Finite element modeling method for reinforcing damaged concrete beam by Fe-SMA

Abstract

The application provides a finite element modeling method for reinforcing a damaged concrete beam by Fe-SMA, and relates to the technical field of reinforcing a damaged concrete beam, and comprises the following steps: applying a pre-damage load to a finite element model of an unreinforced reinforced concrete beam; setting interface stiffness, damage initiation parameters and damage evolution parameters of the interface element according to the bond-slip relationship between the Fe-SMA strip and the concrete; applying a thermal activation effect to the Fe-SMA strip; calculating the transmission of the Fe-SMA recovery stress to the damaged reinforced concrete beam and the loss of prestress caused by the interface slip through the interface element; obtaining load-displacement response, material stress, interface slip and damage evolution results of the reinforced beam; determining the yield bearing capacity, ultimate bearing capacity and failure mode of the reinforced beam; and the application guarantees the state continuity of the whole process of pre-damage, reinforcement, thermal activation and reloading.

Description

Technical Field

[0001] This invention relates to the field of reinforcement of damaged concrete beams, and in particular to a finite element modeling method for Fe-SMA reinforcement of damaged concrete beams. Background Technology

[0002] During long-term service, reinforced concrete bridges commonly experience problems such as concrete cracking, steel bar yielding, decreased cross-sectional stiffness, and reduced load-bearing capacity due to factors such as vehicle loads, environmental erosion, and material deterioration. Applying external iron-based shape memory alloy (Fe-SMA) strips and activating them by heating allows the recovery stress generated by the Fe-SMA to introduce prestress into the damaged beams, providing an effective reinforcement method. Currently, existing modeling methods for finite element analysis of Fe-SMA-strengthened concrete beams typically suffer from the following problems: First, simulating initial damage by directly reducing the elastic modulus or strength of concrete fails to reflect the actual crack distribution, plastic strain, and residual stress within the beam after pre-damage loading and unloading, and also makes it difficult to retain residual deformation after unloading. Second, when applying thermally activated recovery stress, the geometric fact that the Fe-SMA strips are actually installed on the deformed bottom surface of the damaged beam is not considered, resulting in the application of initial strain or initial stress to the undeformed configuration, leading to false internal forces that do not match the actual situation. Third, setting the Fe-SMA strips to be completely bonded to the concrete surface makes it impossible to calculate prestress transfer losses and end peeling phenomena caused by interface slippage. Fourth, each stage—pre-damage, reinforcement activation, thermal activation, and reloading—is calculated independently, lacking a continuous inheritance mechanism for state variables, and thus failing to reflect the complete stress history of the damaged beam from before to after reinforcement. Summary of the Invention

[0003] To address the aforementioned technical problems, the technical solution adopted by this invention is as follows: According to a first aspect of this application, a finite element modeling method for Fe-SMA-reinforced damaged concrete beams is provided, comprising: Obtain the geometric and material parameters of the reinforced concrete beam, reinforcing bars, Fe-SMA strips, and bond interface, and establish a finite element model of the unreinforced reinforced concrete beam. A pre-damage load is applied to the finite element model of the unreinforced concrete beam, and then unloaded after the target damage state is reached to obtain concrete damage variables, plastic strain, residual stress of steel reinforcement, and residual deformation of the beam; displacement control is used in the unloading stage. The concrete damage variables, plastic strain, residual stress of steel bars, and residual deformation of the beam are taken as the initial state of the damaged reinforced concrete beam, and Fe-SMA strip elements and interface elements are activated while maintaining the initial state. The interface stiffness, damage initiation parameters, and damage evolution parameters of the interface unit are set according to the bond-slip relationship between Fe-SMA strips and concrete. The Fe-SMA recovery stress is determined based on the Fe-SMA activation temperature. At least one of the equivalent initial strain method, initial stress method, or temperature field coupling method is used to apply a thermal activation effect to the Fe-SMA strip. The equivalent initial strain or initial stress is applied based on the current deformation configuration of the Fe-SMA strip when it is activated, and geometric nonlinearity is enabled throughout the process. The transfer of Fe-SMA restored stress to the damaged reinforced concrete beam and the prestress loss caused by interface slip are calculated using the interface unit. Based on the inherited damaged state, thermally activated state and interface state variables, an external load is applied to the reinforced concrete beam to obtain the load-displacement response, material stress, interface slip and damage evolution results of the reinforced beam. The yield strength, ultimate bearing capacity, and failure mode of the reinforced beam are determined based on the load-displacement response, material stress, interface slip, and damage evolution results.

[0004] The present invention has at least the following beneficial effects: The finite element modeling method for Fe-SMA-reinforced damaged concrete beams of this invention employs displacement control during the pre-damage unloading stage and retains concrete damage variables, plastic strain, residual stress in the reinforcing steel, and residual deformation of the beam based on the loading-unloading path. This avoids the distortion of damage distribution caused by simply reducing material parameters, making the initial damage state closer to the actual detection results. When activating the Fe-SMA strips, the equivalent initial strain or initial stress is applied based on the current deformation configuration of the beam after unloading, while geometric nonlinearity is enabled. This ensures that the initial geometric position of the strips is consistent with the actual deformation state of the bottom surface of the damaged beam, eliminating additional stress caused by configuration mismatch. The interface element adopts a bond-slip constitutive model, which can calculate the interface shear stress, slip, and prestress loss during the transmission of restoring stress, avoiding the overestimation of the reinforcement effect by the assumption of complete bond. By inheriting displacement, stress, strain, plastic strain, damage variables, and interface states between each analysis stage, the continuity of the state throughout the pre-damage, reinforcement, thermal activation, and reloading processes is ensured, making the predicted load-displacement response, failure mode, and bearing capacity of the reinforced beam more consistent with engineering reality. Attached Figure Description

[0005] To more clearly illustrate the technical solutions in the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0006] Figure 1 A flowchart of the finite element modeling method for Fe-SMA-reinforced damaged concrete beams provided in an embodiment of the present invention; Figure 2 This is a schematic diagram of the overall process framework provided for an embodiment of the present invention; Figure 3 This is a schematic diagram of a Fe-SMA-reinforced damaged reinforced concrete beam structure provided in an embodiment of the present invention; Figure 4 A schematic diagram of the finite element model provided for an embodiment of the present invention. Detailed Implementation

[0007] 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. 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.

[0008] It should be noted that, based on this disclosure, those skilled in the art will understand that one aspect described herein can be implemented independently of any other aspect, and two or more of these aspects can be combined in various ways. For example, any number of aspects set forth herein can be used to implement the device and / or practice the method. Furthermore, this device and / or practice the method can be implemented using other structures and / or functionalities besides one or more of the aspects set forth herein.

[0009] The following will refer to Figure 1 The flowchart shown is a finite element modeling method for Fe-SMA-reinforced damaged concrete beams, which introduces a finite element modeling method for Fe-SMA-reinforced damaged concrete beams.

[0010] The finite element modeling method for Fe-SMA-reinforced damaged concrete beams includes the following steps: S100: Obtain the geometric and material parameters of the reinforced concrete beam, reinforcing bars, Fe-SMA strips, and bond interface, and establish a finite element model of the unreinforced reinforced concrete beam.

[0011] In this embodiment, regarding geometric parameters, for reinforced concrete beams, it is necessary to measure or set the total length of the beam, the calculated span between the support centers, the width and height of the rectangular or T-shaped section, the thickness of the concrete cover, and determine the diameter, number, arrangement position, and corresponding reinforcement ratio of the longitudinal tensile reinforcement and stirrups. Simultaneously, the diameter, spacing, and arrangement range of the stirrups along the beam length direction must be determined. For Fe-SMA strips, their bonding width, thickness, reinforcement length along the bottom of the beam, and whether mechanical anchorage devices are installed at the ends and their anchorage length must be determined. The positions of the supports and external loads also need to be determined at this stage; the overall process is as follows: Figure 2 As shown.

[0012] Regarding material parameters, input the prism compressive strength, axial tensile strength, initial elastic modulus at room temperature, and fracture energy of the concrete. Input the elastic modulus, yield strength, and ultimate strength of the reinforcing steel, and specify its reinforced section modulus. Input the elastic modulus, yield strength, ultimate strength, and initial pre-strain value applied before activation of Fe-SMA. Input the thickness and elastic modulus of the adhesive layer. Additionally, provide experimental data on the recovered stress or recovered strain of Fe-SMA at different activation temperatures, as well as the temperature correction factor for interfacial bond performance.

[0013] When establishing the finite element model, the concrete beam uses three-dimensional solid elements (such as C3D8R), and the longitudinal reinforcement and stirrups use truss elements (such as T3D2). The reinforcement is embedded in the concrete through embedded regions, and the relative slip between the reinforcement and the concrete is not considered, or the relative slip between the two is considered by setting bond-slip elements. The support pads and loading pads use rigid body elements or linear elastic solid elements, and establish normal hard contact and tangential frictional contact with the concrete beam surface through contact relationships.

[0014] In terms of material constitutive model, concrete adopts a damaged plasticity model, defining its expansion angle, flow potential eccentricity, biaxial to uniaxial compressive strength ratio, and tensile-compressive meridian ratio. The compressive stress-inelastic strain relationship and the tensile stress-cracking displacement (or cracking strain) relationship of concrete are also input. Simultaneously, the variation laws of tensile damage variables and compressive damage variables with inelastic strain are set in the material definition, causing stiffness degradation in concrete after tensile cracking and compressive yielding. Reinforcing steel adopts a bilinear or trilinear elastoplastic constitutive model, defining its stress-strain relationship during the strengthening stage.

[0015] For the thermal activation effect of Fe-SMA strips, in addition to defining its room-temperature elastic modulus and Poisson's ratio in the material properties, it is also necessary to pre-set them according to the simulation method used: if the equivalent initial strain method is used, it needs to be defined by applying the initial state in subsequent steps; if the temperature field coupling method is used, the thermal expansion coefficient, specific heat capacity, and thermal conductivity of Fe-SMA need to be defined simultaneously, and the temperature field needs to be applied in subsequent analysis steps. For the adhesive layer, if solid element modeling is used, it is given linear elastic material properties; if only the force is transferred through interface elements, the adhesive layer solid is not built separately, and the initial stiffness and damage parameters of the interface elements are derived from subsequent experimental calibration.

[0016] If the impact of activation temperature on the properties of concrete and steel reinforcement will be considered later, the elastic modulus, strength, and thermal expansion coefficient reduction data of each material at different temperatures should be entered in the material definition so that material degradation can be achieved through the temperature field in the thermal activation analysis step. If a simplified approach is adopted—that is, the impact of activation temperature on the properties of concrete and steel reinforcement is considered negligible when the temperature is below 200℃ and the isothermal time does not exceed 30 minutes—then temperature dependence is not set in the material definition, and only the thermal expansion coefficient of each material is retained in the model to calculate temperature deformation. Fe-SMA reinforced damaged reinforced concrete beam structures are shown below. Figure 3 As shown, the finite element model is as follows Figure 4 As shown.

[0017] S200, apply a pre-damage load to the finite element model of the unreinforced concrete beam, and unload it after reaching the target damage state to obtain concrete damage variables, plastic strain, residual stress of steel bars and residual deformation of beam; wherein, displacement control is adopted in the unloading stage.

[0018] In this step, a pre-damage load is applied to the finite element model of the unreinforced concrete beam, and then unloaded after the target damage state is reached, to obtain concrete damage variables, plastic strain, residual stress of steel bars, and residual deformation of the beam; displacement control is used in the unloading stage.

[0019] Furthermore, the target damage state is determined based on at least one of the following: the ratio of the pre-damage load to the ultimate load of the unreinforced beam, mid-span deflection, crack width, steel reinforcement strain, or concrete damage variable; the pre-damage load is expressed as: ; in, For pre-damage load, This refers to the ultimate load of the unreinforced beam. This represents the pre-damage coefficient.

[0020] Furthermore, the concrete damage variables include concrete tensile damage variables and compressive damage variables, and the plastic strain includes concrete plastic strain and steel plastic strain; the state variables retained after pre-damage unloading also include the unit damage state corresponding to the crack area and the contact state between the support and the loading area.

[0021] Furthermore, during the application of pre-damage loads and unloading: force control or displacement control is used in the loading stage, and displacement control is used in the unloading stage to reduce the displacement of the loading point from the peak displacement to zero or the preset residual displacement value; during unloading, the concrete is unloaded linearly according to the degraded elastic stiffness corresponding to the unloading time, and the steel bars are unloaded according to the initial elastic modulus. After unloading is completed, the state variables of all current integration points are frozen and transferred to the subsequent analysis stage through the state file or restart function.

[0022] In this embodiment, before pre-damage loading, the beam's self-weight is applied to the model as a distributed load, allowing the structure to reach equilibrium under its own weight. If there is support or abutment settlement during the actual construction of the beam, it can also be applied at this stage.

[0023] During pre-damage loading, a vertical load is applied at the loading pad location. Load application can be either force-controlled or displacement-controlled. Force control is suitable for the rising phase of the load-displacement curve, i.e., before the structure enters a significant softening stage. If the pre-damage target is set after the peak load (e.g., above 85% of the ultimate bearing capacity), displacement control must be used to prevent calculation divergence during loading. The loading rate should not be too fast; typically, a displacement increment of 0.5–2 mm per analysis step is preferable. The specific value should be adjusted based on the beam height and calculation span to ensure the entire loading process covers a sufficient number of increment steps to capture damage evolution.

[0024] During loading, the finite element model outputs real-time responses such as mid-span deflection, support reactions, maximum principal strain of the tensile concrete, and longitudinal reinforcement stress. Loading stops when any of these indicators reaches a preset target value. The determination of the target damage state can be based on one of the following methods: one is the ratio of the pre-damaged load to the ultimate load of the unreinforced beam, i.e. ,in Typically, the value is taken between 0.3 and 0.9, corresponding to the progression from mild to severe damage; secondly, the mid-span deflection reaches a predetermined value (such as L / 200, L / 150, or L / 100); and thirdly, the concrete tensile damage variable reaches a certain threshold, or the longitudinal steel reinforcement stress approaches or reaches the yield strength. In actual analysis, it is preferred to use both mid-span deflection and steel reinforcement stress as indicators to avoid the uncertainty caused by the dispersion of concrete strength when relying solely on the load ratio.

[0025] Once the target damage state is reached, the unloading phase begins, with displacement control employed throughout. Specifically, the vertical displacement of the loading point is gradually reduced from the current peak displacement to zero or a preset residual displacement value according to a specified reduction. The unloading step size can be consistent with the loading step size or appropriately increased. During unloading, the concrete damage plasticity model defaults to unloading based on the degraded elastic stiffness corresponding to the current damage state, i.e., the compressive damage variable d. c and tensile damage variable d t The stress remains constant and changes with strain according to The slope of the steel bar decreases linearly without generating new plastic strain or damage increment. The steel bar is based on its initial elastic modulus E. s When unloaded, the stress falls back to near zero, but the plastic strain in the tensile direction is still retained, which is manifested as a positive residual stress.

[0026] After unloading, the state variables at each integration point in the model include: concrete tensile damage variable d. t Variable d of concrete under compressive stress c The equivalent plastic tensile strain and equivalent plastic compressive strain of concrete, the equivalent plastic strain of steel reinforcement (PEEQ) and the current stress tensor, and the displacement field of each node of the beam are also preserved. In addition, the element damage indicators of the tensile cracked area, the contact pressure and contact state between the support pad and the loading pad and the concrete are also preserved.

[0027] To prevent the aforementioned state variables from being overwritten in subsequent analysis steps, a state freeze operation must be performed at the end of the unload analysis step. Specifically, this can be achieved by setting a restart output in the finite element software, writing the stress, strain, state variables, and nodal displacements of all current integration points into a restart file; or by reading the results of the previous step as initial conditions through a state file in subsequent analysis steps. For software platforms that do not support restarting, an "analysis step inheritance" approach can be used, where the next analysis step is directly built upon the convergent configuration at the end of the previous analysis step without any initialization. All three methods described above can achieve complete transfer of the pre-damage state.

[0028] In this step, damage is generated through a physical loading-unloading path, rather than by artificially reducing material parameters. This ensures that the spatial distribution of the damaged area and the magnitude of residual deformation are derived from the actual stress process, eliminating the shortcomings of the parameter reduction method in reflecting stress paths and damage distribution. Displacement control during the unloading stage ensures that the structure can stably converge to the unloading endpoint after passing the peak load and entering the softening phase, avoiding the problem of repeated iteration failures of force control in the descent phase. Concrete and steel reinforcement are unloaded according to their respective constitutive paths, preserving the residual cracking state of the concrete and the residual tensile stress of the steel reinforcement, making the initial state of the damaged beam mechanically self-consistent. Freezing these state variables and passing them to subsequent analysis steps ensures that the calculation results of each stage before reinforcement are not reset to zero by subsequent operations, providing a true initial starting point for evaluating the reinforcement effect.

[0029] S300, the concrete damage variable, plastic strain, steel residual stress and beam residual deformation are taken as the initial state of the damaged reinforced concrete beam, and Fe-SMA strip elements and interface elements are activated while maintaining the initial state. In this step, the concrete damage variables, plastic strain, steel residual stress and beam residual deformation retained after unloading in step S200 are taken as the initial state of the damaged beam. Under the premise that this state remains unchanged, Fe-SMA strip elements and interface elements are activated.

[0030] Furthermore, when a mechanical anchoring device is installed at the end of the Fe-SMA strip, the Fe-SMA strip is divided into a pure bonding zone and an anchoring zone along its length. A nonlinear spring unit or connector unit is added to the anchoring zone to simulate the pull-out resistance provided by the mechanical anchoring. The boundary conditions at the anchoring end are set as follows: ; in, For mechanical anchoring resistance, This represents the end slippage; the total end resistance is the sum of the adhesive layer bonding force and the mechanical anchoring force.

[0031] After the pre-damage loading and unloading phases are completed, the model only contains concrete beams, reinforcement, supports, and loading pads. Fe-SMA strips and adhesive layers are not yet established in the model or, although established, are inactive. If the element birth and death technique is used, the Fe-SMA strip elements and interface elements are set to a "killed" state in the pre-damage analysis step, and the mass, stiffness, and load of these elements do not participate in the overall stiffness integration. If the model modification technique is used, these components are not included in the pre-damage analysis step, and are only added to the model in the current analysis step by "adding components" or "importing components".

[0032] The activation operation is performed at the start of the next analysis step after unloading is complete. Specifically, in the finite element software, the state of the Fe-SMA strip elements and interface elements is switched from "inactive" to "active," and their initial temperature is set to equal to the ambient temperature. After activation, the node coordinates of these elements automatically inherit the deformed coordinates of the current beam bottom surface, rather than the coordinates under the initial undeformed configuration. For example, when residual deflection occurs at mid-span of a damaged beam due to pre-damage, the nodes on the beam bottom surface have already shifted downwards. After the Fe-SMA strip elements are activated, their node coordinates will be the position after this displacement, thus avoiding geometric mismatch caused by installing the strips onto the undeformed beam bottom surface.

[0033] The interface element is positioned between the bottom node of the Fe-SMA strip and the top node of the concrete beam. For the 3D solid model, the interface element is a zero-thickness cohesive element, whose normal and tangential behavior is controlled by the bond-slip constitutive model defined in subsequent steps. Upon activation, the initial damage variable of the interface element is set to zero, assuming that the well-constructed adhesive layer has not yet borne any load.

[0034] If mechanical anchoring devices are considered at both ends of the Fe-SMA strip, the simulation of the anchoring zone needs to be refined. The strip is divided along the beam length into a pure bond section and an anchoring section, with the length of the anchoring section taken as the actual coverage area of ​​the anchor. Within the anchoring section, in addition to the adhesive bonding force transmitted by the interface elements, nonlinear spring elements (Spring2) or connector elements (Connector) are added, with the direction set along the beam length. The force-displacement curve of the spring is determined based on the construction form and preload of the mechanical anchor. The end boundary conditions are thus modified to: the axial force of the Fe-SMA strip at the anchoring end is equal to the force provided by the spring, i.e. The amount of end slip The relative displacement of the interface element at this end is taken. The total end resistance is the sum of the spring resistance and the integral of the adhesive layer shear stress transmitted by the interface element along the anchorage section. For pure bonding conditions without mechanical anchorage, the anchorage end is still treated as a free end, i.e., the axial force is zero.

[0035] Furthermore, the activation temperature of the Fe-SMA strip Below, temperature reduction factors are introduced for the elastic modulus, compressive strength, tensile strength of concrete, and yield strength of steel reinforcement. ;when When each reduction factor is taken as 1.0, The reduction factor mentioned above is determined based on the material performance degradation law at high temperatures.

[0036] In the thermal activation analysis step, if the activation temperature of the Fe-SMA strip does not exceed 100℃, the material properties of concrete and steel reinforcement remain unchanged at room temperature, and only the recovery stress generated by Fe-SMA itself is considered. If the activation temperature is between 100℃ and 250℃, a temperature field dependency is introduced into the material definition. Specifically, in software such as ABAQUS, the elastic modulus, compressive strength, tensile strength of concrete, and yield strength of steel reinforcement are defined as functions of the field variable (temperature), and the corresponding reduction coefficients are input. For routine engineering analysis, if the isothermal time does not exceed 30 minutes and the temperature does not exceed 200℃, the attenuation curves of the mechanical properties of concrete and steel reinforcement at high temperatures given in relevant specifications can be used. If specification data is lacking, simplified linear interpolation can be used, i.e., linear reduction of 5% to 10% within the range from room temperature to 200℃. When the temperature field is applied to the entire model, the thermal expansion deformation of concrete and steel reinforcement due to temperature rise is calculated using their respective thermal expansion coefficients. This deformation is superimposed on the recovery stress of Fe-SMA and participates in the redistribution of internal forces in the structure. If the equivalent initial strain method or initial stress method is used to simulate the recovery stress of Fe-SMA (i.e., no real temperature field is applied), the aforementioned temperature reduction factor is not used, and each material is still calculated according to its room temperature properties.

[0037] In this step, the residual deformation generated by pre-damage is used as the geometric reference for the installation of Fe-SMA strips, ensuring that the initial configuration of the strips is consistent with the actual construction state of being bonded to the bottom of a deflected beam in engineering, thus avoiding false prestress or additional bending moments caused by mismatched configurations. A reinforcement layer is added by activating elements rather than remodeling, ensuring that all state variables in the pre-damage stage are fully preserved without the need for manual data mapping or interpolation. Spring elements are added to the anchorage section to simulate mechanical anchorage resistance, and the end boundary conditions are modified from free ends to spring-constrained ends, allowing the mechanical path of the anchorage device participating in the force to be reflected in the model. This is closer to the actual structure than a pure bond model and provides a calculation basis for subsequent determination of anchorage zone failure. The introduction of a temperature reduction factor quantitatively considers the degradation of concrete and steel properties during thermal activation, avoiding the risk of overestimating structural stiffness and bearing capacity at high temperatures. Furthermore, decoupling the temperature reduction from the thermal activation simulation method allows analysts to flexibly choose whether to enable the reduction and the degree of reduction based on actual heating conditions and material test data, balancing the physical accuracy and ease of use of the model.

[0038] S400, the interface stiffness, damage initiation parameters and damage evolution parameters of the interface unit are set according to the bond-slip relationship between the Fe-SMA strip and the concrete.

[0039] Furthermore, in the aforementioned interfacial bond-slip relationship, the interfacial tangential stress and the amount of slip satisfy the following: ; in, For interfacial tangential bonding stress, This is the relative slip. Peak bond stress, This represents the slip amount corresponding to the peak bond stress. For the interface curve shape parameters, This refers to the interfacial fracture energy. , and The parameters are determined based on the elastic modulus of the adhesive, the thickness of the adhesive layer, the splitting tensile strength of the concrete, the activation temperature, and the number of thermal cycles.

[0040] In this embodiment, the interface unit is positioned between the bottom surface of the Fe-SMA strip and the top surface of the concrete beam. Its constitutive relationship needs to reflect the entire process of deformation, damage, and debonding of the adhesive layer under shear force. The tangential behavior of the interface is controlled by the bond-slip relationship, which is generally expressed as follows: The specific functional form can be a bilinear model, an exponential model, or a Popovic curve model. Taking a typical bilinear model as an example, the curve consists of an elastic segment, a softening segment, and a residual segment: when the slip s is less than s0, the interface is in an elastic state, the shear stress increases linearly with the slip, and the slope is equal to the initial stiffness K. s When the slip reaches s0, the shear stress reaches its peak value τ. max Then it enters the softening stage, where the shear stress gradually decreases with increasing slip; when the slip reaches the limit slip s u At this point, the shear stress drops to zero, and the interface completely debonds. The area enclosed by the curve and the horizontal axis is the interfacial fracture energy G. f Its value is equal to (Under the bilinear model). For operating conditions using exponential or trapezoidal models, s u It can be determined by back-calculation through the relationship between fracture energy and peak stress.

[0041] When setting interface properties in finite element software, normal and tangential behaviors must be defined separately. The normal behavior uses rigid or elastic contact to prevent the Fe-SMA strips from penetrating the concrete surface. The tangential behavior uses the traction-separation constitutive law, and the input parameters include: initial tangential stiffness K. s Peak bond stress τ max Critical slip s0, limiting slip s u (or fracture energy G) f Initial stiffness K s The value of should not be too large, otherwise near-complete adhesion constraint will lead to convergence difficulties; it is usually estimated by dividing the shear modulus of the adhesive layer by the thickness of the adhesive layer, i.e. G at is the shear modulus of the adhesive. a This represents the thickness of the adhesive layer.

[0042] The criterion for initiating interface damage employs either the maximum stress criterion or the second nominal stress criterion. Under the maximum stress criterion, damage is initiated when the interfacial tangential stress reaches the peak bond stress τ. max Damage begins at time; under the secondary nominal stress criterion, when Damage begins at time t s 0 That is, τ max The evolution after damage initiation follows a linear or exponential softening law controlled by fracture energy. The damage variable D increases monotonically from 0 to 1. When D reaches 1, the interface element is deleted, corresponding to complete local debonding.

[0043] τ max , The values ​​of and n cannot be directly taken from the recommended values ​​in the literature, but should be calibrated based on the single shear test results of the Fe-SMA-adhesive-concrete composite. The specific calibration method is as follows: Prepare single shear specimens with Fe-SMA strips adhered to the side of a concrete prism; apply tensile force using a universal testing machine; record the load-end slip curve; convert the curve into an average bond stress-slip relationship; and then obtain the result through nonlinear fitting. , , Parameters such as these are obtained through calibration. These calibrated parameters are then further compared with the adhesive's elastic modulus E. a Adhesive layer thickness t a Splitting tensile strength of concrete f t,sp Establish empirical relationships for direct application in subsequent similar operating conditions. When the activation temperature is higher than room temperature, the interface parameters need to be multiplied by a temperature correction factor, which is obtained by comparing the results of single-shear tests at high temperatures (100℃, 150℃, 200℃, etc.) with those at room temperature. When subjected to multiple thermal cycles, the interface fracture energy G... f The degradation may be due to adhesive aging. The degradation pattern should be determined by post-cycle test data and incorporated into the model.

[0044] By determining interface constitutive parameters through experimental calibration rather than theoretical formulas, the values ​​of peak bond stress, initial stiffness, and fracture energy directly correspond to the actual response of the Fe-SMA-adhesive-concrete composite system, eliminating errors caused by using interface parameters from ordinary steel reinforcement or FRP. Correlating the peak bond stress and peak slip with adhesive modulus, adhesive layer thickness, concrete strength, and activation temperature allows model parameters to be adjusted based on material testing data, covering common working conditions without requiring numerous repeated single-shear tests. Introducing damage initiation and evolution determination mechanisms allows the cumulative effect of interface slip to automatically trigger the debonding process; both end-peeling and mid-debonding failure modes can be identified through the spatial distribution of interface damage variables. The presence of a softening segment prevents a stress drop after the peak bond stress, reducing computational convergence difficulty and accurately reflecting the physical characteristic that the adhesive layer can maintain a certain residual transfer capacity after cracking. The introduction of a mixed-mode damage criterion incorporates the coupling effect of normal and tangential stresses, avoiding the overestimation of shear stress transfer in single-tangential models when normal separation occurs at the interface.

[0045] S500 determines the Fe-SMA recovery stress based on the Fe-SMA activation temperature, applies a thermal activation effect to the Fe-SMA strip using at least one of the equivalent initial strain method, the initial stress method, or the temperature field coupling method, and the equivalent initial strain or initial stress is applied based on the current deformation configuration when the Fe-SMA strip is activated, and geometric nonlinearity is enabled throughout the process.

[0046] Furthermore, the equivalent initial strain method is as follows: the stress is recovered based on the Fe-SMA theory under the fully constrained condition corresponding to the target activation temperature. Determine the equivalent initial strain ,in, The equivalent initial strain is the Fe-SMA elastic modulus; when the equivalent initial strain is applied, the initial geometric coordinates of the Fe-SMA strip are inherited from the current deformation coordinates of the beam bottom node after pre-damage unloading, and the equivalent initial strain increases slowly from zero to the target value in a linear ramp amplitude manner in the early stage of the activated analysis step.

[0047] The thermal activation effect of Fe-SMA can be achieved in three ways in the finite element method, and the appropriate method should be selected based on the functionality and support of the software platform used in actual operation.

[0048] Method 1: Equivalent Initial Strain Method. This method does not require applying a real temperature field; instead, it treats the recovery stress generated by the phase transformation in Fe-SMA as equivalent to the initial mechanical strain. Specifically, based on Fe-SMA material experimental data, the theoretical recovery stress σ corresponding to the target activation temperature (e.g., 160℃, 200℃) under fully constrained conditions is determined. r0 The equivalent initial strain is calculated using the following formula: E f The elastic modulus of Fe-SMA at room temperature is obtained. Then, this initial strain is applied in the material definition of the Fe-SMA strip element using the "Initial State" or "Predefined Field" function. Before applying the strain, it is important to note that the nodal coordinates of the strip element should be inherited from the current deformation coordinates of the beam bottom surface nodes after pre-damage unloading, not the coordinates used in the initial modeling. Specifically, in the analysis step where the Fe-SMA strip is activated, the nodal coordinates of the strip element are set to the current spatial position of the beam bottom surface nodes using "Update Reference Configuration" or "Import Based on Current Configuration," and then the initial strain is applied. The direction of the initial strain is along the strip length, i.e., the longitudinal direction of the beam. To prevent iterative divergence caused by stress impact during activation, the equivalent initial strain is applied with a linear ramp amplitude, i.e., it linearly increases from zero to the target value for the first 10% to 20% of the activated analysis step, and remains constant for the remaining time.

[0049] Method 2: Initial Stress Method. This method directly applies an initial stress field to the Fe-SMA element. Specifically, it involves using the "Initial Stress" or "Predefined Stress Field" function to apply an initial stress field to the Fe-SMA element. r0 The initial stress is applied to all Fe-SMA elements along the length of the strip. Applying the initial stress also requires consideration of the current deformation configuration, i.e., the geometric position of the strip elements is the actual coordinates after unloading the damaged beam. The initial stress method does not require defining strain, but it is important to note that the sum of the initial stress and the stress increment generated by subsequent deformation should not exceed the yield strength of the Fe-SMA. The initial stress is also applied by gradually increasing the slope amplitude.

[0050] Method 3: Temperature Field Coupling Method. This method drives the Fe-SMA to generate phase transformation recovery stress by applying a realistic temperature field. Specifically, the elastic modulus, Poisson's ratio, coefficient of thermal expansion, and the recovery stress-strain relationship as a function of temperature are simultaneously input into the Fe-SMA material definition (this can be achieved through the user material subroutine UMAT or a custom field variable). In the thermal activation analysis step, a heating field from ambient temperature (typically 20°C) to the activation temperature is applied to the Fe-SMA strip, held at that temperature for a period, and then cooled back to ambient temperature. The thermal expansion strain and phase transformation strain generated by the temperature change together constitute the source of the recovery stress. The advantage of this method is that it can simulate the stress evolution throughout the entire heating, isothermal, and cooling process, and the application range of the temperature field can be extended to concrete and reinforcing steel (if a temperature reduction factor is set).

[0051] Regardless of the method used, the geometric nonlinearity switch (NLGEOM) remains active throughout the entire analysis process, from the pre-damage loading stage to the end of the post-strength loading. Its function is to ensure that when the beam undergoes flexural deformation, the equilibrium equations of the finite element model are established based on the current deformed configuration. The stress direction of the Fe-SMA strips adjusts accordingly with the rotation of the beam's bottom surface, avoiding the error caused by the stress vector remaining fixed in the initial direction under the small deformation assumption. Especially when the strengthened beam reaches its ultimate limit state under external loads, the mid-span deflection typically reaches 1 / 100 to 1 / 50 of the span, at which point the geometric nonlinear effect is significant and must be considered.

[0052] The equivalent initial strain method and the initial stress method do not require the establishment of a temperature field, have low computational cost, and good convergence, making them suitable for the analysis of conventional reinforcement schemes. The temperature field coupling method can simulate the stress evolution throughout the entire process of heating-isothermal-cooling, making it suitable for studying the impact of activation process parameters on the reinforcement effect. All three methods share the same premise—the initial strain or initial stress is applied based on the current deformation configuration of the damaged beam after unloading. This ensures that the initial state of the Fe-SMA strips is consistent with the stress state of the deflected beam bottom during actual construction, preventing additional bending moments due to geometric mismatch. The equivalent initial strain is applied slowly with a ramp amplitude, avoiding dynamic effects and iterative divergence caused by instantaneous loading, allowing the activation process to be completed smoothly within the static solution framework. Enabling geometric nonlinearity throughout the process makes the stress calculation results more accurate during the large deflection stage, especially ensuring that the bearing capacity prediction of the reinforced beam under the ultimate state is not biased by linear assumptions.

[0053] S600, through the interface unit, calculates the transfer of Fe-SMA recovering stress to the damaged reinforced concrete beam and the prestress loss caused by interface slippage.

[0054] When the Fe-SMA strip generates restoring stress in step S500, this stress cannot act directly on the concrete beam, but is transmitted point by point through the interface elements. The mechanical mechanism of transmission is as follows: With the Fe-SMA strip free at both ends and constrained in the middle, its axial tensile force is non-uniformly distributed along its length. The tensile force gradually diffuses to the concrete surface through the interface shear stress. Taking a small element dx along the strip length x, the change in the Fe-SMA axial force within this small element is equal to the resultant force of the interface shear stress along the strip width, i.e.: ; in The axial force of the Fe-SMA at position x. For the strip width, The shear stress is at the interface. This differential equation is automatically satisfied by the force balance calculation of the interface elements in the software. In actual operation, the user does not need to manually discretize the equation. As long as the node correspondence between the interface elements, Fe-SMA strip elements, and concrete surface elements is accurate, the software can automatically calculate the axial force, shear stress, and slip at each point along the strip length through iterative solution in each increment step.

[0055] The calculation process for interface transfer is directly coupled with the constitutive behavior of the interface elements. Specifically, in each incremental step, the software first calculates the relative displacements of the interface normal and tangential directions based on the difference between the elongation trend of the Fe-SMA elements under restoring stress and the surface displacement of the concrete beam. Subsequently, the interface element calculates the current shear stress from the relative displacement based on the traction-separation constitutive relation defined in step S400. And the damage variable D. If the shear stress calculated within this incremental step does not exceed the peak bond stress, the interface is in an elastic state, and all recovered stress is transferred to the concrete through the interface; if it exceeds the peak value, the damage variable D increases from 0 to 1, the interface stiffness degrades, some relative displacements are converted into irrecoverable slip, and the transfer efficiency decreases. When D reaches 1, the interface element is deleted, the corresponding local location is completely debonded, and the recovered stress can no longer be transferred.

[0056] Prestress loss is automatically reflected in the model in incremental form, without the need for separately defined loss coefficients. The main sources of loss include: interface slip loss, anchorage deformation loss, concrete compression deformation loss, and temperature recovery loss. Interface slip loss occurs when a large relative slip occurs between the end of the Fe-SMA strip and the concrete surface, indicating that the interface shear stress has entered the softening stage, and the integral value of the strip's axial force along its length is less than the product of the theoretical recovery stress and the cross-sectional area. Anchorage deformation loss is reflected in models with mechanical anchorage through the elongation of the end spring elements. Concrete compression deformation loss occurs when the compression zone of the concrete beam undergoes compression deformation under prestress, causing the beam to arch. This deformation alters the actual strain state of the Fe-SMA strip, correspondingly reducing its axial tensile stress. Temperature recovery loss only occurs when using the temperature field coupling method, manifesting as stress release caused by the thermal expansion and contraction of the Fe-SMA during the cooling phase. All of the above losses are automatically included through the displacement-strain-stress recursive relationship in the finite element model, ultimately calculating the effective prestress of the Fe-SMA. This represents the average value of the current axial stress at each integration point of the element. Analysts can extract the stress distribution curve along the strip length direction in post-processing and compare it with the theoretical recovered stress. Comparison to obtain transmission efficiency The quantitative results.

[0057] In this embodiment, the transfer and loss calculation of restoring stress are automatically completed through mechanical equilibrium calculation and constitutive update of interface elements, eliminating the need for manually specifying the bond efficiency coefficient or empirical reduction rate. This avoids the shortcomings of empirical methods in reflecting the differences in transfer under different reinforcement lengths and damage degrees. The interface shear stress, damage variables, and slip at each point along the strip length are output in the form of distribution cloud maps, allowing analysts to intuitively determine whether debonding has occurred at the ends or whether the middle has entered softening due to shear stress concentration. The components of prestress loss do not need to be estimated separately, but are naturally formed through displacement coordination and stress balance within the model. For example, interface slip loss and concrete compression loss are calculated synchronously and coupled in the same increment step, which is more consistent with actual stress characteristics. By extracting the Fe-SMA stress distribution curve, the attenuation law of effective prestress along the length direction can also be identified, providing a calculation basis for determining the end anchorage length and whether mechanical anchorage is needed.

[0058] S700, based on the inherited damaged state, thermally activated state and interface state variables, applies external loads to the reinforced reinforced concrete beam to obtain the load-displacement response, material stress, interface slip and damage evolution results of the reinforced beam. Furthermore, the recovery stress generated by the thermal activation of the Fe-SMA strips is transferred to the reinforced concrete beam through the interface elements, and the axial force variation along the length of the Fe-SMA strips satisfies: ; in, For Fe-SMA strips at position Axial force at the point, The width of the Fe-SMA strip. For position The interfacial bonding stress at the point; the effective prestress after thermal activation is expressed as: ; For effective prestressing, To theoretically restore stress, This refers to the prestress loss caused by interface slip. This refers to the prestress loss caused by anchorage deformation. This refers to the loss of prestress caused by concrete deformation. This refers to the loss of prestress caused by temperature recovery and material relaxation.

[0059] In this embodiment, after the Fe-SMA thermal activation and cooling, and the establishment of effective prestress are completed in step S600, the model simultaneously contains three state fields: first, the concrete damage distribution, plastic strain, and residual stress of the steel reinforcement inherited from the pre-damage stage; second, the Fe-SMA effective prestress, beam camber, or deflection recovery amount formed during the thermal activation stage; and third, the interface shear stress distribution, slip, and interface damage variables generated during the interface transfer process. All of these state variables are stored at the integration points and nodes of the current model as initial conditions for reloading.

[0060] The reload analysis step is built upon the above state without any initialization or zeroing operations. A vertical load is applied at the loading pad location; the loading method can be force-controlled or displacement-controlled depending on the target. If the full response of the structure from the start of loading to complete failure (including the descent phase) is required, displacement control is used, with the loading rate referring to the pre-damage stage setting, gradually increasing the displacement of the loading point in increments of 0.5–2 mm per analysis step. If only the response under yield load or normal service conditions is required, force control can be used, gradually loading according to the set load increments. Loading continues until the calculation terminates. The conditions for calculation termination include: the load-displacement curve shows a significant peak followed by a sharp decline, and the residual bearing capacity is less than 20% of the peak load; the compressive damage variable in the concrete compression zone reaches a preset threshold (usually 0.85–0.95); or the interface damage variable reaches 1.0 over a large area along the strip length, causing the interface to be unable to continue transmitting shear stress, and the model's calculation is interrupted due to a singular stiffness matrix.

[0061] The core outputs during the loading process include: the relationship curve between the reaction force and displacement at the loading point (i.e., the load-mid-span displacement curve), which can be further refined into the yield load, peak load, and ultimate displacement of the reinforced beam; the curve showing the change in longitudinal steel reinforcement stress with increasing load, used to determine whether the steel reinforcement has reached yield; the curve showing the change in Fe-SMA stress with increasing load, used to determine whether Fe-SMA has reached yield or fracture; the distribution cloud map of concrete tensile damage variables and compressive damage variables, used to identify the crack propagation area and the crushing range of the compression zone; the distribution of interfacial shear stress along the strip length and its change history with increasing load; the distribution of interfacial slip along the strip length, with a focus on whether the end slip increases sharply; and the distribution of interfacial damage variables, used to identify the location and extent of the debonding area.

[0062] The supplementary formula in step S700 further clarifies the quantitative expression of restoring stress transfer and prestress loss. Axial force transfer differential equation. The same applies during the reloading phase, but in this phase, the stress changes in the Fe-SMA strips originate from two aspects: first, the beam deflection caused by external loads further strains the strips; second, the development of interface slip may lead to stress relaxation in some areas. This differential equation is automatically solved in each incremental step through iterative force balance between the interface elements and adjacent Fe-SMA elements, requiring no manual intervention from the analyst. Effective prestress expression. The various losses in the model are not input as independent parameters, but rather naturally formed during the thermal activation and cooling stages through displacement coordination and stress redistribution. Specifically, The decrease in axial stress caused by the slip at the Fe-SMA end and the degradation of the interface element stiffness can be measured by the difference between the stress of the strip end element and the mid-span element. The calculation is only performed in models with mechanical anchoring by the elongation of the end springs. Estimate by multiplying the longitudinal strain in the compression zone of the concrete beam by the elastic modulus of the concrete; In the temperature field coupling method, this is represented by the difference in stress between the Fe-SMA elements before and after cooling. When extracting the results, analysts can directly read the current stress at the integration point of the Fe-SMA mid-span section element through the post-processing function, using it as the measured value of the effective prestress. By comparing this value with the theoretical recovered stress, the comprehensive effect of various losses can be obtained.

[0063] In this embodiment, the model inherits not only geometric displacements during reloading, but also the complete material state history and interface damage history. This ensures that the initial state after reinforcement is not a "zero-start" undamaged beam, but a real structural state that has undergone pre-damage and prestressing, thus guaranteeing the continuity of the loading path. The automatic solution of the axial force transfer differential equation allows the stress distribution along the length of the Fe-SMA strip to be dynamically updated as the external load increases, avoiding deviations caused by using average stress or simplified reduction factors. Especially after local debonding at the interface, the stress redistribution process can be accurately captured. All loss components in the effective prestress expression are naturally formed in the model rather than manually estimated. The determination of the effective prestress value depends on the direct integration of the current stress state and is not limited by the applicability of empirical formulas. The load-displacement curve, combined with the results of material stress, interface slip, and damage evolution, allows analysts to simultaneously obtain four types of information—bearing capacity, ductility, interface performance, and failure mode—within the same model, providing a complete data foundation for the comprehensive evaluation of reinforcement schemes.

[0064] S800, based on the load-displacement response, material stress, interface slip, and damage evolution results, determine the yield capacity, ultimate bearing capacity, and failure mode of the reinforced beam.

[0065] Furthermore, displacement, stress, strain, plastic strain, concrete damage variables, residual stress of steel reinforcement, Fe-SMA stress, interface slip, and interface damage variables are inherited between each analysis stage. When the number of analysis step iterations exceeds the preset maximum value, the automatic bisection method is activated to reduce the current increment step and recalculate. In the initial increment step of activating the Fe-SMA strip, the recovery stress is slowly applied by controlling the step amplitude. The finite element analysis includes the following stages in sequence: self-weight balance stage, pre-damage loading stage, pre-damage unloading stage, Fe-SMA and interface element activation stage, heating activation stage, isothermal holding stage, cooling stage, and post-reinforcement loading stage.

[0066] In this embodiment, the yield strength is determined based on the load-displacement curve and the stress in the reinforcing steel. When the stress in the longitudinal tensile steel first reaches its yield strength, the corresponding load is defined as the yield load, at which point a slope inflection point usually appears on the load-displacement curve. If the curve inflection point is unclear, the "secant stiffness method" can be used to assist in the determination, that is, the load at which the initial tangent stiffness is significantly reduced (e.g., reduced to 70% of the initial stiffness) is taken as the yield load. The ultimate bearing capacity is taken as the load value corresponding to the peak point on the load-displacement curve, which usually appears after the steel yields and before the concrete crushes or the interface debonds. If the curve shows multiple peaks due to interface debonding, the maximum peak load is taken as the ultimate bearing capacity.

[0067] Determining the damage mode requires a comprehensive analysis of multiple calculation results. The criteria for determining each damage mode are as follows: Concrete crushing failure. The load-displacement curve reaches its peak and then drops sharply, and the compressive damage variable d of the concrete in the compression zone in the post-processing contour plot... c When the compressive strain reaches 0.85 or higher, or when the compressive strain of the concrete at the edge of the compression zone exceeds its ultimate compressive strain (usually taken as 0.0033 to 0.0035), it is considered that the concrete in the compression zone is crushed. This failure mode is characterized by a sudden loss of bearing capacity and poor ductility.

[0068] After the tensile reinforcement yields, the concrete crushes (appropriate reinforcement failure). When the longitudinal reinforcement stress first reaches its yield strength, the load-displacement curve continues to rise to its peak value, and then... c The failure is characterized by gradually increasing the load to a threshold, while the tensile damage area in the concrete expands from the mid-span towards both ends. This is the ideal failure mode for reinforced beams, where both load-bearing capacity and ductility are well maintained.

[0069] Fe-SMA strips yield or fracture. When the Mises stress of an Fe-SMA element reaches its yield strength or ultimate strength, it is considered that the Fe-SMA has yielded or fractured. This type of failure often occurs when the interfacial bonding performance is good and the amount of Fe-SMA used is small. It is characterized by a clear plateau or descending segment in the load-displacement curve after the Fe-SMA yields.

[0070] Interface end peeling failure. When the damage variable D of the interface unit near the end of the Fe-SMA strip reaches 1.0, and the end slip s end A sharp increase exceeding the limit slip s u When the failure occurs, it is determined to be end peeling failure. The typical characteristics of this type of failure are: before the bearing capacity reaches the expected value, the end interface debonds first, the Fe-SMA stress rapidly decays to zero along the length direction, and the load-displacement curve suddenly drops before the peak value.

[0071] Mid-section debonding failure. When the D-value of an interface element in the middle of the strip reaches 1.0, and the slip at that point exceeds the limit slip, but the end interface has not completely failed, it is considered mid-section debonding failure. This type of failure is mostly caused by the uneven distribution of the peak shear stress along the length of the interface, and its impact on the bearing capacity is between that of end debonding and complete bonding.

[0072] Anchorage zone failure. For models with mechanical anchorage, when the displacement of the end spring element exceeds the design allowable value, or the spring resistance drops sharply after reaching its ultimate resistance, and local crushing damage occurs in the concrete of the anchorage zone (d... c When the value exceeds 0.8, the anchorage area is considered to have failed.

[0073] Delamination and damage to the concrete cover. Large-area tensile damage (d) occurred in the concrete cover outside the area where the Fe-SMA strip was applied. t When the damage exceeds 0.9 and the damaged area extends along the bottom surface of the beam to the edge of the strip, it is determined to be protective layer peeling.

[0074] Shear failure. When a significant oblique tensile damage zone appears in the web region between the beam end support and the loading point, and the stirrup stress reaches the yield strength, and the load-displacement curve suddenly drops without steel yielding or concrete crushing, it is determined to be shear failure.

[0075] The sequence control and solution stability assurance measures for this step are as follows: The entire finite element analysis is performed in the following stages: First, the beam's self-weight is applied to establish equilibrium within the model. Second, a pre-damage load is applied to the unreinforced beam, using force or displacement control to reach the target state. Third, the pre-damage load is unloaded, with displacement control used throughout to reduce the displacement at the loading points to zero or a preset residual value. After unloading, all variables are frozen using a state file. Fourth, Fe-SMA strip elements and interface elements are activated, with strip node coordinates inheriting the current beam bottom deformation position, achieved through element birth / death or model modification techniques. Fifth, the Fe-SMA strip elements... A heating effect is applied to generate recovery stress through one of three methods: equivalent initial strain, initial stress, or temperature field coupling. The equivalent initial strain is applied slowly with a linear ramp amplitude in the first 10% to 20% of the analysis steps. In the sixth analysis stage, the temperature is maintained at a preset time (e.g., 30 minutes) to keep the Fe-SMA at the activation temperature. During this stage, the stress is redistributed with material relaxation and interface microslip. In the seventh analysis stage, the temperature is cooled to ambient temperature, retaining the effective prestress and interface stress state after cooling. In the eighth analysis stage, an external load is applied to the reinforced beam until the termination condition is met (the bearing capacity drops below 20% of the peak value or the calculation is terminated due to stiffness singularity).

[0076] State variable inheritance between adjacent stages is achieved through restarting or continuous calculation of the analysis step. The inherited variables include: nodal displacement vectors, element integration point stress tensors, strain tensors, equivalent plastic strain (PEEQ), and concrete tensile damage variable d. t and compressive damage variable d c The parameters include residual stress in the reinforcing steel, Fe-SMA stress tensor, interface slip, interface damage variable D, and contact state. In finite element software, as long as the "initial state reset" or "result zeroing" operation is not performed between analysis steps, the software will automatically complete the above inheritance by default using the state at the end of the previous analysis step as the starting state of the next analysis step.

[0077] To ensure the smooth progress of the multi-stage analysis, reasonable solution control parameters need to be set. The maximum number of equilibrium iterations is set to 20-25. If the number of iterations in a certain increment exceeds this value and convergence is still not achieved, the software automatically uses the cut-back method to reduce the load increment or displacement increment of the current increment to 0.25-0.5 times the original value, and re-iterates until convergence or the step size is reduced to the preset lower limit, at which point the analysis terminates. In the initial increment of activating the Fe-SMA strip, the equivalent initial strain or initial stress is gradually increased from zero to the target value through a smooth step amplitude, rather than being applied all at once in the first increment. The typical setting for this amplitude curve is: 5% of the total amplitude is applied in the first sub-step, 15% in the second sub-step, and then gradually accelerated to 100% in the middle of the analysis step, avoiding stress abrupt changes that could cause unrealistic failure of the interface elements at the moment of activation. The pre-damage and reloading phases use an automatic incremental step size, with the initial incremental step size set to 1% to 2% of the estimated total displacement, and subsequently adjusted automatically by the software based on the convergence difficulty.

[0078] In this embodiment, the determination of yield bearing capacity and ultimate bearing capacity is based on the synchronous analysis of load-displacement curves and material stress, avoiding the problem of misjudging the yield point solely based on changes in curve slope. The determination of seven failure modes corresponds to different physical criteria and contour map features, enabling the model to not only output bearing capacity values ​​but also provide clear failure types, providing a basis for identifying weak points in reinforcement schemes. The multi-stage analysis process is set according to the actual construction and loading sequence. At the end of each stage, the complete state is retained as the initial condition for the next stage, allowing pre-damage, reinforcement, thermal activation, and bearing capacity testing to be completed continuously within the same model, avoiding data loss and human error caused by step-by-step modeling and manual mapping of state variables in traditional methods. The maximum iteration count control, combined with the automatic bisection method, prevents the model from immediately terminating when convergence is difficult, instead attempting to continue calculation by reducing the step size, improving the fault tolerance of complex nonlinear analysis. Smoothing amplitude control allows for the slow application of Fe-SMA recovery stress, providing sufficient incremental steps for interface elements to adapt to stress growth, avoiding misjudgment of debonding when the interface shear stress exceeds the peak value at the moment of activation, and also making the calculation in the thermal activation stage easier to converge.

[0079] Furthermore, although the steps of the method in this disclosure are described in a specific order in the accompanying drawings, this does not require or imply that the steps must be performed in that specific order, or that all the steps shown must be performed to achieve the desired result. Additional or alternative steps may be omitted, multiple steps may be combined into one step, and / or a step may be broken down into multiple steps.

[0080] Embodiments of the present invention also provide a non-transitory computer-readable storage medium that can be disposed in an electronic device to store at least one instruction or at least one program related to implementing a method in the method embodiments, wherein the at least one instruction or the at least one program is loaded and executed by the processor to implement the method provided in the above embodiments.

[0081] The program product may employ any combination of one or more readable media. A readable medium may be a readable signal medium or a readable storage medium. A readable storage medium may be, for example, but not limited to, an electrical, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, or device, or any combination thereof. More specific examples of readable storage media (a non-exhaustive list) include: an electrical connection having one or more wires, a portable disk, a hard disk, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), optical fiber, portable compact disk read-only memory (CD-ROM), optical storage devices, magnetic storage devices, or any suitable combination thereof.

[0082] Computer-readable signal media may include data signals propagated in baseband or as part of a carrier wave, carrying readable program code. Such propagated data signals may take various forms, including but not limited to electromagnetic signals, optical signals, or any suitable combination thereof. A readable signal medium may also be any readable medium other than a readable storage medium, capable of sending, propagating, or transmitting programs for use by or in conjunction with an instruction execution system, apparatus, or device.

[0083] The program code contained on the readable medium may be transmitted using any suitable medium, including but not limited to wireless, wired, optical fiber, RF, etc., or any suitable combination thereof.

[0084] Program code for performing the operations of this application can be written in any combination of one or more programming languages, including object-oriented programming languages ​​such as Java and C++, and conventional procedural programming languages ​​such as C or similar languages. The program code can execute entirely on the user's computing device, partially on the user's device, as a standalone software package, partially on the user's computing device and partially on a remote computing device, or entirely on a remote computing device or server. In cases involving remote computing devices, the remote computing device can be connected to the user's computing device via any type of network, including a local area network (LAN) or a wide area network (WAN), or it can be connected to an external computing device (e.g., via the Internet using an Internet service provider).

[0085] Embodiments of the present invention also provide an electronic device, including a processor and the aforementioned non-transitory computer-readable storage medium.

[0086] The electronic device is merely an example and should not impose any limitations on the functionality and scope of use of the embodiments in this application.

[0087] Electronic devices are manifested in the form of general-purpose computing devices. Components of an electronic device may include, but are not limited to: at least one processor, at least one memory, and a bus connecting different system components (including memory and processor).

[0088] The memory stores program code that can be executed by the processor, causing the processor to perform the steps in the various embodiments described in this specification.

[0089] The memory may include readable media in the form of volatile memory, such as random access memory (RAM) and / or cache memory, and may further include read-only memory (ROM).

[0090] The memory may also include programs / utilities having a set (at least one) of program modules, including but not limited to: an operating system, one or more application programs, other program modules, and program data, each or some combination of these examples may include an implementation of a network environment.

[0091] A bus can represent one or more of several types of bus structures, including a memory bus or memory controller, a peripheral bus, a graphics acceleration port, a processor, or a local bus that uses any of the various bus structures.

[0092] Electronic devices can also communicate with one or more external devices (e.g., keyboards, pointing devices, Bluetooth devices, etc.), one or more devices that enable user interaction with the electronic device, and / or any device that enables the electronic device to communicate with one or more other computing devices (e.g., routers, modems, etc.). This communication can be achieved through input / output (I / O) interfaces. Furthermore, electronic devices can communicate with one or more networks (e.g., local area networks (LANs), wide area networks (WANs), and / or public networks, such as the Internet) via network adapters. The network adapter communicates with other modules of the electronic device via a bus. It should be understood that other hardware and / or software modules can be used in conjunction with the electronic device, including but not limited to: microcode, device drivers, redundant processors, external disk drive arrays, RAID systems, tape drives, and data backup storage systems.

[0093] From the above description of the embodiments, those skilled in the art will readily understand that the exemplary embodiments described herein can be implemented by software or by combining software with necessary hardware. Therefore, the technical solutions according to the embodiments of this disclosure can be embodied in the form of a software product, which can be stored in a non-volatile storage medium (such as a CD-ROM, USB flash drive, external hard drive, etc.) or on a network, including several instructions to cause a computing device (such as a personal computer, server, terminal device, or network device, etc.) to execute the methods according to the embodiments of this disclosure.

[0094] Embodiments of the present invention also provide a computer program product including program code, which, when the program product is run on an electronic device, causes the electronic device to perform the steps of the methods described above in various exemplary embodiments of the present invention.

[0095] While specific embodiments of the invention have been described in detail by way of examples, those skilled in the art should understand that the examples are for illustrative purposes only and are not intended to limit the scope of the invention. Those skilled in the art should also understand that various modifications can be made to the embodiments without departing from the scope and spirit of the invention.

Claims

1. A finite element modeling method for Fe-SMA-reinforced damaged concrete beams, characterized in that, include: Obtain the geometric and material parameters of the reinforced concrete beam, reinforcing bars, Fe-SMA strips, and bond interface, and establish a finite element model of the unreinforced reinforced concrete beam. A pre-damage load is applied to the finite element model of the unreinforced concrete beam, and then unloaded after the target damage state is reached to obtain concrete damage variables, plastic strain, residual stress of steel reinforcement, and residual deformation of the beam; displacement control is used in the unloading stage. The concrete damage variables, plastic strain, residual stress of steel bars, and residual deformation of the beam are taken as the initial state of the damaged reinforced concrete beam, and Fe-SMA strip elements and interface elements are activated while maintaining the initial state. The interface stiffness, damage initiation parameters, and damage evolution parameters of the interface unit are set according to the bond-slip relationship between Fe-SMA strips and concrete. The Fe-SMA recovery stress is determined based on the Fe-SMA activation temperature. At least one of the equivalent initial strain method, initial stress method, or temperature field coupling method is used to apply a thermal activation effect to the Fe-SMA strip. The equivalent initial strain or initial stress is applied based on the current deformation configuration of the Fe-SMA strip when it is activated, and geometric nonlinearity is enabled throughout the process. The transfer of Fe-SMA restored stress to the damaged reinforced concrete beam and the prestress loss caused by interface slip are calculated using the interface unit. Based on the inherited damaged state, thermally activated state and interface state variables, an external load is applied to the reinforced concrete beam to obtain the load-displacement response, material stress, interface slip and damage evolution results of the reinforced beam. The yield capacity, ultimate bearing capacity, and failure mode of the reinforced beam are determined based on the load-displacement response, material stress, interface slip, and damage evolution results.

2. The method according to claim 1, characterized in that, The target damage state is determined based on at least one of the following: the ratio of the pre-damage load to the ultimate load of the unreinforced beam, mid-span deflection, crack width, steel reinforcement strain, or concrete damage variable; the pre-damage load is expressed as: ； in, For pre-damage load, This refers to the ultimate load of the unreinforced beam. This represents the pre-damage coefficient.

3. The method according to claim 1, characterized in that, The concrete damage variables include concrete tensile damage variables and compressive damage variables, and the plastic strain includes concrete plastic strain and steel plastic strain; the state variables retained after pre-damage unloading also include the unit damage state corresponding to the crack area and the contact state between the support and the loading area.

4. The method according to claim 1, characterized in that, The activation temperature of the Fe-SMA strip Below, temperature reduction factors are introduced for the elastic modulus, compressive strength, tensile strength of concrete, and yield strength of steel reinforcement. ;when When each reduction factor is taken as 1.0, The reduction factor mentioned above is determined based on the material performance degradation law at high temperatures.

5. The method according to claim 1, characterized in that, In the aforementioned interfacial bonding-slip relationship, the interfacial tangential stress and the amount of slip satisfy the following: ； in, For interfacial tangential bonding stress, This is the relative slip. Peak bond stress, This represents the slip amount corresponding to the peak bond stress. For the interface curve shape parameters, This refers to the interfacial fracture energy. , and The parameters are determined based on the elastic modulus of the adhesive, the thickness of the adhesive layer, the splitting tensile strength of the concrete, the activation temperature, and the number of thermal cycles.

6. The method according to claim 1, characterized in that, The equivalent initial strain method is as follows: stress is recovered based on the Fe-SMA theory under fully constrained conditions corresponding to the target activation temperature. Determine the equivalent initial strain ,in, The equivalent initial strain is the Fe-SMA elastic modulus; when the equivalent initial strain is applied, the initial geometric coordinates of the Fe-SMA strip are inherited from the current deformation coordinates of the beam bottom node after pre-damage unloading, and the equivalent initial strain increases slowly from zero to the target value in a linear ramp amplitude manner in the early stage of the activated analysis step.

7. The method according to claim 1, characterized in that, The recovery stress generated by the thermal activation of Fe-SMA strips is transferred to the reinforced concrete beam through the interface elements. The axial force variation along the length of the Fe-SMA strips satisfies: ； in, For Fe-SMA strips at position Axial force at the point, The width of the Fe-SMA strip. For position The interfacial bonding stress at the point; the effective prestress after thermal activation is expressed as: ； For effective prestressing, To theoretically restore stress, This refers to the prestress loss caused by interface slip. This refers to the prestress loss caused by anchorage deformation. This refers to the loss of prestress caused by concrete deformation. This refers to the loss of prestress caused by temperature recovery and material relaxation.

8. The method according to claim 1, characterized in that, During the application of pre-damage loads and unloading: force control or displacement control is used in the loading stage, and displacement control is used in the unloading stage. The displacement of the loading point is reduced from the peak displacement to zero or the preset residual displacement value. During unloading, the concrete is unloaded linearly according to the degraded elastic stiffness corresponding to the unloading time, and the steel bars are unloaded according to the initial elastic modulus. After unloading is completed, the state variables of all current integration points are frozen and transferred to the subsequent analysis stage through the state file or restart function.

9. The method according to claim 1, characterized in that, When a mechanical anchoring device is installed at the end of the Fe-SMA strip, the Fe-SMA strip is divided into a pure bonding zone and an anchoring zone along its length. A nonlinear spring unit or connector unit is added to the anchoring zone to simulate the pull-out force provided by the mechanical anchoring. The boundary conditions at the anchoring end are set as follows: ； in, For mechanical anchoring resistance, This represents the end slippage; the total end resistance is the sum of the adhesive layer bonding force and the mechanical anchoring force.

10. The method according to claim 1, characterized in that, Displacement, stress, strain, plastic strain, concrete damage variables, steel residual stress, Fe-SMA stress, interface slip, and interface damage variables are inherited between each analysis stage; when the number of analysis step iterations exceeds the preset maximum value, the automatic bisection method is activated to reduce the current increment step and recalculate; In the initial incremental step of activating the Fe-SMA strip, the recovery stress is slowly applied by controlling the smooth step amplitude. The finite element analysis includes the following steps in sequence: self-weight balance stage, pre-damage loading stage, pre-damage unloading stage, Fe-SMA and interface element activation stage, heating activation stage, isothermal holding stage, cooling stage, and post-reinforcement loading stage.

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