing capacity coefficient; Vertical bearing capacity influence coefficient; f T Indicates longitudinal force; f P Indicates lateral force; f u Indicates vertical pull-out force; f d Indicates vertical bearing capacity.

[0028] Furthermore, the flexible joint of the immersed tunnel was simulated, and the stiffness of the joint area was weakened using a local weakening method. The stiffness and local stiffness matrix of the joint were dynamically updated, including: A joint stiffness weakening region is defined, comprising a core weakening region and a weakening transition region; the elastic modulus of the immersed tunnel section is set to... E 1. The elastic modulus of the core weakened region is set to E 2, E 2< E 1. The elastic modulus of the weakened transition region is E 1 and E Interpolation of 2; Set the distance threshold of the core weakened region as l 1. The distance threshold for the weakened transition region is... l 2; the distance from the node to the target weakened region d < l At time 1, the node is in the core weakened zone; when l 1≤ d ≤ l 1+ l At time 2, the node is in the weakened transition region; when d > l 1+ lAt time 2, the node is in the normal range; Calculate the weakening factor w The calculation formula is: ;

[0029] Calculate the local elastic modulus E l The calculation formula is: ;

[0030] Calculate the local shear modulus G l The calculation formula is: ;

[0031] in, v Poisson's ratio of the material; Calculate the in-plane strain energy constant A ip1 The calculation formula is: ;

[0032] Based on weakening factor w and local elastic modulus E l The stiffness and local stiffness matrix of the joint are dynamically updated.

[0033] Furthermore, the motion equations of the PD shell model under static or quasi-static conditions in the global coordinate system are expressed as follows: ;

[0034] in, D Represents the virtual diagonal density matrix; c The damping coefficient; X and U These represent the initial position and displacement of the particle, respectively; force vector. F It consists of the interaction between PD particles and the physical forces exerted on the immersed tunnel by the soil spring.

[0035] Furthermore, the soil spring is an ideal elastoplastic spring. When the load on the soil spring is less than the yield load, the soil spring is in the elastic stage, and the load and deformation are linearly related. When the load reaches the yield load, the soil spring enters the plastic stage, and the deformation continues to increase while the load remains constant.

[0036] Furthermore, the method of applying differential settlement of strata in a quasi-static manner includes: controlling the settlement displacement rate at 0.02 mm / s, adopting a graded loading mode, with a settlement increment of 5 mm for each loading stage, until the total settlement reaches 40 mm.

[0037] The beneficial effects of this invention are: This invention proposes a novel equivalent PD shell model for joint stiffness weakening based on the peri-field dynamics (PD) theoretical framework. This model, combined with a soil spring model, can simulate the longitudinal deformation response of immersed tunnel joints under uneven foundation settlement. Through this model, the deformation of tunnel sections and joints under continuous and discontinuous operating conditions can be more accurately assessed, and potential cracking risks can be effectively predicted. This innovative method not only provides a new theoretical framework for the structural response analysis of immersed tunnels but also offers a more efficient and economical numerical analysis tool for assessing the impact of differential foundation settlement on tunnel safety in practical engineering applications, demonstrating broad application prospects. Attached Figure Description

[0038] Figure 1 A flowchart of a numerical simulation method for immersed tunnel joints considering differential settlement of foundation provided in an embodiment of the present invention; Figure 2 This is a schematic diagram of the degrees of freedom and coordinate system of each mass point in the PD shell model provided in the embodiments of the present invention; Figure 3 PD particles provided in embodiments of the present invention k and j A schematic diagram illustrating the influence of other particles within the same family. Figure 4 A schematic diagram of a Winkler soil spring acting around an immersed tunnel, provided as an embodiment of the present invention; Figure 5 This is a schematic diagram of a connector provided in an embodiment of the present invention; Figure 6 A flowchart of numerical simulation calculation provided for an embodiment of the present invention; Figure 7 This is a diagram illustrating the variation of vertical settlement of pipe sections with differential settlement, provided in an embodiment of the present invention. Figure 8 This is a diagram illustrating the variation of joint opening amount with differential settlement provided in an embodiment of the present invention. Figure 9 The cloud diagrams showing joint opening and pipe section deformation are provided for embodiments of the present invention. Detailed Implementation

[0039] 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 also be modified or changed based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that, unless otherwise specified, the following embodiments and features described therein can be combined with each other.

[0040] The specific embodiments of the present invention will be described in further detail below with reference to the accompanying drawings and examples.

[0041] This invention provides a numerical simulation method for immersed tunnel joints that considers differential foundation settlement, such as... Figure 1 As shown, the method includes the following steps S10-S70.

[0042] S10: Based on the near-field dynamics plate and shell fundamental theory, construct an equivalent PD shell model for weakening the stiffness of the immersed tunnel joint, define the degrees of freedom of each mass point in the PD shell model, and establish the motion equations of the mass points; wherein, the degrees of freedom of each mass point include three displacements and three rotations in the global coordinate system, and three displacements and three rotations in the local coordinate system.

[0043] Specifically, peridynamics (PD) was pioneered by Silling, and Nguyen and Oterkus established a shell model within the state-based peridynamics framework. The method proposed in this patent, based on this shell model, characterizes the flexible joint by parametrically weakening the local stiffness of the immersed tunnel section, and studies the longitudinal deformation response, joint opening behavior, and potential cracking modes of the jointed tunnel section under uneven foundation settlement.

[0044] The method proposed in this invention considers 6 degrees of freedom, i.e., three displacements, for each particle in the plate and shell model. , and ) and three rotational quantities ( , and However, considering that the shell model needs to account for membrane, bending, shear, and rotation in the X, Y, and Z directions during calculation, it is more straightforward to define the six degrees of freedom directly in a local coordinate system. Therefore, the interaction forces are first calculated in their respective local systems, and then superimposed in the global system by transforming the coordinate system. For example... Figure 2 As shown. That is, the six degrees of freedom of the global state (global coordinate system) are represented as: , , , , and The six degrees of freedom of the local state (local coordinate system) are represented as follows: , , , , and ;in, , and These represent the in-plane displacements of the particle in the X, Y, and Z directions, respectively. , and These represent the rotations about the X, Y, and Z axes, respectively.

[0045] like Figure 3 As shown, in peri-field dynamics, the motion of a particle is expressed by an integral-differential equation, i.e., the equation of motion of a point mass: ;

[0046] in, ρ Indicates mass density, u and b These represent displacement and volume force vectors, respectively. δ The range of influence of a certain particle. H X It is a family of particles in the near field. t Representing a point of matter k Acting on a point of matter j Force density on X As a point mass, For point mass X acceleration vector, for X Near field range H X Any particle inside, The volume of each particle, f Let be a bivariate force function, representing a point mass. X 'Acting on a point mass' X Force vector on; In the PD shell model, under the local coordinate system, based on the Lagrange equations and using the principle of virtual work, the equations of motion can be derived as follows: ;

[0047] in q i Indicates degrees of freedom. express q i Time derivative, L The Lagrange function represents the difference between kinetic energy and total potential energy. Specifically, the Lagrange function can be calculated as follows: ;

[0048] in, The strain energy density per unit volume; It is a velocity vector. For volume forces, It is a displacement vector. A (k) For cross-sectional area, This is the quality matrix.

[0049] Considering the interaction states between particles, using the Lagrange equation, the PD equation for the shell structure can be expressed as: ;

[0050] in, Point k and j The interaction state between them; A vector representing force; V j Represents a point mass j The volume.

[0051] Force vector in the equation of motion It can be calculated as: ;

[0052] in, The strain energy density per unit volume; , and These are the particles orbiting each other in the local coordinate system. X , Y and Z The amount of rotation of the shaft, , and The particles are respectively in X , Y and Z In-plane displacement in three directions.

[0053] also, Defined as: ;

[0054] in, This represents the energy release rate of the interaction between particles k and j; g c This represents the average critical energy release rate of the interaction between the corresponding material particles.

[0055] ;

[0056] in, η Indicates the thickness of the shell; Indicates the distance between adjacent particles; and Represented as point masses k and j The micro-potential energy of the interaction between them; V (j) and V(k) They represent point masses respectively. j and k The volume.

[0057] ;

[0058] in, , and These represent the micro-potential energy resulting from plane strain, shear deformation, and bending deformation, respectively.

[0059] Average critical energy release rate corresponding to the interaction between material particles g c The calculation formula is: ,in, G c The critical energy release rate of the immersed tunnel material. N c This represents the total number of interactions traversing a unit crack region. According to existing research, if the near-field range... δ Choose 3 times the interparticle distance, that is ,but N c = 36.

[0060] In addition, to represent the degree of structural damage, the local damage level introduced by silling is referenced. ϕ The calculations were performed, and the specific details are as follows: ;

[0061] in, ( X , t () represents the local damage coefficient. N The total number of interacting particles within the field of view. X (j) and X (k) represent j and k Point mass.

[0062] S20: Perform coordinate transformation between the local and global coordinate systems on the PD shell model, convert the mechanical quantities in the local coordinate system into mechanical quantities in the global coordinate system and superimpose them to obtain the motion equations of the PD shell model in the global coordinate system.

[0063] The immersed tunnel is a three-dimensional rectangular structure, so only the transformation of the planar shell is considered; the curved shell will not be described here. The specific coordinate transformation process from local to global planar shell in step S20 can be achieved through the following steps S201-S202.

[0064] S201: Establish a local triorthogonal basis for each particle.

[0065] A local triorthogonal basis refers to establishing a set of three mutually orthogonal direction vectors (i.e., two tangential vectors) per unit length at a material point φ on the surface of a shell. and ) and a normal direction First, in the global coordinate system ( X , Y , Z Next, give the first k For each particle, define the unit vectors for the local tangent and normal directions, as follows: ;

[0066] in, For point mass k Local coordinate system X Towards the unit vector, For point mass k Local coordinate system Y Towards the unit vector, For point mass k Local coordinate system Z Towards the unit vector, , and Local X To the unit vector in the global X , Y and Z Projected length on the axis , and Local Y To the unit vector in the global X , Y and Z Projected length on the axis , and Local Z To the unit vector in the global X , Y and Z The projected length on the axis.

[0067] Will , and The direction cosine matrix H is formed by stacking rows. k , is represented as: ;

[0068] The coordinate relationship can then be written as: ;

[0069] in, For a particle in local coordinates k , For a point mass in global coordinates k .

[0070] For the shell model, each mass point has a total of 6 degrees of freedom, including displacement and rotation. A 6×6 displacement transformation matrix T is defined. k : ;

[0071] According to the displacement transformation matrix T k get ,on the contrary H k Orthogonal For displacement vector transformation in local coordinate system, This is a displacement vector transformation in the global coordinate system.

[0072] S202: Convert local variables into global variables and then superimpose them.

[0073] According to formula (4), the superposition rule can be obtained as follows: , , ; Based on this superposition rule, the motion equations of the PD shell model in the global coordinate system are further obtained. Among them, The force vector in the global coordinate system. Body strength in the global coordinate system; This is the mass matrix in the global coordinate system.

[0074] S30: Soil-tunnel interaction is represented by an ideal elastic-plastic soil spring.

[0075] For the interaction between the immersed tube and the soil, this embodiment uses empirical springs to represent the soil-tunnel interaction. These springs are all ideally elastic-plastic. Furthermore, these soil response models depend on the type of backfill soil and the tunnel installation conditions. The specific soil spring settings are based on the ALA standard and the research conducted by Liao et al., combined with the actual geological conditions of the immersed tube tunnel. Figure 4 As shown. The specific calculation method is as follows: ;

[0076] in, Indicates the weight per unit soil volume; Indicates the tunnel depth; Indicates the width of the immersed tunnel; This represents the coefficient of lateral pressure on the soil when it is at rest. This represents the frictional resistance between the tunnel surface and the soil. Indicates the internal friction angle of the soil surrounding the tunnel; Indicates the horizontal bearing capacity coefficient; Indicates the vertical lift coefficient; Indicates the vertical bearing capacity coefficient; Vertical bearing capacity influence coefficient; f T Indicates longitudinal force; f P Indicates lateral force; f u Indicates vertical pull-out force; f d Indicates vertical bearing capacity.

[0077] S40: Simulate the flexible joint of the immersed tunnel, use the local weakening method to weaken the stiffness of the joint area, and dynamically update the stiffness and local stiffness matrix of the joint.

[0078] In numerical simulations of immersed tunnels, the treatment of joints is crucial to the accuracy of the overall analysis results. Since the joint region has lower stiffness, typically less than that of the tunnel section itself, its deformation behavior under stress differs from that of the tunnel section. Therefore, accurately simulating the non-uniformity of joint stiffness is key to improving the accuracy of numerical analysis.

[0079] like Figure 5 The diagram shown is a simulation illustration of a joint provided in an embodiment of the present invention. This embodiment employs a near-field dynamic shell theory model, combined with a local weakening method, to simulate the behavior of the joint by weakening the stiffness of the joint region. Specifically, this can be implemented through the following steps S401-S405.

[0080] S401: Setting to weaken joint stiffness.

[0081] The stiffness of the immersed tunnel section remains constant, that is, the elastic modulus of the section is set to... E 1. The joint region is simulated by introducing local weakening. The weakened area of ​​the joint is divided into two parts: the core weakened region and the weakened transition region. The elastic modulus of the core weakened region is... E 2 (lower than the pipe section stiffness) represents the weakened core part of the joint; while the elastic modulus of the weakened transition zone is based on the pipe body stiffness. E 1. Joint stiffness E 2. Perform interpolation transition to ensure a smooth transition of stiffness and guarantee the stability of numerical calculation.

[0082] S402: Selection of the weakening range.

[0083] The weakening range is determined by calculating the distance between the node and the preset target (weakening core area and transition area). Based on the distance from the node to the target area, the weakening factor is calculated. w And based on this factor, the elastic modulus of the material is dynamically adjusted.

[0084] Weakening the core area ( l 1): For distances less than the target area l At node 1, the material completely degrades, that is... w = 0.

[0085] Transition zone ( l 2): For distances from the target area l 1 and l 1+ l At nodes between 2, the stiffness of the material gradually changes. w Linear interpolation is performed based on the distance from the target area.

[0086] Normal area: For areas greater than the target area l 1+ l At node 2, the normal stiffness of the material is maintained, i.e. w = 1.

[0087] Weakening factor w The calculation is based on the distance between the node and the target region, and the specific formula is as follows: ;

[0088] S403: Local stiffness calculation.

[0089] The elastic modulus transitions smoothly between the weakened core region and the normal region, specifically the local elastic modulus. E l The calculation formula is: ;

[0090] shear modulus G From elastic modulus E Poisson's ratio ν The specific local shear modulus was calculated. G l The formula is: ;

[0091] S404: Calculation of energy coefficient.

[0092] elastic modulus E l Changes in the coefficients of thermal expansion affect various energy coefficients of materials, such as in-plane strain energy, bending strain energy, and shear strain energy. The in-plane strain energy constant is used as an example. A ip1For example, the calculation formula is as follows: ;

[0093] Similarly, other energy constants also need to be determined based on... E l Adjustments will be made.

[0094] S405: Update of local mechanical parameters.

[0095] Changes in stiffness directly affect the calculation of mechanical parameters. In the simulation, mechanical parameters such as joint stiffness and the local stiffness matrix are affected by weakening factors. w and local elastic modulus E l It is dynamically updated.

[0096] S50: The immersed tunnel is discretized using a meshless method, which discretizes the immersed tunnel into particles with uniform spacing. The horizontal integral is calculated by summing the centroids of all particles. Soil springs are applied to the material layer near the boundary in the form of volume force to apply differential settlement of the strata in a quasi-static manner.

[0097] This embodiment uses a meshless method for calculation. The rectangular immersed tunnel is discretized into uniformly spaced particles associated with a specific volume. The horizontal integral can be obtained by summing over the centroids of all particles. In PD theory, all external loads are applied as volume forces in the material layer near the boundary. Here, ideal elastic soil springs acting around the immersed tunnel are added to the discretized system by applying specific volume forces in the corresponding directions. Differential ground settlement is applied to the immersed tunnel in a quasi-static manner.

[0098] S60: The motion equations of the PD shell model under static or quasi-static conditions in the global coordinate system are solved using an adaptive dynamic relaxation scheme.

[0099] To accelerate the system's approach to steady state, the Adaptive Dynamic Relaxation (ADR) scheme proposed by Kilic and Madenci is used to solve the peri-field dynamic equations in the static or quasi-static cases in global coordinates. The specific equations are: ;

[0100] in, D Represents the virtual diagonal density matrix; c The damping coefficient; X and U These represent the initial position and displacement of the particle, respectively; force vector. F It consists of the interaction between PD particles and the physical forces exerted on the immersed tunnel by the soil spring.

[0101] In addition, referring to silling's settings, the near-field range δ Set to 3 times the particle spacing Δ X It can efficiently save computation time and achieve high computational accuracy.

[0102] S70: Output the simulation results of longitudinal deformation, joint opening, and potential cracking risk of the immersed tunnel segment based on the solution results.

[0103] The above numerical solution process can be implemented in a self-written C++ program, and the flowchart of the calculation program is as follows: Figure 6 As shown.

[0104] To verify the feasibility of the method proposed in this invention, this embodiment uses a model test conducted by (Li et al. 2025) for numerical verification. The test model consists of four pipe section units, each with a length of 1000 mm, for a total length of 4000 mm. The experiment was carried out using a graded loading method. Relying on a synchronous control system, vertical displacement was applied at a displacement rate of 0.02 mm / s by precisely controlling an electro-hydraulic jack, and differential settlement was induced step by step. The settlement increment for each loading level was 5 mm, ultimately reaching 40 mm.

[0105] The concrete used in the experiment was C30 fine aggregate concrete, with a mix ratio of cement:sand:crushed stone:water = 1:1.96:2.4:0.48, and its compressive strength was measured to be 28.6 MPa. Geometrically, the cross-sectional dimensions were scaled down from the standard pipe section dimensions of a certain immersed tunnel in Guangzhou at a 1:44 ratio, with a pipe section cross-sectional dimension of 0.5 m × 0.16 m (width × height) and a soil cover thickness of 0.2 m. The shear stiffness of the immersed tube ranged from 1 / 76 to 1 / 24; in this verification, a shear stiffness value of 1 / 60 was selected for analysis. During the numerical analysis, the soil parameters and material parameters of the immersed tube sections involved in the model are detailed in Table 1. These parameters will be used to calculate the deformation response of the pipe section under different working conditions and the influence of differential settlement with the foundation, to verify the effectiveness and feasibility of the proposed near-field dynamics-based joint stiffness weakening equivalent PD shell model.

[0106] Table 1 Material Parameters

[0107] Figure 7 This diagram illustrates the variation of vertical deformation along the tunnel under different foundation differential settlement conditions when the joint stiffness is weakened to 1 / 60th of the actual pipe section. The diagram shows a linear relationship between the vertical deformation along the tunnel and the differential foundation settlement. In the weakened joint zone of the tunnel (… J 1. J 2.J 3) Due to the influence of the soil cover on the left side and the self-weight of the pipe section, uneven settlement occurs at the joint. J The deformation response of the two joints J The impact on section 1 and the leftmost pipe section is relatively small. (Except for the joint) J 1. External, connector J 2 and J There are significant differences in vertical deformation between the left and right pipe sections. Specifically: J The right-side pipe section of joint 2 is affected by uneven foundation settlement, resulting in downward settlement; due to the stiffness of the joint, J The left side of the joint also experienced some vertical deformation, but due to the foundation and J 1. Constraints on the left side pipe section J 2. The vertical deformation of the left pipe section is smaller than that of the right section, as reflected by the curve in the figure. J 2. Vertical deformation and rotation angle of the left and right pipe sections at the joint. J 3. The right side is mainly affected by the resistance of the surrounding soil, resulting in J The three locations showed signs of subsidence, and this phenomenon became more pronounced as the uneven settlement of the foundation increased.

[0108] Furthermore, by comparing the results of existing scaled-down model tests, it can be found that the results obtained by this method are basically consistent with the conclusions of the model tests, further verifying the rationality and effectiveness of the method proposed in this invention.

[0109] Figure 8 The relationship between joint opening and differential settlement is demonstrated. Joint opening is an important indicator for assessing the leakage risk of immersed tube joints and is of great significance to the sealing performance of the structure. Overall, the calculation results of this invention are basically consistent with the results obtained from model tests, both showing that joint J2 has the largest opening, followed by J3 and finally J1. This is due to the effect of differential settlement on J2. In summary, this analysis further demonstrates the reliability of the calculation values ​​calculated by the method proposed in this invention.

[0110] Figure 9 Differential settlement δ = 30 mm deformation cloud diagram of the immersed tube body and joints. As shown in the figure, under differential settlement, joint J2 deforms first and produces a certain angle, manifested as compression at the bottom and tension at the top. Subsequently, joint J3 undergoes the opposite deformation to J2, manifested as tension at the bottom and compression at the top. Throughout the process, the deformation at joint J1 is very small, which also intuitively shows that the calculation results of the proposed method are basically consistent with the experimental results, verifying the reliability and efficiency of the method.

[0111] The above embodiments are only used to illustrate the present invention and are not intended to limit the present invention. Those skilled in the art can make various changes and modifications without departing from the spirit and scope of the present invention. Therefore, all equivalent technical solutions also fall within the scope of the present invention, and the patent protection scope of the present invention should be defined by the claims.

Claims

1. A numerical simulation method for immersed tunnel joints considering differential foundation settlement, characterized in that, The method includes: Based on the fundamental theory of near-field dynamics plate and shell, a PD shell model equivalent to weakening the stiffness of the immersed tunnel joint is constructed. The degrees of freedom of each mass point in the PD shell model are defined, and the motion equations of the mass points are established. The degrees of freedom of each mass point include three displacements and three rotations in the global coordinate system, and three displacements and three rotations in the local coordinate system. The PD shell model is subjected to coordinate transformation between the local coordinate system and the global coordinate system. The mechanical quantities in the local coordinate system are converted into mechanical quantities in the global coordinate system and superimposed to obtain the motion equation of the PD shell model in the global coordinate system. Soil-tunnel interaction is represented by a soil spring with ideal elastic-plastic properties; The flexible joint of the immersed tunnel was simulated, and the stiffness of the joint area was weakened by the local weakening method. The stiffness and local stiffness matrix of the joint were dynamically updated. The immersed tunnel is discretized using a meshless method, which discretizes the immersed tunnel into particles with uniform spacing. The horizontal integral is calculated by summing the centroids of all particles. Soil springs are applied to the material layer near the boundary in the form of volume forces to apply differential settlement of the strata in a quasi-static manner. An adaptive dynamic relaxation scheme is used to solve the motion equations of the PD shell model in the static or quasi-static case in the global coordinate system. Based on the solution results, the simulation results of longitudinal deformation, joint opening, and potential cracking risk of the immersed tunnel segment are output.

2. The numerical simulation method for immersed tunnel joints considering differential foundation settlement as described in claim 1, characterized in that, The equation of motion for the particle is expressed as: ； in, ρ Indicates mass density, u and b These represent displacement and volume force vectors, respectively. δ The range of influence of a certain particle. H X It is a family of particles in the near field. t Representing a point of matter k Acting on a point of matter j Force density on X As a point mass, For point mass X acceleration vector, for X Near field range H X Any particle inside, The volume of each particle, f Let be a bivariate force function, representing a point mass. X 'Acting on a point mass' X Force vector on; In the PD shell model, under the local coordinate system, based on the Lagrange equations and using the principle of virtual work, the following equations of motion are derived: ； in q i Indicates degrees of freedom. express q i Time derivative, L The Lagrangian function, representing the difference between kinetic energy and total potential energy, is expressed as: ； in, The strain energy density per unit volume; It is a velocity vector. For volume forces, It is a displacement vector. A (k) For cross-sectional area, This is the quality matrix; Considering the interaction states between particles, the PD equation for the shell structure can be expressed using the Lagrange equation as follows: ； in, Point k and j The interaction state between them; A vector representing force; V j Represents a point mass j Volume; Force vectors in the equations of motion The calculation formula is: ； in, The strain energy density per unit volume; , and These are the particles orbiting each other in the local coordinate system. X , Y and Z The amount of rotation of the shaft, , and The particles are respectively in X , Y and Z In-plane displacement in three directions, Defined as: ； in, This represents the energy release rate of the interaction between particles k and j; g c This represents the average critical energy release rate of the interaction between the corresponding material particles; ； in, η Indicates the thickness of the shell; Indicates the distance between adjacent particles; and Represented as point masses k and j The micro-potential energy of the interaction between them; V (j) and V (k) They represent point masses respectively. j and k Volume; ； in, , and These represent the micro-potential energy resulting from plane strain, shear deformation, and bending deformation, respectively. Average critical energy release rate corresponding to the interaction between material particles g c The calculation formula is: ,in, G c The critical energy release rate of the immersed tunnel material. N c This represents the total number of interactions traversing a unit crack region. Local damage degree is introduced to characterize the degree of damage to the structure. The formula for calculating the local damage degree is as follows: ； in, ( X , t () represents the local damage coefficient. N The total number of interacting particles within the field of view. X (j) and X (k) represent j and k Point mass.

3. The numerical simulation method for immersed tunnel joints considering differential foundation settlement as described in claim 2, characterized in that, The PD shell model undergoes a coordinate transformation between the local and global coordinate systems. The mechanical quantities in the local coordinate system are converted to those in the global coordinate system and then superimposed to obtain the motion equations of the PD shell model in the global coordinate system, including: Establish a local triorthogonal basis for each particle; Based on the local triorthogonal basis and the superposition rule determined by formula (4), the motion equations of the PD shell model in the global coordinate system are obtained.

4. The numerical simulation method for immersed tunnel joints considering differential foundation settlement as described in claim 3, characterized in that, Establish a local triorthogonal basis for each particle, including: In the global coordinate system ( X , Y , Z Next, give the first k For each particle, define the unit vectors for the local tangent and normal directions, as follows: ； in, For point mass k Local coordinate system X Towards the unit vector, For point mass k Local coordinate system Y Towards the unit vector, For point mass k Local coordinate system Z Towards the unit vector, , and Local X To the unit vector in the global X , Y and Z Projected length on the axis , and Local Y To the unit vector in the global X , Y and Z Projected length on the axis , and Local Z To the unit vector in the global X , Y and Z Projected length on the axis; Will , and The direction cosine matrix H is formed by stacking rows. k , represented as: ； According to the direction cosine matrix H k The coordinate relationship is expressed as: ； in, For a particle in local coordinates k , For a point mass in global coordinates k ; Each particle has 6 degrees of freedom, and a 6×6 displacement transformation matrix T is defined. k for: ； According to the displacement transformation matrix T k get ,on the contrary H k Orthogonal For displacement vector transformation in local coordinate system, This is a displacement vector transformation in the global coordinate system.

5. The numerical simulation method for immersed tunnel joints considering differential foundation settlement as described in claim 4, characterized in that, The superposition rule is expressed as follows: ， ， ； in, The force vector in the global coordinate system. Body strength in the global coordinate system; This is the mass matrix in the global coordinate system.

6. The numerical simulation method for immersed tunnel joints considering differential foundation settlement as described in claim 1, characterized in that, The relevant calculation formula for the soil spring is as follows: ； in, Indicates the weight per unit soil volume; Indicates the tunnel depth; Indicates the width of the immersed tunnel; This represents the coefficient of lateral pressure on the soil when it is at rest. This represents the frictional resistance between the tunnel surface and the soil. Indicates the internal friction angle of the soil surrounding the tunnel; Indicates the horizontal bearing capacity coefficient; Indicates the vertical lift coefficient; Indicates the vertical bearing capacity coefficient; Vertical bearing capacity influence coefficient; f T Indicates longitudinal force; f P Indicates lateral force; f u Indicates vertical pull-out force; f d Indicates vertical bearing capacity.

7. The numerical simulation method for immersed tunnel joints considering differential foundation settlement as described in claim 1, characterized in that, A simulation of the flexible joint in an immersed tunnel was conducted. A local weakening method was used to reduce the stiffness of the joint area, and the stiffness and local stiffness matrix of the joint were dynamically updated, including: A joint stiffness weakening region is defined, comprising a core weakening region and a weakening transition region; the elastic modulus of the immersed tunnel section is set to... E 1. The elastic modulus of the core weakened region is set to E 2, E 2< E 1. The elastic modulus of the weakened transition region is E 1 and E Interpolation of 2; Set the distance threshold of the core weakened region as l 1. The distance threshold for the weakened transition region is... l 2; the distance from the node to the target weakened region d < l At time 1, the node is in the core weakened zone; when l 1≤ d ≤ l 1 + l At time 2, the node is in the weakened transition region; when d > l 1 + l At time 2, the node is in the normal range; Calculate the weakening factor w The calculation formula is: ； Calculate the local elastic modulus E l The calculation formula is: ； Calculate the local shear modulus G l The calculation formula is: ； in, v For the material Poisson's ratio; Calculate the in-plane strain energy constant A ip1 The calculation formula is: ； Based on weakening factor w and local elastic modulus E l The stiffness and local stiffness matrix of the joint are dynamically updated.

8. The numerical simulation method for immersed tunnel joints considering differential foundation settlement as described in claim 1, characterized in that, The motion equations of the PD shell model under static or quasi-static conditions in the global coordinate system are expressed as follows: ； in, D Represents the virtual diagonal density matrix; c The damping coefficient; X and U These represent the initial position and displacement of the particle, respectively; force vector. F It consists of the interaction between PD particles and the physical forces exerted on the immersed tunnel by the soil spring.

9. The numerical simulation method for immersed tunnel joints considering differential foundation settlement as described in claim 1, characterized in that, The soil spring is an ideal elastic-plastic spring. When the load on the soil spring is less than the yield load, the soil spring is in the elastic stage, and the load and deformation are linearly related. When the load reaches the yield load, the soil spring enters the plastic stage, and the deformation continues to increase while the load remains constant.

10. The numerical simulation method for immersed tunnel joints considering differential foundation settlement as described in claim 1, characterized in that, The methods for applying differential settlement of strata in a quasi-static manner include: controlling the settlement displacement rate at 0.02 mm / s, adopting a graded loading mode, with a settlement increment of 5 mm for each loading stage, until the total settlement reaches 40 mm.

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