A prestressed beam simulation method based on non-perfect bonding theory and vector finite element
By combining incomplete bonding theory with vector finite element method, the bond slip and tensile stiffening effects of prestressed beams are accurately simulated, solving the problem of inaccurate simulation in existing technologies and realizing high-precision analysis from loading to failure.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- ZHEJIANG UNIV
- Filing Date
- 2026-04-13
- Publication Date
- 2026-07-31
AI Technical Summary
Existing technologies struggle to accurately simulate the bond-slip behavior between steel bars/prestressed tendons and concrete in prestressed concrete beams under load, especially the tensile stiffening effect after cracking, leading to inaccurate simulations of structural stress behavior.
Using the incomplete bonding theory and vector finite element method, the mechanical response of the prestressed beam section is calculated through the incomplete bonding theory model. The beam structure is discretized into a mass system by combining the vector finite element method, and the displacement response is calculated by explicit time integration. This accurately simulates the complete behavior of the beam from loading to failure, especially the tensile stiffening effect after cracking.
It achieves high-precision simulation of the process from elastic to ultimate failure of prestressed beams, avoids dependence on empirical formulas, and has high consistency and robustness. It is applicable to the analysis of prestressed beams with different steel reinforcement and material strengths.
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Figure CN122021196B_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the fields of structural engineering and computational mechanics, and in particular relates to a method for simulating prestressed beams based on incomplete bonding theory and vector finite element method. Background Technology
[0002] Accurate simulation of the bond-slip behavior of prestressed concrete beams under load, including the steel reinforcement / prestressing tendons and concrete, and the impact of decreased material bond strength on the overall structural stress behavior, especially the degradation of beam stiffness after cracking, the development of concrete cracks, and the decrease in the beam's ultimate bearing capacity, is crucial for structural safety assessment and design optimization.
[0003] Traditional analytical methods, such as section analysis based on the moment-curvature (M-χ) relationship, are the foundation of design codes in various countries. This method, based on the plane section assumption, can provide relatively accurate results before concrete cracking. However, for post-cracking stress behavior, this method heavily relies on empirical effective bending stiffness formulas, such as the Branson formula, for simulation. More importantly, the M-χ method cannot account for the bond-slip effect between steel reinforcement and concrete from a mechanical perspective, and therefore cannot accurately reflect the "tensile stiffening" phenomenon caused by bond-slip.
[0004] In numerical simulation, while the traditional finite element (FE) method can handle complex geometries and material nonlinearities, it has limitations in practical engineering applications. First, the accuracy of finite element analysis is highly dependent on modeling techniques, such as the selection of element type, mesh density, and boundary conditions. To achieve sufficient accuracy, high-density meshes are typically required, leading to enormous computational costs, especially in nonlinear iterative analyses. Second, traditional finite element methods also struggle to directly simulate tensile stiffening effects using simple constitutive models, often necessitating empirical adjustments.
[0005] Vector Form Intrinsic Finite Element (VFIFE) is an emerging numerical method based on vector mechanics principles. It discretizes the structure as a system of point masses and describes the structural behavior by solving the equations of motion of these particles. VFIFE demonstrates unique advantages in handling highly nonlinear problems such as large deformation, cracking, and collisions. However, accurately incorporating the complex cross-sectional mechanical behaviors of prestressed concrete beams after cracking (such as bond slip and tensile stiffening) into VFIFE frames remains a challenge for current technologies.
[0006] Patent document CN120163009A discloses a method, system, and device for simulating prestressed beams based on vector finite element method. The method includes: establishing a numerical model of the prestressed beam using the vector finite element method; discretizing the numerical model of the prestressed beam into multiple nodes and multiple prestressed beam elements; calculating the stiffness matrix of each prestressed beam element; determining the governing equations of each node based on the stiffness matrix; solving the governing equations of each node using the central difference method; and updating the state of each node according to the solution results to complete the simulation of the prestressed beam.
[0007] Patent document CN121031238A discloses a method for constructing a coefficient prediction model, a coefficient prediction method, and a storage medium. The construction method includes the following steps: modeling a prestressed beam of a bridge using finite element numerical software and simulating the tensioning process; constructing a bridge parameter coefficient array, dividing it into a training dataset and a validation dataset; performing data standardization and tensor data transformation; constructing a data loader; constructing a fully connected multilayer multitasking neural network based on and , which includes a standardizer, a forward propagation function, a loss function, a learning rate scheduler, and an optimizer; performing model training; encapsulating the prediction function; and integrating the standardizer and the trained prediction model. Summary of the Invention
[0008] The purpose of this invention is to provide a prestressed beam simulation method based on incomplete bonding theory and vector finite element method. This method can overcome the dependence of traditional methods on empirical formulas and accurately simulate the complete stress behavior of beams from elasticity, cracking, yielding to ultimate failure, especially the tensile stiffening effect after cracking.
[0009] To achieve the objectives of this invention, the following technical solution is provided: a method for simulating prestressed beams based on incomplete bonding theory and vector finite element method, comprising the following steps:
[0010] The mechanical response of prestressed beam sections under different loads was calculated based on the incomplete bonding theoretical model, so as to construct the moment-rotation relationship of prestressed beam sections under load.
[0011] The vector finite element method is used to discretize the prestressed beam structure containing the prestressed beam section into multiple mass points, and the multiple mass points are connected by massless beam elements to construct a global structural analysis model.
[0012] The moment-rotation relationship is used as the force-deformation relationship of the massless beam element in the global structural analysis model. The displacement response of each mass point in the global structural analysis model is calculated by explicit time integration. During the calculation process, the internal forces are updated according to the deformation of the massless beam element and the moment-rotation relationship until the termination condition is met, and the simulation results of the prestressed beam structure are output.
[0013] This invention is based on the bond-slip mechanics mechanism of the steel-concrete interface, thus accurately deriving the tensile stiffening effect at the section level. Simultaneously, it couples PI section analysis and VFIFE global structural analysis, enabling precise simulation of the complete load-displacement curve of a prestressed beam from the start of loading, concrete cracking, steel yielding, to final failure. Comparison with experimental data confirms a high degree of consistency. Finally, the VFIFE method employs explicit integration and point-value description, eliminating the need to assemble and solve the tangent stiffness matrix. This method possesses inherent advantages and robustness in handling material nonlinearity, large geometric deformation, and abrupt behavioral changes after cracking.
[0014] Specifically, the bending moment-rotation relationship includes both before and after the section cracks;
[0015] Before the section cracks, the strain of concrete, non-prestressed steel bars and prestressed tendons along the beam height is calculated using linear strain distribution. The stress at the corresponding beam height is calculated based on the stress-strain relationship of the material. Then, the neutral axis position is solved by iterative equilibrium of internal forces in the section to calculate the moment-rotation relationship under load.
[0016] After the section cracks, the tension of the corresponding steel bars and prestressed tendons is calculated by using the pre-constructed tension-slip relationship and the slip at the crack section. Then, the neutral axis position is solved by iteratively balancing the internal forces of the section to calculate the moment-rotation relationship under load.
[0017] Specifically, the tension-slip relationship is obtained by performing tensile stiffening analysis on a concrete prism encasing reinforcing bars or prestressed tendons.
[0018] Specifically, the process of iteratively solving for the neutral axis position through cross-sectional internal force equilibrium is as follows:
[0019] Given a section rotation angle, estimate the initial value of the neutral axis depth;
[0020] Based on the calculated stress or tension, determine whether the axial force of the section is balanced. If it is not balanced, adjust the initial value and recalculate the corresponding stress or tension.
[0021] If the condition is determined to be in equilibrium, the sum of the moments of stress or tension about the neutral axis of the cross section is calculated to obtain the resisting bending moment corresponding to the rotation angle of the cross section.
[0022] Repeat the above process to establish the moment-rotation relationship.
[0023] Specifically, the virtual inverse motion method is used to decouple the total displacement of the massless beam element in the current time step into rigid body motion and pure deformation, and the pure deformation is extracted to calculate the corresponding internal forces in the deformation and moment-rotation relationship.
[0024] Specifically, in the global structural analysis model, multiple mass points are subject to motion constraints using Newton's second law.
[0025] Specifically, explicit time integration is performed using the central difference method.
[0026] Specifically, the termination conditions include meeting preset calculation time or loading conditions.
[0027] Compared with the prior art, the beneficial effects of the present invention are as follows:
[0028] Based on mechanical mechanism: The incomplete bond theory (PI) analysis adopted in this application is based on the bond-slip mechanical mechanism of the steel bar and concrete interface, thereby accurately deriving the tensile stiffening effect at the section level, completely avoiding the dependence on empirical formulas in traditional methods;
[0029] High precision throughout the entire process: This application couples PI section analysis and VFIFE global structural analysis, which can accurately simulate the complete load-displacement curve of prestressed beams from the start of loading, concrete cracking, steel yielding to final failure. The results have been verified to be highly consistent with experimental data.
[0030] Good numerical stability: The VFIFE method uses explicit integration and point value description, without the need to assemble and solve the tangent stiffness matrix. It has a natural advantage and robustness in dealing with material nonlinearity, large geometric deformation and abrupt behavior after cracking.
[0031] Wide applicability: The method of this application can be applied to the analysis of prestressed beams with different steel bars / prestressing tendons, different reinforcement ratios, prestress levels, and material strengths, providing a more reliable and superior analytical tool for the design and performance evaluation of prestressed structures. Attached Figure Description
[0032] Figure 1 This is a flowchart of the prestressed beam simulation method based on incomplete bonding theory and vector finite element method provided in this embodiment;
[0033] Figure 2 This is a schematic diagram illustrating the specific calculations provided in this embodiment;
[0034] Figure 3 This is a schematic diagram of the example model provided in this embodiment;
[0035] Figure 4 This is a comparison chart of the calculation results provided in this embodiment. Detailed Implementation
[0036] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. The components of the embodiments of the present invention described and shown in the accompanying drawings can generally be arranged and designed in various different configurations. Therefore, the following detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to illustrate selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention.
[0037] like Figure 1 As shown, this embodiment provides a method for simulating prestressed beams based on incomplete bonding theory and vector finite element method, including the following specific steps:
[0038] S1. Perform cross-sectional mechanical analysis, using the incomplete bond theory (PI) model to calculate the mechanical response of the prestressed beam cross-section under different loads, thereby establishing the moment-rotation (M-θ) relationship characterizing the behavior of the cross-section;
[0039] S2. Perform global structural analysis. Using the Vector Finite Element Method (VFIFE), the prestressed beam structure is discretized into a system composed of multiple mass points and massless beam elements connecting the mass points.
[0040] S3. Perform coupling analysis, using the moment-rotation (M-θ) relationship established in step 1 as the force-deformation relationship of the massless beam element in step 2, and use it to calculate the internal forces of the prestressed beam element; Step 4. Solve the dynamic control equations of the global structural analysis model, calculate the displacement response of each mass point by explicit time integration method, and update its internal forces according to the deformation of the beam element and the moment-rotation (M-θ) relationship until the preset calculation time or loading condition is reached, and output the simulation results of the prestressed beam.
[0041] More specifically, such as Figure 2 The diagram shows the specific calculation process of the method provided in this embodiment.
[0042] Establish the constitutive relation of the beam element, namely the moment-rotation (M-θ) relationship. The goal of this step is to accurately describe the complete mechanical behavior of the section during the stress process, from elasticity to cracking and finally to failure, especially to capture the "tensile stiffening" effect from the mechanical mechanism.
[0043] S1.1 Pre-cracking behavior: such as Figure 2As shown on the left side of the flowchart, for a given section rotation angle (θ), it is assumed that the section strain is linearly distributed before concrete cracking. Based on the stress-strain relationship of the material, the strain of the concrete and the compressed steel reinforcement are calculated. ), tensile reinforcement ( ) and prestressed tendons ( The stress in the section is determined by iteratively adjusting the neutral axis depth (c) until the section meets the condition for internal force equilibrium. Once the internal forces in the section are in equilibrium, the moments of all internal force components about the neutral axis can be calculated, yielding the resisting bending moment (M) corresponding to that rotation angle (θ). This process is repeated to obtain the portion of the section before cracking in the M-θ curve.
[0044] S1.2 Post-cracking behavior and tensile stiffening: When the tensile stress in concrete exceeds its tensile strength, the concrete cracks. At this time, the tensile force at the crack section is entirely borne by the reinforcing steel and prestressing tendons. However, between the cracks, due to the presence of bond stress, the tensile force is transferred from the reinforcing steel back to the concrete, allowing the concrete between the cracks to still bear some of the tensile force, i.e., the "tensile stiffening" effect.
[0045] To simulate this effect, this application employs a mechanics-based sub-model. The analysis extracts a concrete prism enclosing the reinforcing bars / prestressing tendons. The goal of this analysis is to establish a direct relationship between the tensile force (P) of the reinforcing bars / prestressing tendons at the crack section and the slip (Δ) of the reinforcing bars / prestressing tendons relative to the concrete, i.e., the P-Δ curve.
[0046] The P-Δ relationship was obtained through numerical methods. The prism was discretized into infinitesimal segments ( Starting from the cross-section at the crack, given an initial slip ( ) and trial tension ( Within each infinitesimal segment, the bond force (B) is calculated based on the known bond-slip constitutive model (τ-Δ), and the changes in steel reinforcement and concrete forces are then calculated accordingly. Simultaneously, the change in slip is calculated based on the strain difference between the steel reinforcement and concrete. This process iterates along the length of the prism, continuously adjusting the initial calculated tension. This continues until the physical boundary conditions are met (slip at the midpoint between the two cracks is zero). For prestressed tendons, the strain term related to prestress must also be considered when calculating the slip increment. Through this analysis, a series of (P, Δ) corresponding points can be obtained, forming a complete P-Δ curve.
[0047] S1.3 Internal Force Equilibrium after Cracking: After cracking of the section, for a given section rotation angle (θ), first calculate the neutral axis depth (c). Concrete compression zone ( ) and compression reinforcement ( The internal forces of the reinforcement bars are still based on strain calculations. However, the key difference lies in the fact that the tensile reinforcement bars (…) ) and prestressed tendons ( The tensile force is no longer based on strain, but on the amount of slip at the crack section (e.g., Calculate the slippage. Substitute this slippage amount into the P-Δ curve obtained in S1.2 to find the corresponding tension (F).
[0048] Next, check the axial force balance of the cross-section. If unbalanced, adjust the neutral shaft depth (c) and recalculate until balance is achieved. After balance, calculate all internal forces ( The resultant torque (M) of ).
[0049] By repeating steps 1.1 to 1.3 above for a continuous rotation angle (θ) from 0 to the limit state, a complete M-θ curve incorporating the tensile stiffening effect can be obtained. This curve is then recorded and used as input for global structural analysis.
[0050] A global structural analysis model of the beam is established, and the VFIFE method is used to discretize the continuous prestressed beam into a series of mass points, which are connected by massless beam elements.
[0051] The entire mass of the structure is concentrated and distributed among the individual point masses. The motion of these point masses follows Newton's second law: ,in As an external force, For internal force, This is the damping force.
[0052] In this embodiment, the core lies in coupling steps S1 and S2. That is, in S2, the internal force on the particle ( The internal forces of these beam elements are provided by the massless beam elements that connect them. The internal forces of these beam elements are not based on traditional linear elastic or elastoplastic constitutive models, but are obtained directly from the complete, highly nonlinear M-θ constitutive curves established in S1.
[0053] like Figure 2 As shown on the right, after the structure is discretized, the global structure analysis starts from time t=0 and enters the time step cycle.
[0054] S4.1 Solving the governing equations: At each time step, the external loads are updated first. Then, the governing equations for the motion of all particles are solved using the explicit time integration method to obtain the new positions of the particles at that time.
[0055] S4.2 Obtaining Pure Deformation: Since particle motion includes rigid body displacement and element deformation, they must be decoupled. This application employs a virtual inverse motion method. Through translation and rotation, the element's position at the current time (t) is "inversely" moved back to the reference time (t). a The location of the element is determined to extract its pure deformation components, which include, for example, axial elongation. and the relative corners at both ends , .
[0056] S4.3 Update internal forces: The relative rotation angle obtained in step 4.2 is used as the internal force. Substituting into the M-θ curve established in step one, we find the resisting bending moment M corresponding to the current deformation. This bending moment M is then converted into equivalent nodal internal forces ( ), applied to the particle, for calculation in the next time step.
[0057] S4.4 Loop: Repeat the above process, continuously updating the particle positions and element internal forces until the final calculation time or load step is reached. Output the particle displacements and element internal forces.
[0058] To demonstrate the reliability and accuracy of the method provided in this embodiment, the following were selected: Figure 3 The example shown is used for verification. The calculation model has a length of 2440 mm, a width of 380 mm, and a height of 266 mm. The concrete compressive strength is 44.5 MPa, the tensile strength is 3.16 MPa, and the elastic modulus is 29.9 GPa. The tensile strength of ordinary steel reinforcement is 420 MPa, and the elastic modulus is 210 GPa. The tensile strength of prestressed steel reinforcement is 1860 MPa, and the elastic modulus is 195 GPa. The initial prestress is 965 MPa. The calculation model has 17 nodes and 16 elements, with a single node mass of 4.92 kg. The loading method is displacement control. The boundary condition is simply supported at both ends.
[0059] Figure 4 This is a comparison graph of the load-displacement curves for this example. The dotted lines in the graph represent experimental data, and the dashed lines represent the calculation results using the method of this application.
[0060] from Figure 4 As can be clearly seen, the predicted results (dashed line) of the method in this application show a high degree of consistency with the experimental data (dotted line), whether in the initial elastic stage stiffness, stiffness degradation after cracking, yield point, or in the final ultimate bearing capacity and declining failure stage. This fully demonstrates that the method in this application, by considering bond slip and tensile stiffening from a mechanical mechanism perspective, can more accurately and robustly simulate the entire stress behavior of prestressed concrete beams.
[0061] Furthermore, the terms "upper," "lower," "inner," "outer," "front," and "rear" are used for descriptive purposes only and should not be construed as indicating or implying relative importance. Unless otherwise specifically stated, the relative steps, numerical expressions, and values of the components and steps set forth in these embodiments do not limit the scope of the invention.
[0062] Of course, the above description is only a specific embodiment of the present invention and is not intended to limit the scope of the present invention. All equivalent changes or modifications made to the structure, features and principles described in the claims of the present invention should be included in the scope of the claims of the present invention.
[0063] Finally, it should be noted that the above-described embodiments are merely specific implementations of the present invention, used to illustrate the technical solutions of the present invention, and not to limit it. The scope of protection of the present invention is not limited thereto. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that any person skilled in the art can still modify or easily conceive of changes to the technical solutions described in the foregoing embodiments within the technical scope disclosed in the present invention, or make equivalent substitutions for some of the technical features; and these modifications, changes, or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention, and should all be covered within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.
Claims
1. A method for simulating prestressed beams based on incomplete bonding theory and vector finite element method, characterized in that, Includes the following steps: The mechanical response of prestressed beam sections under different loads was calculated based on the incomplete bonding theoretical model, so as to construct the moment-rotation relationship of prestressed beam sections under load. The bending moment-rotation relationship includes both before and after the section cracks; Before the section cracks, the strain of concrete, non-prestressed steel bars and prestressed tendons along the beam height is calculated using linear strain distribution. The stress at the corresponding beam height is calculated based on the stress-strain relationship of the material. Then, the neutral axis position is solved by iterative equilibrium of internal forces in the section to calculate the moment-rotation relationship under load. After the section cracks, the pre-constructed tension-slip relationship is used in conjunction with the slip at the crack section to calculate the tension of the corresponding steel bars and prestressed tendons. Then, the neutral axis position is solved by iterative equilibrium of internal forces in the section to calculate the moment-rotation relationship under load. The process of iteratively solving for the neutral axis position through cross-sectional internal force equilibrium is as follows: Given a section rotation angle, estimate the initial value of the neutral axis depth; Based on the calculated stress or tension, determine whether the axial force of the section is balanced. If it is not balanced, adjust the initial value and recalculate the corresponding stress or tension. If the condition is determined to be in equilibrium, the sum of the moments of stress or tension about the neutral axis of the cross section is calculated to obtain the resisting bending moment corresponding to the rotation angle of the cross section. Repeat the above process to establish the moment-rotation relationship; The tension-slip relationship was obtained by performing tensile stiffening analysis on a concrete prism encasing reinforcing bars or prestressed tendons. The vector finite element method is used to discretize the prestressed beam structure containing the prestressed beam section into multiple mass points, and the multiple mass points are connected by massless beam elements to construct a global structural analysis model. The moment-rotation relationship is used as the force-deformation relationship of the massless beam element in the global structural analysis model. The displacement response of each mass point in the global structural analysis model is calculated by explicit time integration. During the calculation process, the internal forces are updated according to the deformation of the massless beam element and the moment-rotation relationship until the termination condition is met, and the simulation results of the prestressed beam structure are output.
2. The prestressed beam simulation method based on incomplete bonding theory and vector finite element method according to claim 1, characterized in that, The virtual inverse motion method is used to decouple the total displacement of the massless beam element in the current time step into rigid body motion and pure deformation, and extract the internal forces corresponding to the pure deformation in the deformation and moment-rotation relationship.
3. The prestressed beam simulation method based on incomplete bonding theory and vector finite element method according to claim 1, characterized in that, In the global structural analysis model, multiple mass particles are subject to motion constraints based on Newton's second law.
4. The prestressed beam simulation method based on incomplete bonding theory and vector finite element method according to claim 1, characterized in that, In the global structural analysis model, multiple mass particles are subject to motion constraints based on Newton's second law.
5. The prestressed beam simulation method based on incomplete bonding theory and vector finite element method according to claim 1, characterized in that, The termination conditions include meeting preset calculation time or loading conditions.