Multi-process driven horizontal broken stratum excavation reserved deformation amount determination method
Patent Information
- Application Number
- CN202310929446.1
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2023-07-26
- Publication Date
- 2026-08-21
- Estimated Expiration
- 2043-07-26
AI Technical Summary
[0006]本发明为了解决上述问题,提出了一种多过程驱动的水平破碎地层开挖预留变形量确定方法,本发明解决了如何确定地下工程开挖预留变形量这一难题,克服了以前简单以围岩等级为依据的缺点,基于地质分析、地表和洞内物探、数值模拟和人工智能算法多过程驱动确定地下工程开挖预留变形量,基于专特尔菲理论邀请专家团队对上述结果进行分析,最终综合获取地下工程开挖预留变形量,评价信息丰富,结果准确率高
本发明利用多过程驱动,能够准确确定水平破碎地层地下工程开挖预留变形量,解决了如何确定地下工程开挖预留变形量这一难题,克服了以前简单以围岩等级为单一依据的缺点。
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Abstract
Description
Technical Field
[0001] This invention belongs to the technical field of underground engineering excavation pre-deformation calculation, and relates to a method for determining the pre-deformation of horizontal fractured strata excavation driven by multiple processes. Background Technology
[0002] The statements in this section are merely background information related to the present invention and do not necessarily constitute prior art.
[0003] When underground engineering is excavated in horizontally fractured strata, due to the poor integrity and low strength of the strata, it is very easy to induce collapse and large deformation under the action of groundwater and excavation. This can cause the initial support of the underground engineering structure to exceed the building limit. In order to ensure that the excavation of underground engineering does not exceed the limit, a reserved deformation amount is added to the excavation outline during the design stage to ensure that the deformation of the surrounding rock will not cause the initial support to encroach on the limit.
[0004] The allowable deformation of the surrounding rock is of great significance in underground engineering construction and is a very important indicator. If the allowable deformation is set too small, the engineering problem of initial support encroachment cannot be avoided, and arch replacement will be required later, increasing construction risks and project investment. If the allowable deformation is set too large, it will increase the excavation size of the underground project, reduce the stability of the underground project, and increase the amount of initial shotcrete used, significantly increasing the construction cost.
[0005] At present, the pre-deformation allowance for excavation is mainly based on the surrounding rock grade. However, due to the complex and variable nature of the surrounding rock and the geological structure, it cannot effectively guide on-site construction. Therefore, how to reasonably reserve the pre-deformation allowance for underground engineering excavation has always been a difficult problem in the field of underground engineering. Summary of the Invention
[0006] To address the aforementioned problems, this invention proposes a multi-process driven method for determining the allowable deformation amount for excavation in horizontally fractured strata. This invention solves the difficult problem of determining the allowable deformation amount for underground engineering excavation, overcoming the shortcomings of previous methods that simply relied on the surrounding rock grade. It determines the allowable deformation amount for underground engineering excavation through a multi-process driven approach, based on geological analysis, surface and underground geophysical exploration, numerical simulation, and artificial intelligence algorithms. An expert team is invited to analyze the results based on Delphi theory, ultimately obtaining a comprehensive assessment of the allowable deformation amount for underground engineering excavation. The method provides rich evaluation information and high accuracy.
[0007] According to some embodiments, the present invention adopts the following technical solution: A method for determining the allowable deformation amount in multi-process driven horizontal fractured strata excavation includes the following steps: Based on the location of the anomaly and its development pattern, potential areas of large deformation were initially delineated, and the risk level of large deformation was initially determined. Based on the risk level, the implementation areas for surface drilling and underground geophysical exploration were determined, and test sections and application sections were distinguished. Geophysical exploration was conducted within the application section to establish a numerical model. Based on the on-site excavation procedures, multiple types of excavation conditions were designed. Combined with the designed support parameters, numerical simulations of multiple excavation conditions were conducted to determine the optimal reserved deformation amount for the application section obtained from the numerical simulation. Establish an evaluation index system for reserved deformation based on geophysical parameters, geological parameters and construction parameters. Obtain data through each construction cycle of the test section to predict the reserved deformation of the tunnel face when passing through a certain stratum. The construction allowance deformation is determined by combining the optimal allowance deformation and the predicted construction allowance deformation.
[0008] As an alternative implementation, the anomaly detection location is obtained based on surface analysis, surface geophysical exploration, and semi-airborne transient electromagnetic detection.
[0009] Furthermore, the specific process of obtaining the location of the anomaly detection includes: based on geological exploration, using surface analysis to preliminarily identify the fractured strata and their occurrence from the surface conditions; If the surface conditions are flat, surface geophysical exploration is used for detection. If the surface conditions are uneven and exceed the set value, semi-airborne transient electromagnetic exploration is used. The low-resistivity area is identified as a water-bearing fractured area.
[0010] As an alternative implementation method, when establishing a numerical model, the process of determining the rock mass material parameters of the mathematical model includes obtaining rock cores through surface drilling, conducting rock mechanics experiments and geophysical experiments to obtain parameters such as the rock mass's elastic modulus, Poisson's ratio, cohesion, friction angle, density, rock mass wave velocity, and rock mass resistivity. After obtaining the relationship between rock mass wave velocity and resistivity and rock mass strength parameters through fitting, rock mass wave velocity and resistivity are obtained through geophysical exploration inside the tunnel. Based on the relationship between the obtained geophysical parameters and rock mass strength parameters, the rock mass parameters within the geophysical exploration area are inverted within a set length interval to obtain the rock mass material parameters within the geophysical exploration area in the strata ahead. The parameters outside the geophysical exploration area are based on the rock mass parameters obtained from borehole core sampling.
[0011] As an alternative implementation method, when establishing a numerical model, the process of determining the constitutive relation of the mathematical model includes: conducting creep tests on the rock mass material of the test section, obtaining the creep parameters of the rock mass material, and then inputting them into the creep damage model to obtain the constitutive relation.
[0012] As an alternative implementation method, when establishing a numerical model, the underground engineering excavation of the test section is simulated, the displacement data collected from the test section is compared with the data from the numerical simulation, and the parameters of the numerical model are calibrated or revised.
[0013] As an alternative implementation method, the specific process of conducting numerical simulation of multiple excavation conditions includes obtaining the deformation of the surrounding rock under different excavation methods, using vertical displacement as the rating index, and dividing the numerical model according to a set length interval after it stabilizes. The maximum vertical position in each interval is the optimal reserved deformation amount in that application segment interval.
[0014] As an alternative implementation method, the reserved deformation evaluation index system includes geophysical parameters, geological parameters, and construction parameters. The geophysical parameters include seismic wave velocity and resistivity values. The geological parameters include surrounding rock grade, rock mass integrity, and the angle between the rock strata dip angle and the tunnel axis. The construction parameters include tunnel depth, tunnel closure time, support parameters, and arch settlement.
[0015] As an alternative implementation method, the specific process of predicting the construction allowance deformation amount through a certain stratum in front of the tunnel face includes constructing a neural network model, which takes an index system as input and outputs the crown settlement, i.e. the construction allowance deformation amount determined based on artificial intelligence algorithms.
[0016] A multi-process driven system for determining the allowable deformation during excavation in horizontally fractured strata includes: The area division module is configured to initially delineate potential large deformation areas based on the location and development pattern of anomalies, and to initially determine the risk level of large deformation. Based on the risk level, it will target and guide the determination of the implementation areas for surface drilling and underground geophysical exploration, and distinguish between test sections and application sections. The numerical model calculation module is configured to obtain the results of geophysical exploration within the application section, establish a numerical model, design multiple types of excavation conditions based on the on-site excavation procedures, and conduct numerical simulations of multiple excavation conditions in combination with the designed support parameters to determine the optimal reserved deformation amount of the application section based on the numerical simulation. The intelligent prediction module is configured to establish an evaluation index system for reserved deformation based on geophysical parameters, geological parameters and construction parameters. It obtains data sets through each construction cycle of the test section and predicts the reserved deformation of the tunnel face when passing through a certain stratum. The fusion module is configured to fuse the optimal reserved deformation amount and the predicted construction reserved deformation amount to comprehensively determine the construction reserved deformation amount.
[0017] Compared with the prior art, the beneficial effects of the present invention are as follows: This invention utilizes a multi-process driven approach to accurately determine the allowable deformation amount for underground engineering excavation in horizontally fractured strata, solving the difficult problem of how to determine the allowable deformation amount for underground engineering excavation and overcoming the shortcomings of the previous method that simply relied on the surrounding rock grade as the sole basis.
[0018] This invention utilizes a neural network-based model for predicting the pre-deformation amount of underground engineering excavation. This model comprehensively considers geophysical parameters, geological parameters, and construction parameters, and is rich in data, contains a wide range of content, is highly adaptable, representative, and has broad application value.
[0019] This invention determines the pre-excavation deformation allowance for underground engineering projects from multiple perspectives, including geology, geophysics, and artificial intelligence, and comprehensively obtains the final pre-excavation deformation allowance. Moving from a broad overview to specific details, and from coarse to fine analysis, this method improves effectiveness while ensuring the evaluation results, and has broad application value.
[0020] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, preferred embodiments are described below in detail with reference to the accompanying drawings. Attached Figure Description
[0021] The accompanying drawings, which form part of this invention, are used to provide a further understanding of the invention. The illustrative embodiments of the invention and their descriptions are used to explain the invention and do not constitute an improper limitation of the invention.
[0022] Figure 1 This is a flowchart of this embodiment. Detailed Implementation
[0023] The present invention will be further described below with reference to the accompanying drawings and embodiments.
[0024] It should be noted that the following detailed description is illustrative and intended to provide further explanation of the invention. Unless otherwise specified, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains.
[0025] It should be noted that the terminology used herein is for the purpose of describing particular embodiments only and is not intended to limit the scope of exemplary embodiments according to the invention. As used herein, the singular form is intended to include the plural form as well, unless the context clearly indicates otherwise. Furthermore, it should be understood that when the terms "comprising" and / or "including" are used in this specification, they indicate the presence of features, steps, operations, devices, components, and / or combinations thereof.
[0026] Example 1 A multi-process driven method for determining the allowable deformation for underground engineering excavation in horizontally fractured strata overcomes the shortcomings of previous methods that relied solely on the surrounding rock grade. This method determines the allowable deformation from multiple perspectives—geological, geophysical, and human-intelligent approaches—and finally, an expert team uses the Delphi expert scoring method to comprehensively obtain the final allowable deformation. The method provides rich evaluation information and high accuracy.
[0027] like Figure 1 As shown, it specifically includes: Step 1: First, based on the location and development patterns of anomalies delineated by surface analysis, surface geophysical exploration, and semi-airborne transient electromagnetic mapping, potential large deformation areas are preliminarily delineated, and the risk level of large deformation is preliminarily determined. Based on the risk level, the implementation areas for surface drilling and underground geophysical exploration are determined, and test sections and application sections are distinguished.
[0028] When underground engineering is excavated in horizontally fractured strata, it is very easy to exceed the limits. Therefore, the primary task is to delineate the underground engineering area where large deformation may occur during the design phase and to assess the risk of large deformation.
[0029] Field investigations are conducted using methods such as geological reconnaissance, surface geophysical exploration, and semi-airborne transient electromagnetic methods. Surface analysis, based on geological exploration, preliminarily identifies fractured strata and their occurrence from surface outcrops. If the surface conditions are gentle, surface geophysical exploration (e.g., EH4 magnetotelluric instrument) can be used for detection. If the surface conditions are poor, semi-airborne transient electromagnetic methods can be used. Low-resistivity areas can be identified as water-bearing fractured regions.
[0030] By combining commonly used risk classification methods, the risk level of large deformations can be preliminarily determined, and the implementation areas for surface drilling and underground geophysical exploration can be determined based on the risk level.
[0031] In this embodiment, the risk of large deformation is divided into three levels: A, B, and C, with the risk level decreasing sequentially. In the A and B risk areas, on the basis of surface exploration, surface drilling is further carried out in a reasonable and denser manner, and geophysical exploration is carried out in the tunnels in the A and B risk areas.
[0032] In this embodiment, when distinguishing between test sections and application sections, the first 300m of the A and B-level risk sections within the tunnel can be designated as test sections, and the subsequent A and B-level risk sections with large deformations can be designated as application sections. All test sections and application sections are A and B-level risk areas.
[0033] Step 2: Conduct geophysical exploration within the application section, and build a numerical model based on the design documents. Based on the on-site excavation procedures, design multiple excavation conditions, and combine them with the designed support parameters to conduct numerical simulations of various excavation conditions. Determine the optimal allowable deformation D1 for the application section based on the numerical simulation results.
[0034] In this embodiment, multiple types of excavation conditions are designed, including three-step excavation, three-step 7-step left-side excavation, and three-step 7-step right-side excavation.
[0035] A reasonable and effective numerical simulation includes four aspects: first, determining the model material parameters; second, selecting a reasonable and effective constitutive relation for the rock mass; third, verifying the effectiveness of the model; and fourth, a reasonable and effective simulation scheme.
[0036] Regarding the determination of model material parameters, rock cores are obtained through surface drilling, and rock mechanics and geophysical experiments are conducted to obtain parameters such as the rock mass's elastic modulus, Poisson's ratio, cohesion, friction angle, density, rock mass wave velocity, and rock mass resistivity. After obtaining the relationship between rock mass wave velocity and resistivity and rock mass strength parameters through fitting, rock mass wave velocity and resistivity can be obtained through geophysical exploration inside the tunnel. Based on the relationship between the obtained geophysical parameters and rock mass strength parameters, each excavation cycle in underground engineering is generally 0.5m-1.5m. Therefore, with 1.5m as an interval, the rock mass parameters in the geophysical exploration area are inverted to obtain the rock mass material parameters in the geophysical exploration area in the strata ahead. The parameters outside the geophysical exploration area are based on the rock mass parameters obtained from borehole core sampling.
[0037] Regarding the constitutive relationship of the rock mass: creep tests can be conducted on the rock mass material of the test section to obtain the creep parameters of the rock mass material, which can then be incorporated into the existing Burgers creep damage model.
[0038] Regarding the verification of model validity: the underground engineering excavation of the test section is simulated, and the displacement data collected in the test section is compared with the data from the numerical simulation, which can calibrate the accuracy of the numerical model.
[0039] The approach to setting up simulation conditions for the numerical model is as follows: based on the on-site excavation process, design multiple types of excavation conditions, and combine them with the designed support parameters to conduct numerical simulations of multiple excavation conditions, thereby obtaining the deformation of the surrounding rock under different excavation methods.
[0040] After the numerical simulation is completed, the vertical displacement is used as the rating index. Once the numerical model stabilizes (e.g., the ratio of the maximum unbalanced force is less than 1e), the rating is determined. -6 Divide the application into 1.5m intervals, and the maximum vertical position within each interval is the optimal reserved deformation amount D1 within that application segment.
[0041] Step 3: Establish an evaluation index system for reserved deformation based on geophysical parameters, geological parameters, and construction parameters. Obtain datasets through each construction cycle of the test section and use artificial intelligence algorithms (such as GNN neural networks) to predict the reserved deformation D2 for construction in front of the tunnel face through a certain stratum.
[0042] The process for carrying out the prediction of deformation allowance in underground engineering excavation based on artificial intelligence is as follows: 1) Through theoretical analysis, establish an evaluation index system that encompasses geophysical parameters, geological parameters, and construction parameters. This system includes seismic wave velocity, resistivity value, surrounding rock grade, rock mass integrity, rock stratum dip angle and tunnel axis angle, tunnel depth, tunnel closure time, support parameters, and arch settlement.
[0043] 2) Data Acquisition: Data sets are acquired for each construction cycle (typically 0.5m-1.5m) of the test section, serving as the sample set for the artificial intelligence algorithm. Indicator data for the unexcavated section can be obtained through geological reconnaissance, surface geophysical exploration, geological drilling, tunnel geophysical exploration, and construction organization design, serving as the prediction set for the artificial intelligence algorithm.
[0044] 3) Model Training and Prediction: The data from the test section is fed into the GNN neural network model for training. The test set collected from the application section (unexcavated section) is then fed into the trained model to predict the output index, arch settlement. This value is the construction allowance deformation D2 determined based on the artificial intelligence algorithm.
[0045] Step 4: Establish an expert team including construction units, geological experts, and design experts. Use the Delphi scoring method to score, evaluate, and assign weights to the deformation amounts D1 and D2 obtained based on numerical simulation and artificial intelligence, and finally comprehensively determine the construction allowance deformation amount D.
[0046] Of course, in other embodiments, based on experience, corresponding weights can be added to the optimal reserved deformation amount D1 and the construction reserved deformation amount D2 to obtain the final construction reserved deformation amount D.
[0047] The parameter values in the above embodiments are merely examples; other parameters may be used in other embodiments.
[0048] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.
[0049] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.
[0050] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.
[0051] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.
[0052] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.
[0053] While the specific embodiments of the present invention have been described above in conjunction with the accompanying drawings, this is not intended to limit the scope of protection of the present invention. Those skilled in the art should understand that various modifications or variations that can be made by those skilled in the art without creative effort based on the technical solutions of the present invention are still within the scope of protection of the present invention.
Claims
1. A method for determining the allowable deformation amount in excavation of horizontally fractured strata driven by multiple processes, characterized in that, Includes the following steps: Based on the location of the anomaly and its development pattern, potential areas of large deformation were initially delineated, and the risk level of large deformation was initially determined. Based on the risk level, the implementation areas for surface drilling and underground geophysical exploration were determined, and test sections and application sections were distinguished. Geophysical exploration was conducted within the application section to establish a numerical model. Based on the on-site excavation procedures, multiple types of excavation conditions were designed. Combined with the designed support parameters, numerical simulations of multiple excavation conditions were conducted to determine the optimal reserved deformation amount for the application section obtained from the numerical simulation. Establish an evaluation index system for reserved deformation based on geophysical parameters, geological parameters and construction parameters. Obtain data through each construction cycle of the test section to predict the reserved deformation of the tunnel face when passing through a certain stratum. The optimal allowable deformation amount and the predicted construction allowable deformation amount are combined to comprehensively determine the construction allowable deformation amount.
2. The method for determining the allowable deformation amount in multi-process driven horizontal fractured strata excavation as described in claim 1, characterized in that, The location of the anomaly was determined based on surface analysis, surface geophysical exploration, and semi-airborne transient electromagnetic detection.
3. The method for determining the allowable deformation amount in multi-process driven horizontal fractured strata excavation as described in claim 2, characterized in that, The specific process of obtaining the location of anomalies includes: based on geological exploration, using surface analysis to preliminarily identify fractured strata and their occurrence from the surface conditions; If the surface conditions are flat, surface geophysical exploration is used for detection. If the surface conditions are uneven and exceed the set value, semi-airborne transient electromagnetic exploration is used. The low-resistivity area is identified as a water-bearing fractured area.
4. The method for determining the allowable deformation amount in multi-process driven horizontal fractured strata excavation as described in claim 1, characterized in that, When establishing a numerical model, the process of determining the rock mass material parameters of the mathematical model includes obtaining rock cores through surface drilling, conducting rock mechanics and geophysical experiments to obtain parameters such as the rock mass's elastic modulus, Poisson's ratio, cohesion, friction angle, density, rock mass wave velocity, and rock mass resistivity. After obtaining the relationship between rock mass wave velocity and resistivity and rock mass strength parameters through fitting, rock mass wave velocity and resistivity are obtained through geophysical exploration inside the tunnel. Based on the relationship between the obtained geophysical parameters and rock mass strength parameters, the rock mass parameters within the geophysical exploration area are inverted within a set length interval to obtain the rock mass material parameters within the geophysical exploration area in the strata ahead. The parameters outside the geophysical exploration area are based on the rock mass parameters obtained from borehole core sampling.
5. The method for determining the allowable deformation amount in multi-process driven horizontal fractured strata excavation as described in claim 1 or 4, characterized in that, When establishing a numerical model, the process of determining the constitutive relation of the mathematical model includes: conducting creep tests on the rock mass material of the test section, obtaining the creep parameters of the rock mass material, and then substituting them into the creep damage model to obtain the constitutive relation.
6. The method for determining the allowable deformation amount in multi-process driven horizontal fractured strata excavation as described in claim 1, characterized in that, When establishing a numerical model, the underground engineering excavation of the test section is simulated, the displacement data collected from the test section is compared with the data from the numerical simulation, and the parameters of the numerical model are calibrated or revised.
7. The method for determining the allowable deformation amount in multi-process driven horizontal fractured strata excavation as described in claim 1, characterized in that, The specific process of conducting numerical simulation of multiple excavation conditions includes obtaining the deformation of the surrounding rock under different excavation methods, using vertical displacement as the rating index, and dividing the numerical model into intervals according to a set length after the model stabilizes. The maximum vertical position in each interval is the optimal reserved deformation amount in that application segment interval.
8. The method for determining the allowable deformation amount in multi-process driven horizontal fractured strata excavation as described in claim 1, characterized in that, The evaluation index system for reserved deformation includes geophysical parameters, geological parameters, and construction parameters. Geophysical parameters include seismic wave velocity and resistivity values. Geological parameters include surrounding rock grade, rock mass integrity, and the angle between the rock strata dip angle and the tunnel axis. Construction parameters include tunnel depth, tunnel closure time, support parameters, and arch settlement.
9. The method for determining the allowable deformation amount in multi-process driven horizontal fractured strata excavation as described in claim 8, characterized in that, The specific process of predicting the construction allowance deformation amount in a certain stratum in front of the tunnel face includes constructing a neural network model. The model takes the allowance deformation amount evaluation index system as input and outputs the crown settlement, that is, the construction allowance deformation amount determined based on artificial intelligence algorithm.
10. A system for determining the allowable deformation amount in horizontally fractured strata excavation driven by multiple processes, characterized in that, include: The area division module is configured to initially delineate potential large deformation areas based on the location and development pattern of anomalies, and to initially determine the risk level of large deformation. Based on the risk level, it will target and guide the determination of the implementation areas for surface drilling and underground geophysical exploration, and distinguish between test sections and application sections. The numerical model calculation module is configured to obtain the results of geophysical exploration within the application section, establish a numerical model, design multiple types of excavation conditions based on the on-site excavation procedures, and conduct numerical simulations of multiple excavation conditions in combination with the designed support parameters to determine the optimal reserved deformation amount of the application section based on the numerical simulation. The intelligent prediction module is configured to establish an evaluation index system for reserved deformation based on geophysical parameters, geological parameters and construction parameters. It obtains data sets through each construction cycle of the test section and predicts the reserved deformation of the tunnel face when passing through a certain stratum. The fusion module is configured to fuse the optimal reserved deformation amount and the predicted construction reserved deformation amount to comprehensively determine the construction reserved deformation amount.
Citation Information
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