A method for calculating limit stress of non-circular tunnel surrounding rock

Abstract

This invention discloses a method for calculating the ultimate stress of surrounding rock in a non-circular tunnel, belonging to the field of tunnel engineering. The method includes the following steps: establishing a coordinate system with the center of the invert arch as the origin and defining the plastic zone based on fundamental parameters, wherein the fundamental parameters include tunnel geometric parameters, altered rock physical and mechanical parameters, and in-situ stress parameters; deriving the characteristic stresses and rotation angles of the invert arch boundary and the side arch boundary based on the fundamental parameters, the coordinate system, and the Mohr-Coulomb yield criterion to obtain the boundary conditions; deriving the differential and integral equations of the slip line in the plastic zone based on the altered rock physical and mechanical parameters to obtain the basic equations; constructing a computational grid using the finite difference method based on the boundary conditions and the basic equations and iteratively solving the problem to obtain the converged characteristic stresses and rotation angles of the grid nodes; and calculating the ultimate stress distribution of the plastic zone based on the converged characteristic stresses and rotation angles of the grid nodes.

Description

Technical Field

[0001] This invention belongs to the field of tunnel engineering technology, and in particular relates to a method for calculating the ultimate stress of surrounding rock in non-circular tunnels. Background Technology

[0002] As transportation infrastructure extends into areas with complex geological conditions, the number of deep-buried ultra-long tunnel projects is increasing, making the stability control of altered surrounding rock sections a key technical challenge. For example, in non-circular tunnels like the Daliangshan Tunnel, which are buried at depths exceeding 1200m, the axis of the tunnel intersects obliquely with the altered rock zone. Due to the action of thermal fluids, the altered rock exhibits low strength (cohesion 388kPa, friction angle 15°) and high plasticity. Under high ground stress (horizontal ground stress reaching 14.46MPa), it is extremely prone to large deformation disasters such as invert heave and cracks. In some areas, the heave can reach 1.2m, seriously threatening construction safety and project quality.

[0003] Current methods for calculating the ultimate stress of surrounding rock are mostly applicable to circular tunnels, exhibiting poor adaptability to non-circular cross-sections. Furthermore, they generally rely on empirical criteria, using only the ratio of uniaxial compressive strength of the rock mass to in-situ stress as the deformation criterion, without fully considering the influence of key physical and mechanical parameters such as cohesion and friction angle of altered rocks. While the traditional slip line method is used for stress analysis, it fails to derive suitable equations for the boundary conditions combining curved surfaces in non-circular tunnels, resulting in insufficient accuracy in ultimate stress prediction. This makes it difficult to accurately guide the design of anti-deformation measures such as counterpressure treatment and steel pipe pile reinforcement, potentially leading to wasted engineering costs or support failure. Therefore, a method for calculating the ultimate stress of surrounding rock that is adaptable to non-circular tunnels and takes into account the influence of multiple parameters is urgently needed. Summary of the Invention

[0004] To address the aforementioned technical problems, this invention proposes a method for calculating the ultimate stress of surrounding rock in non-circular tunnels, thereby resolving the issues present in the prior art.

[0005] To achieve the above objectives, the present invention provides a method for calculating the ultimate stress of surrounding rock in a non-circular tunnel, comprising: A coordinate system with the center of the inverted arch as the origin is established based on the basic parameters, and the range of the plastic zone is defined. The basic parameters include tunnel geometric parameters, altered rock physical and mechanical parameters, and in-situ stress parameters. Based on the aforementioned basic parameters, coordinate system, and Mohr-Coulomb yield criterion, the characteristic stresses and rotation angles of the invert arch boundary and the side arch boundary are derived to obtain the boundary conditions. Based on the physical and mechanical parameters of altered rocks, the differential and integral equations of slip lines in the plastic zone are derived, and the basic equations are obtained. Based on the boundary conditions and basic equations, the finite difference method is used to construct a computational mesh and iteratively solve the problem to obtain the convergent mesh node characteristic stresses and rotation angles. Based on the characteristic stresses and rotation angles of the converged mesh nodes, the ultimate stress distribution in the plastic zone is calculated.

[0006] Optionally, the physical and mechanical parameters of the altered rock include: cohesion, friction angle, and gravimetric density; The ground stress parameters include: vertical ground stress and horizontal ground stress.

[0007] Optionally, the process of deriving the characteristic stresses and rotation angles of the invert arch boundary and the side arch boundary to obtain the boundary conditions includes: Based on the conditions of different curved boundaries and the tangent angle of the boundary curves, the boundary normal stress and shear stress are obtained; Based on the boundary normal stress, shear stress, cohesion of altered rock, and friction angle, the characteristic stress and rotation angle of the invert arch boundary are obtained by solving the equation using the Mohr-Coulomb yield criterion. Based on the zero-stress condition and the tangent angle of the side arch curve boundary, the direction of the first principal stress of the side arch boundary is obtained, and then the characteristic stress and rotation angle of the side arch boundary are calculated.

[0008] Optionally, after obtaining the characteristic stresses and rotation angles of the invert arch boundary and the side arch boundary, the process also includes handling singularities. The process of handling singularities includes: treating the singularity as a transition point between the zero-length α line and the zero-length β line, solving the integral equations of the invert arch boundary and the side arch boundary simultaneously, and solving for the rotation angle of the singularity.

[0009] Optionally, the process of deriving the differential equation of the slip line in the plastic zone includes: the angle between the α line and the β line and the direction of the first principal stress is π / 4 minus half of the friction angle of the altered rock.

[0010] Optionally, in the process of constructing the computational grid using the finite difference method, the average spacing between grid nodes is 0.15 meters.

[0011] Optionally, the process of constructing a computational grid and iteratively solving using the finite difference method includes: Based on the known coordinates, rotation angle, and characteristic stress of node A on line α, and the known coordinates, rotation angle, and characteristic stress of node B on line β, an iterative calculation is performed using an integral equation in the form of finite difference to obtain the coordinates, rotation angle, and characteristic stress of the internal node M; the iteration is repeated until the change in the calculated parameters of node M is less than a set threshold.

[0012] Optionally, after calculating the ultimate stress distribution in the plastic zone, the method further includes: determining the maximum depth of the plastic zone based on the converged grid node coordinates and slip line distribution; and determining the reinforcement length of the steel pipe pile based on the maximum depth of the plastic zone.

[0013] Compared with the prior art, the present invention has the following advantages and technical effects: The method for calculating the ultimate stress of surrounding rock in non-circular tunnels provided by this invention accurately adapts to complex curved surface boundaries such as inverts and side arches by establishing a targeted non-circular cross-sectional coordinate system and plastic zone model, overcoming the limitation of traditional methods that are only applicable to circular tunnels. This method integrates key mechanical parameters such as cohesion, friction angle, and in-situ stress of altered rocks, and combines the Mohr-Coulomb criterion and slip line theory to construct a boundary condition and basic equation system that fits actual working conditions. Employing the finite difference method and iterative solution technology, the accuracy of the prediction of the ultimate stress and plastic zone slip system is significantly improved, providing a reliable basis for the quantitative judgment of large deformation of the invert under compression. Finally, this method can provide accurate design parameters for engineering measures such as steel pipe pile reinforcement and counterpressure treatment, effectively guiding the design and optimization of support structures, ensuring construction safety while avoiding support failure or over-support, and has significant engineering application value and promising prospects for promotion. Attached Figure Description

[0014] The accompanying drawings, which form part of this application, are used to provide a further understanding of this application. The illustrative embodiments and descriptions of this application are used to explain this application and do not constitute an undue limitation of this application. In the drawings: Figure 1 This is a schematic diagram of the sliding line model of the non-circular tunnel invert arch according to an embodiment of the present invention; Figure 2 This is a schematic diagram of the nodal solution using the finite difference method according to an embodiment of the present invention; Figure 3 This is a schematic diagram of the ultimate stress distribution in the surrounding rock according to an embodiment of the present invention; Figure 4 This is a schematic diagram of the engineering anti-deformation measures in an embodiment of the present invention. Detailed Implementation

[0015] It should be noted that, unless otherwise specified, the embodiments and features described in this application can be combined with each other. This application will now be described in detail with reference to the accompanying drawings and embodiments.

[0016] It should be noted that the steps shown in the flowchart in the accompanying drawings can be executed in a computer system such as a set of computer-executable instructions, and although a logical order is shown in the flowchart, in some cases the steps shown or described may be executed in a different order than that shown here.

[0017] Example 1 This invention aims to provide a method for calculating the ultimate stress of surrounding rock in non-circular tunnels, specifically addressing the shortcomings of existing technologies such as poor adaptability to non-circular tunnels, incomplete consideration of key parameters, and insufficient accuracy in ultimate stress prediction. This provides support for the safety and stability of tunnel construction in deeply buried altered surrounding rock sections. The method needs to adapt to the curved cross-sectional characteristics of the main arch, invert arch, and side arches of non-circular tunnels. Considering the engineering reality that such tunnels often intersect with altered rock zones, it derives boundary condition equations that fit the actual working conditions, overcoming the limitation of traditional methods being only applicable to circular tunnels. This method meets the stress calculation needs of projects with burial depths exceeding 1000 meters, such as the Daliangshan Tunnel. Simultaneously, it integrates key physical and mechanical parameters such as cohesion, friction angle, and weight density of altered rock with the geostress environment, establishing a multi-factor coupled ultimate stress calculation model to compensate for the shortcomings of existing methods that ignore the influence of core parameters. Furthermore, it optimizes the numerical solution logic of the slip line method, improving calculation accuracy through the finite difference method and iterative algorithms. Figure 2 As shown, it can accurately predict the slip system and stress distribution in the plastic zone of the surrounding rock, providing quantitative criteria for large compression deformation; ultimately, it provides a scientific design basis for anti-deformation measures such as counter-pressure treatment and steel pipe pile reinforcement, and clarifies the matching relationship between support parameters and plastic zone depth. While effectively controlling project costs and avoiding support failure or over-support, it ensures the construction safety and project quality of deeply buried non-circular tunnels in altered surrounding rock sections.

[0018] This invention constructs a calculation system for the ultimate stress of surrounding rock in non-circular tunnels based on the slip line method. First, considering the curved cross-sectional characteristics of the main arch, invert arch, and side arch combination of non-circular tunnels, a suitable coordinate system is established. The geometric parameters of the cross-section are determined with the center of the invert arch as the origin. The research scope of the plastic zone is clarified, and the tunnel cross-section symmetry is utilized to analyze only half of the invert arch area. Core assumptions such as plane strain, altered rock in the plastic zone being in a yielding state, and rigid-plastic materials conforming to the Mohr-Coulomb yield criterion are set, laying the foundation for stress calculation.

[0019] By combining the physical and mechanical parameters of the altered surrounding rock (cohesion, friction angle, and mass density) with the geostress environment (vertical and horizontal geostress), boundary condition equations combining curved surfaces are derived. For the invert arch curved boundary, considering the back pressure or zero stress after excavation, the relationship between normal stress and shear stress is established through the Mohr-Coulomb yield criterion, and the boundary characteristic stress and rotation angle are solved. For the lateral arch boundary, the expressions for characteristic stress and rotation angle are determined based on the zero stress condition, while the transition relationship of the slip line at the singularity is handled to improve the boundary condition system.

[0020] Based on the analysis of the stress state at a point in the plastic zone using the Mohr stress circle, the angular relationship between the α-line, the β-line, and the principal stress directions is clarified. The differential equation of the slip line is derived, and a complete system of fundamental equations is established by combining the calculation formulas for normal stress and shear stress with the differential equilibrium equations. The fundamental equations are solved using the finite difference method, with the intersection of the α-line and the β-line as the mesh nodes. Based on the parameters of known points on adjacent boundaries, the coordinates, rotation angles, and characteristic stresses of the nodes are calculated through an iterative algorithm. An error threshold is set to control the iteration accuracy, ensuring the accuracy of the stress distribution and the calculation of the slip system in the plastic zone.

[0021] The optimal mesh size was determined by mesh sensitivity analysis, and parameter analysis was carried out by integrating influencing factors such as gravity and counterpressure. A quantitative criterion for large extrusion deformation was established, and the correlation between the uniaxial compressive strength, cohesion, friction angle and ultimate stress of the rock mass was clarified. At the same time, the maximum depth of the plastic zone was calculated to provide data support for the design of anti-deformation measures. By optimizing the counterpressure parameters and matching them with the length of the steel pipe piles, the scientific design of the support measures was achieved.

[0022] This invention provides a method for calculating the ultimate stress of surrounding rock in non-circular tunnels. This embodiment uses a tunnel (non-circular cross-section) on the Yunxian-Lincang Expressway in Yunnan Province, China, as an example. Based on the slip line method, the ultimate stress of the surrounding rock is calculated, thereby serving the prevention and control of large deformation due to arch compression within altered surrounding rock. Figure 4 This is a schematic diagram of the engineering anti-deformation measures according to an embodiment of the present invention. The specific steps are as follows: Step 1: Obtain basic parameters.

[0023] (1) Obtain the physical and mechanical parameters of the altered rock: The density of the altered rock was measured to be γ=2200kg / m³ (equivalent to 22kN / m³) through indoor physical property tests; the cohesion of the altered rock was measured to be c=388kPa and the friction angle was measured to be φ=15° through direct shear tests; the uniaxial compressive strength of the altered rock was calculated to be R=1.01MPa according to the formula R=2Cc0.

[0024] (2) Obtaining ground stress parameters: The ground stress in the tunnel area was measured on-site using the hydraulic fracturing method to obtain the vertical ground stress σ. v =10.06MPa, horizontal ground stress σ h =14.46MPa.

[0025] (3) Obtaining tunnel geometric parameters: Determining the non-circular tunnel cross-sectional structure (composed of a main arch, an invert arch, and two side arches), wherein the coordinates of the center O of the invert arch are (0, 5.99m), the radius r = 11.09m, the distance between the two ends of the invert arch (point S on the right) and the x-axis is 3.6m, and the width of the invert arch is w = 11.14m; the equation of the ST segment of the side arch is: (x ranges from 5.57 to 6.26m).

[0026] Step 2: Construct a sliding line model of the non-circular tunnel invert arch, such as... Figure 1 As shown.

[0027] (1) Model assumptions: The model follows four core assumptions: the problem is a plane strain problem; the altered rock in the plastic zone is in a yielding state; the altered rock is a rigid-plastic material that conforms to the Mohr-Coulomb yield criterion (τ=C+σntanφ); the slip line model can accurately predict the stress distribution of the invert arch.

[0028] (2) Establish a coordinate system: With the center O of the invert arch as a reference, establish a plane rectangular coordinate system (x-axis horizontal, y-axis vertical), and only study the plastic zone on the right side of the invert arch (using the symmetry of the tunnel cross section, the plastic zone on the right side is surrounded by Q STU) to simplify the calculation.

[0029] (3) Define the scope of the plastic zone: After the tunnel is excavated, the altered surrounding rock will undergo plastic deformation in the area near the exposed surface under the action of earth pressure. Focus on the plastic zone around the invert arch and ignore the interference of non-core areas.

[0030] Step 3: Derive the boundary conditions and basic equations.

[0031] (1) Derive the boundary conditions.

[0032] Invert arch QS segment (curved boundary): Apply counterpressure q (initially set to 0), the shear angle δ of QS segment satisfies: .

[0033] Calculate boundary stress: (normal stress) Substituting (shear stress) into the Mohr-Coulomb yield criterion, we can solve for the characteristic stress: Since the inverted arch is in the passive region, the larger root of the equation is selected.

[0034] Solve for the rotation angle θ: .

[0035] Side arch ST segment (curved boundary): Apply zero stress (σ3=0). Based on the fact that the direction of σ1 is consistent with the tangential direction of the side arch, we obtain... satisfy: .

[0036] Calculate the characteristic stress of segment ST: Singularity S treatment: Treat point S as both a "zero-length α line (QS segment transition)" and a "zero-length β line (ST segment transition)", and solve the simultaneous integral equations: ; Substitute the boundary parameters of QS segment and ST segment to determine the key parameters of point S (θ_S=69.49°) and connect the two boundary segments.

[0037] (2) Derive the basic equation.

[0038] Differential equation of slip line: The angle between the α line, the β line and the first principal stress σ1 is μ = π / 4 − φ / 2 (μ = 30° when φ = 15°), differential equation: ; Stress equilibrium equations: Calculation of normal stress and shear stress: ; σ is the characteristic stress. Substituting into the differential equilibrium equation and considering the gravitational load (X=0, Y=-γ), we obtain the integral equation: ; Step 4: Solve for the ultimate stress using the finite difference method.

[0039] (1) Construct a computational grid.

[0040] Node definition: The intersection of the α line and the β line is the finite difference node.

[0041] Grid size: Select the optimal average size of 0.15m for document verification (ensuring maximum). The pressure tends to stabilize at 1.5269 MPa.

[0042] (2) Iteratively solve the node parameters.

[0043] Initial settings: Use boundary segment QS (point Q), segment ST (point T), and singularity S as initial nodes, and input known parameters (such as point S). ).

[0044] Calculation of internal node M (based on known point A on line α and known point B on line β): Coordinate calculation: ; in, : ; ; , which is the initial iteration value.

[0045] Calculation of characteristic stress σ: .

[0046] (3) Calculate the ultimate stress. For example... Figure 3 As shown, substituting the converged σ (characteristic stress) and θ (rotation angle) into step three... The formula is used to calculate the ultimate stress at all nodes.

[0047] Key Result Extraction: Maximum Horizontal Ultimate Stress Maximum vertical ultimate stress .

[0048] Step 5: Results Analysis and Engineering Applications.

[0049] (1) Judgment of large compression deformation: By comparing the calculated maximum ultimate stress (e.g., σx=1.5269MPa) with the uniaxial compressive strength of the altered rock (1.01MPa), and combined with the on-site ground stress (horizontal ground stress 14.46MPa is much greater than σx), it is judged that if no measures are taken, the invert arch will undergo large compression deformation.

[0050] (2) Determination of plastic zone depth: Extract the slip line distribution data and determine the maximum depth of the slip line around the invert arch (e.g., the depth is 3.95m at x=5.57m). Based on this, design the reinforcement length of the steel pipe pile (which needs to be greater than the maximum depth, take 6m).

[0051] (3) Optimization of back pressure parameters: Set different back pressure values ​​q (0.1~0.5MPa), repeat step four to calculate the corresponding maximum σx, and find that for every 0.1MPa increase in q, σx increases linearly by about 0.2MPa; combined with engineering requirements, select 10m high gravel (providing 0.2MPa back pressure) as a temporary back pressure measure to control the deformation of the invert arch.

[0052] Step Six: Method Validation.

[0053] (1) Verification object: strip foundation shallow foundation (width 1m, applied load q=1).

[0054] (2) Verification process: The bearing coefficient Nq under different friction angles φ (5°~30°) was calculated using this method.

[0055] (3) Verification results: Compared with Prandtl's theoretical results, the relative errors are all less than 5%. For example, when φ=15°, the Nq of this method is 3.9080, the theoretical value is 3.9412, and the error is 0.81%.

[0056] (4) Conclusion: The accuracy and reliability of the ultimate stress calculation method of this patent are verified.

[0057] This invention achieves significant technical results through an innovative calculation system for the ultimate stress of surrounding rock in non-circular tunnels, comprehensively solving many shortcomings of traditional methods. It is adaptable to the curved cross-sectional characteristics of the main arch, invert arch, and side arch combinations in non-circular tunnels, breaking through the limitation of traditional methods being only applicable to circular tunnels. It successfully adapts to complex engineering scenarios such as the Daliangshan Tunnel, which has a burial depth exceeding 1,000 meters and intersects with alteration zones, achieving accurate calculation of the ultimate stress of surrounding rock in non-circular cross-section tunnels.

[0058] By integrating key physical and mechanical parameters such as cohesion, friction angle, and weight density of altered rocks with the geostress environment, a multi-factor coupled calculation model is established. This model compensates for the shortcomings of existing methods that neglect core parameters. By combining the optimized slip line method and finite difference iterative algorithm, the calculation error is controlled within 10^-15, which greatly improves the prediction accuracy of the ultimate stress and plastic zone slip system. The deviation between the calculated maximum stress value and the actual engineering monitoring data is less than 5%.

[0059] The quantitative criterion for large compression deformation constructed by this method clarifies the correlation between the uniaxial compressive strength, cohesion, friction angle, and ultimate stress of the rock mass, replacing traditional empirical criteria and providing a scientific basis for anti-deformation measures. By accurately calculating the maximum depth of the plastic zone, precise matching of steel pipe pile length can be achieved. Combined with the optimization of counterpressure parameters, the problems of support failure or over-support are effectively avoided, reducing project costs by 15%-20%.

[0060] Meanwhile, this method has good versatility and can be extended to deep-buried non-circular tunnel projects under different geological conditions. It significantly improves the construction safety factor of complex surrounding rock sections, reduces the probability of disasters such as invert arch heave and cracks, and ensures project quality and construction progress. It has important engineering application value and promotion prospects.

[0061] The above are merely preferred embodiments of this application, but the scope of protection of this application is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in this application should be included within the scope of protection of this application. Therefore, the scope of protection of this application should be determined by the scope of the claims.

Claims

1. A method for calculating the ultimate stress of surrounding rock in a non-circular tunnel, characterized in that, Includes the following steps: A coordinate system with the center of the inverted arch as the origin is established based on the basic parameters, and the range of the plastic zone is defined. The basic parameters include tunnel geometric parameters, altered rock physical and mechanical parameters, and in-situ stress parameters. Based on the aforementioned basic parameters, coordinate system, and Mohr-Coulomb yield criterion, the characteristic stresses and rotation angles of the invert arch boundary and the side arch boundary are derived to obtain the boundary conditions. Based on the physical and mechanical parameters of altered rocks, the differential and integral equations of slip lines in the plastic zone are derived, and the basic equations are obtained. Based on the boundary conditions and basic equations, the finite difference method is used to construct a computational mesh and iteratively solve the problem to obtain the convergent mesh node characteristic stresses and rotation angles. Based on the characteristic stresses and rotation angles of the converged mesh nodes, the ultimate stress distribution in the plastic zone is calculated.

2. The method for calculating the ultimate stress of surrounding rock in a non-circular tunnel according to claim 1, characterized in that, The physical and mechanical parameters of the altered rock include: cohesion, friction angle, and gravimetric density; The ground stress parameters include: vertical ground stress and horizontal ground stress.

3. The method for calculating the ultimate stress of surrounding rock in a non-circular tunnel according to claim 1, characterized in that, The process of deriving the characteristic stresses and rotation angles of the invert arch boundary and the side arch boundary, and obtaining the boundary conditions, includes: Based on the conditions of different curved boundaries and the tangent angle of the boundary curves, the boundary normal stress and shear stress are obtained; Based on the boundary normal stress, shear stress, cohesion of altered rock, and friction angle, the characteristic stress and rotation angle of the invert arch boundary are obtained by solving the equation using the Mohr-Coulomb yield criterion. Based on the zero-stress condition and the tangent angle of the side arch curve boundary, the direction of the first principal stress of the side arch boundary is obtained, and then the characteristic stress and rotation angle of the side arch boundary are calculated.

4. The method for calculating the ultimate stress of surrounding rock in a non-circular tunnel according to claim 3, characterized in that, After obtaining the characteristic stresses and rotation angles of the invert arch boundary and the side arch boundary, the process also includes dealing with singularities. The process of dealing with singularities includes: treating the singularity as a transition point between the zero-length α line and the zero-length β line, solving the integral equations of the invert arch boundary and the side arch boundary simultaneously, and solving for the rotation angle of the singularity.

5. The method for calculating the ultimate stress of surrounding rock in a non-circular tunnel according to claim 4, characterized in that, The process of deriving the differential equation of the slip line in the plastic zone includes: the angle between the α line and the β line and the direction of the first principal stress is π / 4 minus half of the friction angle of the altered rock.

6. The method for calculating the ultimate stress of surrounding rock in a non-circular tunnel according to claim 1, characterized in that, In the process of constructing the computational grid using the finite difference method, the average spacing between grid nodes is 0.15 meters.

7. The method for calculating the ultimate stress of surrounding rock in a non-circular tunnel according to claim 1, characterized in that, The process of constructing a computational grid and iteratively solving using the finite difference method includes: Based on the known coordinates, rotation angle, and characteristic stress of node A on line α, and the known coordinates, rotation angle, and characteristic stress of node B on line β, an iterative calculation is performed using an integral equation in the form of finite difference to obtain the coordinates, rotation angle, and characteristic stress of the internal node M; the iteration is repeated until the change in the calculated parameters of node M is less than a set threshold.

8. The method for calculating the ultimate stress of surrounding rock in a non-circular tunnel according to claim 1, characterized in that, After calculating the ultimate stress distribution in the plastic zone, the method further includes: determining the maximum depth of the plastic zone based on the converged grid node coordinates and slip line distribution; and determining the reinforcement length of the steel pipe pile based on the maximum depth of the plastic zone.

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