Composite beam plastic flexural capacity correction calculation method considering interface slippage

Abstract

The invention discloses a composite beam plastic flexural capacity correction calculation method considering interface slippage. The method is suitable for a hogging moment area of a steel-concrete composite beam reinforced by an external prestress carbon fiber reinforced composite material rib. The method comprises the following steps: firstly, determining basic static boundary parameters and a shear connector constitutive model; in the iterative calculation, solving a temporary bearing capacity by utilizing an axial force and moment balance equation, and calculating interface slippage along the beam length and a deformation loss item at the support based on internal force distribution; then determining a rigidity degradation coefficient according to the average slip amount, and dynamically reducing the static yield bending moment to reconstruct the plastic hinge length; and finally, constructing a corrected deformation coordination equation, deducting a deformation loss item in geometric deformation calculation to obtain the real elongation of the in-vitro CFRP rib, and updating a stress parameter until a result is converged. According to the method, by introducing a deformation loss item and a rigidity degradation mechanism, deviation caused by traditional rigid connection assumption is corrected, and accurate prediction of the ultimate bearing capacity is achieved.

Description

Technical Field

[0001] This invention relates to the field of civil engineering structural calculation technology, specifically to a corrected calculation method for the plastic flexural bearing capacity of composite beams considering interface slip. Background Technology

[0002] Steel-concrete composite beams fully utilize the compressive strength of concrete and the tensile strength of steel, and are widely used in bridge engineering and building structures. With the increase in structural service life or load levels, strengthening the negative moment zone of composite beams using external prestressing technology has become a common engineering measure. Carbon fiber reinforced polymer (CFRP) reinforcement, due to its high strength, lightweight, and corrosion resistance, is frequently used as external prestressing material. When assessing the load-bearing capacity of such strengthened structures, ensuring the accuracy of the calculation model is crucial for the safe design of the structure.

[0003] Current techniques for calculating the flexural capacity of externally prestressed composite beams typically rely on classical plasticity theory and deformation compatibility conditions. During the calculation process, designers often assume a tight connection between the steel beam and the concrete slab, satisfying the plane section assumption, and assuming that rotational deformation of the section is entirely converted into axial elongation of the external CFRP reinforcement. Based on this assumption, the stress increment of the external CFRP reinforcement can be directly derived by calculating the ultimate rotation angle or section curvature of the plastic hinge region, and then superimposed to obtain the ultimate bearing capacity value of the structure.

[0004] However, existing calculation methods generally suffer from a significant drawback: they neglect the weakening effect of steel-concrete interface slip on deformation compatibility. In the negative bending moment region, shear connectors bear substantial shear forces, inevitably leading to relative slippage between the steel beam and the concrete slab, causing their deformations to no longer be perfectly coordinated. This interface slip consumes some of the section rotation deformation, resulting in the actual elongation of the external CFRP reinforcement being less than the theoretical value calculated based on the rigid connection assumption. If the traditional deformation compatibility equation is directly applied without deducting the deformation loss caused by slippage, the true stress level of the external prestressing tendons will be overestimated, resulting in a calculated value higher than the actual load-bearing capacity of the structure, posing a safety hazard to the engineering structure. Summary of the Invention

[0005] To address the shortcomings of existing technologies, this invention provides a method for correcting the calculation of the plastic flexural bearing capacity of composite beams by considering interface slip, thus solving the problem of overestimation of bearing capacity caused by neglecting interface slip.

[0006] The first aspect of this invention provides a corrected calculation method for the plastic flexural capacity of composite beams considering interface slip, applicable to the negative bending moment zone of steel-concrete composite beams reinforced with externally prestressed carbon fiber reinforced composite (CFRP) tendons. This method solves the problem that traditional calculation models neglect the influence of steel-concrete interface slip, thus overestimating the stress increment and ultimate bearing capacity of the externally prestressed CFRP tendons.

[0007] The calculation method includes the following steps: Obtain the geometric dimensions and material properties of the steel-concrete composite beam, and determine the load-slip constitutive model of the shear connector. Based on this, calculate the static yield moment and the effective length of the external CFRP reinforcement. Define the above parameters as basic static boundary parameters. Based on the initial state of the iterative calculation set by the basic static boundary parameters, the stress parameters of the current external prestressed carbon fiber reinforced composite tendon are assigned as the effective prestress after deducting the prestress loss, and the initial interface state is set to a no-slip state. The force balance solution is performed, using the current stress parameters as input, the height of the plastic neutral shaft is determined by the axial force balance equation and the internal force distribution data of the section is generated, the moment balance equation is established by taking moments on the plastic neutral shaft, and the provisional ultimate bearing capacity is calculated. Perform the calculation of interface slip distribution and cumulative amount, extract the cross-sectional internal force distribution data and call the load-slip constitutive model, calculate the interface slip distribution along the beam length in the negative bending moment region, and integrate to obtain the average slip amount in the plastic hinge region and the deformation loss term at the support section; Dynamic yield moment feedback and plastic hinge length reconstruction are performed. The stiffness degradation coefficient is determined based on the average slip. The static yield moment is reduced using the stiffness degradation coefficient to obtain the dynamic yield moment. The plastic hinge length is then recalculated using the dynamic yield moment and the provisional ultimate bearing capacity. Perform stress updates under non-coordinated deformation conditions, construct modified deformation compatibility equations, subtract deformation loss terms from geometric deformation calculations to obtain the true elongation, and update the current stress parameters accordingly. If the updated stress parameters do not meet the convergence conditions, return to the force balance solution step.

[0008] The second aspect of this invention further explains the key technical principles involved in the above calculation method.

[0009] In determining the basic static boundary parameters, the static yield moment is defined as the smaller of the yield moment when the bottom compression flange of the steel beam reaches its yield strength and the yield moment when the longitudinal reinforcement in the concrete slab reaches its yield strength. The effective length of the external CFRP reinforcement is calculated as follows: take twice the total geometric length of the external prestressed carbon fiber reinforced composite reinforcement between the two end anchorage points, and divide it by the sum of the number of plastic hinges formed when the component fails and the sum of the two.

[0010] In the calculation of interface slip distribution and accumulation, the interface slip strain is defined as the difference between the longitudinal tensile strain of the concrete slab bottom surface and the longitudinal tensile strain of the steel beam upper flange. The calculation process uses a method of discretizing the shear span length into multiple infinitesimal segments, and employs a backward difference scheme to perform inverse integration from the inflection point towards the support direction, thereby determining the relative interface slip at any location. The deformation loss term is the relative interface slip value at the support section obtained through integration.

[0011] In dynamic yield moment feedback and plastic hinge length reconstruction, the stiffness degradation coefficient is negatively correlated with the average slip. The dynamic yield moment is calculated by multiplying the difference between the static yield moment and the yield moment of the pure steel beam section acting independently by the stiffness degradation coefficient, and then adding the yield moment of the pure steel beam section acting independently. The calculation of the plastic hinge length is based on the assumption that the bending moment within the shear span follows a linear distribution, and its value is equal to the shear span length multiplied by a correction factor, which is a subtraction of the ratio of the dynamic yield moment to the provisional ultimate bearing capacity.

[0012] In stress updates under non-coordinated deformation conditions, the modified deformation compatibility equation describes the geometric relationship that the true elongation of the externally prestressed carbon fiber reinforced composite reinforcement is equal to the product of the ultimate rotation angle and the effective lever arm minus the deformation loss term. The ultimate rotation angle is equal to the ultimate curvature controlled by the ultimate compressive strain of the steel multiplied by the updated plastic hinge length. If the calculation object is a continuous beam, the calculation of the true elongation also includes a deformation superposition step: that is, obtaining the ultimate compressive strain of the concrete in the positive bending moment zone, calculating the rotation angle and corresponding additional elongation at the mid-span section, and superimposing the additional elongation into the true elongation in the negative bending moment zone.

[0013] This invention introduces a deformation loss term and a dynamic stiffness degradation mechanism into the calculation model, which can quantify the weakening effect of interface slip on the overall stiffness and deformation coordination of the structure, thereby accurately predicting the ultimate bearing capacity of steel-concrete composite beams reinforced with externally prestressed carbon fiber reinforced composite material.

[0014] This invention provides a corrected calculation method for the plastic flexural capacity of composite beams considering interface slip. It has the following beneficial effects: 1. This invention constructs a modified deformation compatibility equation in the stress update step under non-coordinated deformation conditions, combines the load-slip constitutive model of the shear connector to quantify the deformation loss term at the support section, and introduces this deformation loss term into the calculation logic of the elongation of the external CFRP reinforcement. By subtracting the part consumed by interface slip from the theoretical elongation generated by geometric rotation, the bias of overestimating the strain increment of the external CFRP reinforcement under the traditional rigid connection assumption is corrected, and the accurate assessment of the true stress state of the external prestressed carbon fiber reinforced composite reinforcement is achieved, thereby ensuring the safety of the ultimate bearing capacity calculation results.

[0015] 2. This invention establishes a negative correlation between the stiffness degradation coefficient and the average slip in the plastic hinge region. This coefficient is used to dynamically reduce the static yield moment in the static boundary parameters of the foundation. The physical phenomenon of the reduction in the combined effect caused by interface slip is transformed into a numerical adjustment of the section's bending capacity. Based on this, the plastic hinge length is reconstructed in real time. This solves the problem of inaccurate plastic hinge length values ​​caused by neglecting stiffness degradation in traditional methods. It achieves accurate simulation of the nonlinear mechanical behavior caused by slip in the negative bending moment region of steel-concrete composite beams.

[0016] 3. This invention, through the design of a general iterative calculation architecture that includes a deformation superposition step, is not only applicable to simply supported beam systems, but also to continuous beam systems. It can obtain the ultimate compressive strain of concrete in the positive bending moment zone, calculate the additional elongation caused by the mid-span section rotation angle, and superimpose it into the actual elongation in the negative bending moment zone. Through this regional deformation coupling mechanism, it breaks the limitations of a single negative bending moment zone calculation model, and achieves comprehensive coverage of composite beam systems reinforced with external prestressed carbon fiber reinforced composite material under different boundary conditions, effectively improving the engineering applicability of the calculation method. Attached Figure Description

[0017] Figure 1 This is an overall flowchart of the present invention; Figure 2 This is a schematic diagram of the elastic theory calculation model of the steel-concrete composite beam used to calculate the static yield bending moment in an embodiment of the present invention; Figure 3 This is a schematic diagram of the plastic theory calculation model for calculating the ultimate bearing capacity of the steel-concrete composite beam in an embodiment of the present invention; Figure 4 This is a schematic diagram of the moment-curvature relationship and the distribution of plastic hinge length in an embodiment of the present invention; Figure 5 This is a schematic diagram of the plastic hinge deformation geometric model of the composite beam reinforced by external prestressing in an embodiment of the present invention. Detailed Implementation

[0018] The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only 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 are within the scope of protection of the present invention.

[0019] Please see the appendix Figure 1-5This invention provides a method for correcting the calculation of the plastic flexural capacity of composite beams considering interface slip. It is applicable to steel-concrete composite beams reinforced with externally prestressed carbon fiber reinforced polymer (CFRP) bars, particularly for the negative moment region (i.e., the support region) of steel-concrete composite beams. In this embodiment, the steel beam section is limited to either Class 1 or Class 2 sections as defined in European standard EN1993-1-1. Class 1 or Class 2 sections as defined in EN1993-1-1 have sufficient rotational capacity, allowing the steel-concrete composite beam to undergo plastic analysis under ultimate limit conditions and form plastic hinges.

[0020] This embodiment is based on the theory of plastic hinges, while abandoning the idealized assumption in traditional calculations that there is a "complete shear connection (no slip) between the steel beam and the concrete slab". In the physical model, it is assumed that the concrete slab in the negative bending moment zone cracks and withdraws from the tensile work. The tensile force is borne by the longitudinal reinforcement inside the concrete slab and the external CFRP reinforcement, while the compressive force is borne by the bottom compression flange and part of the web of the steel beam, and the steel in the compression zone reaches the yield strength. More importantly, this embodiment introduces the layered plane section assumption, which assumes that the steel beam and the concrete slab each follow plane section deformation, but there is relative slip at the interface between the steel beam and the concrete slab. This causes the strain distribution of the steel beam and the strain distribution of the concrete slab to change abruptly at the interface. Based on the principle of energy conservation, the shear slip potential energy generated by the relative slip at the interface between the steel beam and the concrete slab is regarded as the internal energy dissipation of the system, which directly affects the construction of the deformation compatibility equation.

[0021] The specific implementation process is constructed as a system flow that includes parameter initialization, path selection, and closed-loop iteration.

[0022] In step S100, the system first determines the static boundary parameters required for the calculation. This step includes calculating the static yield moment based on the geometry and material properties of the steel-concrete composite beam. and the effective length of external CFRP reinforcement ; For static yield bending moment The determination of this not only involves calculating the bending moment when the longitudinal reinforcement in the concrete slab yields in tension. Simultaneously calculate the bending moment when the bottom compression flange of the steel beam reaches its yield strength. .

[0023] Based on the mechanical model of this embodiment, the yield moment of the bottom compression flange of the steel beam is... Calculate using the following formula:

[0024] In the formula: The yield strength of the steel beam; This is the distance from the outer edge of the bottom compression flange of the steel beam to the neutral axis of the equivalent section; This refers to the tension force of the prestressed tendons; This refers to the distance from the CFRP bar to the elastic neutral axis; the last term. It characterizes the contribution of the axial pressure generated by prestress to the stress in the cross section.

[0025] Longitudinal reinforcement yield moment Calculate using the following formula:

[0026] In the formula: The yield strength of the steel reinforcement; This is the distance from the longitudinal reinforcement bars within the concrete slab to the neutral axis of the converted section; For the conversion factor of CFRP reinforcement section; This represents the area of ​​the CFRP reinforcement.

[0027] Final decision and The smaller value in the range is used as the initial threshold for determining whether the section has entered the plastic stage, i.e. .

[0028] In calculation and At that time, the effective section moment of inertia that takes into account the effect of concrete cracking was adopted. And the contribution of the reverse bending moment generated by prestressing. ,in = For the effective length of external CFRP reinforcement Based on the full length of the external CFRP tendon and the number of intermediate steering blocks or support points The correction is made, and the calculation formula is as follows:

[0029] In the formula: This indicates the total geometric length of the external CFRP reinforcement between the two end anchorage points; This indicates the number of plastic hinges formed when a component fails. For simply supported beams, this represents the number of hinges formed upon failure. Take 1; for continuous beam failure (plastic hinges are formed at the points of maximum positive and negative bending moments). Generally, a value of 2 or 3 is used. This formula reflects the influence of the plastic hinge distribution on the effective deformation length of the external CFRP reinforcement.

[0030] In step S200, based on engineering accuracy requirements or calculation conditions, this embodiment provides two optional calculation paths.

[0031] The first approach is a simplified calculation method for engineering applications. For preliminary design estimations or scenarios where high-precision slip analysis is not required, the stress of the external CFRP reinforcement under the limit state is directly determined using empirical formulas based on statistical corrections. Under this path, the stress of the external CFRP reinforcement in the limiting state. The value is taken as the effective prestress. The sum of the fixed increment value of 105 MPa, i.e. The ultimate bearing capacity can be solved by directly substituting this value into the moment equilibrium equation of the cross section. No iterative calculations are required.

[0032] The second approach is a high-precision iterative approach based on slip correction. Before entering the first iteration calculation in step S300, the initial state of the iteration is first set: let the number of iteration steps be... Assume the initial tensile stress of the external CFRP reinforcement. Equal to the effective prestress after deducting prestress loss Assume the initial interface state is a no-slip state, i.e., the stiffness degradation coefficient is... Set to 1.0, interface slip deformation loss term Setting it to 0 is the core of this embodiment. This path solves the nonlinear problem of stress and deformation being mutually causal through the iterative cycle of steps S300 to S600.

[0033] Please see Figure 3 In step S300, a force balance solution is performed. Based on the external CFRP tendon stress assumed in the current iteration step, the axial force balance equation is used. Determine the height of the plastic neutralization shaft Based on this, a torque balance equation is established by taking moments about the plastic neutral axis. Calculate the provisional ultimate bearing capacity In this equilibrium equation, the contribution of the compression zone comes from the compression flange and part of the web of the steel beam, while the contribution of the tension zone comes from the longitudinal reinforcement inside the concrete slab and the external CFRP reinforcement.

[0034] In step S400, the interface slip distribution and accumulation are calculated. Using the internal force distribution obtained in step S300, the interface shear flow along the length of the steel-concrete composite beam is established. Combined with the pre-defined shear pin load-slip constitutive model The interface slip distribution along the beam length in the negative bending moment region was obtained by numerical integration. Here, This represents the ultimate shear capacity of a single stud. As the slip stiffness coefficient, the system further studies the interface slip distribution. Integral processing is performed to determine the average slip within the plastic hinge region. At this point, the slip value at the support section obtained from the above calculation is extracted and defined as the deformation loss term, which is used to correct the deformation compatibility equation in the subsequent step S600.

[0035] In step S500, dynamic yield moment feedback and plastic hinge length reconstruction are performed, which is a key feature distinguishing this embodiment from existing technologies. This embodiment does not use static yield moment... Instead of considering it a constant, it is the average slip calculated based on step S400. Introducing a stiffness degradation coefficient Stiffness degradation coefficient The system characterizes the reduction in sectional bending stiffness caused by the degradation of combined effects due to interface slip, and utilizes the stiffness degradation coefficient. With respect to the initial static yield moment After reduction, the dynamic yield moment is obtained. Based on the geometric relationship of the bending moment diagram, the reduced dynamic yield bending moment is used. and current ultimate bearing capacity Recalculate the plastic hinge length Due to dynamic yield moment The reduction in the plastic hinge length was calculated. It will exhibit nonlinear expansion characteristics, thus conforming to the physical fact that stiffness degradation leads to the diffusion of the plastic zone.

[0036] In step S600, stress updates under non-coordinated deformation conditions are performed. This is achieved using the extended plastic hinge length. Calculate the limit angle ,in To the ultimate compressive strain of steel Controlled limiting curvature. A modified deformation compatibility equation is used when calculating the elongation of the external CFRP tendon: In the formula This represents the effective height distance from the centroid of the external CFRP reinforcement section to the outer edge of the bottom compression flange of the steel beam. The deformation compatibility equation explicitly subtracts the deformation loss term caused by interface slip from the geometric elongation. This process yields the true elongation of the external CFRP reinforcement. If the object of calculation is a continuous beam, deformation superposition is also required in this step: based on the ultimate compressive strain of the concrete in the positive bending moment zone (-0.0033) set in step S100, the rotation angle of the mid-span section and the corresponding elongation of the external CFRP reinforcement are calculated and superimposed onto the true elongation in the negative bending moment zone. In order to obtain the total elongation of the external CFRP tendon, the actual elongation is then used. and the effective length of external CFRP reinforcement Update the stress values ​​of the external CFRP reinforcement.

[0037] Furthermore, for continuous beam structures, this embodiment also requires initializing the calculation parameters for the positive bending moment zone at mid-span. Since, under the ultimate limit state, in addition to the formation of a plastic hinge in the negative bending moment zone, the concrete compression edge in the positive bending moment zone at mid-span will also reach its ultimate compressive strain (based on relevant specifications and engineering experience), the ultimate compressive strain of the concrete in the positive bending moment zone at mid-span is set to -0.0033. This parameter will be used in subsequent step S600 to calculate the cumulative contribution of deformation in the positive bending moment zone to the total elongation of the external CFRP reinforcement. After initializing the basic static parameters, the method provided in this embodiment proceeds to the path selection step S200. Step S200 establishes the specific algorithm logic for solving the ultimate bearing capacity based on the specific engineering calculation accuracy requirements and the completeness of the known conditions. This embodiment includes two calculation paths: an iterative calculation path based on interface slip potential energy correction (i.e., the first calculation path) and a simplified calculation path based on engineering experience values ​​(i.e., the second calculation path).

[0038] The first calculation path is a preferred embodiment of the present invention, applicable to high-precision bearing capacity analysis of the negative bending moment zone of steel-concrete composite beams, especially for cases with low shear connection strength or requiring precise assessment of the impact of interface slip on the plastic hinge length extension. Under the first calculation path, the calculation process enters a closed-loop iteration module, where the system calculates the ultimate stress of the externally prestressed CFRP reinforcement. Treating the variables to be solved, an interface shear-slip constitutive model and a dynamic yield moment feedback mechanism are introduced. The equilibrium equations of the combined forces, the geometric deformation compatibility equations, and the energy dissipation equations are then iteratively solved. The first computational path addresses the ultimate bearing capacity. Ultimate stress of external CFRP tendons The specific implementation steps for the nonlinear coupling problem, where each element is causally related, will be detailed in subsequent embodiments.

[0039] The second calculation path is a simplified engineering implementation method provided by this invention, applicable to engineering scenarios involving preliminary design estimation, rapid verification, or lack of detailed interface slip constitutive parameters. Based on statistical analysis results or optimization suggestions from relevant design specifications, the second calculation path employs a non-iterative direct calculation method. After entering the second calculation path, step S210 directly assigns values ​​to the stress of the externally prestressed CFRP tendon under the ultimate limit state.

[0040] Specifically, in the second calculation path, the stress parameters of the external CFRP reinforcement... Determine according to the following empirical formula:

[0041] In the formula The value represents the effective prestress of the external CFRP bar after applying prestress and deducting all prestress losses, in megapascals (MPa); the value 105 is a stress increment characteristic value derived from a large amount of experimental data and parameter analysis, characterizing the stress growth level of the external CFRP bar under the ultimate state.

[0042] After confirming After obtaining the specific numerical values, the second calculation path performs the following steps to solve for the bearing capacity: based on the axial force equilibrium condition. Determine the height of the plastic neutral axis of the cross section. In this equilibrium equation, the compression term includes the yield force of the bottom compression flange and compression web of the steel beam, and the tension term includes the yield force of the longitudinal reinforcement in the concrete slab and the tensile force of the external CFRP reinforcement after assignment. ,in This represents the cross-sectional area of ​​the external CFRP reinforcement, using the calculated plastic neutral axis height. Determine the lever arms of the internal forces in each part and establish the moment equilibrium equations for the cross section. The ultimate bearing capacity was calculated. .

[0043] The second calculation path ignores the specific interface slip distribution calculation and dynamic correction of the plastic hinge length, treating the steel-concrete composite beam as a rigid connection system, thus simplifying the calculation process. The first and second calculation paths share the same set of basic parameter initialization data, but differ in their stress solution mechanisms. In practical applications, path switching is completed in step S200 according to the pre-set calculation mode or accuracy requirements.

[0044] In order to accurately simulate the mechanical behavior of the composite beam in the negative bending moment zone under the ultimate state, the following constitutive models were established for the steel beam, the steel reinforcement in the concrete slab, the external CFRP reinforcement, and the shear connection of the steel-concrete interface in this embodiment.

[0045] For the longitudinal reinforcement in steel beams and concrete slabs, this embodiment adopts an ideal elastoplastic model. The ideal elastoplastic model neglects the strain hardening effect of steel, assuming that the steel obeys Hooke's law before reaching its yield strength, and that after yielding, the stress remains constant while the strain continues to increase. The stress of the longitudinal reinforcement in the steel beams and concrete slabs is... With strain The relationship is expressed as follows:

[0046] In the formula: The elastic modulus of steel; For yield strain ; Yield strength (corresponding to steel beam) Reinforcing bars ); As a sign function, in this model, the ultimate failure criterion for the compression zone (bottom of the negative bending moment zone) of the steel beam is set as follows: when the compressive strain of the bottom compression flange of the steel beam reaches the ultimate compressive strain... When the cross-section reaches its ultimate bearing capacity, the ultimate compressive strain in this embodiment is determined. The value is -0.014, which applies to Class 1 or Class 2 sections as defined in EN1993-1-1.

[0047] For externally prestressed CFRP tendons, this embodiment employs a linear elastic brittle failure model. The stress in the externally prestressed CFRP tendons... With strain Maintain a linear relationship until fracture:

[0048] In the formula: The elastic modulus of CFRP reinforcement; This represents the ultimate tensile strength. In the calculation, if the stress exceeds... Then it is determined to be a broken structure.

[0049] For concrete materials, since this embodiment mainly calculates the bearing capacity of the negative bending moment zone, the concrete slab in the negative bending moment zone is in a tensile state. Therefore, the contribution of the tensile strength of the concrete is ignored in the model, and the tensile force is entirely borne by the longitudinal steel bars inside the concrete slab and the external CFRP bars.

[0050] For the interface between the steel beam and the concrete slab, this embodiment defines a load-slip constitutive model for the shear connector to describe the nonlinear relationship between the shear force and relative slip of a single shear connector. This embodiment uses an exponential constitutive equation:

[0051] In the formula: The relative slip is Shear force (N) of a single shear connector; This represents the relative slippage (mm) between the upper flange of the steel beam and the bottom surface of the concrete slab. The ultimate shear capacity (N) of a single shear connector; The slip stiffness coefficient (mm) -1 ).

[0052] To apply the discrete constitutive model of the aforementioned single shear-resistant connector to the calculation of continuous shear flow integrals along the beam length, this embodiment further defines the interface distributed shear stiffness. Assume that the steel-concrete composite beam is arranged laterally with Shear connectors are arranged in rows, with longitudinal spacing along the beam length as follows: The distributed tangent stiffness per unit length is calculated as follows:

[0053] In the formula: The unit is N / mm 2 The distributed stiffness function This will be directly substituted into the subsequent slip differential equation to calculate the interface slip distribution. The distribution of these components allows for a mechanical mapping from the constitutive model of micro-connectors to the deformation of macro-components.

[0054] Step S100 establishes the reference physical quantities required for subsequent iterative calculations based on the geometry, material properties, and prestressing state of the steel-concrete composite beam. In this embodiment, the initialization process is specifically divided into a sub-step for calculating the effective length of the external CFRP reinforcement and a sub-step for calculating the initial static yield moment.

[0055] In sub-step S110, which calculates the effective length of the external CFRP reinforcement, the physical length used for deformation compatibility analysis is modified based on the arrangement of the external prestressed CFRP reinforcement. Since the external prestressed CFRP reinforcement only undergoes displacement compatibility with the steel-concrete composite beam at the anchorage points and turning blocks, while the strain distribution within the free section is approximately uniform, the local deformation compatibility conditions at the cross-section cannot be directly used. This embodiment uses the following formula to calculate the effective length of the external CFRP reinforcement. :

[0056] In the formula: This indicates the total geometric length of the external CFRP reinforcement between the two end anchor points, in millimeters (mm). This represents the total number of steering blocks or intermediate support points set between the two end anchorage points. This formula reflects the influence of the frictional effect of the steering blocks and the effect of multi-point constraints on the strain averaging effect of the external CFRP reinforcement. If there are no intermediate steering blocks, then... =0, at which point the effective length is 0. Take as the total geometric length .

[0057] In the initial static yield moment calculation sub-step S120, the geometric properties of the cracked section in the negative moment zone are calculated based on the principle of ignoring the contribution of the concrete in the tension zone. Since the concrete slab in the negative moment zone is in tension and is assumed to be cracked, only the steel beam section and the longitudinal reinforcement section in the concrete slab are considered when calculating the section stiffness.

[0058] During the calculation, the interface between the upper flange of the steel beam and the concrete slab is set as the reference datum plane (i.e., the coordinates in the section height direction). =0, downwards is positive).

[0059] Calculate the position of the elastic neutral axis of the equivalent section. :

[0060] Based on the position of the elastic neutralization axis Calculate the effective section moment of inertia considering the effect of cracking. :

[0061] in: This represents the cross-sectional area of ​​the steel beam. This represents the vertical distance from the centroid of the steel beam section to the reference datum plane. The moment of inertia of the steel beam's cross section; This represents the total cross-sectional area of ​​the longitudinal reinforcement bars within the concrete slab. This represents the vertical distance (absolute value) from the centroid of the longitudinal reinforcement to the reference datum plane. This represents the moment of inertia of the longitudinal reinforcement group itself. Indicates the elastic modulus of longitudinal reinforcement With the elastic modulus of steel beams The ratio, i.e. ; Please see Figure 2 In obtaining the effective cross-sectional moment of inertia Then, the yield moment of the bottom compression flange of the steel beam was calculated separately. and the tensile yield moment of longitudinal reinforcement The bottom compression flange of the steel beam has reached its yield strength. Yield bending moment at time The calculation is as follows:

[0062] Longitudinal reinforcement reaches yield strength Yield bending moment at time The calculation is as follows:

[0063] In the formula: It represents the vertical distance from the outer edge of the bottom compression flange of the steel beam (i.e., the lower flange of the steel beam in the negative bending moment zone) to the neutral axis of the converted section; It represents the vertical distance from the centroid of the longitudinal reinforcement to the neutral axis of the converted section; This indicates the effective preload of the external CFRP reinforcement. The term characterizes the gain contribution of the prestressing effect to the bending resistance of the section.

[0064] Comparing the two bending moment values, the smaller value is taken as the initial static yield moment for determining whether the section enters the plastic state. :

[0065] The initial static yield moment This will serve as the benchmark value for dynamic yield moment correction in subsequent step S500. If the external load moment is less than... If the steel-concrete composite beam is in the elastic stage, then it is determined that the beam is in the elastic stage; if it is greater than... If the condition is met, the plastic stage is determined and the calculation process for the plastic hinge length is initiated.

[0066] Step S300 is the starting step of the iterative calculation cycle. Based on the stress value of the externally prestressed CFRP tendon set in the current iteration step, this step solves for the plastic neutral axis position and ultimate flexural bearing capacity of the section.

[0067] In step S310, the force balance equations along the axial direction of the steel-concrete composite beam are established. Under the ultimate state in the negative bending moment region, it is assumed that the strain distribution on the cross section satisfies the plane section assumption, and that each part of the material fully utilizes its plastic properties. The specific mechanical model is set as follows: the entire cross section of the concrete slab is under tension and cracking, with its tensile contribution neglected; the steel beam is divided into two parts by a plastic neutral axis, with the area below the neutral axis being the compression zone, where the stress reaches the yield strength. (Under compression), the area above the neutral axis is under tension, where the stress reaches the yield strength. (Tension); The longitudinal reinforcement in the concrete slab is under tension, and the stress reaches the yield strength. Externally prestressed CFRP tendons provide tensile force, and their stress value is taken as the input value of the current iteration step. .

[0068] Based on the above stress state, calculate the area of ​​the compression zone of the steel beam required to maintain axial balance. :

[0069] In the formula: This represents the area of ​​a steel beam section under compression, calculated from the bottom of the beam upwards, and expressed in square millimeters (mm). 2 ); This represents the total cross-sectional area of ​​the steel beam; This represents the total cross-sectional area of ​​the longitudinal reinforcement bars within the concrete slab; This indicates the cross-sectional area of ​​the external CFRP reinforcement.

[0070] Calculate the required area of ​​the pressure zone Then, the height of the plastic neutralization axis is determined based on the specific geometric dimensions of the steel beam. , defined here Let be the vertical distance from the plastic neutral axis to the outer edge of the bottom compression flange of the steel beam (i.e., the bottom compression flange of the steel beam in the negative bending moment region). Less than or equal to the area of ​​the bottom compression flange of the steel beam The plastic neutralization axis is located inside the bottom flange:

[0071] like If the area is greater than the area of ​​the bottom compression flange of the steel beam, then the plastic neutralization axis is located within the web:

[0072] In the formula: This indicates the width of the bottom compression flange of the steel beam; This indicates the thickness of the bottom compression flange of the steel beam; This indicates the thickness of the web of the steel beam.

[0073] In step S320, the determined plastic neutral shaft height is used. With the plastic neutral axis as the moment center, the moment equilibrium equation is established. To solve for the ultimate bearing capacity ultimate bearing capacity of cross section It is composed of the superposition of the following four bending moment contributions:

[0074] The specific calculation formulas for each item are as follows:

[0075] in, This indicates the bending moment resistance provided by the compression zone of the steel beam. This is the distance from the centroid of the compression zone section of the steel beam to the bottom edge of the steel beam; This indicates the bending moment resistance provided by the tension zone of the steel beam. This is the distance from the centroid of the tension zone section of the steel beam to the bottom edge of the steel beam; This indicates the bending moment resisted by the longitudinal reinforcement within the concrete slab. This indicates the bending moment resistance provided by the external CFRP reinforcement; Indicates the total height of the steel beam; It represents the vertical distance from the centroid of the longitudinal reinforcement to the interface between the steel beam and the concrete slab; This indicates the effective height distance from the centroid of the external CFRP reinforcement section to the outer edge of the lower surface (or bottom edge) of the bottom compression flange of the steel beam.

[0076] For the centroid of the compression zone Calculation: If the plastic neutralization axis is located inside the bottom flange ( ),but If the plastic neutralization axis is located within the web ( Then calculate according to the following formula:

[0077] For the centroid of the tension zone The calculation involves removing the compression zone from the entire cross-section of the steel beam. The distance from the geometric centroid of the remaining portion (i.e., the tension portion) to the bottom edge of the steel beam. Those skilled in the art can determine this based on the determined height of the compression zone. The geometric dimensions of the steel beams are obtained directly using the principle of area moment. The numerical values ​​are not detailed here, and the specific geometric derivation process will not be repeated.

[0078] Through steps S310 and S320, this embodiment achieves a given CFRP tendon stress. Below, the deformation compatibility reference point of the cross section (i.e., the position of the plastic neutral axis) was uniquely determined. ) and corresponding bearing capacity .Should This value will be used in subsequent steps to back-calculate the plastic hinge length, while The value will be used to construct the modified deformation compatibility equation.

[0079] Step S400, based on the cross-sectional internal force state determined in step S300, quantifies the incompatible deformation at the interface between the steel beam and the concrete slab, and solves for the interface slip distribution along the beam length. Average slip within the plastic hinge region and deformation loss due to slip This embodiment divides the process into the following sub-steps: In step S410, the source of the driving interface slippage is calculated, namely the longitudinal strain difference between the upper flange of the steel beam and the bottom surface of the concrete slab. Although the layered plane section assumption is introduced in the section force balance calculation, the curvature at the interface between the steel beam and the concrete slab is set. Coordination and consistency, based on the plastic neutralization shaft height determined in step S300. Ultimate compressive strain of the bottom compression flange of the steel beam Calculate the cross-sectional curvature under the current limiting state. :

[0080] Using this curvature Based on the constitutive relations of the materials, calculate the longitudinal strain of the materials on both sides of the interface. Calculate the longitudinal tensile strain of the upper flange of the steel beam (below the interface). The calculation is as follows:

[0081] Longitudinal tensile strain on the bottom surface of the concrete slab (upper side of the interface) The tensile strain is primarily controlled by the deformation of the longitudinal reinforcement within the concrete slab. Considering the tensile stiffening effect after concrete cracking, an equivalent calculation method based on the stress of the longitudinal reinforcement is adopted. Since the longitudinal reinforcement is located above the bottom surface of the concrete slab (i.e., the distance of the reinforcement from the neutral axis is greater than the distance of the interface from the neutral axis), the tensile strain at the interface should be obtained by subtracting the geometric increment caused by curvature from the reinforcement strain.

[0082] This allows us to obtain the interfacial slip strain per unit length along the beam's length (i.e., the derivative of the slip distribution function). To ensure that the slip value is positive (representing a greater elongation of the concrete slab relative to the steel beam), it is defined as follows:

[0083] in: It represents the absolute value of the vertical distance from the centroid of the longitudinal reinforcement to the reference plane (the interface between the upper flange of the steel beam and the concrete slab).

[0084] In step S420, the numerical difference method is used to solve for the length along the shear span. Interface slip distribution slit length Defined as the horizontal distance from the point of maximum negative bending moment (support center) to the inflection point (point of zero bending moment), the shear span length. Discretize into There are infinitesimal segments, each with a length of _ . ,definition The section at this point is the support section. The point is the inflection point. The boundary conditions are set as follows: at the inflection point, since the bending moment is zero and the shear force transfer accumulation ends, the slip is zero, i.e. =0, during the integration process, assume the bending moment distribution within the shear span is 0. It follows a linear distribution law. For any infinitesimal node... According to its corresponding bending moment The relationship between the cross-section yield moment and the yield moment determines whether the cross-section has entered the plastic state, thus allowing the selection of the corresponding slip strain derivative. For the length of the plastic hinge For the infinitesimal segment within the range, the slip strain derivative is taken as the limit value calculated in step S410; for the elastic segment, it is reduced according to elastic theory. Using the backward difference scheme, the inverse integration is performed from the inflection point towards the support direction to calculate the value at any position. Relative slip at point :

[0085] In the formula This is the known slip of the previous node (the side closer to the inflection point).

[0086] In step S430, based on the calculated interface slip distribution... Two key physical quantities were extracted for subsequent correction calculations: mean slip and deformation loss term. Mean slip Defined as the average value of interface slip within the currently defined plastic hinge length, it characterizes the overall degree of degradation of interface stiffness. Its calculation formula is:

[0087] In the formula Given the assumed plastic hinge length in the current iteration step, the deformation loss term is defined at the support section (i.e. The relative sliding value of the interface at the location:

[0088] The deformation loss term This represents the cumulative misalignment displacement between the longitudinal reinforcement in the concrete slab and the upper flange of the steel beam due to insufficient shear connection at the interface (incomplete interaction). This misalignment displacement causes a shift in the rotation center of the steel beam section, resulting in the actual tensile elongation of the external CFRP reinforcement being less than the geometric elongation calculated based on the plane section assumption. As a correction term, it is directly substituted into the subsequent deformation compatibility equations.

[0089] Step S500 is a dynamic parameter feedback step, which is based on the average interface slip calculated in step S400. The bending stiffness characteristics of the steel-concrete composite beam are modified, and the plastic hinge length and dynamic yield moment are updated. Step S500 modifies the ideal model of "complete interaction" to the actual model of "partial interaction" by introducing a stiffness degradation mechanism, which specifically includes the following sub-steps: In step S510, the degree to which interface slip weakens the overall bending capacity of the composite beam is quantified. Because the slippage between the steel beam and the concrete slab disrupts the strain compatibility of the cross-section, the effective bending stiffness of the composite beam is lower than the theoretical value under perfect interaction. This embodiment defines a stiffness degradation coefficient. Its value is the ratio of the effective stiffness to the ideal stiffness under the current slip state. Based on the load-slip constitutive relationship of the shear connector, a... With average slip Functional relationship:

[0090] In the formula: This represents the stiffness degradation coefficient, with a value range of (0,1]. This represents the average interface slip within the plastic hinge region extracted in step S430; This represents the shear connection stiffness sensitivity coefficient, in mm. -1 , This reflects the rate of decay of the interfacial slip resistance to bending stiffness. For studded connections, those skilled in the art can calculate and determine this rate based on the shear stiffness measured by the pull-out test and the stud density. In the absence of experimental data, empirical values ​​can be used (e.g., in this embodiment). A value between 0.5 and 1.5 is acceptable.

[0091] In step S520, the stiffness degradation coefficient is used. The yield moment of the composite beam is modified. The initial static yield moment calculated in step S120 is an upper limit value obtained based on the plane section assumption and neglecting the effect of slip. As slip increases, the moment when the section enters the plastic state is earlier, which is manifested as a decrease in the yield moment. The modified dynamic yield moment... The calculation formula is as follows:

[0092] This represents the yield moment of a pure steel beam section (without considering the contribution of the concrete slab) under independent action. To ensure closed-loop calculation parameters, this embodiment... Calculate using the following formula:

[0093] in Let be the moment of inertia of the steel beam section. The yield strength of the steel beam It is the distance from the centroid of the steel beam section to the outer edge of the bottom compression flange (bottom flange) of the steel beam; The contribution of external prestress to the flexural strength of the section (consistent with step S120) is defined in this formula, which clarifies the physical limits: when (Without slip) ;when (During complete slip) → That is, it degenerates into the stress state of a prestressed pure steel beam.

[0094] Please see Figure 4 In step S530, based on the gradient bending moment distribution theory, combined with the modified dynamic yield bending moment... and current ultimate bearing capacity Redefine the physical range of the plastic hinge, and the length of the plastic hinge. Defined as the bending moment within the shear span being greater than the dynamic yield moment. The length of the region, assuming the cut segment If the bending moment distribution within is approximately linear, then the updated plastic hinge length... The calculation is as follows:

[0095] In the formula: This indicates the shear span length (i.e., the distance from the support section to the inflection point), consistent with the definition in step S420; This represents the current ultimate flexural capacity calculated in step S320, and the updated value. The plastic hinge length value used in the previous iteration will be compared with the value used in the previous iteration. If the relative error between the two exceeds the preset convergence tolerance (e.g., 5%), the iteration is considered unconverged. In this case, the value calculated in this step should be used. Update the new integration cap and return to step S430 to recalculate the average slip. Until the convergence condition is met, the converged... This will be used as the final geometric parameter and substituted into the subsequent deformation compatibility equations to calculate the plastic hinge rotation angle.

[0096] Step S600 is the non-coordinated deformation and stress update step, which updates the interface slip loss term calculated in the previous step S400. The plastic hinge length updated in step S500 In combination, a modified deformation compatibility equation incorporating slip effects is established to solve for the actual stress response of the externally prestressed CFRP tendon, and an iterative convergence determination is performed. This embodiment specifically divides this process into the following sub-steps: In step S610, the macroscopic rotational capacity of the support section is determined based on the updated plastic hinge length. In the limit state of the negative bending moment region, it is assumed that plastic deformation is mainly concentrated at a length of... Within the plastic hinge region, the curvature is approximately uniformly distributed, and the value is the limiting curvature. Based on this, the total rotation angle generated by the plastic hinge region is calculated. :

[0097] Please see Figure 5 In step S620, a deformation compatibility equation incorporating an interface slip correction term is constructed. In the fully interactive theoretical model, the elongation of the external CFRP reinforcement is entirely determined by the geometric displacement caused by the section rotation. However, considering the slip effect, the interface slip between the upper flange of the steel beam and the concrete slab is considered. This leads to inconsistencies in the deformation of the composite section, causing the actual elongation between the anchorage points of the external CFRP reinforcement to be less than the theoretical geometric elongation. Therefore, the true elongation of the external CFRP reinforcement under the ultimate condition is... The calculation is as follows:

[0098] The formula is obtained by subtraction. The geometric elongation calculated based on the plane section assumption was corrected, thus reflecting the weakening effect of incomplete interaction on the stress growth of external CFRP tendons.

[0099] In step S630, the stress value of the external CFRP reinforcement is updated based on the actual elongation and material constitutive relation, and it is determined whether the calculation process has converged. The stress increment of the external CFRP reinforcement considering the slip effect is calculated. and total stress calculation value :

[0100]

[0101] In the formula This represents the elastic modulus of external CFRP reinforcement.

[0102] Perform a convergence check on the stress balance and apply the stress values ​​calculated in this round. Compared with the assumed stress value input at the beginning of this iteration Compare and calculate the relative error :

[0103] Set the preset convergence tolerance (For example, in this embodiment, take) ),like If the iterative calculation has converged, then the current value is output. The actual stress of the external CFRP reinforcement under the ultimate state And simultaneously output the ultimate bearing capacity corresponding to step S320 as the final calculation result. If If the iteration has not yet converged, the input parameters need to be updated. Let the input stress value for the next iteration be... Then return to step S300, and use the updated stress values ​​to re-solve the cross-sectional force balance, calculate the interface slip distribution, and feed back the dynamic parameters until the convergence condition is met.

[0104] For the numerical oscillation problem in the above iteration process, those skilled in the art can use the relaxation method to smooth the update of the input value; For example, let:

[0105] (in Let be the relaxation factor, taking values ​​of . This numerical calculation optimization method is a well-known technology in this field and will not be described in detail here.

[0106] After completing the iterative calculations of steps S100 to S600 and satisfying the convergence condition, this method enters the result output and engineering applicability determination stage. The calculation method described in this embodiment is based on specific mechanical boundary conditions and applicable scenarios. The following provides a detailed explanation of the applicable scope logic of this method and the engineering application of the final calculation results.

[0107] Regarding the definition of applicable scenarios, the ultimate bearing capacity calculation method described in this invention is mainly applied to the negative bending moment section of steel-concrete composite beams. The negative bending moment section usually exists at the intermediate support of continuous composite beams or at the root of cantilever composite beams. In these areas, the concrete slab is under tension, and the lower flange of the steel beam is under compression. The assumptions in this method regarding "ignoring the tensile contribution of concrete in the tension zone", "considering the tensioning and stiffening of longitudinal reinforcement", and "external CFRP reinforcement is arranged in the tension zone of the section or acts on the beam through a steering device" are all customized for the negative bending moment stress characteristics. For the specific form of the steel beam, this method is not only applicable to I-beams, but also to box beams, channel beams, and other open or closed section forms. The corresponding section geometric properties (such as area, moment of inertia, and plastic neutral axis position) can be calculated based on the actual section shape. For concrete slabs, it covers cast-in-place concrete slabs, precast assembled concrete slabs, and concrete slabs using ultra-high performance concrete. Different materials only reflect differences in constitutive relation parameters and do not change the overall calculation logic of this method.

[0108] Regarding the implementation of the interaction of the core logic components, this method breaks through the default assumption of complete interaction in traditional design. In actual engineering, especially in the reinforcement of existing bridges or the evaluation of old composite beam structures, due to corrosion, fatigue damage, or insufficient initial design of shear connectors (such as studs), the steel-concrete interface often cannot maintain ideal deformation compatibility under ultimate loads, resulting in non-negligible interface slip. This method, through the closed-loop feedback system constructed in steps S400 to S600, quantifies this interface slip into specific physical parameters and directly corrects the deformation compatibility equation. Therefore, this method is particularly suitable for evaluating the true load-bearing capacity of steel-concrete composite beams with low shear connection. When applied to newly built composite beams with sufficient shear connection, the slip calculated by this method will approach zero, and the stiffness degradation coefficient will approach 1. At this point, the calculation result will automatically degenerate and converge to the classical complete interaction theory solution, thus proving that this method has universal downward compatibility in mechanical logic and can cover the entire spectrum of working conditions from complete slip to complete connection.

[0109] In step S700, when step S630 determines the iteration error... Less than the preset convergence tolerance When the calculation process terminates, the key performance indicators under the limit state are output. The output parameters include: the ultimate flexural capacity of the composite beam in the negative bending moment zone; the ultimate true stress of the externally prestressed CFRP reinforcement; and the maximum or average value of the interface slip distribution.

[0110] Based on the above output results, those skilled in the art can perform verification of the ultimate limit state of bearing capacity. Define the structural importance coefficient. (Selected according to relevant bridge structural design specifications, for example, take 1.1) and the combined design value of load effects. (i.e., the design value of the maximum negative bending moment generated by the external load), if the following formula is satisfied:

[0111] If the steel-concrete composite beam meets the design requirements under the current external prestressing strengthening scheme, then the flexural bearing capacity is deemed to meet the design requirements. Conversely, if... If the value is less than the design requirements, it indicates that the current reinforcement scheme is insufficient. In this case, the reinforcement parameters need to be adjusted, such as increasing the cross-sectional area of ​​the external CFRP reinforcement, increasing the initial prestress level, or changing the arrangement height of the external CFRP reinforcement. The entire calculation process should then be restarted from step S100 until a reinforcement design scheme that meets the requirements is obtained.

[0112] Furthermore, the interface slip output by this method can also be used as a control indicator. In some high-performance design requirements, in addition to meeting load-bearing capacity requirements, it is also necessary to limit the interface slip to prevent premature shearing of the connector. If the calculated slip exceeds the ultimate deformation capacity of the shear connector (this value is determined by the descending segment of the load-slip curve from the connector push-out test), even... Even if the requirements are met, it should still be determined that the structure is at risk of brittle failure at the interface, and the maximum allowable bearing capacity should be limited or it should be reinforced by adding shear connectors.

[0113] This embodiment also provides a simplified formula calculation mode suitable for rapid engineering estimation. This mode is mainly used to quickly assess the ultimate bearing capacity of externally prestressed steel-concrete composite beams in the early stages of scheme design, or as the initial value input for the aforementioned iterative calculation. This simplified mode introduces an "interface slip reduction coefficient" to equivalently replace the aforementioned complex interface slip field integral and deformation coordination iteration process, specifically including the following sub-steps: In step S810, the shear connection degree index within the shear span zone is defined and calculated. This shear connection strength index reflects the degree to which the number of shear connections between the steel beam and the concrete slab meets the fully plastic resistance requirements of the section. In the negative bending moment zone, the concrete slab cracks under tension, and the longitudinal shear resistance requirement of the section is mainly controlled by the yield strength of the longitudinal reinforcement within the slab. Therefore, the shear connection strength index... Defined as the ratio of the total shear capacity of the shear connectors actually arranged within the shear span to the yield strength of the longitudinal reinforcement in the concrete slab:

[0114] In the formula: Indicates the length of the shear span The total number of shear connectors (such as studs) arranged within the range from the support section to the inflection point; It can be calculated based on the formulas in the current steel-concrete composite structure design code; when When, it is defined as a fully shear-resistant connection; when When this occurs, it is defined as a partial shear connection.

[0115] In step S820, the interface slip reduction coefficient is constructed. Degree of connection with shear strength The mapping relationship between them, the coefficient This is a dimensionless correction factor used to characterize the proportion of stress increment loss in external CFRP tendons due to interface slip. Considering the nonlinear relationship between interface slip and connectivity, this embodiment uses a piecewise function to describe this law. As a specific implementation method, it can be determined using the following formula. :

[0116] for For extremely low connection conditions with a shear stiffness <0.5, due to severely insufficient interfacial shear stiffness, it is recommended to use conservative values ​​in simplified calculations (e.g., ...). = Alternatively, a detailed evaluation can be performed using the iterative methods described in steps S100 to S600. The physical significance lies in the fact that it parameterizes the complex interface non-coordinated deformation effect into a reduction coefficient that can be directly multiplied, thus preserving the simplicity of the plane section assumption in mathematical form, while correcting the calculation error caused by slip in numerical form.

[0117] In step S830, the ultimate stress of the external CFRP tendon under the ultimate state is directly calculated using the aforementioned coefficients. and cross-sectional flexural bearing capacity Instead of iterative iteration, it is based on the empirical plastic hinge length. Directly calculate the stress increment of the external CFRP reinforcement and the ultimate stress of the external CFRP reinforcement. The calculation formula is as follows:

[0118] Among them, the extreme angle The following can be simplified from the geometric relationship of the cross section:

[0119] In the formula To simplify the calculation, the assumed plastic neutral axis height, By establishing a simplified axial force equilibrium equation for the cross section, and neglecting the compressive effect of concrete in the negative bending moment region, it is assumed that the steel beam yields in the compression zone, the longitudinal reinforcement yields, and the external CFRP reinforcement reaches the estimated stress, satisfying the following:

[0120] Determined based on this equilibrium condition Then, the simplified ultimate bearing capacity was obtained using the principle of moment balance. :

[0121] In the formula: This indicates the effect of axial force on the plastic neutralization axis of the steel beam section. Plastic resistance to bending moment; , These represent the centroids of the longitudinal reinforcement and the external CFRP reinforcement, respectively, from the plastic neutral axis. The vertical distance.

[0122] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A corrected calculation method for the plastic flexural capacity of composite beams considering interface slippage, characterized in that, Includes the following steps: Step S100: Obtain the geometric dimensions and material properties of the steel-concrete composite beam. By introducing the load-slip constitutive model of the shear connector, calculate the static yield moment and the effective length of the external CFRP reinforcement. Define the static yield moment, the effective length of the external CFRP reinforcement, and the load-slip constitutive model as the basic static boundary parameters. Step S200: Based on the basic static boundary parameters determined in step S100, set the initial state of the iterative calculation, assign the stress parameters of the current external prestressed carbon fiber reinforced composite reinforcement to the effective prestress after deducting the prestress loss, set the initial interface state to a no-slip state, set the stiffness degradation coefficient to one and the deformation loss term to zero. Step S300: Perform force balance solution. Construct axial force balance equation using the current stress parameters of the externally prestressed carbon fiber reinforced composite tendon as input conditions. Use the axial force balance equation to determine the height of the plastic neutral axis and generate the internal force distribution data of the section. Take the moment of the plastic neutral axis to establish the moment balance equation and calculate the provisional ultimate bearing capacity. Step S400: Calculate the interface slip distribution and cumulative amount, extract the cross-sectional internal force distribution data generated in step S300, call the load-slip constitutive model determined in step S100, calculate the interface slip distribution from the inflection point to the support section along the beam length in the negative bending moment region, perform integral calculation on the interface slip distribution within the plastic hinge length range to obtain the average slip amount, and extract the slip value at the support section as the deformation loss term; Step S500: Perform dynamic yield moment feedback and plastic hinge length reconstruction. Determine the stiffness degradation coefficient based on the average slip obtained in step S400. Use the stiffness degradation coefficient to reduce the static yield moment calculated in step S100 to obtain the dynamic yield moment. Recalculate the plastic hinge length using the dynamic yield moment and the provisional ultimate bearing capacity obtained in step S300. Step S600: Perform stress update under non-coordinated deformation conditions, calculate the ultimate rotation angle using the plastic hinge length updated in step S500, construct a modified deformation compatibility equation, which describes the geometric relationship between the actual elongation of the externally prestressed carbon fiber reinforced composite tendon and the ultimate rotation angle and deformation loss term, calculate the actual elongation using the modified deformation compatibility equation, and update the stress parameters of the current externally prestressed carbon fiber reinforced composite tendon using the actual elongation. Determine whether the updated stress parameters meet the convergence condition with the input values ​​from the previous round. If the convergence condition is met, output the provisional ultimate bearing capacity as the final ultimate bearing capacity. If the convergence condition is not met, return to step S300.

2. The method for correcting the plastic flexural bearing capacity of composite beams considering interface slip according to claim 1, characterized in that, In step S100, the static yield moment is determined as follows: Calculate the yield moment when the bottom compression flange of the steel beam reaches its yield strength and the yield moment when the longitudinal reinforcement in the concrete slab reaches its yield strength, respectively. Compare the magnitudes of the yield moment when the bottom compression flange of the steel beam reaches its yield strength and the yield moment when the longitudinal reinforcement in the concrete slab reaches its yield strength, and select the smaller value as the static yield moment. The effective length of the external CFRP reinforcement is calculated as follows: take twice the total geometric length of the external prestressed carbon fiber reinforced composite reinforcement between the two end anchorage points, and divide it by the sum of the number of plastic hinges formed when the component fails and the sum of the two.

3. The method for correcting the plastic flexural bearing capacity of composite beams considering interface slip according to claim 1, characterized in that, In step S100, the load-slip constitutive model adopts an exponential constitutive equation, which describes the nonlinear relationship between the shear force and the relative slip of a single shear connector. The exponential constitutive equation includes two characteristic parameters: the ultimate shear capacity of a single shear connector and the slip stiffness coefficient.

4. The method for correcting the plastic flexural bearing capacity of composite beams considering interface slip according to claim 1, characterized in that, In step S300, the height of the plastic neutralization shaft is determined as follows: Based on the cross-sectional area of ​​the steel beam, the cross-sectional area of ​​the longitudinal steel bars, and the cross-sectional area of ​​the externally prestressed carbon fiber reinforced composite reinforcement, combined with their respective yield strengths and the stress parameters of the externally prestressed carbon fiber reinforced composite reinforcement in the current iteration step, the area of ​​the compression zone of the steel beam required to maintain axial equilibrium is calculated. If the area of ​​the compression zone of the steel beam required to maintain axial balance is less than or equal to the area of ​​the bottom compression flange of the steel beam, then the plastic neutralization axis is determined to be located within the bottom compression flange of the steel beam; if the area of ​​the compression zone of the steel beam required to maintain axial balance is greater than the area of ​​the bottom compression flange of the steel beam, then the plastic neutralization axis is determined to be located within the web of the steel beam.

5. The method for correcting the plastic flexural bearing capacity of composite beams considering interface slip according to claim 1, characterized in that, In step S300, the provisional ultimate bearing capacity is composed of the superposition of four bending moment contributions, namely the resistance bending moment provided by the compression zone of the steel beam, the resistance bending moment provided by the tension zone of the steel beam, the resistance bending moment provided by the longitudinal reinforcement in the concrete slab, and the resistance bending moment provided by the external prestressed carbon fiber reinforced composite reinforcement. The effective lever arm of the bending moment provided by the externally prestressed carbon fiber reinforced composite reinforcement is taken as the effective height distance from the centroid of the externally prestressed carbon fiber reinforced composite reinforcement section to the outer edge of the bottom compression flange of the steel beam minus the height of the plastic neutralization axis.

6. The method for correcting the plastic flexural bearing capacity of composite beams considering interface slip according to claim 1, characterized in that, In step S400, the calculation process of the interface slip distribution includes: The interface slip strain per unit length along the beam length is calculated based on the cross-sectional curvature. The interface slip strain is defined as the difference between the longitudinal tensile strain of the bottom surface of the concrete slab and the longitudinal tensile strain of the upper flange of the steel beam. The shear span length is discretized into multiple infinitesimal segments. Using the backward difference scheme, the relative slip of the interface at any position is calculated by performing inverse integration from the inflection point to the support direction. The deformation loss term is defined as the relative slip value of the interface at the support section obtained by integration.

7. The method for correcting the plastic flexural bearing capacity of composite beams considering interface slip according to claim 1, characterized in that, In step S500, the stiffness degradation coefficient is negatively correlated with the average slip; the larger the average slip, the smaller the stiffness degradation coefficient. The dynamic yield moment is calculated by multiplying the difference between the static yield moment and the yield moment of the pure steel beam section under independent action by the stiffness degradation coefficient, and then adding the yield moment of the pure steel beam section under independent action.

8. The method for correcting the plastic flexural bearing capacity of composite beams considering interface slip according to claim 1, characterized in that, In step S500, the calculation of the plastic hinge length is based on the assumption that the bending moment within the shear span follows a linear distribution. The value of the plastic hinge length is equal to the shear span length multiplied by a correction factor, which is a factor minus the ratio of the dynamic yield moment to the provisional ultimate bearing capacity.

9. The method for correcting the plastic flexural bearing capacity of composite beams considering interface slip according to claim 1, characterized in that, In step S600, the specific construction method of the modified deformation compatibility equation is as follows: The actual elongation of the externally prestressed carbon fiber reinforced composite tendon is set to be equal to the product of the limit angle and the effective lever arm minus the deformation loss term; Wherein, the ultimate rotation angle is equal to the ultimate curvature controlled by the ultimate compressive strain of the steel multiplied by the plastic hinge length updated in step S500; the effective lever arm is the effective height distance from the centroid of the external prestressed carbon fiber reinforced composite material section to the outer edge of the bottom compression flange of the steel beam minus the height of the plastic neutral axis.

10. The method for correcting the plastic flexural bearing capacity of composite beams considering interface slip according to claim 1, characterized in that, Step S600, when calculating the actual elongation of the externally prestressed carbon fiber reinforced composite tendon, also includes a deformation superposition step: Obtain the ultimate compressive strain of concrete in the positive bending moment zone, and calculate the rotation angle of the mid-span section and the corresponding additional elongation of the externally prestressed carbon fiber reinforced composite reinforcement based on the ultimate compressive strain of concrete in the positive bending moment zone. Then, add the additional elongation to the actual elongation in the negative bending moment zone to obtain the total elongation of the externally prestressed carbon fiber reinforced composite reinforcement.

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