A design method for strengthening corrosion-damaged concrete structures based on load path

Abstract

This invention discloses a load path-based design method for strengthening corroded concrete structures, comprising: determining a topology optimization formula for the load path search problem of the damaged structure; establishing a numerical model of the corroded structure considering the effects of material performance loss and bond degradation under corrosion damage; obtaining the elastic and plastic strain energy of each element through finite element analysis, calculating the element sensitivity based on the sensitivity formula of the given corrosion damage strain energy form, updating the element sensitivity using a fuzzy filtering scheme, and clarifying the element design variables and element types; updating the stiffness of the minimum density element based on the proposed automatic stiffness adjustment method of the minimum density element; clarifying the convergence error in the load path search process by combining convergence criteria, generating the optimal load path of the damaged structure, and guiding the structural strengthening design. This invention can overcome the numerical instability problem caused by the distortion of the minimum density element, can reasonably generate the optimal load path of the damaged structure, and can guide the layout of strengthening materials.

Description

Technical Field

[0001] This invention relates to the field of nonlinear topology optimization technology for concrete structures, and in particular to a reinforcement design method for corrosion-damaged concrete structures based on load paths. Background Technology

[0002] Concrete bridges constitute a significant proportion of existing bridges in my country. When subjected to corrosion damage, concrete structures are prone to defects such as reduced reinforcement cross-sectional area, rust expansion and cracking of the protective layer, and bond degradation between concrete and reinforcement. This leads to disordered stress distribution within the concrete structure, no longer conforming to the plane section assumption; this area is known as the stress disturbance zone (D-zone) of the concrete structure. The D-zone of a concrete structure is a critical load-bearing component, and its crack resistance and load-bearing capacity affect the durability and safety of the entire structure. Currently, the force transmission mechanism of the D-zone of corroded concrete structures under complex boundary conditions remains difficult to accurately characterize. The tension-compression bar model is considered the most effective method for designing the D-zone of concrete structures. When using the tension-compression bar model to design the D-zone, the most critical issue is determining the configuration of the model. Currently, the stress tracing method, load path method, and topology optimization method have been successfully applied to the design of tension-compression bar models for the D-zone of concrete structures. However, the tension-compression bar models obtained based on the stress tracing method and load path method are not unique; they often rely on the designer's intuition and experience, exhibiting significant subjectivity.

[0003] To overcome the subjectivity of stress tracing and load path methods, some scholars have conducted preliminary research on tension-compression bar models of concrete structures using topology optimization. Existing methods focus on searching load paths for sound, elastic concrete structures to determine the corresponding tension-compression bar models. However, searching load paths for corrosion-damaged concrete structures involves material deterioration and bond degradation, and is a nonlinear topology optimization problem, making it more complex than for sound, elastic concrete structures. First, the conversion of material density during the load path search process for corrosion-damaged concrete structures causes distortion of the minimum density element, leading to numerical instability in the load path search process. Then, the material deterioration, bond degradation, and concrete cracking caused by corrosion, along with their coupling effects, significantly increase the complexity of the load path search process. Currently, no research has been reported on load path searching for corrosion-damaged concrete structures. The load path of corrosion-damaged concrete structures is of great significance for describing the force transmission mechanism in its D-zone, evaluating structural performance, and guiding maintenance and reinforcement decisions.

[0004] Existing methods for strengthening concrete structures include bonding steel plates, increasing cross-sections, external prestressing, fiber composite reinforcement, new material reinforcement, and combined reinforcement. However, most of these methods involve full-section reinforcement, resulting in issues such as high self-weight, high cost, and long development cycles. Ideal reinforcement design should follow the structural load transfer path, fully utilize the performance of reinforcement materials, and improve their utilization efficiency. The degree of structural damage and the type of external loads can alter the structural load path and the trajectory of reinforcement material placement. Further research is needed on how to place reinforcement materials for corrosion-damaged concrete structures based on load paths. Therefore, a load path-based reinforcement design method for corrosion-damaged concrete structures is urgently needed. This method should not only rationally generate load paths for corrosion-damaged concrete structures but also effectively overcome numerical instability caused by the distortion of minimum density elements, and guide the placement of reinforcement materials based on the load paths of the damaged concrete structure. Summary of the Invention

[0005] To address the aforementioned technical problems, this invention proposes a load path-based reinforcement design method for corroded and damaged concrete structures. This method can reasonably generate load paths for corroded and damaged concrete structures, effectively overcome numerical instability caused by the distortion of minimum density elements, and guide the placement of reinforcement materials based on the load paths of damaged reinforced concrete (RC) structures.

[0006] To achieve the above objectives, this invention provides a method for strengthening corrosion-damaged concrete (RC) structures based on load paths, the main steps of which include:

[0007] S1: Determine the nonlinear topology optimization formula for the load path search problem of RC structure with corrosion damage;

[0008] S2: Calculate the properties of the corroded material, bond strength, and spring stiffness;

[0009] S3: Establish a numerical model of RC structure with corrosion damage considering material performance loss and adhesion degradation;

[0010] S4: Nonlinear topology optimization analysis of the RC structure with corrosion damage: First, perform finite element analysis on the numerical model of the RC structure with corrosion damage established in step S3; then, obtain the elastic and plastic strain energy of each element, and calculate the sensitivity of each element based on the sensitivity formula of the strain energy form of the corrosion damage element; finally, use a fuzzy filtering scheme to update the element sensitivity information, determine the target volume of the next optimization step, and update the design variables and element types.

[0011] S5: Adjust the stiffness of the minimum density element based on the proposed automatic adjustment method for minimum density element stiffness;

[0012] S6: Calculate the convergence error during the load path search process of the RC structure with corrosion damage;

[0013] S7: Generate the load path for the RC structure with corrosion damage and verify the rationality of the load path;

[0014] S8: Based on the load path of RC structure with corrosion damage, guide the layout of reinforcement materials.

[0015] Furthermore, in step S1, the nonlinear topology optimization formula for the load path search problem of the damaged structure is as follows:

[0016] The optimization objective of the load path search problem for RC structures with corrosion damage is to find the load path within the volume constraint V. * The load path that generates the minimum structural flexibility (i.e., maximum stiffness) is then used, with the concrete element material density as the design variable. The mathematical expression for the nonlinear optimization problem of the RC structure with corrosion damage is as follows:

[0017]

[0018]

[0019] in, The objective function is the structural compliance. Design a vector of variables for the unit; Let be the material density of the i-th element, which can be taken as 1 or in the Two-Way Progressive Structure Optimization (BESO) method. f ext U is the external load vector; η v is the structural displacement vector; i and V * These represent the volume of the i-th element and the volume fraction of the specified material, respectively. n is the number of concrete elements in the design domain. The minimum density of the unit cell; This represents the residual force vector of the RC structure subjected to nonlinear corrosion damage.

[0020] Furthermore, in step S2, the calculation methods for the properties of the corroded material, the bond strength, and the spring stiffness are as follows:

[0021] 1) The properties of corroded materials include the calculation of the compressive strength of corroded concrete and the ultimate tensile strength of corroded steel bars, and the compressive strength f′ of corroded concrete. cm It can be represented as:

[0022]

[0023] Among them, f cm k represents the compressive strength of uncorroded concrete. r ε is an empirical coefficient; cu ε1 represents the maximum compressive strain; ε2 represents the average tensile strain of the cracked concrete, which can be expressed as:

[0024] ε1=n bs ·w cr / b w (4)

[0025] Among them, b w n represents the initial width of the uncorroded RC beam; bs The quantity of reinforcing steel in the compression zone of the RC beam; w cr The total crack width of the RC beam damaged by corrosion can be calculated as follows:

[0026] w cr =2πX(u) rs -1) (5)

[0027] Among them, u rs X represents the volumetric expansion rate; X represents the depth of steel corrosion, which can be expressed as:

[0028]

[0029] Where η and r s These are the steel reinforcement corrosion rate and the steel reinforcement radius, respectively.

[0030] 2) The ultimate tensile strength of corroded steel bars can be expressed as:

[0031] f y，c =(1-α) y η)f y (7)

[0032] f u，c =(1-α) u η)f u (8)

[0033] ε u，c =(1-α1η)ε u (9)

[0034] Among them, f y，c f u，c With ε u，c These represent the yield strength, ultimate tensile strength, and ultimate strain of the corroded steel reinforcement, respectively; f y f u With ε u These represent the yield strength, ultimate strength, and ultimate strain of the uncorroded steel reinforcement, respectively; α y α u α1 is the correlation coefficient;

[0035] 3) Bond strength τ between corroded steel bars and concrete a，η It can be represented as:

[0036]

[0037] Among them, C c D represents the thickness of the concrete protective layer. l The diameter of the longitudinal reinforcement; A s，η λ is the cross-sectional area of ​​the corroded stirrup; ζ, k are empirical coefficients; s is the stirrup spacing; f ct The tensile strength of concrete, f cm f is the compressive strength of concrete; sy，η The yield strength of the corroded stirrup can be expressed as:

[0038]

[0039] Where, η s f represents the average corrosion rate of the stirrups. sy Let be the yield strength of the uncorroded stirrup. The stiffness of the spring element can be expressed as:

[0040] k sp =π·D·l r ·τ a，η / S1 (12)

[0041] Among them, l r S1 is the length of the concrete unit; S1 is the maximum slip value corresponding to the maximum bond stress of the steel reinforcement.

[0042] Furthermore, in step S4, the derivation process of the sensitivity formula for the strain energy form of the corrosion damage element is as follows:

[0043] Unit sensitivity is the objective function For design variables The partial derivative of is expressed as follows:

[0044]

[0045] Among them, K t，η This is the tangent stiffness matrix under structural equilibrium conditions; These are the internal forces under structural equilibrium. In formula (13)... In the nonlinear finite element analysis of RC structures with corrosion damage, the term can be approximately expressed as:

[0046]

[0047] Therefore, the end compliance of a corrosion-damaged RC structure can be approximately expressed as:

[0048]

[0049] When a rust-damaged RC structure is in equilibrium, the internal force f int Equal to external load f extThis item This can be viewed as the work done by the external load (i.e., strain energy). Sensitivity formula It can be approximately rewritten as:

[0050]

[0051] Among them, w i，η The strain energy of element i in each iteration step. The corrosion-damaged RC structure is composed of concrete, a weakly elastic material, and steel reinforcement. Internal loads of the structure. It can be written as:

[0052]

[0053] Among them, f co，η f we and f cr，η These are the internal loads of the concrete element, the weakly elastic material element, and the reinforced steel element, respectively. The internal load of the reinforced steel element is independent of the material's virtual density. Therefore, in equation (17), this term... Equals zero. Sensitivity formula for RC structure. It can be rewritten as:

[0054]

[0055] Among them, w co，η and w we These represent the strain energies of concrete and weakly elastic material elements, respectively.

[0056] In nonlinear topology optimization, the strain energy of a corrosion-damaged RC structure in equilibrium is easier to manipulate, and the strain energy of each element can be directly extracted through nonlinear finite element analysis. The strain energy w of element i in the design domain... i，η for:

[0057]

[0058] The formula for the sensitivity of RC structures to corrosion damage can be expressed as:

[0059]

[0060] Where, λ o It is a very small constant, which can be set as λ. o =10 -10 λ is added to the denominator of the second term on the right to avoid [the following]. Ambiguity may arise.

[0061] Furthermore, in step S5, the proposed automatic adjustment method for the stiffness of the minimum density element includes the following steps:

[0062] First, extract the maximum strain of the minimum density element in each optimization step. Then, determine the maximum strain. With strain threshold ε * Size; if A multi-proportional stiffness growth strategy is used to increase the stiffness of the minimum density element in the next iteration step to suppress numerical instability caused by the distortion of the minimum density element during the load path search process; if Then reduce the stiffness of the minimum density element in the next iteration step to suppress the optimization error caused by the increase in element stiffness;

[0063] The numerical instability problem caused by the distortion of the minimum density element is as follows:

[0064] In the BESO method, cell deletion is achieved by converting the relative density from 1.0 to... This is achieved through [method / method], rather than complete removal. The stiffness matrix k of each element... i With relative density The relationship is:

[0065]

[0066] Where B is the element strain matrix; k c is the stiffness matrix of the concrete element; P is the penalty factor. Design variables. This indicates that the i-th element is a "hollow" element, that is, a minimum density element, and its stiffness matrix is... Minimum density is usually set to The penalty factor is set to P = 3. Therefore, the stiffness of the minimum density element is artificially reduced by 10. -9 This infinitesimal stiffness causes distortion in the minimum density element, leading to numerical instability in the load path search process.

[0067] The minimum density element stiffness multi-proportional growth strategy is as follows:

[0068] The minimum density element has extremely low stiffness and can be considered a weakly elastic material. Its constitutive relation can be expressed as:

[0069]

[0070] Among them, D e E represents the constitutive matrix of a weakly elastic material. min With v e These are the elastic modulus and Poisson's ratio, respectively. Therefore, the stiffness of the unit can be adjusted by changing the material's elastic modulus.

[0071] In the process of nonlinear topology optimization, the elastic modulus E minA value that is too small can lead to numerical instability, but excessively increasing the elastic modulus E... e This can introduce optimization errors. Therefore, a multi-proportional stiffness growth strategy for the minimum density element is proposed. In each optimization iteration, the element stiffness can be adjusted by different proportions according to the maximum strain increment of the minimum density element. This can overcome the numerical instability problem caused by the distortion of the minimum density element and suppress the optimization error caused by increasing the element stiffness. In each optimization iteration, the elastic modulus of the minimum density element is automatically adjusted by tracking the maximum strain of the minimum density element. In the (k+1)th optimization step, The following criteria can be used for automatic adjustment:

[0072]

[0073] in, and These are the elastic moduli of the minimum density element in the k-th and k+1-th optimization steps, respectively; α1, α2, α3, and α4 are coefficients used to control the rate of stiffness increase in the unit; α4 is the maximum strain of the smallest density element in the k-th optimization step; α1, α2, α3, and α4 are coefficients used to control the rate of stiffness increase in the unit; ε is the threshold value in each optimization step. * Updated to f(x) is the sigmoid function, which can be expressed as:

[0074]

[0075] in, The sigmoid function can be used to gradually adjust the elastic modulus. like Figure 3 As shown.

[0076] In the process of nonlinear topology optimization, if This means that some of the minimum density elements in the structure may be distorted. In the next optimization iteration, the elastic modulus of the minimum density elements should be increased. This is to enhance the stiffness of its elements, thereby preventing distortion of the minimum density elements. If Then it should be reduced in the next optimization iteration step. To suppress optimization errors caused by increasing the stiffness of the minimum density element. During the load path search process, the elastic modulus of the weakly elastic material... Eventually approaches the threshold E max This setup can overcome the numerical instability problem during the load path search process of rust-damaged RC structures.

[0077] Furthermore, in step S4, the total strain energy w of each unit i It can be represented as:

[0078]

[0079] in, and These represent the elastic and plastic strain energies of each unit, respectively.

[0080] Furthermore, in step S4, the method for determining the fuzzy filtering scheme and the target volume for the next optimization step is as follows:

[0081] A fuzzy filtering scheme is used to obtain a mesh-independent topology optimization solution. This filtering scheme requires further processing of the original element sensitivity. The filtering scheme can be expressed as follows:

[0082]

[0083] w(r ij ) = max(0, r min -r ij (27)

[0084] Where, r ij η is the distance between the centers of units i and j; w is the weighting function for the average original sensitivity; η j r is the weighting factor; min The filter radius. The sensitivity of the current iteration step. Sensitivity compared to the previous iteration step Further averaging is performed to obtain a convergent solution. Cell sensitivity. It can be represented as:

[0085]

[0086] Based on the volume V of the current iteration step k Compared with evolutionary ratio ER, the target volume V for the next iteration can be obtained. k+1 .

[0087] V k+1 =V k (1±ER) (29)

[0088] Furthermore, in step S4, the method for updating design variables and unit types is as follows:

[0089] If the sensitivity of the concrete unit Less than the sensitivity threshold Then the element density (i.e., the design variable) Switch from 1 to For weakly elastic body elements, if Unit density from Switch to 1; then adjust according to cell density. The cross-sectional properties of concrete and weakly elastic materials are redistributed.

[0090] Furthermore, in step S4, the convergence criterion for the load path search process of the rust-damaged RC structure is:

[0091]

[0092] Among them, Er k denoted as , where is the convergence error of the RC structure damaged by corrosion; ∈ represents the allowable convergence error; and N is the number of iteration steps.

[0093] Furthermore, before performing topology optimization analysis on the rust-damaged RC structure in step S4, it is necessary to define initial optimization parameters, such as the filter radius r. min Penalty factor P, target volume fraction V * Evolutionary rate ER, initial strain threshold ε * Minimum density of material Elastic modulus threshold E max and initial elastic modulus

[0094] Initialize design variables and concrete section properties: Set the section properties for "solid" and "hollow" materials. "Solid" material is concrete; "hollow" material is an elastic material with a Poisson's ratio of 0.3 and an elastic modulus of [missing value]. Where E c For the elastic modulus of concrete; set all initial design variables to All element cross-sectional properties within the design domain are set to "solid" material cross-sectional properties. Subsequently, the cross-sectional properties will be transformed based on the material density value in each optimization step.

[0095] Furthermore, in step S8, the method for guiding the placement of reinforcement materials based on the load path of the RC structure with corrosion damage is as follows:

[0096] Ideally, the placement of reinforcement materials should follow the structural load transfer path, fully utilize the performance of the reinforcement materials, and improve their utilization efficiency. Based on the load path analysis of the damaged structure, the stress characteristics of each component are considered. For tension members, reinforcement materials with high tensile strength are placed at the corresponding locations; for compression members, reinforcement materials with high compressive strength are required. The placement of reinforcement materials should follow the load path of the corrosion-damaged RC structure.

[0097] Because the present invention adopts the above technical solutions, the beneficial effects of the present invention are as follows:

[0098] This invention provides a load path-based design method for strengthening corroded concrete structures. It considers the impact of material performance loss and steel bond degradation under corrosion damage, proposes an automatic stiffness adjustment method for minimum density elements, and provides a design sensitivity formula for the strain energy form of corroded damaged elements. This method overcomes the numerical instability problem caused by the distortion of minimum density elements and can reasonably generate the optimal load path for the damaged structure, guiding the placement of strengthening materials. The load path generated by this invention for corroded RC structures provides theoretical guidance for describing the force transmission mechanism in zone D of corroded RC structures, evaluating structural performance, and guiding maintenance and strengthening decisions. Attached Figure Description

[0099] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the structures shown in these drawings without creative effort.

[0100] Figure 1 The stress-strain curve of the concrete CDP model in this invention;

[0101] Figure 2 This is a schematic diagram of the bilinear constitutive model of the corroded steel bars in this invention;

[0102] Figure 3 This is a schematic diagram of stiffness adjustment based on the sigmoid function in this invention;

[0103] Figure 4 This is a flowchart of the load path search process for the RC structure suffering from corrosion damage in this invention.

[0104] Figure 5 This is a schematic diagram of the optimized model of the reinforced concrete beam in this invention;

[0105] Figure 6 This is the load path for the RC beam with a corrosion rate η = 5% in this invention;

[0106] Figure 7 The load paths of the RC beam under different load conditions in this invention are: (a) BESO method; (b) F = 5N; (c) F = 50N; (d) F = 100N; (e) F = 150N; (f) F = 200N; (g) F = 250N and (h) F = 300N;

[0107] Figure 8The load paths for RC beams with different corrosion rates under an external load of F = 150 N in this invention are: (a) η = 0%; (b) η = 1.5%; (c) η = 3%; (d) η = 5%; (e) η = 10%; (f) η = 15% and (g) η = 20%. Detailed Implementation

[0108] The technical solutions of the present invention will now be clearly and completely described with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, and not all of them. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0109] The technical solutions of the various embodiments of the present invention can be combined with each other, but this must be based on the ability of those skilled in the art to implement them. When the combination of technical solutions is contradictory or cannot be implemented, it should be considered that such combination of technical solutions does not exist and is not within the scope of protection claimed by the present invention.

[0110] Reference Figures 1-8 This invention provides a method for strengthening corroded concrete structures based on load paths, the steps of which include:

[0111] S1: Determine the nonlinear topology optimization formula for the load path search problem of rust-damaged RC structures;

[0112] The optimization objective of the load path search problem for RC structures with corrosion damage is to find the load path within the volume constraint V. * The load path that generates the minimum structural flexibility (i.e., maximum stiffness) is generated. Taking the concrete element material density as the design variable, the mathematical expression for the nonlinear optimization problem of the RC structure with corrosion damage is:

[0113]

[0114]

[0115] in, The objective function is the structural compliance. Design a vector of variables for the unit; Let be the material density of the i-th element, which can be taken as 1 or in the Two-Way Progressive Structure Optimization (BESO) method. f ext U is the external load vector; η v is the structural displacement vector; i and V * These represent the volume of the i-th element and the volume fraction of the specified material, respectively. n is the number of concrete elements in the design domain. The minimum density of the unit cell; This represents the residual force vector of the RC structure subjected to nonlinear corrosion damage.

[0116] S2: Calculate the properties of the corroded material, bond strength, and spring stiffness;

[0117] 1) Steel reinforcement corrosion reduces the compressive strength of concrete. Assuming uniform corrosion of longitudinal steel reinforcement, the compressive strength f′ of concrete under corrosion... cm It can be represented as:

[0118]

[0119] Among them, f cm k represents the compressive strength of uncorroded concrete. r ε is an empirical coefficient; cu ε1 represents the maximum compressive strain; ε2 represents the average tensile strain of the cracked concrete, which can be expressed as:

[0120] ε1=n bs ·w cr / b w (4)

[0121] Among them, b w n represents the initial width of the uncorroded RC beam; bs The quantity of reinforcing steel in the compression zone of the RC beam; w cr The total crack width of the RC beam damaged by corrosion can be calculated as follows:

[0122] w cr =2πX(u) rs -1) (5)

[0123] Among them, u rs X represents the volumetric expansion rate; X represents the depth of steel corrosion, which can be expressed as:

[0124]

[0125] Where η and r s These are the steel reinforcement corrosion rate and the steel reinforcement radius, respectively.

[0126] 2) The ultimate tensile strength of corroded steel bars can be expressed as:

[0127] f y，c =(1-α) y η)f y (7)

[0128] f u，c =(1-α) u η)f u (8)

[0129] εu，c =(1-α1η)ε u (9)

[0130] Among them, f y，c f u，c With ε u，c These represent the yield strength, ultimate tensile strength, and ultimate strain of the corroded steel reinforcement, respectively; f y f u With ε u These represent the yield strength, ultimate strength, and ultimate strain of the uncorroded steel reinforcement, respectively; α y α u α1 is the correlation coefficient.

[0131] 3) The bond-slip behavior between corroded reinforcing bars and concrete can be simulated using spring elements. The stiffness of the spring element can be determined by the bond strength of the reinforcing bars and the corresponding slip. Each pair of overlapping reinforcing bars is connected to the concrete node using a spring element of zero length to simulate the bond-slip behavior. The bond strength τ between the corroded reinforcing bars and concrete... a，η It can be represented as:

[0132]

[0133] Among them, C c D represents the thickness of the concrete protective layer. l The diameter of the longitudinal reinforcement; A s，η λ represents the cross-sectional area of ​​the corroded stirrups; λ, ζ, and k are empirical coefficients; s is the stirrup spacing; and f is the tensile strength of the concrete. ct Its compressive strength f cm Related, can be expressed as f sy，η The yield strength of the corroded stirrup can be expressed as:

[0134]

[0135] Where, η s f represents the average corrosion rate of the stirrups. sy Let be the yield strength of the uncorroded stirrup. The stiffness of the spring element can be expressed as:

[0136] k sp =π·D·l r ·τ a，η / S1 (12)

[0137] Among them, l r S1 is the length of the concrete unit; S1 is the maximum slip value corresponding to the maximum bond stress of the steel reinforcement.

[0138] S3: Establishing a finite element model of a corroded concrete structure

[0139] 1) Establishing a geometric model: Based on the actual engineering structure, determine the geometric dimensions of each component in the concrete structure. In this embodiment, the ABAQUS software Part module is used to construct the geometric model of components such as concrete, steel bars and pads, and the Assembly module is used to assemble each component.

[0140] 2) Defining the constitutive model of corroded materials: The asymmetric response of concrete under tension and compression is simulated using the Concrete Damaged Plasticity model in ABAQUS. The Property module is used to define the corroded concrete and reinforcing steel. The hardening and softening behavior of concrete is simulated using the constitutive relations of concrete in the Chinese standard GB50010. The mechanical responses of concrete under tension and compression are shown below. Figure 1 As shown, a bilinear constitutive model is used to characterize the mechanical properties of corroded steel bars, such as... Figure 2 As shown.

[0141] 3) Define mesh generation and loading method: Use the Mesh module to mesh materials such as concrete and steel reinforcement, using regular quadrilateral elements as much as possible, and keeping the element sizes of steel reinforcement and concrete as consistent as possible. Numerical simulations of concrete and corroded steel reinforcement are performed using four-node CPS4R elements and two-node linear two-dimensional truss T2D2 elements, respectively.

[0142] 4) Use the Load module to define boundary conditions for the concrete structure. Concentrated forces and constraints are applied to the rigid plate to prevent excessive deformation of local concrete elements. The boundary conditions can be determined according to the actual engineering situation.

[0143] 5) Setting up spring elements between concrete and corroded steel bars: Using the Special tool in the Interaction module and its Springs / Dashpots tool, two spring elements of zero length are used to connect the coincident nodes of the corroded steel bars and concrete elements. In the two springs, the vertical spring deformation is negligible compared to the longitudinal deformation, so the stiffness coefficient K of the vertical spring can be taken as infinite. The bond slip between the steel bars and concrete is mainly simulated by the longitudinal spring, and the stiffness of the spring element is determined by k in equation (12). sp Confirmed. Assume the stirrups and top longitudinal reinforcement are fully bonded to the surrounding concrete. Use embedded equations in the Interaction module to connect the concrete element nodes to the top longitudinal reinforcement and stirrup element nodes respectively. Use tie constraints to connect the rigid steel plate to the contact surface of the design domain.

[0144] 6) Set up the solver and define the finite element output: Define the output of the finite element module in the Step module, namely element stress, strain, elastic and plastic strain energy, etc. To improve computational efficiency, use multiple processors and GPU acceleration in the Job module; keep other settings at their defaults.

[0145] S4: Perform topology optimization analysis on the rust-damaged RC structure. The specific steps are as follows:

[0146] 1) Define initial optimization parameters: filter radius r min Penalty factor P, target volume fraction V * Evolutionary rate ER, initial strain threshold ε * Minimum density of material Elastic modulus threshold E max and initial elastic modulus

[0147] 2) Initialize design variables and concrete section properties: Set the section properties for "solid" and "hollow" materials. "Solid" material is concrete; "hollow" material is an elastic material with a Poisson's ratio of 0.3 and an elastic modulus of [missing value]. Where E c For the elastic modulus of concrete. Set all initial design variables to... All element section properties within the design domain are set to "solid" material section properties.

[0148] 3) Nonlinear finite element analysis based on ABAQUS. Extract the elastic and plastic strain energies of each element in this optimization iteration step, and the total strain energy w of element i. i It can be represented as:

[0149]

[0150] in, and These represent the elastic and plastic strain energies of each unit, respectively.

[0151] 4) Calculate the sensitivity of RC structural elements to corrosion damage, and calculate the sensitivity of elements in the structural design domain;

[0152] The formula for the sensitivity of RC structures to corrosion damage can be expressed as:

[0153]

[0154] Where, λ o It is a very small constant, which can be set as λ. o =10 -10 λ is added to the denominator of the second term on the right to avoid [the following]. Ambiguity may arise.

[0155] 5) Update element sensitivity information: A fuzzy filtering scheme is used to obtain a mesh-independent topology optimization solution. The original element sensitivity is further processed, and the element sensitivity can be expressed as:

[0156]

[0157] w(r ij ) = max(0, r min -r ij (27)

[0158] Where, r ij η is the distance between the centers of units i and j; w is the weighting function for the average original sensitivity; η j r is the weighting factor; min The filter radius; the sensitivity of the current iteration step. Sensitivity compared to the previous iteration step Averaging; unit sensitivity It can be represented as:

[0159]

[0160] 6) Determine the target volume for the next optimization step: based on the volume V of the current iteration step. k Compared with evolutionary ratio ER, the target volume V for the next iteration can be obtained. k+1 .

[0161] V k+1 =V k (1±ER) (29)

[0162] 7) Update design variables and element types: If the sensitivity of the concrete element... Less than the sensitivity threshold Then the unit density Switch from 1 to For weakly elastic body elements, if Unit density from Switch to 1. Then adjust according to the cell density. The cross-sectional properties of concrete and weakly elastic materials are redistributed.

[0163] S5: Automatic adjustment of stiffness for minimum density elements: First, extract the maximum strain of the minimum density element. Then, determine the maximum strain in this iteration step. With strain threshold ε * The size. If A multi-proportional stiffness growth strategy is used to increase the elastic modulus of the minimum density element in the next iteration. To suppress numerical instability caused by the distortion of the minimum density element during the load path search process; if Then reduce the value in the next iteration step. To suppress the optimization error caused by the increase in element stiffness, see formula (23).

[0164] S6: Calculate the convergence error Er during the load path search process for the RC beam with corrosion damage. k The optimization iteration process is repeated until the target volume and convergence criterion are met. A convergence criterion is introduced to terminate the load path search process. This convergence criterion is:

[0165]

[0166] Where ∈ represents the allowable convergence error; N is the number of iteration steps. The optimization process terminates when the structural flexibility remains stable, generating the optimal load path for the corrosion-damaged RC structure.

[0167] S7: Verify the rationality of the load path for the RC structure with corrosion damage. A numerical example of a RC beam with corrosion damage is used to illustrate the rationality of the generated load path.

[0168] S8: Reinforcement Material Layout Guided by Load Path of Corrosion-Damaged Reinforced Concrete Structures. Ideally, reinforcement material layout should follow the structural load transfer path, fully utilizing the performance of the reinforcement materials and improving their utilization efficiency. Based on the load path analysis of the damaged structure, the stress characteristics of each component are considered. For tension members, reinforcement materials with higher tensile strength are placed at appropriate locations; for compression members, reinforcement materials with higher compressive strength are required. The layout of reinforcement materials should follow the load path of the corrosion-damaged reinforced concrete structure.

[0169] To further illustrate the operational steps of the load path topology search method for rust-damaged concrete structures provided by this invention, the following are the attached... Figure 5 The load path search and reinforcement material layout of the rust-damaged RC beam shown are investigated. The geometric dimensions of the specimen are 600×100mm, and the size of the pad block is 20×10mm. The relevant geometric parameters of the RC beam are shown in Table 1, and the mechanical properties of the concrete and steel reinforcement are shown in Table 2.

[0170] Table 1 Geometric parameters of RC beams

[0171]

[0172] Table 2 Mechanical properties of concrete and steel reinforcement

[0173]

[0174] The specific process of the reinforcement design method for corrosion-damaged concrete structures based on load path in this example is as follows:

[0175] S1: Determine the nonlinear topology optimization formula for the load path search problem of rust-damaged RC beams;

[0176] The optimization objective of the load path search problem for rust-damaged RC beams is to find the load path within the volume constraint V. * The mathematical expression for generating the load path with the minimum structural flexibility is:

[0177]

[0178]

[0179] S2: Calculate the properties of the corroded material, bond strength, and spring stiffness. The corrosion rate of the reinforcing steel in the RC beam is 5%. According to the literature "Failure analysis of corroded RC beams subjected to shear-flexural actions, Antonino Recupero, Nino Spinella, Francesco Tondolo, Engineering Failure Analysis, 2018, 93: 26-37", the empirical coefficient k is given. r =0.1, volume expansion rate u rs =2, number of reinforcing bars in the compression zone n bs =2, empirical coefficient α y α u α1 can be taken as 0.012, 0.011 and 0.03 respectively; based on formulas (3)-(9), the steel corrosion depth X is 0.304 mm and the total crack width w is 0.304 mm. cr The average tensile strain ε1 is 1.909 mm, and should be less than or equal to the ultimate strain ε of the concrete. s，u ε1 can be taken as 0.0033; therefore, the compressive strength f′ of concrete under corrosion is... cm The maximum strength of the corroded steel bar is 36.36 MPa, the yield strength of the corroded steel bar is 367 MPa, the ultimate strength of the corroded steel bar is 548.1 MPa, and the ultimate strain of the corroded steel bar is 0.425.

[0180] According to the literature "Shear and flexural strength prediction of corroded R.C. beams, G. Campione, F. Cannella, L. Cavaleri, Construction and Building Materials, 2017, 149: 395-405", the empirical coefficients ζ, k, and λ can be taken as 0.1, 0.16, and 0.4, respectively, and the protective layer thickness C cThe stirrup spacing is 50mm, and the concrete unit length is l. (Note: The original text contains some inconsistencies and unclear grammatical structure. A more accurate translation would require the full context.) r The maximum slip value S1 is 1 mm, and the maximum slip value S1 is 5 mm. Based on formulas (10)-(12), the tensile strength f of the corroded concrete can be obtained. ct The stress is 3.54 MPa, and the stirrup yield strength f is... sy，η The pressure is 300 MPa, and the cross-sectional area of ​​the stirrups is A. s，η It is 50.24mm. 2 Bond strength τ between corroded steel bars and concrete a，η The spring unit stiffness is 3.83 MPa, and the spring unit stiffness is k. sp It is 1443.14 kN / mm.

[0181] S3: Establish a finite element model of a concrete structure with corrosion damage.

[0182] 1) Establish the geometric model. Create concrete, reinforcing steel, and spacer components based on the RC beam and reinforcement dimensions. Use the Assembly module to assemble the components.

[0183] 2) Define the constitutive model for the corroded materials. The concrete and reinforced steel materials are defined using the Property module > Mechanical > Plasticity > Concrete Damage Plasticity. In the CDP model, parameters include expansion angle ψ = 36°, flow potential eccentricity ∈ = 0.1, and the ratio of biaxial compressive strength to uniaxial compressive strength. The ratio k of the distance between the hydrostatic pressure axis and the compression and tension meridians. ζ =0.6667 and viscosity parameter μ=0.0005. For detailed instructions on using the concrete CDP model, please refer to the ABAQUS manual.

[0184] 3) Define the mesh generation and loading method. The concrete design domain is discretized into 120×20 meshes using quadrilateral element meshes. The concrete element type is CPS4R, and the element size is 5mm. The reinforcement element type is T2D2, and the element length is 5mm. The concentrated load F = 150N is vertically applied to the rigid pad node in the middle of the RC beam using the Concentrated Force option in the Load module. All displacement and rotation options of the left rigid pad node are constrained, and the vertical displacement of the right rigid pad node is constrained using the Displacement / Rotation option.

[0185] 4) Set up spring elements between the concrete and the corroded reinforcing steel. Using the Interaction module > Special > Springs / Dashpots tool, connect the coincident nodes of the corroded reinforcing steel and the concrete elements with two spring elements of zero length, and set the stiffness k of the spring elements. sp The value is 1443.14 kN / mm. The reinforced steel components are embedded into the concrete components using embedded equations in the Interaction module. Tie constraints are used to connect the rigid steel plate to the element nodes of the contact surface of the design domain.

[0186] 5) Set up the solver and define the finite element output. In the Step module > Static, under General, define the analysis step. In this case, geometric nonlinearity is not considered, so the Nlgeom option is turned off. In Incrementation, set the Maximum number of increments to 10000, the Initial load step size to 0.001, and the Minimum load step size to 10. -8 The maximum load step size is set to 1, and the Unsymmetric solution option is enabled. Define the finite element outputs: Create Field Output and Create History Output. Enable the following options in Stresses: S (Stress components and invariants), Strains, E (Total strain components), Energy: ENER (All energy magnitudes), and Energy: ALLEN (All energy totals). Create a working step in the Job module, enable the Parallelization option, set Use multiple processors to 8, set Use GPGPU acceleration to 6, and keep other settings at their defaults. The corrosion-damaged RC beam optimization model is as follows: Figure 5 As shown.

[0187] S4: Topology optimization analysis of the RC beam with corrosion damage. The BESO method was used for topology optimization of the RC beam with corrosion damage. For the Python program of the BESO method, please refer to the reference "A simple and compact Python code for complex 3D topology optimization, Zhi Hao Zuo, Yi Min Xie, Advances in Engineering Software, 2015, 85: 1-11". In addition, the automatic stiffness adjustment method for the minimum density element was written in Python.

[0188] 1) Define initial optimization parameters: filter radius r min The diameter is 20 mm, the penalty factor P is 3, and the target volume fraction V is... * The evolution rate ER is 0.02, and the initial strain threshold ε is 50%. * The minimum density of the material is 0.0015. The elastic modulus threshold E is 0.001. max The initial elastic modulus is 10 MPa. 10 -4 E c .

[0189] 2) Initialize design variables and concrete section properties. Set the section properties for "solid" and "hollow" materials. "Solid" material is concrete; "hollow" material is an elastic material with a Poisson's ratio of 0.3 and an elastic modulus of [missing value]. Set all design variables to All element section properties within the design domain are set to "solid" material section properties.

[0190] 3) Nonlinear finite element analysis was performed on the RC beam with corrosion damage. The elastic and plastic strain energies of each element in each optimization iteration step were extracted, and the total strain energy w of element i was also analyzed. i It can be calculated using formula (25).

[0191] 4) Calculate the sensitivity of each unit of the RC structure to corrosion damage based on formula (20).

[0192] 5) Update the unit sensitivity information based on formulas (26)-(28).

[0193] 6) Determine the target volume for the next optimization step based on formula (29).

[0194] 7) Update design variables and element types. Use a bisection method to determine the sensitivity threshold for this iteration step. If the sensitivity of the concrete unit Less than the sensitivity threshold Then the unit density Switch from 1 to For weakly elastic body elements, if Unit density from Switch to 1. Then adjust according to the cell density. The cross-sectional properties of concrete and weakly elastic materials are redistributed.

[0195] S5: Automatic stiffness adjustment for minimum density elements. In the automatic stiffness adjustment method, coefficients α1, α2, α3, and α4 can be set to 1.2, 0.5, 1, and 0.9, respectively. First, the maximum strain of the minimum density element in this iteration step is extracted. Then, determine the maximum strain. With strain threshold ε * Size; if Based on formula (23), increase the elastic modulus of the minimum density element. if Then, based on formula (23), the elastic modulus of the minimum density element is reduced.

[0196] S6: Convergence criterion Er for calculating the load path search process of rust-damaged RC beams based on formula (30) k According to the literature "Bi-directional evolutionary topology optimization of continuum structures with one or multiple materials, Huang X, Xie YM, Computational Mechanics, 2009, 43(3): 393-401," the convergence error is 0.01%, and the number of iterations is N = 5. The optimization iteration process is repeated until the target volume and convergence criteria are met. When the structural flexibility remains stable for at least 10 consecutive iterations, the optimization process terminates, generating the optimal load path for the corrosion-damaged RC structure.

[0197] S7: Verify the rationality of the optimal load path for the RC structure with corrosion damage. Based on the invented method, a load path was generated for an RC beam with an external load of F = 150N and a corrosion rate of η = 5%, as follows: Figure 6 As shown, the load path extends from the loading point and support point to 1 / 4 of the beam, forming two almost parallel load path trajectories, which reflect the shear behavior of the RC beam's bending and shear segments and can reasonably describe the shear force transfer mechanism of the RC beam with corrosion damage.

[0198] To further verify the rationality and correctness of the analytical method of this invention, the load path development law of RC beams with corrosion damage under different loads and corrosion rates was studied using the method of this invention. Figure 7 The load path development law of RC beams under different loads (BESO method; F=5N; F=50N; F=100N; F=150N; F=200N; F=250N and F=300N) is presented. Figure 7 (a) shows the load path of the elastic RC beam based on the BESO method. Under the minimum load F = 5 N, the load path of the RC beam is almost identical to the elastic load paths in existing literature. When the load F increases to 100 N, the load path of the RC beam begins to change, extending directly from the loading point to the support point, consistent with the internal force trajectory of the classic tension-compression model in existing literature. Figure 7 As shown in (c), when the external load increases further, the end of the load path near the support point gradually moves towards the mid-span of the RC beam, as... Figure 7 As shown in (c)-(f). Subsequently, as the external load further increases to near the limit state, the load path extends from the loading point and support point towards the 1 / 4 mark of the beam, forming two almost parallel load path trajectories, demonstrating the shear behavior of the RC beam, as shown... Figure 7 As shown in (g)-(h), the method of this invention effectively solves the numerical instability problem in the nonlinear topology optimization process and successfully generates the load path of the RC beam. The load path of the damaged concrete structure can guide the placement of reinforcement materials.

[0199] Figure 8 The load path variations of RC beams with different corrosion rates (0–20%) under an external load of F = 150 N are presented. The load path of the RC beams with corrosion damage gradually changes with increasing corrosion rate. The variation law of the load path of RC beams with corrosion damage under different corrosion levels and different load conditions is consistent. Corrosion accelerates the transition of the load path of the RC beam from the non-limit state to the limit state.

[0200] S8: Reinforcement material placement guided by load path analysis of corrosion-damaged RC structures. Based on the load path analysis of the damaged structure, the stress characteristics of each component are determined. For tension members, reinforcement materials with high tensile strength are placed at the appropriate locations; for compression members, reinforcement materials with high compressive strength are required. The placement of reinforcement materials should follow the load path of the corrosion-damaged RC structure.

[0201] The above are merely preferred embodiments of the present invention and do not limit the patent scope of the present invention. All equivalent structural transformations made using the contents of the present invention's specification and drawings under the inventive concept of the present invention, or direct / indirect applications in other related technical fields, are included within the patent protection scope of the present invention.

Claims

1. A method for strengthening corroded concrete structures based on load path, characterized in that, Includes the following steps: Step S1: Determine the nonlinear topology optimization formula for the load path search problem of the corroded RC structure; Step S2: Calculate the properties of the corroded material, bond strength, and spring stiffness; Step S3: Establish a numerical model of the RC structure for corrosion damage that considers material property loss and bond degradation; Step S4: Perform nonlinear topology optimization analysis on the rust-damaged RC structure: First, perform nonlinear finite element analysis on the numerical model of the rust-damaged RC structure established in Step S2; then, obtain the elastic and plastic strain energy of each element, and calculate the sensitivity of each element based on the sensitivity formula of the strain energy form of the rust-damaged element; finally, use a fuzzy filtering scheme to update the element sensitivity information, determine the target volume of the next optimization step, and update the design variables and element types. Step S5: Adjust the stiffness of the minimum density element based on the automatic adjustment method of minimum density element stiffness; Step S6: Calculate the convergence error during the load path search process of the RC structure with corrosion damage; Step S7: Generate the load path for the rust-damaged RC structure and verify the rationality of the load path; Step S8: Guide the placement of reinforcement materials based on the load path of the RC structure with corrosion damage; Step S5 specifically includes: First, extracting the maximum strain of the minimum density element obtained in step S4. Then, determine the maximum strain. With strain threshold Size; if The elastic modulus of the minimum density element in the next iteration step is increased by using a multi-proportional growth strategy for the minimum density element stiffness. ;if Then the decrease in the next iteration step ; In step S1, the nonlinear topology optimization formula is as follows: The optimization objective of the load path search problem for corroded RC structures is to find the load path within volume constraints. The load path with the minimum structural flexibility is generated below. Taking the concrete element material density as the design variable, the mathematical expression of the nonlinear optimization problem of the corroded RC structure is: Minimize: (1) Subject to: (2) in, The objective function is the structural compliance. Design a vector of variables for the unit; Let be the material density of the i-th unit; External load vector; The structural displacement vector; and These represent the volume of the i-th element and the volume fraction of the specified material, respectively; n is the number of concrete elements in the design domain. The minimum density of the unit cell; This represents the residual force vector of the nonlinear rusted RC structure.

2. The method for strengthening corroded concrete structures based on load path according to claim 1, characterized in that, In step S2, the calculation methods for the properties of the corroded material, bond strength, and spring stiffness are as follows: 1) Steel reinforcement corrosion reduces the compressive strength of concrete. Assuming uniform corrosion of longitudinal steel reinforcement, what is the compressive strength of concrete under corrosion? It can be represented as: (3) in, The compressive strength of uncorroded concrete; This is an empirical coefficient; For maximum compressive strain; The average tensile strain of cracked concrete can be expressed as: (4) in, The initial width of the uncorroded RC beam; This refers to the number of reinforcing bars in the compression zone of the RC beam. The total crack width of the corroded RC beam can be calculated as follows: (5) in, The volume expansion rate; The depth of steel reinforcement corrosion can be expressed as: (6) in, and These are the steel reinforcement corrosion rate and the steel reinforcement radius, respectively. 2) The ultimate tensile strength of corroded steel bars can be expressed as: (7) (8) (9) in, , and These represent the yield strength, ultimate tensile strength, and ultimate strain of the corroded steel bars, respectively. , and These represent the yield strength, ultimate tensile strength, and ultimate strain of the uncorroded steel bars, respectively. , and For relevant empirical coefficients; 3) The bond-slip behavior between corroded reinforcing bars and concrete can be simulated using spring elements. The stiffness of the spring elements can be determined by the bond strength of the reinforcing bars and the corresponding slippage. Each pair of overlapping reinforcing bars is connected to the concrete node using spring elements of zero length to simulate the bond-slip behavior between them. The bond strength between the corroded reinforcing bars and concrete... It can be represented as: (10) in, This refers to the thickness of the concrete protective layer. The diameter of the longitudinal reinforcing bar; This represents the cross-sectional area of ​​the corroded stirrup; , , This is an empirical coefficient; It refers to the spacing of the stirrups; the tensile strength of the concrete. Its compressive strength Related, can be expressed as ; The yield strength of the corroded stirrup can be expressed as: (11) in, The average corrosion rate of the stirrups; The yield strength of the uncorroded stirrups; the stiffness of the spring element can be expressed as: (12) in, The length of the concrete unit; This represents the maximum slip value corresponding to the point of maximum bond stress in the reinforcing steel.

3. The method for strengthening corroded concrete structures based on load path according to claim 1, characterized in that, In step S4, the design sensitivity formula for the strain energy form of the corrosion damage element is as follows: (13) in, Let be a constant, and let it be... = p is the penalty factor. and These represent the total strain energy of concrete and weakly elastic material elements, respectively.

4. The method for strengthening corroded concrete structures based on load path according to claim 1, characterized in that, In step S5, the automatic adjustment method for the stiffness of the minimum density unit further includes the following steps: In each optimization iteration, the elastic modulus of the minimum density element is automatically adjusted by tracking the maximum strain of the minimum density element. In the (k+1)th optimization step, Automatic adjustment is achieved using a multi-proportional stiffness growth strategy for minimum density elements: (14) in, and These are the elastic moduli of the minimum density element in the k-th and k+1-th optimization steps, respectively; It is the maximum strain of the smallest density element in the kth optimization step; , , and It is a coefficient used to control the rate of increase in the stiffness of the control unit; the threshold is set in each optimization step. Updated to , The sigmoid function can be represented as: (15) in, .

5. The method for strengthening corroded concrete structures based on load path according to claim 1, characterized in that, In step S8, the placement of reinforcement materials guided by the load path of the RC structure affected by corrosion damage is as follows: Based on the load path analysis of the damaged structure, the stress characteristics of each component are analyzed. For tension components, reinforcement materials with high tensile strength are placed at the corresponding locations, and for compression components, reinforcement materials with high compressive strength are placed. The placement of reinforcement materials should follow the load path of the corrosion-damaged RC structure.

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