A method for predicting the flexural capacity of a plate column joint and related equipment

Abstract

The application discloses a kind of plate column node bending capacity prediction method and its related equipment, comprising: the bending characteristics of the establishment unit width plastic resistance bending moment expression of orthogonal two-way reinforced plate are combined;Establish the initial virtual work equation of the equal of external force work and internal plastic work, form characteristic yield line mode library;Establish the target virtual work equation of the equal of target external force work and target internal plastic work;For the geometric compatibility equation between yield line relative rotation and virtual displacement of characteristic yield line mode;Plate section geometric parameters, material mechanics parameters and unit width plastic resistance bending moment expression are substituted into target virtual work equation in the combination of boundary conditions and geometric compatibility equation, and energy balance expression is constructed;Based on energy balance expression, the bending capacity calculation model of target column section type is derived, the partial derivative of yield line direction angle in bending capacity calculation model is solved, and the partial derivative is equal to zero, to obtain the ultimate bending capacity of target column section node.

Description

Technical Field

[0001] This invention belongs to the field of slab-column structure design and relates to a method for predicting the bending bearing capacity of slab-column joints and related equipment. Background Technology

[0002] Slab-column structures are widely used in buildings due to their superior space utilization and economy. However, slab-column joints are critical load-bearing areas and also weak points in the structure. Their failure, especially brittle punching shear failure, often leads to catastrophic progressive collapse.

[0003] Traditional research and practice typically use the smaller of punching shear capacity and flexural capacity to assess the safety of slab-column joints. However, this method has two fundamental flaws: First, the classical yield line theory upon which the calculation of flexural capacity relies may fail when the slab reinforcement ratio is low, failing to provide a true safety upper limit; second, and more critically, existing theories and major domestic and international structural design codes (such as GB50010, ACI 318, and Eurocode 2) severely lack specific analysis methods for irregularly shaped column joints such as cruciform, T-shaped, and L-shaped joints.

[0004] With the diversification of modern building functions and forms, the application of irregular-shaped columns is increasing. Compared with regular square columns, the complex cross-sectional shape of irregular-shaped columns leads to a fundamental change in the yield line development mode (i.e., failure mode) of the surrounding slabs. Directly applying the yield line mode and calculation formula of square columns to the joints of irregular-shaped columns will produce unacceptable errors and constitute potential safety hazards. Summary of the Invention

[0005] The purpose of this invention is to overcome the shortcomings of the prior art and provide a method for predicting the flexural bearing capacity of slab-column joints and related equipment, which can accurately reflect the unique failure mechanism of different column sections (especially irregular columns).

[0006] To achieve the above objectives, the present invention employs the following technical solution: A method for predicting the flexural bearing capacity of a slab-column joint includes: Establish the relationship between the bending moment and curvature of the plate section considering the nonlinear constitutive properties of concrete and the yield strength of steel reinforcement, and establish an expression for the plastic resistance bending moment per unit width in combination with the bending characteristics of orthogonal two-way reinforced plates; Establish an initial virtual work equation that equals the work done by external forces and the plastic work done by internal forces, determine the characteristic yield line patterns corresponding to different types of column sections, and form a characteristic yield line pattern library; Based on the characteristic yield line patterns corresponding to the target column section type in the characteristic yield line pattern library, a target virtual work equation is established to make the target external force work equal to the target internal force plastic work. A geometric compatibility equation is established between the relative rotation angle of the yield line and the virtual displacement for the characteristic yield line mode. By combining boundary conditions and geometric compatibility equations, the geometric parameters of the plate section, the mechanical parameters of the material, and the expression for the plastic resistance bending moment per unit width are substituted into the target virtual work equation to construct an energy balance expression; Based on the energy balance expression, a calculation model for the flexural bearing capacity of the target column section type is derived. The partial derivative of the yield line direction angle in the flexural bearing capacity calculation model is calculated and set to zero to obtain the ultimate flexural bearing capacity of the target column section node.

[0007] Optionally, the process of determining the characteristic yield line patterns corresponding to different types of column sections and forming a characteristic yield line pattern library includes: determining the basic geometric shape of the yield line of various column sections based on the plate deformation compatibility conditions, optimizing the yield line direction angle in the basic geometric shape by combining numerical optimization, establishing characteristic yield line patterns corresponding to square columns, cross-shaped columns, T-shaped columns and L-shaped columns respectively, and saving them in the characteristic yield line pattern library.

[0008] Optionally, the expression for the plastic resistance moment per unit width is:

[0009] in, This represents the plastic resistance to bending moment per unit width. A s This represents the area of ​​tensile reinforcement per unit width. f y This represents the design value of the tensile strength of the reinforcing steel. h 0 indicates the effective height of the cross-section; f cm This represents the design value of the concrete compressive strength; r This indicates the ratio of tensile reinforcement bars within the slab.

[0010] Optionally, after obtaining the ultimate flexural capacity of the target column section node, a model verification process is also included. The model verification process includes: A slab-column joint bending test database was constructed, which includes various column cross-section types and various working conditions. The database contains multiple sets of test data, and each set of test data includes at least structural geometric parameters, material mechanical parameters and measured bending bearing capacity values. By using a flexural bearing capacity calculation model and inputting the structural geometric parameters and material mechanical parameters of the corresponding groups, the predicted flexural bearing capacity of each group of test data is obtained. The accuracy of the flexural bearing capacity calculation model was verified using the predicted values ​​of flexural bearing capacity.

[0011] Optionally, the process of verifying the accuracy of the flexural bearing capacity calculation model using the predicted flexural bearing capacity values ​​includes: The predicted flexural bearing capacity of each set of test data is compared with the corresponding measured flexural bearing capacity value to calculate the ratio; Statistical analysis was performed on the ratios of all experimental data to calculate the mean and coefficient of variation of the ratios; When the mean of the ratio is within a preset range and the coefficient of variation of the ratio is less than or equal to a preset threshold, the flexural bearing capacity calculation model is deemed to meet the engineering prediction accuracy requirements.

[0012] Optionally, in the process of establishing the target virtual work equation where the target external force work and the target internal force plastic work are equal, the assumptions of isotropic reinforcement and symmetry are adopted to determine the quantitative relationship of the ultimate bending moment in the orthogonal direction and the mathematical angular constraint relationship of each yield line direction angle in the characteristic yield line mode.

[0013] Optionally, in the process of calculating the partial derivative of the yield line direction angle in the flexural bearing capacity calculation model and setting the partial derivative equal to zero, the partial derivatives are calculated for the multiple mutually coupled yield line direction angles corresponding to different types of column sections. The analytical constraint range of the yield line direction angle is determined by combining the section geometric constraint conditions and the plastic hinge development conditions, and then the minimum upper limit solution of the flexural bearing capacity calculation model is obtained.

[0014] A system for predicting the flexural bearing capacity of a slab-column joint includes: The expression derivation module is used to establish the relationship between the bending moment and curvature of the plate section considering the nonlinear constitutive properties of concrete and the yield strength of steel reinforcement, and to establish the expression for the plastic resistance bending moment per unit width by combining the bending characteristics of orthogonal two-way reinforced plates. The pattern library construction module is used to establish an initial virtual work equation that equals the work done by external forces and the plastic work done by internal forces, determine the characteristic yield line patterns corresponding to different types of column sections, and form a characteristic yield line pattern library. The target equation establishment module is used to establish a target virtual work equation that equals the target external force work and the target internal force plastic work based on the characteristic yield line patterns of the corresponding target column section types in the characteristic yield line pattern library. The geometric compatibility module is used to establish the geometric compatibility equations between the relative rotation angle and virtual displacement of the yield line for the characteristic yield line mode. The equilibrium construction module is used to combine boundary conditions and geometric compatibility equations, and substitute the plate section geometric parameters, material mechanical parameters and the expression for the plastic resistance bending moment per unit width into the target virtual work equation to construct an energy balance expression; The bearing capacity prediction module is used to derive the flexural bearing capacity calculation model of the target column section type based on the energy balance expression, calculate the partial derivative of the yield line direction angle in the flexural bearing capacity calculation model and set the partial derivative equal to zero, and solve for the ultimate flexural bearing capacity of the target column section node.

[0015] A computer device includes a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the steps of the method for predicting the flexural bearing capacity of a slab-column joint.

[0016] A computer-readable storage medium storing a computer program that, when executed by a processor, implements the steps of the method for predicting the flexural bearing capacity of a slab-column joint.

[0017] Compared with the prior art, the present invention has the following beneficial effects: This invention, based on yield line theory and the principle of virtual work, establishes a characteristic yield line pattern library covering different types of column cross-sections. It solves the problem of uncertain yield lines due to the geometric irregularities of irregular column cross-sections, resulting in asymmetric and complex directional yield lines. By deriving a closed-form analytical model for calculating flexural bearing capacity and obtaining a minimum upper bound solution by taking partial derivatives with respect to the yield line direction angle, the complex problem of flexural bearing capacity of irregular column joints is simplified into an explicit analytical model with well-defined parameters. This accurately reflects the unique bending failure mechanism of different column cross-sections without requiring tedious finite element iterative calculations, significantly lowering the computational threshold and improving the efficiency and safety of engineering design.

[0018] Furthermore, in determining the characteristic yield line pattern library, instead of simple geometric classification, the basic geometric shape of the yield line is determined through theoretical analysis, and then the yield line direction angle is optimized by numerical optimization. Each irregular column cross-section has a corresponding characteristic yield line pattern, effectively overcoming the systematic errors and potential safety hazards caused by directly applying empirical formulas for regular square columns in traditional methods, and greatly improving the accuracy and stability of the model's prediction results.

[0019] Furthermore, a specific explicit expression for the plastic resistance to bending moment per unit width was clarified, mechanically coupling conventional engineering parameters such as the area of ​​tensile reinforcement, design value of yield strength, effective height of section, design value of concrete compressive strength, and reinforcement ratio. This formula, devoid of free parameters, accurately quantifies the mechanical response of orthogonally reinforced slabs under conditions where tensile reinforcement and the yield line are orthogonal. It provides a parameter basis for the application of the virtual work principle and the construction of an energy balance expression, ensuring the rigor and non-replicability of the explicit calculation model in its theoretical derivation.

[0020] Furthermore, by collecting measured data on bending failures of various column cross-section types and material geometric parameters, the prediction results of the explicit calculation model are uniformly calibrated, enabling the purely theoretical analytical model to form a two-way calibration with the actual experimental results. This effectively avoids the risk of theoretical assumptions deviating from engineering reality and ensures, from the perspective of practical verification, that the model has excellent generalization adaptability and real predictive value under varying working conditions.

[0021] Furthermore, the mean of the ratio of predicted to measured values ​​is limited to a preset range that approaches the ideal value, and the coefficient of variation is required to be below the allowable upper limit to eliminate random errors, verifying that the explicit calculation model has both unbiasedness and low dispersion. This ensures that the prediction results have no systematic bias and good repeatability, providing an intuitive and quantitative accuracy judgment standard for the design and safety assessment of slab-column structures.

[0022] Furthermore, in establishing the target virtual work equation and geometric compatibility equation, the assumptions of isotropic reinforcement and symmetry of the yield curve about the coordinate axes are introduced, clarifying the quantitative relationship of the ultimate bending moment in orthogonal directions and the mathematical constraints between the direction angles of each yield line. While ensuring the rationality of the mechanical analysis, the complex energy dissipation boundary conditions are effectively simplified, reducing the high-degree-of-freedom three-dimensional spatial deformation problem to a geometrically mathematical model with clearly defined parameter constraints. This significantly reduces the difficulty of analytical derivation and enhances the solvability of the model in practical calculations.

[0023] Furthermore, the range of values ​​for the multiple coupled yield line direction angles in the expression for the bearing capacity of irregular column joints is jointly determined by the cross-sectional geometric constraints and the development conditions of the plastic hinge. By taking the partial derivatives of each direction angle and setting them to zero, following the principle of finding the minimum upper limit solution in the plastic limit analysis theorem, the most unfavorable critical state with minimum energy consumption under the bending failure mode of the structure can be accurately captured through analytical optimization, ensuring that the ultimate bending bearing capacity obtained is true and reliable. Attached Figure Description

[0024] Figure 1 This is a schematic diagram of the moment-curvature relationship of the present invention; Figure 2 This is a schematic diagram of the yield line mode of the plate-square column node of the present invention; Figure 3 This is a schematic diagram of the yield line mode of the plate-cross-shaped column node of the present invention; Figure 4 This is a schematic diagram of the yield line mode of the plate-L-shaped column node of the present invention; Figure 5 This is a schematic diagram of the yield line mode of the plate-T-shaped column node of the present invention; Figure 6 This is a schematic diagram of the evaluation results of the present invention.

[0025] in: c The length of the column section; c 1 represents the length of the column section leg; c 2 represents the thickness of the column section; L Let be the side length of the board; L 0 represents the effective side length of the plate; a、b、l、i、g、n The direction angle of the yield line. Detailed Implementation

[0026] To enable those skilled in the art to better understand the present invention, the technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort should fall within the scope of protection of the present invention.

[0027] It should be noted that the terms "first," "second," etc., in the specification, claims, and accompanying drawings of this invention are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged where appropriate so that the embodiments of the invention described herein can be implemented in orders other than those illustrated or described herein. Furthermore, the terms "comprising" and "having," and any variations thereof, are intended to cover a non-exclusive inclusion; for example, a process, method, system, product, or apparatus that comprises a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such processes, methods, products, or apparatus.

[0028] Example 1 This embodiment provides a method for predicting the flexural bearing capacity of slab-column joints based on characteristic yield line modes. This method is particularly suitable for predicting the flexural bearing capacity of slabs and columns with different cross-sections such as square, cross, T-shaped, and L-shaped, providing a key theoretical basis for the design and safety assessment of irregularly shaped slab-column structures. The method includes the following steps: Step 1: Based on yield line theory and internal force analysis, establish the moment-curvature relationship of the slab section considering the nonlinear constitutive properties of concrete and the yield strength of steel reinforcement; based on the bending characteristics of orthogonal two-way reinforced slabs, derive an explicit expression for the plastic resistance moment per unit width; according to the deformation compatibility conditions of the slab, establish a virtual work equation that equals the work done by external forces and the plastic work done by internal forces; through theoretical analysis, determine the basic geometric shape of the yield line, and establish standardized characteristic yield line patterns corresponding to square columns, cross-shaped columns, T-shaped columns, and L-shaped columns, forming a systematic characteristic yield line pattern library; Step 2: Establishing the explicit calculation model: For each column section type, based on its corresponding standardized characteristic yield line mode, establish a virtual work equation that equals the work done by external forces and the plastic work done by internal forces; based on the analysis of the yield line geometry and boundary conditions, describe the coordination relationship between the relative rotation angle and displacement of the yield line through geometric equations; combined with the plastic resistance bending moment per unit width, derive the closed analytical explicit calculation model of the bending capacity of each column section type. Step 3: Model Validation Step: Construct a slab-column joint bending test database containing various column cross-section types such as square columns, cross-shaped columns, T-shaped columns, and L-shaped columns, as well as various working conditions. Each set of data should include at least the column cross-section geometric parameters, slab geometric parameters, material parameters, and measured bending capacity values. Use this database to uniformly validate all explicit bending capacity calculation models, compare the predicted values ​​with the measured values, and calculate the mean and coefficient of variation to complete the accuracy assessment.

[0029] In step one, the bending failure test data of the plate-column joint is systematically collected from existing technical literature to construct the bending failure test database of the plate-column joint.

[0030] Each set of data in the experimental database should include the column cross-sectional shape and column thickness. c 1. Column height c 2. Effective height of plate section h 0. Effective side length of the board L 0. Tensile longitudinal reinforcement ratio of slab r Yield strength of longitudinal reinforcement in slab f y and axial compressive strength of concrete f c Information.

[0031] The specific technical approach for determining the characteristic yield line mode in step one is as follows: Step 1-1: Based on the actual stress state of the slab section under ultimate load, establish the total moment-curvature relationship of the slab section that comprehensively considers the nonlinear constitutive relationship of concrete and the yield strength of steel reinforcement, so as to provide a theoretical basis for accurately describing the nonlinear bending behavior of the slab section and determining the ultimate bending moment.

[0032] Steps 1-2: Based on the bending characteristics of orthogonal two-way reinforced slabs, derive the explicit expression for the plastic resistance to bending moment per unit width under the condition that the yield line is orthogonal to the tensile reinforcement, and determine the key mechanical parameters required for subsequent bearing capacity calculations.

[0033] Steps 1-3: Based on the deformation compatibility conditions of the plate, establish a virtual work equation in which the work done by external forces is equal to the plastic work done by internal forces, construct the energy balance relationship of the plate-column joint system under the limit state, and associate the external load with the internal plastic dissipation.

[0034] Steps 1-4: Determine the basic geometric shape of the yield line of various column sections through theoretical analysis, and optimize key geometric parameters such as the yield line direction angle by combining numerical optimization. Establish standardized characteristic yield line patterns corresponding to square columns, cross-shaped columns, T-shaped columns and L-shaped columns respectively, forming a systematic characteristic yield line pattern library, as shown in Table 1.

[0035] Table 1 Characteristic Yield Line Pattern Library

[0036] The moment-curvature constitutive relation of the plate section established in step 1-1 specifically reflects the nonlinear material behavior throughout the entire process from the elastic stage to yielding, providing a theoretical basis for accurately predicting the nodal limit state. Its moment-curvature relationship is as follows: Figure 1 As shown.

[0037] The plastic resistance moment per unit width determined in steps 1-2 provides key mechanical parameters for establishing the subsequent explicit calculation model. The resistance moment per unit width generated by the tension reinforcement is as follows:

[0038] in, This represents the plastic resistance to bending moment per unit width. A s This represents the area of ​​tensile reinforcement per unit width. f y This represents the design value of the tensile strength of the reinforcing steel. h 0 indicates the effective height of the cross-section; f cm This represents the design value of the concrete compressive strength; r This indicates the ratio of tensile reinforcement bars within the slab.

[0039] The virtual work equation established in steps 1-3 fully expresses the energy conservation relationship between the work done by external forces and the plastic work done by internal forces. The virtual work equation is as follows:

[0040] in, P i Indicates a unit concentrated load; d ( x , y () indicates the displacement generated at a suitable point within the plate; m u Indicates the ultimate bending moment per unit width; i n Indicates the relative angle between the two sides of the yield line; Indicates the length of the yield line.

[0041] The yield line modes determined through optimization analysis in steps 1-4 can accurately reflect the unique failure characteristics of various column sections, and their yield line modes are as follows: Figure 2-Figure 5 As shown.

[0042] The specific technical approach for the explicit calculation model of the flexural bearing capacity of each node determined in step two is as follows: Step 2-1: For each type of column section, based on its corresponding standardized characteristic yield line mode, establish a virtual work equation in which the external force work and the internal force plastic work are equal, and determine the energy balance relationship of the plate-column joint under ultimate load.

[0043] Step 2-2: Based on the analysis of the geometric shape of the yield line, establish the geometric compatibility equation between the relative rotation angle of the yield line and the virtual displacement, and clarify the mechanical relationship between the deformation compatibility condition and the virtual work equation.

[0044] Steps 2-3: Combining boundary conditions and geometric compatibility equations, substitute the plate cross-sectional geometric parameters, material mechanical parameters, and plastic resistance bending moment per unit width into the virtual work equation to construct a complete energy balance expression.

[0045] Steps 2-4: Based on the energy balance expression, derive the explicit calculation model of closed analytical flexural bearing capacity for each column section type; for each column node, further calculate the partial derivative with respect to the yield line direction angle and set it to zero to obtain the minimum upper limit solution of the flexural bearing capacity. This solution is used to characterize the ultimate flexural bearing capacity of the node under the bending failure mode. P flex .

[0046] The specific technical path for step three, model validation, is as follows: Step 3-1: Systematically collect measured data on slab-column joint bending failure from existing literature or experiments, and construct a slab-column joint bending test database containing various column cross-section types (square columns, cruciform columns, T-shaped columns, and L-shaped columns) and various working conditions; each set of test data should include at least: column limb thickness. c 1. Column height c 2. Effective height of plate section h 0. Effective side length of the board L 0. Tensile longitudinal reinforcement ratio of slab r Yield strength of tensile longitudinal reinforcement f y axial compressive strength of concrete f c and measured flexural bearing capacity value P test .

[0047] Step 3-2: For each set of test samples in the database from Step 3-1, use the explicit calculation model established in Steps 2-1 to 2-4, and input the corresponding structural geometric parameters and material mechanical parameters to solve for the predicted value of the flexural bearing capacity of the sample. P flex .

[0048] Step 3-3: Calculate the predicted values ​​for each group of samples. P fle Compared with measured values P test Compare and calculate the ratio. l = P flex / Ptest ; and perform statistical analysis on the ratios of all samples to calculate their mean. m λ With coefficient of variation COV λ ; Steps 3-4: When m λ Located in the interval [0.95, 1.05] and COV λ If the value is ≤0.15, the explicit calculation model is deemed to have acceptable engineering prediction accuracy; otherwise, the explicit calculation model in step two needs to be corrected or recalibrated.

[0049] As verified in steps 3-4, the prediction accuracy of the explicit calculation model satisfies the following condition: the average ratio of the predicted value to the measured value of all verification samples is [value missing]. m λ =1.03, coefficient of variation is COV λ =0.11, indicating that the model has both unbiasedness and low discreteness, and has practical engineering value.

[0050] The comparison results between the punching shear capacity of the slab-column joint predicted by this method and the measured punching shear capacity are as follows: Figure 6 As shown.

[0051] Example 2 In this embodiment, steps 1-4 to 2-4 are performed for the plate-square column node, and the other steps and parameters are the same as in embodiment 1.

[0052] Steps 1-4, which determine the standardized yield line mode for the plate-square column joint, exhibit typical radial failure mechanism characteristics, as shown in the following figure. Figure 2 As shown.

[0053] Step 2-1 establishes the internal force work equation based on the yield line mode of the square column joint, which accurately describes the sum of the internal work done by each plastic hinge line. The internal force work equation is as follows:

[0054] in, Indicates the plate ENTF Internal energy virtual skill, Indicates the plate AHNE The internal energy is merely a form of empty skill.

[0055]

[0056]

[0057] Step 2-2, in establishing the calculation model for the square column joint, adopts the assumptions of isotropic reinforcement and symmetry, as follows: m x =m y= m u , α+β=π / 4 in, display board x , y Ultimate bending moment per unit width of cross section in the direction; α、β Indicates the direction angle of the yield line.

[0058] The geometric equations established in steps 2-3 accurately describe the mathematical relationships between the geometric parameters in the yield line mode, providing the necessary geometric constraints for the application of the virtual work principle. The geometric equations are as follows:

[0059] in, display board N Click K Distance between points.

[0060] Steps 2-4 establish, for the first time, an explicit flexural bearing capacity calculation model for a plate-square column joint by simultaneously solving geometric equations, internal force work equations, and virtual work equations. The calculation model is as follows:

[0061] in, P flex Indicates the flexural bearing capacity of the slab-column joint; c Indicates the side length of the column section; L Indicates the side length of the board; α Indicates the direction angle of the yield line. .

[0062] Steps 2-4 α By taking the partial derivatives and setting them to zero, we can obtain the minimum upper limit solution for the flexural capacity of the plate-square column joint. ,

[0063] in, Indicates the direction angle α Find the partial derivatives.

[0064] Example 3 In this embodiment, steps 1-4 to 2-4 are performed for the plate-cross-shaped column node. Other steps and parameters are the same as in embodiment 1.

[0065] Steps 1-4 determine the yield line mode of the plate-cross-shaped column joint under concentrated load, as follows: Figure 3 As shown.

[0066] Step 2-1: Based on the yield line model of the plate-cross-shaped column joint under concentrated load, the internal force work equation is established as follows:

[0067]

[0068]

[0069] Step 2-2 establishes the following assumptions based on the yield line model of the plate-cross-shaped column joint under concentrated load: Assuming the plate is isotropically reinforced and the yield curve under the ultimate bending state is about x、y The axes each possess a certain degree of symmetry: m x =m y= m u , α+β=π / 4 Step 2-3 establishes the geometric equations based on the yield line model of the plate-cross-shaped column joint under concentrated load as follows:

[0070] Step 2-4: Establish a calculation model for the flexural bearing capacity of the plate-cross-column joint based on the yield line mode of the plate-cross-column joint under concentrated load:

[0071] in, c 1 indicates the length of the column section; c 2 indicates the thickness of the column section. α Indicates the direction angle of the yield line. .

[0072] Steps 2-4 α By taking the partial derivatives and setting them to zero, we can obtain the minimum upper limit solution for the flexural capacity of the plate-cross-column joint. ,

[0073] Example 4 In this embodiment, steps 1-4 to 2-4 are performed for the plate-T-shaped column node. Other steps and parameters are the same as in embodiment 1.

[0074] Steps 1-4 determine the yield line mode of the plate-T-column joint under concentrated load, as follows: Figure 4 As shown.

[0075] Step 2-1: Based on the yield line model of the plate-T-column joint under concentrated load, establish the internal force work equation as follows:

[0076]

[0077]

[0078]

[0079]

[0080]

[0081] Step 2-2 establishes the following assumptions based on the yield line model of the plate-T-column joint under concentrated load: Assuming the plate is isotropically reinforced and the yield curve under the ultimate bending state is about x、y The axes each possess a certain degree of symmetry: m x =m y= m u , α+β= π / 4、η=θ、λ+θ=φ、 tan φ= 2 、 tan （ π / 2 -φ)= 1 / 2 Step 2-3 establishes the geometric equations based on the yield line model of the plate-T-column joint under concentrated load as follows:

[0082]

[0083] Step 2-4: Establish a calculation model for the flexural bearing capacity of the plate-T-column joint based on the yield line mode of the plate-T-column joint under concentrated load:

[0084]

[0085]

[0086] in, c 1 indicates the length of the column section; c 2 indicates the thickness of the column section; a、i、n Indicates the direction angle of the yield line. , .

[0087] Steps 2-4 a, iBy taking the partial derivatives and setting them to zero, we can obtain the minimum upper limit solution for the flexural bearing capacity of the plate-T-column joint. ,

[0088] ,

[0089] Example 5 In this embodiment, steps 1-4 to 2-4 are performed for the plate-L-shaped column node. Other steps and parameters are the same as in embodiment 1.

[0090] Steps 1-4 determine the yield line mode of the plate-L-shaped column joint under concentrated load, as follows: Figure 5 As shown.

[0091] Step 2-1: Based on the yield line model of the plate-L-shaped column joint under concentrated load, establish the internal force work equation as follows:

[0092]

[0093]

[0094]

[0095]

[0096]

[0097] Step 2-2 establishes the following assumptions based on the yield line model of the plate-L-shaped column joint under concentrated load: Assuming the plate is isotropically reinforced and the yield curve under the ultimate bending state is about x、y The axes each possess a certain degree of symmetry: m x =m y= m u , α+β= π / 4 ,γ+η= π / 4 ,λ+θ= π / 4.

[0098] Step 2-3 establishes the geometric equations based on the yield line model of the plate-L-shaped column joint under concentrated load as follows:

[0099]

[0100]

[0101] Step 2-4: Establish a calculation model for the flexural bearing capacity of the plate-L-shaped column joint based on the yield line mode of the plate-L-shaped column joint under concentrated load.

[0102]

[0103] in, a、i、n Indicates the direction angle of the yield line. ; .

[0104] Steps 2-4 α By taking the partial derivatives and setting them equal to zero, we can obtain the minimum upper limit solution for the flexural bearing capacity of the plate-L-shaped column joint. ,

[0105] ,

[0106] ,

[0107] Based on embodiments 1-5, the present invention has the following advantages.

[0108] This invention systematically establishes a characteristic yield line pattern library covering square, cruciform, T-shaped, and L-shaped columns for the first time, fundamentally overcoming the technical bottleneck of the inability to determine the yield line due to the geometric irregularity of irregular column cross-sections, which results in asymmetrical, multi-branched, and complex morphologies with varying directions. The construction of this pattern library is not a simple geometric classification, but rather a combination of theoretical analysis and numerical optimization to determine a mechanically unique standardized characteristic yield line pattern for each type of irregular column cross-section.

[0109] This invention follows the yield line theory and the principle of virtual work, deriving an explicit calculation model with a closed analytical expression for each column section independently, rather than relying on empirical regression based on experimental data, thus ensuring the model's non-reproducibility. Its scientific rigor is reflected in three aspects: First, the analytical expression for the bearing capacity of each column section is uniquely determined by the virtual work equation and the mechanical equilibrium conditions of the yield line mode; the formula structure has no free parameters. Second, multiple yield line direction angles (such as those for T-shaped columns and L-shaped columns) are included in the bearing capacity expression for irregularly shaped column joints. a、i、n The values ​​of the angles are mutually coupled, and their range is determined by the cross-sectional geometric constraints and the development conditions of the plastic hinge. This invention provides analytical constraints for each direction angle through theoretical derivation, forming a complete mathematical model. Third, after unified verification by a multi-condition test database, the average ratio of the predicted value to the measured value is... m λThe coefficient of variation (COV) is 1.03. λ It is 0.11.

[0110] This invention simplifies the complex problem of flexural bearing capacity of irregularly shaped column joints into an explicit analytical model with well-defined parameters. Engineering designers can directly apply this model without finite element analysis, significantly lowering the computational threshold. The innovation of this simplification lies in the fact that while the conventional approach for those skilled in the art is to solve irregularly shaped column problems through numerical simulation, this invention achieves a closed analytical expression by revealing the evolution law of the yield line of irregularly shaped column joints—a path that is not readily apparent. The model input parameters are all conventional parameters used in engineering design (column dimensions, plate thickness, reinforcement ratio, material strength), and their completeness has been verified through sensitivity analysis of a large amount of experimental data. Compared to the finite element method, the explicit computational model of this invention can complete a single calculation within seconds, with definite and unique results and no numerical convergence issues, offering an irreplaceable efficiency advantage in multi-round iterative design or large-scale parameter analysis.

[0111] Existing standards (such as ACI 318, Eurocode 5, and GB 50010-2015) only provide calculation methods for circular or square columns. For irregularly shaped column nodes such as cross-shaped, T-shaped, and L-shaped columns, the engineering community has long lacked scientific methods and usually uses approximations such as equivalent square columns, which introduces systematic errors. This invention verifies through experiments that the mean of the ratio of predicted value to measured value is... m λ =1.03, close to the ideal value of 1.0, proving that the explicit calculation model established in this invention has no systematic bias. This accuracy stems from the characteristic yield line mode tailored to each irregular column section and the mechanical derivation strictly based on the principle of virtual work, rather than empirical data regression, ensuring that the prediction results can truly reflect the actual flexural bearing capacity of the slab-column joint. Coefficient of Variation (COV) λ =0.11, significantly lower than the 15% allowable upper limit for discreteness in engineering practice, indicating that the model's calculation results have good repeatability and low discreteness. This stability is attributed to the uniqueness and optimality of the standardized yield line mode determined by theoretical analysis and numerical optimization in this invention. This invention provides a scientifically reliable calculation method for irregular column joints for the first time, effectively filling a technological gap in this field and directly improving the design safety level of irregular column-slab-column structures.

[0112] The established series of explicit computational models are expressed in closed analytical formulas. The input parameters are all conventional parameters in engineering design (column dimensions, slab thickness, reinforcement ratio, material strength, etc.). They require no iterative calculations or finite element analysis and can be directly embedded into the engineering design process and safety assessment system. Compared to numerical simulation methods, the models of this invention can complete a single calculation within seconds, with definite and unique results and no numerical convergence issues. They possess irreplaceable efficiency advantages in multi-round iterative design or large-scale parameter analysis, providing a reliable theoretical basis and convenient computational tools for the design and reinforcement of slab-column structures.

[0113] Example 6 In this embodiment, a slab-column joint flexural bearing capacity prediction system is provided. This slab-column joint flexural bearing capacity prediction system can be used to implement the above-mentioned slab-column joint flexural bearing capacity prediction method. Specifically, the slab-column joint flexural bearing capacity prediction system includes an expression derivation module, a pattern library construction module, an objective equation establishment module, a geometric coordination module, a balance construction module, and a bearing capacity prediction module.

[0114] The expression derivation module is used to establish the relationship between the bending moment and curvature of the plate section considering the nonlinear constitutive properties of concrete and the yield strength of steel reinforcement, and to establish the expression for the plastic resistance bending moment per unit width by combining the bending characteristics of orthogonal two-way reinforced plates.

[0115] The pattern library construction module is used to establish an initial virtual work equation that equals the work done by external forces and the plastic work done by internal forces, determine the characteristic yield line patterns corresponding to different types of column sections, and form a characteristic yield line pattern library.

[0116] The target equation establishment module is used to establish a target virtual work equation that makes the target external force work equal to the target internal force plastic work based on the characteristic yield line pattern of the target column section type in the characteristic yield line pattern library.

[0117] The geometric compatibility module is used to establish geometric compatibility equations between the relative rotation angle and virtual displacement of the yield line for characteristic yield line modes.

[0118] The equilibrium construction module is used to combine boundary conditions and geometric compatibility equations, and substitute the plate section geometric parameters, material mechanical parameters, and the expression for the plastic resistance bending moment per unit width into the target virtual work equation to construct an energy balance expression.

[0119] The load-bearing capacity prediction module is used to derive the flexural load-bearing capacity calculation model of the target column section type based on the energy balance expression. It calculates the partial derivative of the yield line direction angle in the flexural load-bearing capacity calculation model and sets the partial derivative equal to zero to obtain the ultimate flexural load-bearing capacity of the target column section node.

[0120] Example 7 In this embodiment, a computer device is provided, comprising a processor and a memory. The memory stores a computer program, which includes program instructions. The processor executes the program instructions stored in the computer storage medium. The processor may be a Central Processing Unit (CPU), or other general-purpose processors, digital signal processors (DSPs), application-specific integrated circuits (ASICs), field-programmable gate arrays (FPGAs), or other programmable logic devices, discrete gate or transistor logic devices, discrete hardware components, etc. It is the computing and control core of the computer device, suitable for implementing one or more instructions, specifically suitable for loading and executing one or more instructions to achieve a corresponding method flow or corresponding function. The processor described in this embodiment can be used in the operation of a method for predicting the flexural bearing capacity of slab-column joints, including: establishing a relationship between the slab section bending moment and curvature considering the nonlinear constitutive structure of concrete and the yield strength of steel reinforcement; establishing an expression for the plastic resistance bending moment per unit width based on the bending characteristics of orthogonal two-way reinforced slabs; establishing an initial virtual work equation where the external work and the internal plastic work are equal; and determining the characteristics corresponding to different types of column sections. Characteristic yield line patterns are identified, forming a characteristic yield line pattern library. Based on the characteristic yield line patterns corresponding to the target column section types in the characteristic yield line pattern library, a target virtual work equation is established to ensure that the target external force work and the target internal force plastic work are equal. A geometric compatibility equation is established between the relative rotation angle of the yield line and the virtual displacement for the characteristic yield line patterns. Combining the boundary conditions and the geometric compatibility equation, the geometric parameters of the plate section, the material mechanical parameters, and the expression for the plastic resistance bending moment per unit width are substituted into the target virtual work equation to construct an energy balance expression. Based on the energy balance expression, a calculation model for the flexural bearing capacity of the target column section type is derived. The partial derivative of the yield line direction angle in the flexural bearing capacity calculation model is calculated and set to zero to obtain the ultimate flexural bearing capacity of the target column section node.

[0121] Example 8 This embodiment provides a computer-readable storage medium (Memory), which is a memory device in a computer device used to store programs and data. It is understood that the computer-readable storage medium here can include both the built-in storage medium in the computer device and extended storage media supported by the computer device. The computer-readable storage medium provides storage space that stores the operating system of the computer device. Furthermore, the storage space also stores one or more instructions suitable for loading and execution by a processor. These instructions can be one or more computer programs (including program code). It should be noted that the computer-readable storage medium here can be high-speed RAM or non-volatile memory, such as at least one disk storage device.

[0122] One or more instructions stored in a computer-readable storage medium can be loaded and executed by the processor to implement the corresponding steps of the method for predicting the flexural bearing capacity of slab-column joints in the above embodiments; one or more instructions in the computer-readable storage medium are loaded and executed by the processor in the following steps: establishing the relationship between the bending moment and curvature of the slab section considering the nonlinear constitutive properties of concrete and the yield strength of steel reinforcement, and establishing an expression for the plastic resistance bending moment per unit width in combination with the bending characteristics of orthogonal two-way reinforced slabs; establishing an initial virtual work equation in which the work done by external forces is equal to the plastic work done by internal forces, determining the characteristic yield line patterns corresponding to different types of column sections, and forming a characteristic yield line pattern library; based on the characteristic yield line pattern library... Based on the characteristic yield line pattern corresponding to the target column section type, a target virtual work equation is established to ensure that the target external force work and the target internal force plastic work are equal. A geometric compatibility equation is established between the yield line relative rotation angle and the virtual displacement for the characteristic yield line pattern. Combining the boundary conditions and the geometric compatibility equation, the geometric parameters of the plate section, the material mechanics parameters, and the expression for the plastic resistance bending moment per unit width are substituted into the target virtual work equation to construct an energy balance expression. Based on the energy balance expression, a calculation model for the flexural bearing capacity of the target column section type is derived. The partial derivative of the yield line direction angle in the flexural bearing capacity calculation model is calculated and set to zero to obtain the ultimate flexural bearing capacity of the target column section node.

[0123] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, this application can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, this application 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, optical storage, etc.) containing computer-usable program code.

[0124] This application is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of this application. 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... Figure one One or more processes and / or boxes Figure one A device that provides the functions specified in one or more boxes.

[0125] 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 one One or more processes and / or boxes Figure one The function specified in one or more boxes.

[0126] 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 one One or more processes and / or boxes Figure one The steps of the function specified in one or more boxes.

[0127] In the above embodiments of this application, the descriptions of each embodiment have different focuses. For parts not described in detail in a certain embodiment, please refer to the relevant descriptions of other embodiments.

[0128] The above description is only a preferred embodiment of this application. It should be noted that for those skilled in the art, several improvements and modifications can be made without departing from the principle of this application, and these improvements and modifications should also be considered within the scope of protection of this application.

[0129] It should be understood that the above description is for illustrative purposes and not for limitation. Many embodiments and applications beyond the examples provided will become apparent to those skilled in the art upon reading the above description.

Claims

1. A method for predicting the flexural bearing capacity of a slab-column joint, characterized in that, include: Establish the relationship between the bending moment and curvature of the plate section considering the nonlinear constitutive properties of concrete and the yield strength of steel reinforcement, and establish an expression for the plastic resistance bending moment per unit width in combination with the bending characteristics of orthogonal two-way reinforced plates; Establish an initial virtual work equation that equals the work done by external forces and the plastic work done by internal forces, determine the characteristic yield line patterns corresponding to different types of column sections, and form a characteristic yield line pattern library; Based on the characteristic yield line patterns corresponding to the target column section type in the characteristic yield line pattern library, a target virtual work equation is established to make the target external force work equal to the target internal force plastic work. A geometric compatibility equation is established between the relative rotation angle of the yield line and the virtual displacement for the characteristic yield line mode. By combining boundary conditions and geometric compatibility equations, the geometric parameters of the plate section, the mechanical parameters of the material, and the expression for the plastic resistance bending moment per unit width are substituted into the target virtual work equation to construct an energy balance expression; Based on the energy balance expression, a calculation model for the flexural bearing capacity of the target column section type is derived. The partial derivative of the yield line direction angle in the flexural bearing capacity calculation model is calculated and set to zero to obtain the ultimate flexural bearing capacity of the target column section node.

2. The method for predicting the flexural bearing capacity of a slab-column joint according to claim 1, characterized in that, The process of determining the characteristic yield line patterns corresponding to different types of column sections and forming a characteristic yield line pattern library includes: determining the basic geometric shape of the yield line of various column sections based on the plate deformation compatibility conditions, optimizing the yield line direction angle in the basic geometric shape by combining numerical optimization, establishing characteristic yield line patterns corresponding to square columns, cross-shaped columns, T-shaped columns and L-shaped columns respectively, and saving them in the characteristic yield line pattern library.

3. The method for predicting the flexural bearing capacity of a slab-column joint according to claim 1, characterized in that, The expression for the plastic resistance moment per unit width is: in, This represents the plastic resistance to bending moment per unit width. A s This represents the area of ​​tensile reinforcement per unit width. f y This represents the design value of the tensile strength of the reinforcing steel. h 0 indicates the effective height of the cross-section; f cm This represents the design value of the concrete compressive strength; ρ This indicates the ratio of tensile reinforcement bars within the slab.

4. The method for predicting the flexural bearing capacity of a slab-column joint according to claim 1, characterized in that, After obtaining the ultimate flexural capacity of the target column section joint, the process also includes a model verification process, which includes: A slab-column joint bending test database was constructed, which includes various column cross-section types and various working conditions. The database contains multiple sets of test data, and each set of test data includes at least structural geometric parameters, material mechanical parameters and measured bending bearing capacity values. By using a flexural bearing capacity calculation model and inputting the structural geometric parameters and material mechanical parameters of the corresponding groups, the predicted flexural bearing capacity of each group of test data is obtained. The accuracy of the flexural bearing capacity calculation model was verified using the predicted values ​​of flexural bearing capacity.

5. The method for predicting the flexural bearing capacity of a slab-column joint according to claim 4, characterized in that, The process of verifying the accuracy of the flexural bearing capacity calculation model using the predicted flexural bearing capacity values ​​includes: The predicted flexural bearing capacity of each set of test data is compared with the corresponding measured flexural bearing capacity value to calculate the ratio; Statistical analysis was performed on the ratios of all experimental data to calculate the mean and coefficient of variation of the ratios; When the mean of the ratio is within a preset range and the coefficient of variation of the ratio is less than or equal to a preset threshold, the flexural bearing capacity calculation model is deemed to meet the engineering prediction accuracy requirements.

6. The method for predicting the flexural bearing capacity of a slab-column joint according to claim 1, characterized in that, In the process of establishing the target virtual work equation where the target external force work and the target internal force plastic work are equal, the assumptions of isotropic reinforcement and symmetry are adopted to determine the quantitative relationship of the ultimate bending moment in the orthogonal direction and the mathematical angular constraint relationship of each yield line direction angle in the characteristic yield line mode.

7. The method for predicting the flexural bearing capacity of a slab-column joint according to claim 1, characterized in that, In the process of calculating the partial derivative of the yield line direction angle in the flexural bearing capacity calculation model and setting the partial derivative equal to zero, the partial derivatives are calculated for multiple mutually coupled yield line direction angles corresponding to different types of column sections. The analytical constraint range of the yield line direction angle is determined by combining the section geometric constraint conditions and the plastic hinge development conditions, and then the minimum upper limit solution of the flexural bearing capacity calculation model is obtained.

8. A system for predicting the flexural bearing capacity of a slab-column joint, characterized in that, include: The expression derivation module is used to establish the relationship between the bending moment and curvature of the plate section considering the nonlinear constitutive properties of concrete and the yield strength of steel reinforcement, and to establish the expression for the plastic resistance bending moment per unit width by combining the bending characteristics of orthogonal two-way reinforced plates. The pattern library construction module is used to establish an initial virtual work equation that equals the work done by external forces and the plastic work done by internal forces, determine the characteristic yield line patterns corresponding to different types of column sections, and form a characteristic yield line pattern library. The target equation establishment module is used to establish a target virtual work equation that equals the target external force work and the target internal force plastic work based on the characteristic yield line patterns of the corresponding target column section types in the characteristic yield line pattern library. The geometric compatibility module is used to establish the geometric compatibility equations between the relative rotation angle and virtual displacement of the yield line for the characteristic yield line mode. The equilibrium construction module is used to combine boundary conditions and geometric compatibility equations, and substitute the plate section geometric parameters, material mechanical parameters and the expression for the plastic resistance bending moment per unit width into the target virtual work equation to construct an energy balance expression; The bearing capacity prediction module is used to derive the flexural bearing capacity calculation model of the target column section type based on the energy balance expression, calculate the partial derivative of the yield line direction angle in the flexural bearing capacity calculation model and set the partial derivative equal to zero, and solve for the ultimate flexural bearing capacity of the target column section node.

9. A computer device comprising a memory, a processor, and a computer program stored in the memory and executable on the processor, characterized in that, When the processor executes the computer program, it implements the steps of the method for predicting the flexural bearing capacity of a plate-column joint as described in any one of claims 1 to 7.

10. A computer-readable storage medium storing a computer program, characterized in that, When the computer program is executed by the processor, it implements the steps of the method for predicting the flexural bearing capacity of a plate-column joint as described in any one of claims 1 to 7.

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