Nonlinear random analysis method for longitudinal mechanical performance of shield tunnel

Abstract

The application relates to the technical field of tunnel and underground structure analysis, and discloses a nonlinear random analysis method for longitudinal mechanical properties of a shield tunnel, which comprises the following steps: assembling a structure stiffness matrix and a load array; generating a double-parameter coupled random field of a foundation; generating a nonlinear random foundation support constraint; adaptively and iteratively solving a nonlinear system; and outputting a shield longitudinal mechanical property index system. The core is as follows: a space correlation matrix of an initial stiffness of a foundation and a limit counterforce of the foundation is constructed, a double-parameter coupled random field capable of reflecting the deterioration characteristics of a soft and weak stratum is generated through joint random disturbance, a tail joint probability triggering mechanism and exponential decay mapping, an adaptive relaxation control strategy is adopted, an equivalent stiffness matrix and a residual load are dynamically updated, and the nonlinear system is iteratively solved. The method can represent the nonlinear characteristics of multiple-source uncertainty of a foundation and tunnel structure-foundation interaction, and improves the accuracy and robustness of longitudinal mechanical property analysis of the shield tunnel under complex stratum conditions.

Description

Technical Field

[0001] This invention relates to the field of mechanical analysis technology for tunnels and underground structures, specifically to a nonlinear stochastic analysis method for the longitudinal mechanical properties of shield tunnels. Background Technology

[0002] During the construction and operation of shield tunnels, the longitudinal stress and deformation patterns of the structure are important bases for evaluating the overall safety of the tunnel, the durability of the segments, and the accuracy of construction posture control.

[0003] Due to the influence of multiple sources of disturbance, such as geological heterogeneity, complex groundwater environment, non-uniform buoyancy, voiding, and assembly posture adjustments, shield tunnels are prone to adverse responses along their longitudinal direction, including uneven settlement, bending distortion, shear deformation, and joint cracking. If the longitudinal mechanical behavior cannot be accurately simulated and identified, risks such as structural stress imbalance, segment joint failure, and degradation of waterproofing function may gradually accumulate and evolve during construction or operation, posing potential hazards to project safety. Therefore, establishing a longitudinal mechanical performance analysis method for tunnel structures that can accurately reflect the multi-source uncertainties of the foundation and the longitudinal nonlinear characteristics of the tunnel structure-foundation interaction is of significant engineering importance for the safety control of shield tunnels.

[0004] Existing longitudinal analyses of shield tunnels often employ simplified elastic foundation beam models, treating the strata as a linear or homogeneous support system and assuming that the initial stiffness and ultimate reaction force are independent. This analytical framework struggles to reflect the nonlinear characteristics of foundation reaction force variations with displacement in weak strata, and cannot describe actual behaviors such as ultimate reaction force attenuation, coupling degradation, and correlations among foundation parameters. Furthermore, traditional random field methods typically only handle single-parameter perturbations and cannot characterize the coupling effects of initial foundation stiffness and ultimate reaction force in terms of spatial correlation and probability distribution, limiting their ability to simulate the dynamic degradation of weak foundations. Simultaneously, existing solution methods generally employ iterative strategies with fixed step sizes or fixed relaxation factors, which are prone to convergence instability or low computational efficiency when dealing with strong nonlinearities in the foundation and random input perturbations.

[0005] In summary, existing technologies still have significant shortcomings in the simulation of two-parameter correlation of foundations, modeling of nonlinear foundation supports, and longitudinal nonlinear iterative solutions under random disturbance conditions. These limitations make it difficult to meet the demands of refined longitudinal mechanical analysis under the combined effects of randomness, nonlinearity, and coupling in practical engineering projects. Therefore, it is necessary to propose a nonlinear stochastic analysis method for the longitudinal mechanical properties of shield tunnels that comprehensively considers the coupled random field of the foundation's two-parameter structure, the nonlinear support characteristics of the foundation, and an adaptive iterative solution strategy. This method aims to improve the adaptability and reliability of the analysis results to complex soft strata and load conditions. Summary of the Invention

[0006] The main objective of this invention is to provide a nonlinear stochastic analysis method for the longitudinal mechanical properties of shield tunnels, aiming to address the significant shortcomings of existing technologies in areas such as dual-parameter coupled simulation of the foundation, nonlinear foundation support modeling, and longitudinal nonlinear iterative solutions under stochastic disturbance conditions. The specific technical solution is as follows:

[0007] A nonlinear stochastic analysis method for the longitudinal mechanical properties of a shield tunnel includes the following steps:

[0008] Step 1: Obtain shield tunnel parameters; divide finite element node coordinate information;

[0009] Step 2: Assemble the structural stiffness matrix and load array based on the shield tunnel parameters obtained in Step 1;

[0010] Step 3: Generate a two-parameter coupled random field for the foundation, specifically including:

[0011] Step 3.1: Construct the spatial covariance matrix and perform Cholesky decomposition. Specifically, based on the finite element node coordinate information obtained in Step 1, form a spatial correlation matrix; and obtain the lower triangular matrix through Cholesky decomposition.

[0012] Step 3.2: Construct a standard normal joint random field, specifically: establish a linear correlation matrix between parameters; generate a standard normal white noise matrix and introduce spatial correlation; obtain a spatially correlated and parameter-coupled joint random field through the parameter coupling matrix;

[0013] Step 3.3: Map and calculate the log-normal coupled random field of the foundation's two parameters;

[0014] Step 3.4: Calculate the coupling triggering factor and implement the exponential decay mapping of the two-parameter coupled random field. Specifically, the tail joint probability of each node is calculated using a two-dimensional normal distribution, and the coupling triggering factor is constructed. The corrected random field of the initial stiffness and ultimate reaction force of the foundation is obtained by using the exponential decay mapping.

[0015] Step 3.5: Perform mean correction and lower bound protection on the modified random field to obtain the final coupled random field of initial foundation stiffness and ultimate foundation reaction force;

[0016] Step 4: Generate nonlinear stochastic foundation support constraints;

[0017] Step 5: Adaptively iteratively solve the nonlinear system and output the longitudinal mechanical performance index system of the shield.

[0018] Preferably, the shield tunnel parameters in step one include the section bending stiffness, the total length of the shield tunnel structure analysis, the total number of nodes in the Timoshenko beam model of the shield tunnel, the section shear stiffness, the element length, the perimeter of the outer surface of the tunnel, and the load characteristics; the load characteristics include buoyancy, equivalent void force, the amplitude of time-varying loads, and the application location.

[0019] The following is the output system of longitudinal mechanical performance indicators for the tunnel boring machine in step five:

[0020] ;

[0021] in: , , , , as well as These are the longitudinal positions of the shield tunnel. Vertical deformation, rotation, bending moment, shear force, longitudinal curvature, and actual structural load.

[0022] Preferably, step two specifically includes:

[0023] Based on the parameters determined in step one, the total number of nodes in the Timoshenko beam model of the shield tunnel is determined. Given the longitudinal coordinates of the tunnel A finite element model of a shield tunnel was established using Timoshenko beam theory; the stiffness matrix of the beam element in the Timoshenko beam tunnel was established. ,in: Here is the element bending stiffness matrix; This is the element shear stiffness matrix; through the matrix mapping of element stiffness to global stiffness, a basic stiffness matrix for shield tunnels that can superimpose boundary constraints is formed. ,in: Let T be the mapping matrix from the unit to the global coordinates, and T be the transpose.

[0024] Preferably, in step 3.1: the vertical coordinate... Location With the Location covariance It is expressed as follows:

[0025] ;

[0026] in: The horizontal correlation length;

[0027] Lower triangular matrix The expression is as follows:

[0028] .

[0029] Preferably, in step 3.2: the linear correlation matrix between parameters The expression is as follows:

[0030] ;

[0031] in: Belongs to the initial stiffness of the foundation ultimate reaction force of foundation The linear correlation coefficient between them; For Cholesky decomposition function;

[0032] Space related The expression is as follows:

[0033] ;

[0034] in: The matrix is ​​a standard normal white noise matrix with a matrix scale of . , The number of random simulations; This is a function used to construct a block diagonal matrix;

[0035] Spatially correlated and parameter-coupled joint random fields The expression is as follows:

[0036] ;

[0037] in: Initial stiffness of the foundation The standard normal random field; ultimate reaction force of the foundation The standard normal random field; and The matrix scale is ; It is the identity matrix, and its matrix scale is ; The notation for calculating the Kronecker product.

[0038] Preferably, the mapping calculation of the log-normal coupled random field of the foundation two parameters in step 3.3 is as follows:

[0039] The initial foundation parameters are log-normally mapped as follows:

[0040] ; ;

[0041] in: ; ; ; ; , and Let represent the mean, coefficient of variation, and log-normal random field of the initial stiffness of the foundation, respectively; , and Let represent the mean, coefficient of variation, and base log-normal random field of the ultimate reaction force of the foundation, respectively.

[0042] Preferably, the coupling triggering factor in step 3.4 The expression is as follows:

[0043] ;

[0044] in: To scale, ; The shape index, ; This is the trigger threshold; This represents the upper limit of the coupling triggering factor. The joint probability of the tail of a node;

[0045] Corrected random field for initial stiffness of foundation Modified random field of foundation ultimate reaction force The expression is as follows:

[0046] ; ;

[0047] in: The attenuation strength of the initial stiffness of the foundation; The attenuation intensity of the ultimate reaction force of the foundation; This is the natural exponential function in mathematics.

[0048] Preferably, the final coupled random field of the initial stiffness of the foundation in step 3.5 The final coupled random field with the ultimate reaction force of the foundation The expression is as follows:

[0049] ; ;

[0050] in: The minimum allowable value for the initial stiffness of the foundation set for the project; The minimum allowable value of the ultimate reaction force of the foundation set for the project; The symbol for the function that retrieves the maximum value; This is a function for taking the average value.

[0051] The total number of random simulations required is [number]. Next, the first of them The final coupled stochastic field of the initial foundation stiffness and ultimate foundation reaction obtained in this step is denoted as follows: and .

[0052] Preferably, step four, generating nonlinear stochastic foundation support constraints, specifically includes the following steps:

[0053] Constructing the energy expression of a nonlinear foundation under displacement perturbation :

[0054] ;

[0055] pass The first-order and second-order derivatives yield the expression for the foundation reaction force. Jacobi expression for foundation support As shown below:

[0056] ; ;

[0057] in: This represents the vertical displacement value of the tunnel node; and These represent the undetermined values ​​of the initial stiffness and ultimate reaction force of the foundation, respectively.

[0058] The first The final coupled random field of the initial stiffness and ultimate reaction of the foundation obtained in this step and Substitute and In the middle, it won the first The vertical coordinate position during the random calculation is Scalar functions of foundation reaction force and Jacobi scalar functions of foundation support and ,as follows:

[0059] ;

[0060] ;

[0061] Based on the node division information of the shield tunnel finite element model from step two, the nonlinear foundation reaction force and foundation support of the shield tunnel are arrayed and matrixed to establish the first... Nonlinear stochastic foundation reaction function array during sub-random calculation and the Jacobi function matrix of nonlinear stochastic foundation support ,as follows:

[0062] ;

[0063] ;

[0064] in: Let the vertical displacement vectors of the nodes be column matrices. ; The perimeter of the tunnel's outer surface; The unit length;

[0065] Regarding the already obtained and In subsequent iterative solutions, it is only necessary to update the column matrix of vertical displacement vectors of the nodes in real time. By incorporating these parameters into the calculation, the Jacobi matrix of random foundation support and the array of random foundation reaction forces can be updated instantly.

[0066] Preferably, step five, the adaptive iterative solution of the nonlinear system, specifically includes the following steps:

[0067] Initialize nodal displacements Vertical displacement of nodes Random foundation reaction function array and the Jacobi function matrix of random foundation support Generate the initial equivalent stiffness matrix With initial residual load And calculate the initial nodal displacement increment. ;

[0068] During the iterative solution process, i.e., at the th... In the next iteration, the calculation is based on the current node displacement. Vertical displacement of nodes Dynamically update the Jacobi function matrix of random foundation support Equivalent stiffness matrix Calculate the displacement increment And correct nodal displacements Simultaneously, the random foundation reaction function array is recalculated. With residual load When the residuals satisfy the convergence condition or number of iterations Reaching the maximum set value At that time, output the final node displacement result. ;

[0069] In each iteration, according to the... Rate of change of residual in the next iteration Adaptive adjustment of relaxation factor To improve the convergence stability and computational efficiency of nonlinear stochastic systems;

[0070] in: This represents the load array obtained in step two; and They represent the first Second and third The residual load generated by the next iteration update; Represents the index of a vector or column matrix, starting from the first number and going every two numbers until the last number; This is the convergence limit; It is a norm function; To prevent division by zero of small constants; The step size adjustment index; and These represent the lower and upper limits of the relaxation factor, respectively. This is the amplitude limiting operator function.

[0071] The effect of applying the technical solution of this invention is:

[0072] The nonlinear stochastic analysis method for the longitudinal mechanical properties of shield tunnels disclosed in this invention includes: acquiring shield tunnel parameters; dividing finite element node coordinate information; assembling the structural stiffness matrix and load array based on the obtained shield tunnel parameters; generating a two-parameter coupled random field of the foundation (including: forming a spatial correlation matrix; constructing a standard normal joint random field; mapping and calculating the log-normal coupled random field of the foundation's two parameters; obtaining the corrected random field of the foundation's initial stiffness and ultimate reaction force; obtaining the final coupled random field of the foundation's initial stiffness and ultimate reaction force); generating nonlinear stochastic foundation support constraints; adaptively iteratively solving the nonlinear system; and outputting the shield tunnel's longitudinal mechanical performance index system. This method first establishes an equivalent longitudinal finite element model of the shield tunnel based on Timoshenko beam theory and constructs a spatial correlation matrix of the foundation's initial stiffness and ultimate reaction force. Through joint random perturbation, a tail-end joint probability triggering mechanism, and exponential decay mapping, a two-parameter coupled log-normal random field reflecting the deterioration characteristics of weak strata is generated. Based on this, a nonlinear stochastic foundation reaction function and a Jacobi support function matrix are constructed to achieve real-time updates of the foundation support effect under displacement disturbances. Through an adaptive relaxation control-based nonlinear iterative solution strategy, the equivalent stiffness matrix and residual loads are dynamically updated to obtain key performance indicators of the shield tunnel, including vertical displacement, rotation, bending moment, shear force, longitudinal curvature, and actual structural loads. This method can simultaneously characterize the multi-source uncertainties of the foundation and the nonlinear characteristics of the tunnel structure-foundation interaction, improving the accuracy and robustness of the longitudinal mechanical performance analysis of shield tunnels under complex geological conditions. Attached Figure Description

[0073] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the structures shown in these drawings without creative effort.

[0074] Figure 1 This is a flowchart of a stochastic analysis method for the longitudinal mechanical properties of shield tunnels based on adaptive nonlinear iterative solution in an embodiment of the present invention;

[0075] Figure 2 This is the final coupled random field result of the initial stiffness of the foundation in the embodiment of the present invention;

[0076] Figure 3 This is the final coupled random field result of the foundation ultimate reaction force in the embodiment of the present invention;

[0077] Figure 4 This is the iterative curve of the adaptive iterative solution process in the embodiments of the present invention;

[0078] Figure 5 This is a comparative analysis of the longitudinal mechanical performance indicators (taking vertical deformation as an example) obtained from the solutions in the embodiments of the present invention;

[0079] The realization of the objective, functional features and advantages of the present invention will be further explained in conjunction with the embodiments and with reference to the accompanying drawings. Detailed Implementation

[0080] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without creative effort are within the scope of protection of the present invention.

[0081] Example:

[0082] A nonlinear stochastic analysis method for the longitudinal mechanical properties of shield tunnels, the process is detailed in [link to flowchart]. Figure 1 It includes the following steps:

[0083] First, obtain the shield tunnel parameters; divide the finite element node coordinate information;

[0084] Obtaining shield tunnel parameters involves acquiring and inputting key parameters, as detailed below:

[0085] Through real-world engineering cases or virtual engineering scenarios, identify and input the key parameters required for the finite element modeling, foundation two-parameter coupled stochastic simulation, nonlinear foundation constraint generation, and nonlinear system adaptive iterative solution steps:

[0086] The parameters required for assembling the structural stiffness matrix and load array include: section bending stiffness. The total number of nodes in the Timoshenko beam model of a shield tunnel. Section shear stiffness ; Element length ; Perimeter of the outer surface of the tunnel Load characteristics include buoyancy, equivalent free-floating force, amplitude and application location of other time-varying loads.

[0087] The parameters required for generating a two-parameter coupled random field for a foundation include: the type of random variable analysis (foundation ultimate reaction force). or initial stiffness of foundation ); Horizontal correlation length of random fields Mean value of initial stiffness of foundation Coefficient of variation of initial stiffness of foundation Mean value of ultimate reaction force of foundation Coefficient of variation of ultimate reaction force of foundation Number of random simulations Initial stiffness of foundation ultimate reaction force of foundation Linear correlation coefficient between Scaling Shape index Trigger threshold Upper limit of coupling triggering factor The attenuation strength of the initial stiffness of the foundation The attenuation intensity of the ultimate reaction force of the foundation The minimum allowable value of the initial stiffness of the foundation as specified in the project. Minimum allowable value of ultimate reaction force of foundation .

[0088] Parameters required for adaptive iterative solution of nonlinear systems: maximum set value for the number of iterations Convergence limit Small constants to prevent division by zero Step size adjustment index Lower limit of relaxation factor and upper limit .

[0089] See Table 1 for details:

[0090] Table 1 Key Parameters

[0091]

[0092] Second, assemble the structural stiffness matrix and load array, specifically:

[0093] The total number of nodes in the Timoshenko beam model of the shield tunnel. Given the longitudinal coordinates of the tunnel Considering the bending deformation, shear deformation, and their bending-shear coupling effect in the longitudinal mechanical behavior of shield tunnels, a finite element model of the shield tunnel is established using Timoshenko beam theory. The stiffness matrix of the beam element in the Timoshenko beam tunnel is then established. By mapping element stiffness to overall stiffness, a basic stiffness matrix for shield tunnels with superimposed boundary constraints is formed. This is used to support subsequent nonlinear solutions.

[0094] In this embodiment, the element bending stiffness matrix and element shear stiffness matrix It can be expressed as follows:

[0095] ;

[0096] ;

[0097] ;

[0098] in: The mapping matrix from the unit to the global coordinates; The shear-bending ratio is dimensionless. The flexural stiffness of the cross section; Shear stiffness of the cross section; The unit length is denoted as .

[0099] Considering the buoyancy of the shield tunnel Equivalent vacancy force Or other time-varying loads The loads are distributed to the finite element nodes of the shield tunnel through spatial interpolation and node mapping, forming a structural load array. This enables high-precision spatial loading representation of structural loads.

[0100] In this embodiment, the element stiffness matrix The specific values ​​are shown in Table 2:

[0101] Table 2 Element Stiffness Matrix Specific numerical table

[0102]

[0103] Third, generate a two-parameter coupled random field for the foundation, specifically:

[0104] To achieve the initial stiffness of the foundation in a nonlinear foundation ultimate reaction force of foundation Robust coupled stochastic simulations are mainly achieved through the following steps:

[0105] Step 3.1: Construct the spatial covariance matrix and perform Cholesky decomposition, specifically:

[0106] Based on the coordinate information of the partitioned finite element nodes, a spatial correlation matrix is ​​formed. The vertical coordinate is the first Location With the Location covariance The following is an expression:

[0107] ;

[0108] The lower triangular matrix is ​​obtained through Cholesky decomposition. :

[0109] ;

[0110] in: The horizontal correlation length.

[0111] In this embodiment, this decomposition can be quickly calculated using the `chol` function in the commercial software MATLAB, which is a conventional technique in this field.

[0112] Step 3.2: Construct a standard normal joint random field, specifically:

[0113] Establish the linear correlation matrix between parameters :

[0114] ;

[0115] Generate a standard normal white noise matrix The matrix scale is And introduce spatial correlation :

[0116] ;

[0117] A spatially correlated and parametrically coupled joint random field is obtained through the parametric coupling matrix:

[0118] ;

[0119] in: The number of random simulations; Initial stiffness of the foundation ultimate reaction force of foundation The linear correlation coefficient between them ; and These are the initial stiffness of the foundation. ultimate reaction force of foundation A standard normal random field with a matrix scale of ; For the scale The identity matrix; The Cholesky decomposition function can be calculated using the chol function in the commercial MATLAB software in this embodiment, which is a conventional method. The function used to construct the block diagonal matrix can be the blkdiag function in the commercial software MATLAB, which is a conventional method. For the Kronecker product calculation symbol, this embodiment can use the kron function in the commercial software MATLAB for calculation, which is a conventional method. Assuming a standard normal distribution, this embodiment uses the randn function from the commercial MATLAB software for calculation, which is a conventional method.

[0120] Step 3.3: Mapping and calculating the log-normal coupled random field of the foundation's two parameters, specifically:

[0121] Log-normal mapping of initial foundation parameters:

[0122] ; ;

[0123] ; ; ; ;

[0124] in: , and Let represent the mean, coefficient of variation, and log-normal random field of the initial stiffness of the foundation, respectively; , and Let represent the mean, coefficient of variation, and base log-normal random field of the ultimate reaction force of the foundation, respectively.

[0125] Step 3.4: Calculate the coupling triggering factor and implement the exponential decay mapping of the two-parameter coupled random field, specifically:

[0126] The tail joint probability of each node is calculated using a two-dimensional normal distribution. And construct coupling triggering factors :

[0127] ;

[0128] The corrected random field for the initial stiffness and ultimate reaction of the foundation is obtained by using an exponential decay mapping. and :

[0129] ; ;

[0130] in: To scale; The shape index; This is the trigger threshold; This represents the upper limit of the coupling triggering factor. and These represent the attenuation intensity of the initial stiffness of the foundation and the ultimate reaction force of the foundation, respectively. This is the natural exponential function in mathematics.

[0131] In this embodiment, the tail joint probability Using the mvncdf function in the commercial MATLAB software for calculation is a standard method.

[0132] Step 3.5, mean preservation and lower bound protection, specifically:

[0133] To ensure consistent marginal expectations, the modified random field... and Mean correction and lower bound protection are performed to obtain the final coupled random field of initial foundation stiffness and ultimate foundation reaction force. and :

[0134] ; ;

[0135] in: and The minimum allowable values ​​of initial foundation stiffness and ultimate reaction force of the foundation as specified in the project; The symbol for the function that retrieves the maximum value; This is a function for taking the average value.

[0136] Because the total number of random simulations needs to be performed Next, the first of them The final coupled random field record of the initial foundation stiffness and ultimate foundation reaction obtained in this step is simplified as follows: and .

[0137] In this embodiment, the initial stiffness of the foundation is ultimately coupled with a random field. and foundation ultimate reaction force The final coupled random field The results are as follows Figure 2 and Figure 3 As shown, 500 sets of random samples were generated that were unevenly distributed along the longitudinal direction of the tunnel. Each location along the longitudinal direction had different initial foundation stiffness and ultimate foundation reaction force. By using the method proposed in this invention, a coupled random field considering the synergistic amplification effect of the foundation's two parameters can be conveniently generated at the same time.

[0138] Fourth, generate nonlinear stochastic foundation support constraints, as follows:

[0139] Constructing the energy expression of a nonlinear foundation under displacement perturbation :

[0140] ;

[0141] pass The first-order and second-order derivatives yield the expression for the foundation reaction force. Jacobi expression for foundation support As shown below:

[0142] ; ;

[0143] in: This represents the vertical displacement value of the tunnel node; and These represent the undetermined values ​​of the initial stiffness and ultimate reaction force of the foundation, respectively.

[0144] The first The final coupled random field of the initial stiffness and ultimate reaction of the foundation obtained in this step and Substitute and In the middle, it won the first The vertical coordinate position during the random calculation is Scalar functions of foundation reaction force and Jacobi scalar functions of foundation support and ,as follows:

[0145] ;

[0146] ;

[0147] Based on the node division information of the finite element model of the shield tunnel, the nonlinear foundation reaction force and foundation support of the shield tunnel are arrayed and matrixed to establish the first... Nonlinear stochastic foundation reaction function array during sub-random calculation and the Jacobi function matrix of nonlinear stochastic foundation support ,as follows:

[0148] ;

[0149] ;

[0150] in: Let the vertical displacement vectors of the nodes be column matrices. ; It is the perimeter of the outer surface of the tunnel.

[0151] Regarding the already obtained and In the following iterative solution, it is only necessary to update the column matrix of vertical displacement vectors of the nodes in real time. By inputting these values ​​into the calculation, the Jacobi matrix of random foundation support and the array of random foundation reaction forces can be updated instantly.

[0152] Fifth, adaptive iterative solution of nonlinear systems, specifically:

[0153] Initialize nodal displacements Vertical displacement of nodes Random foundation reaction function array and the Jacobi function matrix of random foundation support Generate the initial equivalent stiffness matrix With initial residual load And calculate the initial nodal displacement increment. .

[0154] During the iterative solution process, i.e., at the th... In the next iteration, the calculation is based on the current node displacement. Vertical displacement of nodes Dynamically update the Jacobi function matrix of random foundation support Equivalent stiffness matrix Calculate the displacement increment And correct nodal displacements Simultaneously, the random foundation reaction function array is recalculated. With residual load When the residuals satisfy the convergence condition or number of iterations Reaching the maximum set value At that time, output the final node displacement result. .

[0155] In each iteration, according to the... Rate of change of residual in the next iteration Adaptive adjustment of relaxation factor This aims to improve the convergence stability and computational efficiency of nonlinear stochastic systems.

[0156] In the formula: This represents the load array obtained in step two; and They represent the first Second and third The residual load generated by the next iteration update; Represents the index of a vector or column matrix, starting from the first number and going every two numbers until the last number; This is the convergence limit; It is a norm function; To prevent division by zero of small constants; The step size adjustment index; and These represent the lower and upper limits of the relaxation factor, respectively. This is the amplitude limiting operator function.

[0157] The iterative curve of the adaptive iterative solution process in this embodiment is as follows: Figure 4 As shown in the iterative curves, the adaptive iterative solution of the nonlinear system using this invention can achieve results less than [a certain value] in no more than 11 iterations across 500 random calculations. The convergence residual condition is such that some results satisfy the convergence condition in the 9th iteration.

[0158] Sixth, output the longitudinal mechanical performance index system of the tunnel boring machine, specifically:

[0159] Extract and indirectly calculate the longitudinal position of the shield tunnel Vertical deformation Corner Bending moment Shear force Longitudinal curvature and actual structural load To form a longitudinal mechanical performance index system , It is used to evaluate the longitudinal mechanical properties of tunnel structures under multi-source disturbance conditions.

[0160] The comparative analysis results of the longitudinal mechanical performance indicators (taking vertical deformation as an example) calculated in this embodiment are as follows: Figure 5 As shown, Figure 5The results show that, using the deterministic nonlinear analysis results as a benchmark, and under the premise of adopting the relevant parameters of this embodiment, the vertical deformation variation range of the shield tunnel under random disturbances of initial foundation stiffness and ultimate foundation reaction force is only 0.6 mm and 1.2 mm, respectively. The method of this invention can better reflect the non-uniformity of the longitudinal foundation of the shield tunnel, with a variation range reaching 2.1 mm, and the random sample calculation results can better reflect the characteristics of the soft foundation.

[0161] Based on the above, this embodiment discloses a nonlinear stochastic analysis method for the longitudinal mechanical properties of shield tunnels, particularly a method based on a two-parameter coupled random field and adaptive nonlinear iteration for analyzing the longitudinal mechanical properties of shield tunnels. This method establishes a three-dimensional equivalent longitudinal finite element model of the shield tunnel based on Timoshenko beam theory and constructs a spatial correlation matrix between the initial stiffness and ultimate reaction force of the foundation. Through joint random perturbation, a tail-end joint probability triggering mechanism, and exponential decay mapping, a two-parameter coupled log-normal random field reflecting the deterioration characteristics of weak strata is generated. On this basis, a nonlinear stochastic foundation reaction function and a Jacobi support function matrix are constructed to achieve real-time updates of the foundation support effect under displacement perturbation. Through an adaptive relaxation control-based nonlinear iterative solution strategy, the equivalent stiffness matrix and residual load are dynamically updated to obtain the responses of key indicators such as vertical displacement (i.e., vertical deformation), rotation angle, bending moment, shear force, longitudinal curvature, and actual structural load of the shield tunnel. This method can simultaneously characterize the multi-source uncertainty of the foundation and the nonlinear characteristics of the structure-foundation interaction, thereby improving the accuracy and robustness of the longitudinal mechanical performance analysis of shield tunnels under complex geological conditions.

[0162] The above description is only a preferred embodiment of the present invention and does not limit the scope of the present invention. All equivalent structural transformations made under the inventive concept of the present invention using the contents of the present invention specification and drawings, or direct / indirect applications in other related technical fields, are included within the protection scope of the present invention.

Claims

1. A nonlinear stochastic analysis method for the longitudinal mechanical properties of a shield tunnel, characterized in that, Includes the following steps: Step 1: Obtain shield tunnel parameters; divide finite element node coordinate information; Step 2: Assemble the structural stiffness matrix and load array based on the shield tunnel parameters obtained in Step 1; Step 3: Generate a two-parameter coupled random field for the foundation, specifically including: Step 3.1: Construct the spatial covariance matrix and perform Cholesky decomposition. Specifically, based on the finite element node coordinate information obtained in Step 1, form a spatial correlation matrix; and obtain the lower triangular matrix through Cholesky decomposition. Step 3.2: Construct a standard normal joint random field, specifically: establish a linear correlation matrix between parameters; generate a standard normal white noise matrix and introduce spatial correlation; obtain a spatially correlated and parameter-coupled joint random field through the parameter coupling matrix; Step 3.3: Map and calculate the log-normal coupled random field of the foundation's two parameters; Step 3.4: Calculate the coupling triggering factor and implement the exponential decay mapping of the two-parameter coupled random field. Specifically, the tail joint probability of each node is calculated using a two-dimensional normal distribution, and the coupling triggering factor is constructed. The corrected random field of the initial stiffness and ultimate reaction force of the foundation is obtained by using the exponential decay mapping. Step 3.5: Perform mean correction and lower bound protection on the modified random field to obtain the final coupled random field of initial foundation stiffness and ultimate foundation reaction force; Step 4: Generate nonlinear stochastic foundation support constraints; Step 5: Adaptively iteratively solve the nonlinear system and output the longitudinal mechanical performance index system of the shield.

2. The nonlinear stochastic analysis method for the longitudinal mechanical properties of shield tunnels as described in claim 1, characterized in that, The shield tunnel parameters in step one include the flexural stiffness of the cross section, the total number of nodes in the Timoshenko beam model of the shield tunnel, the shear stiffness of the cross section, the element length, the perimeter of the outer surface of the tunnel, and the load characteristics; the load characteristics include buoyancy, equivalent void force, amplitude and application location of time-varying loads; The following is the output system of longitudinal mechanical performance indicators for the tunnel boring machine in step five: ； in: , , , , as well as These are the longitudinal positions of the shield tunnel. Vertical deformation, rotation, bending moment, shear force, longitudinal curvature, and actual structural load.

3. The nonlinear stochastic analysis method for the longitudinal mechanical properties of shield tunnels as described in claim 2, characterized in that, Step two specifically includes: Based on the parameters determined in step one, the total number of nodes in the Timoshenko beam model of the shield tunnel is determined. Given the longitudinal coordinates of the tunnel A finite element model of a shield tunnel was established using Timoshenko beam theory; the stiffness matrix of the beam element in the Timoshenko beam tunnel was established. ,in: Here is the element bending stiffness matrix; This is the element shear stiffness matrix; through the matrix mapping of element stiffness to global stiffness, a basic stiffness matrix for shield tunnels that can superimpose boundary constraints is formed. ,in: Let T be the mapping matrix from the unit to the global coordinates, and T be the transpose.

4. The nonlinear stochastic analysis method for the longitudinal mechanical properties of shield tunnels as described in any one of claims 1-3, characterized in that, In step 3.1: the vertical coordinate... Location With the Location covariance It is expressed as follows: ； in: The horizontal correlation length; Lower triangular matrix The expression is as follows: 。 5. The nonlinear stochastic analysis method for the longitudinal mechanical properties of shield tunnels as described in claim 4, characterized in that, In step 3.2: the linear correlation matrix between parameters The expression is as follows: ； in: Belongs to the initial stiffness of the foundation ultimate reaction force of foundation The linear correlation coefficient between them ; For Cholesky decomposition function; Space related The expression is as follows: ； in: The matrix is ​​a standard normal white noise matrix with a matrix scale of . , The number of random simulations; This is a function used to construct a block diagonal matrix; Spatially correlated and parameter-coupled joint random fields The expression is as follows: ； in: Initial stiffness of the foundation The standard normal random field; ultimate reaction force of foundation The standard normal random field; and The matrix scale is ; It is the identity matrix, and its matrix scale is ; The notation for calculating the Kronecker product.

6. The nonlinear stochastic analysis method for the longitudinal mechanical properties of shield tunnels as described in claim 5, characterized in that, The mapping calculation of the log-normal coupled random field of the foundation's two parameters in step 3.3 is as follows: The initial foundation parameters are log-normally mapped as follows: ； ； in: ; ; ; ; , and Let represent the mean, coefficient of variation, and log-normal random field of the initial stiffness of the foundation, respectively; , and Let represent the mean, coefficient of variation, and base log-normal random field of the ultimate reaction force of the foundation, respectively.

7. The nonlinear stochastic analysis method for the longitudinal mechanical properties of shield tunnels as described in claim 6, characterized in that, Coupling triggering factor in step 3.4 The expression is as follows: ； in: To scale, ; The shape index, ; This is the trigger threshold; This represents the upper limit of the coupling triggering factor; The joint probability of the tail of a node; Corrected random field for initial stiffness of foundation Modified random field of foundation ultimate reaction force The expression is as follows: ； ； in: The attenuation strength of the initial stiffness of the foundation; The attenuation intensity of the ultimate reaction force of the foundation; This is the natural exponential function in mathematics.

8. The nonlinear stochastic analysis method for the longitudinal mechanical properties of shield tunnels as described in claim 7, characterized in that, The final coupled random field of the initial stiffness of the foundation in step 3.5 The final coupled random field with the ultimate reaction force of the foundation The expression is as follows: ； ； in: The minimum allowable value for the initial stiffness of the foundation set for the project; The minimum allowable value of the ultimate reaction force of the foundation set for the project; This is a function for taking the average value; The total number of random simulations required is [number]. Next, the first of them The final coupled stochastic field of the initial foundation stiffness and ultimate foundation reaction obtained in this step is denoted as follows: and .

9. The nonlinear stochastic analysis method for the longitudinal mechanical properties of shield tunnels as described in claim 8, characterized in that, Step four, generating nonlinear stochastic foundation support constraints, specifically includes the following steps: Constructing the energy expression of a nonlinear foundation under displacement perturbation : ； pass The first-order and second-order derivatives yield the expression for the foundation reaction force. Jacobi expression for foundation support As shown below: ； ； in: This represents the vertical displacement value of the tunnel node; and These represent the undetermined values ​​of the initial stiffness and ultimate reaction force of the foundation, respectively. The first The final coupled random field of the initial stiffness and ultimate reaction of the foundation obtained in this step and Substitute and In the middle, it won the first The vertical coordinate position during the random calculation is Scalar functions of foundation reaction force and Jacobi scalar functions of foundation support and ,as follows: ； ； Based on the node division information of the shield tunnel finite element model from step two, the nonlinear foundation reaction force and foundation support of the shield tunnel are arrayed and matrixed to establish the first... Nonlinear stochastic foundation reaction function array during sub-random calculation and the Jacobi function matrix of nonlinear stochastic foundation support ,as follows: ； ； in: Let the vertical displacement vectors of the nodes be a column matrix. ; The outer perimeter of the tunnel; The unit length; Regarding the already obtained and In subsequent iterative solutions, it is only necessary to update the column matrix of vertical displacement vectors of the nodes in real time. By incorporating these parameters into the calculation, the Jacobi matrix of random foundation support and the array of random foundation reaction forces can be updated instantly.

10. The nonlinear stochastic analysis method for the longitudinal mechanical properties of shield tunnels as described in claim 9, characterized in that, Step five, the adaptive iterative solution of the nonlinear system, specifically includes the following steps: Initialize nodal displacements Vertical displacement of nodes Random foundation reaction function array and the Jacobi function matrix of random foundation support Generate the initial equivalent stiffness matrix With initial residual load And calculate the initial nodal displacement increment. ; During the iterative solution process, i.e., at the th... In the next iteration, the displacement of the current node is used for calculation. Vertical displacement of nodes Dynamically update the Jacobi function matrix of random foundation support Equivalent stiffness matrix Calculate the displacement increment And correct nodal displacements Simultaneously, the random foundation reaction function array is recalculated. With residual load When the residuals satisfy the convergence condition or number of iterations Reaching the maximum set value At that time, output the final node displacement result. ; In each iteration, according to the... Rate of change of residual in the next iteration Adaptive adjustment of relaxation factor ; in: This represents the load array obtained in step two; and They represent the first Second and third The residual load generated by the next iteration update; Represents the index of a vector or column matrix, starting from the first number and going every two numbers until the last number; This is the convergence limit; It is a norm function; To prevent division by zero of small constants; The step size adjustment index; and These represent the lower and upper limits of the relaxation factor, respectively; This is the amplitude limiting operator function.

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