A deteriorating bridge anti-seismic reinforcement opportunity decision method, system and device based on real options and a medium

Abstract

The application relates to a deterioration bridge anti-seismic reinforcement opportunity decision method, system, equipment and medium based on a real option. The method obtains a degradation parameter sequence, constructs a damage state probability model driven by historical cumulative damage, makes the damage probability reflect the path dependence of the seismic load, discretizes the continuous degradation process into multiple degradation levels, simulates the reinforcement yield fluctuation characteristics under each level, establishes a dynamic mapping between the degradation level and the asset volatility, constructs a degradation state coupled binary tree, drives the asset value evolution according to the volatility corresponding to the degradation level of each node, obtains the reinforcement real option value reflecting the degradation risk difference through backward recursion, and finally takes the cumulative yield calculated based on the damage probability and the reinforcement real option value as the evaluation dimension, and determines the optimal reinforcement opportunity through multi-criteria evaluation.

Description

Technical Field

[0001] This invention belongs to the field of bridge engineering, and in particular relates to a method, system, equipment and medium for determining the timing of seismic reinforcement of deteriorated bridges based on real options. Background Technology

[0002] During long-term service in chloride ion-corrosion environments or de-icing salt environments, corrosion degradation and seismic risk are two core factors affecting the structural safety and economical operation and maintenance of reinforced concrete bridges. To scientifically determine the timing of seismic strengthening during the bridge planning period, real option-based decision-making methods have gained increasing attention. These methods treat strengthening decisions as future-exercise options, guiding the selection of the optimal timing by quantifying the flexible value of postponing the decision. However, existing real option-based methods for determining the timing of seismic strengthening of deteriorated bridges still suffer from two fundamentally coupled defects in their technical implementation, causing their output of the optimal strengthening timing to often deviate from engineering realities.

[0003] The first drawback lies in the fact that existing methods implicitly assume that the damage state transition of structures between different seismic events follows Markov properties in their seismic vulnerability models. Specifically, these models typically describe the damage probability of a structure using a log-normal distribution vulnerability function. The construction logic of this function dictates that it only uses the current seismic intensity as an input variable, without recording or reflecting the seismic load history that the structure has experienced before the current earthquake. However, reinforced concrete bridges exhibit significant load sequence effects under seismic sequences. Even if the initial small and medium-sized earthquakes do not directly cause the structure to enter a state of severe damage, they will still cause irreversible damage such as microcrack propagation, bond slip degradation between steel reinforcement and concrete, and local concrete crushing within the components. These damages will significantly alter the structure's response characteristics and damage development path under subsequent seismic events, making the structure's seismic vulnerability exhibit path dependence. Log-normal distribution vulnerability functions cannot characterize the bias effect of cumulative damage caused by historical earthquakes on current and future vulnerability levels. This leads to a systematic underestimation of the true probability of structural damage after multiple small and medium earthquakes, which in turn makes reinforcement decisions based on such vulnerability assessments overly aggressive.

[0004] The second drawback lies in the fact that existing real option-based reinforcement timing decision-making frameworks typically employ the constant volatility assumption in binomial option pricing models. This assumption holds that the volatility of asset value remains constant throughout the service life, unaffected by time or changes in structural condition. However, for reinforced concrete bridges subjected to chloride ion corrosion, the structural resistance degradation caused by steel corrosion is an irreversible process that intensifies over time. The deepening corrosion not only directly reduces the structure's load-bearing capacity but also significantly increases the uncertainty of the structure's damage state under seismic loading. This means that the volatility of reinforcement returns itself dynamically changes with time and the state of degradation. The constant volatility assumption fails to reflect the time-varying nature of the cumulative return uncertainty caused by corrosion degradation, causing option pricing parameters to deviate from actual risk and return levels, thus affecting the accuracy of option value calculations.

[0005] The two aforementioned shortcomings are not isolated but rather coupled. On one hand, the Markov assumption of log-normal vulnerability leads to distorted probabilities of damaged states, causing bias in the cumulative returns used by the binomial tree model to determine the initial asset value. On the other hand, the constant volatility assumption prevents the binomial tree model from sensing the dynamic impact of degradation states on the risk structure, further amplifying pricing errors. The combined effect of these two factors makes it impossible to accurately balance the economic waste caused by premature reinforcement with the safety risks associated with delayed reinforcement in reinforcement timing decisions.

[0006] Therefore, there is an urgent need to develop a method that can simultaneously reflect the path dependence effect of earthquake load history on vulnerability and the dynamic impact of corrosion degradation on asset value volatility, so as to form a complete and synergistic improvement scheme from damage probability modeling to option pricing and timing decision-making. Summary of the Invention

[0007] Therefore, it is necessary to provide a method, system, equipment, and medium for determining the timing of seismic reinforcement of deteriorated bridges based on real options, in order to address the above-mentioned technical problems.

[0008] Firstly, this application provides a method for determining the timing of seismic reinforcement of deteriorated bridges based on real options, including:

[0009] S1. Obtain the degradation parameter sequence of the bridge over time during its service life; wherein, the degradation parameter sequence includes parameters reflecting the corrosion state and mechanical performance degradation of the structure;

[0010] S2. Based on the degradation parameter sequence, a damage state probability model driven by historical cumulative damage is constructed for the bridge under various degradation states, and the output depends on the historical seismic loads borne by the bridge structure.

[0011] S3. Based on the parameters in the degradation parameter sequence, the continuous degradation process of the bridge is discretized into multiple degradation levels; by simulating the fluctuation characteristics of the expected returns of the reinforcement project under each degradation level, the asset volatility corresponding to each degradation level is obtained, and a mapping relationship between degradation level and asset volatility is established.

[0012] S4. Construct a degenerate state coupled binary tree with the planning period as the expiration time and the decision point as the start time; wherein, each node of the degenerate state coupled binary tree corresponds to a different decision point and stores the asset value representing the expected return of the reinforcement project;

[0013] S5. Based on the mapping relationship, obtain the asset volatility corresponding to the degradation level of each node in the degenerate state coupled binary tree; based on the asset volatility corresponding to each node, evolve the asset value of each node in the degenerate state coupled binary tree, and calculate the root node option value by backward recursion based on the asset value of each node to obtain the reinforced real option value.

[0014] S6. Based on the probability of damage state, calculate the cumulative risk value for different candidate reinforcement years; determine the cumulative benefit for each candidate reinforcement year based on the cumulative risk value;

[0015] S7. Using cumulative returns and the value of the real option for reinforcement as evaluation dimensions, a multi-criteria evaluation function is constructed to weight and score each candidate reinforcement year to obtain a comprehensive score; based on the comprehensive score, the optimal reinforcement time is determined from all candidate reinforcement years.

[0016] Secondly, this application also provides a real options-based decision-making system for seismic reinforcement of deteriorated bridges, used to implement the method described in the first aspect, the system comprising:

[0017] The degradation parameter acquisition module is used to acquire the degradation parameter sequence of the bridge over time during its service life; the degradation parameter sequence includes parameters reflecting the corrosion state and mechanical performance degradation of the structure.

[0018] The damage probability modeling module is used to construct a damage state probability model of the bridge under various degradation states driven by historical cumulative damage based on the degradation parameter sequence, and output the damage state probability that depends on the historical seismic loads that the bridge structure has borne.

[0019] The volatility mapping module is used to discretize the continuous degradation process of a bridge into multiple degradation levels based on the parameters in the degradation parameter sequence; by simulating the volatility characteristics of the expected returns of the reinforcement project under each degradation level, the asset volatility corresponding to each degradation level is obtained, and a mapping relationship between degradation level and asset volatility is established.

[0020] The coupled binary tree construction module is used to construct a degenerate state coupled binary tree with the planning period as the expiration time and the decision point as the start time; wherein, each node of the degenerate state coupled binary tree corresponds to a different decision point and stores the asset value representing the expected return of the reinforcement project;

[0021] The option value calculation module is used to obtain the asset volatility corresponding to the degradation level of each node in the degenerate state coupled binary tree based on the mapping relationship; based on the asset volatility corresponding to each node, the asset value of each node in the degenerate state coupled binary tree is evolved, and the option value of the root node is calculated backward recursively based on the asset value of each node to obtain the value of the reinforced real option.

[0022] The risk-return calculation module is used to calculate the cumulative risk value for different candidate reinforcement years based on the probability of damage status; and to determine the cumulative return for each candidate reinforcement year based on the cumulative risk value.

[0023] The decision optimization module is used to evaluate the cumulative returns and the value of the real option for reinforcement by constructing a multi-criteria evaluation function, weighting each candidate reinforcement year to obtain a comprehensive score, and then determining the optimal reinforcement time from all candidate reinforcement years based on the comprehensive score.

[0024] Thirdly, this application also provides a computer device, including a memory and a processor, wherein the memory stores a computer program, and the processor executes the computer program to implement a real options-based decision-making method for seismic reinforcement of deteriorated bridges as described in the first aspect.

[0025] Fourthly, this application also provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements a real options-based method for determining the timing of seismic reinforcement of deteriorated bridges as described in the first aspect.

[0026] The aforementioned method, system, equipment, and medium for seismic reinforcement timing of deteriorated bridges based on real options, acquires a sequence of degradation parameters reflecting structural corrosion and mechanical performance degradation, constructs a damage state probability model driven by historical cumulative damage, and enables the damage probability to reflect the path-dependent impact of the seismic load sequence on the structure. Simultaneously, based on the degradation parameters, the continuous degradation process is discretized into multiple degradation levels, and a dynamic mapping relationship between degradation levels and asset volatility is established by simulating the fluctuation characteristics of the expected returns of reinforcement projects under each degradation level. After constructing a coupled binary tree of degradation states with the planning period as the maturity date... The method drives asset value evolution based on asset volatility corresponding to the degradation level of each node, and calculates the option value of the root node through backward recursion to obtain the reinforcement real option value that reflects the risk differences of different degradation stages. Finally, using the cumulative return calculated based on the probability of damage state and the reinforcement real option value as evaluation dimensions, the method performs a weighted score on each candidate reinforcement year through multi-criteria evaluation to determine the optimal reinforcement time. This method overcomes the problem of distortion in reinforcement time judgment caused by ignoring path dependence effect and adopting constant volatility assumption in traditional methods, and improves the accuracy and reliability of seismic reinforcement time decision-making for deteriorated bridges. Attached Figure Description

[0027] To more clearly illustrate the technical solutions in the embodiments or related technologies of this application, the accompanying drawings used in the description of the embodiments or related technologies will be briefly introduced below. Obviously, the accompanying drawings described below are only some embodiments of this application. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0028] Figure 1 A flowchart illustrating a seismic reinforcement timing decision method for deteriorated bridges based on real options, provided by this invention.

[0029] Figure 2 This is a schematic diagram illustrating the process of obtaining reinforced real option value in one optional embodiment of the present invention;

[0030] Figure 3 This invention provides a structural schematic diagram of a decision-making system for seismic reinforcement of deteriorated bridges based on real options. Detailed Implementation

[0031] To make the objectives, technical solutions, and advantages of this application clearer, the following detailed description is provided in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the scope of this application.

[0032] To facilitate a better understanding of the technical solution of this invention, the following explanations are provided for the technical terms involved in this invention:

[0033] (1) Fick's second law: describes the variation of concentration with time and space during unsteady diffusion. The basic governing equation in the one-dimensional diffusion of chloride ions in concrete is: ,in Indicates chloride ion concentration. Indicates time, Indicates the chloride ion diffusion coefficient. This indicates the depth from the concrete surface. The corrosion initiation time determination in this invention is based on Fick's second law.

[0034] (2) XGBoost (eXtreme Gradient Boosting): An ensemble learning algorithm based on the gradient boosting framework. It iteratively trains multiple CART regression trees, fitting the residuals of the previous iteration in each iteration, and finally obtains the prediction result by weighted summation. An explicit regularization term is introduced into the objective function, and a second-order Taylor expansion is used to approximate the loss function. In this invention, XGBoost is used to establish a nonlinear mapping between degradation parameters and bridge mechanical response.

[0035] (3) Park-Ang damage model: a two-parameter earthquake damage assessment model. The damage index is composed of the superposition of the maximum displacement term and the hysteretic energy dissipation term. The former reflects the damage component caused by the maximum deformation, and the latter reflects the damage component caused by the cumulative energy dissipation and is multiplied by the energy dissipation contribution weighting coefficient. This invention uses the Park-Ang model to calculate the damage increment of the bridge structure under each earthquake action, and introduces a damage-related displacement angle degradation function to consider the influence of cumulative damage on the mechanical properties of the component.

[0036] (4) Inverse Gaussian Process (IG Process): A Levy subadditive stochastic process whose increment follows an inverse Gaussian distribution. The sample path has monotonically non-decreasing and non-negative characteristics, making it naturally suitable for describing irreversible damage accumulation phenomena. In this invention, the inverse Gaussian process is used to characterize the probabilistic evolution of the damage state of bridge structures during long-term service.

[0037] (5) CRR binomial tree model (Cox-Ross-Rubinstein binomial tree): A classic discrete pricing method for financial options, it models the underlying asset price as having two possible states—upward and downward—over an equal time step, and recursively calculates the risk-neutral value of the option at each node by progressively backwards from the option's expiration date. This method can effectively handle the early exercise characteristics of American options. In this invention, the CRR binomial tree model is used to calculate the real option value of the reinforcement project.

[0038] (6) Real Option: A decision-making framework that analogizes real asset investment and financing decisions to financial options. Investors have the right, but not the obligation, to execute the investment at a future point in time. The value of the option comes from the uncertainty of the underlying asset price and the flexibility of management. In this invention, the decision to strengthen bridge seismic resistance is modeled as a call option, with the underlying asset being the expected economic benefits generated by the strengthening project.

[0039] (7) Non-Markov process: A stochastic process in which the state transition probability depends on the complete historical path, and the future evolution of the system is affected by the cumulative influence of all past experiences. In this invention, the damage state transition of the bridge is a non-Markov process. The cumulative damage caused by historical earthquake loads changes the mechanical properties of the components, thereby affecting the calculation of damage increment under subsequent earthquake action.

[0040] (8) Risk-neutral pricing: A no-arbitrage framework for pricing financial derivatives, where the expected rate of return of all tradable assets equals the risk-free rate, and the option value equals the conditional expected value of future returns discounted at the risk-free rate. In this invention, it is used to calculate the value of real options on reinforcement projects that do not depend on the manager's personal risk preferences, providing a stable value benchmark for multi-criteria decision-making.

[0041] The present invention provides a method for determining the timing of seismic reinforcement of deteriorated bridges based on real options. This method operates on a central server of a bridge management system, which connects to a corrosion monitoring database, a seismic hazard database, a structural finite element analysis workstation, and a user decision terminal. The corrosion monitoring database is deployed on edge computing nodes at the bridge site, collecting monitoring data such as chloride ion concentration distribution, resistivity, half-cell potential, and carbonation depth within the concrete, and periodically pushing this data to the central server. The seismic hazard database stores historical earthquake catalog data for the target area and provides a query interface for the central server to retrieve earthquake events. The structural finite element analysis workstation is configured with high-performance computing nodes, deploys a refined three-dimensional finite element model of the bridge structure, receives batch tasks from the central server, performs nonlinear time history analysis or static pushover analysis, and transmits the results back. The user decision terminal provides a human-computer interactive graphical interface, receives the optimal reinforcement timing decision results pushed by the central server, and displays them in a visual format. The overall data flow is as follows: corrosion monitoring data and earthquake catalog data are imported into the central server; the central server calibrates the parameters of the Fick diffusion model and corrosion rate model, calculates the degradation parameters at each service time point and integrates them into a multi-dimensional feature vector, calls the pre-trained XGBoost model to predict the mechanical response, and generates a degradation parameter sequence; based on the degradation parameter sequence, the central server constructs a non-homogeneous Poisson earthquake model and a Park-Ang damage increment calculation module, and establishes a damage state probability model in combination with the inverse Gaussian process; the central server calls K-means clustering to discretize the degradation parameters into degradation levels, calibrates the asset volatility of each level through Monte Carlo simulation and establishes a mapping relationship; the asset volatility is input into the CRR binomial tree model, and the reinforcement real option value is calculated under the risk-neutral pricing framework; finally, the optimal reinforcement timing is determined based on multi-criteria decision-making and output to the user terminal.

[0042] Example 1:

[0043] refer to Figure 1 The document presents a flowchart illustrating a real options-based method for determining the timing of seismic reinforcement of deteriorated bridges, which includes the following steps:

[0044] S1. Obtain the degradation parameter sequence of the bridge over time during its service life; wherein, the degradation parameter sequence includes parameters reflecting the corrosion state and mechanical performance degradation of the structure.

[0045] Specifically, the process of obtaining the degradation parameter sequence consists of four sequential stages: corrosion initiation time determination, corrosion evolution calculation, mechanical property degradation calculation, and XGBoost model mapping. The calculation results of the previous stage provide input data for the next stage. Finally, the XGBoost model integrates all degradation parameters and outputs the predicted values ​​of the structural mechanical response of the bridge at each service time point, forming a complete degradation parameter sequence.

[0046] Phase 1: Determining the Corrosion Initiation Time. In the initial service phase after a bridge is built, the reinforcing steel within the concrete cover is in a passivated state maintained by an alkaline environment, and activation corrosion has not yet occurred on the steel surface. As service time progresses, chloride ions from the atmosphere or marine environment migrate inwards through the pore network of the concrete cover via diffusion, gradually accumulating on the surface of the reinforcing steel. The one-dimensional unsteady-state diffusion process of chloride ions in concrete is governed by Fick's second law, and the partial differential equation is: .in, Indicates service time Distance from concrete surface at any time The mass percentage of chloride ions at the location. Indicates time dependence Ambient temperature and relative humidity The time-varying chloride ion diffusion coefficient. The time-varying diffusion coefficient reflects three types of physical effects that occur over time in concrete hydration: decreased porosity due to increased ion migration caused by rising temperature, and changes in humidity affecting the connectivity of diffusion paths. Time-varying diffusion coefficient The expression is in the form of a product of three independent correction factors, specifically: .in, This represents the baseline chloride ion diffusion coefficient measured under reference age (usually 28 days), reference temperature, and reference humidity conditions, in units of... ; This represents the age decay factor of concrete. For reference age (28 days) The age decay index typically ranges from 0.2 to 0.6. This factor reflects the monotonically decreasing diffusion coefficient over time due to the densification of the pore structure caused by the continuous hydration of cementitious materials. This represents the temperature correction factor, which describes the thermal activation process based on the Arrhenius formula. The ion migration rate increases exponentially with increasing temperature. The humidity correction factor is represented by an S-shaped function that describes the nonlinear effect of relative humidity on pore water connectivity. When the relative humidity is too low, the pore water becomes discontinuous, causing the diffusion path to be interrupted. When the relative humidity approaches saturation, the diffusion channel is fully connected.

[0047] When the chloride ion concentration on the surface of the reinforcing steel reaches a level exceeding the critical chloride ion concentration At this point, the passivation film on the surface of the reinforcing steel is broken down in a localized area, and the depassivation process occurs, marking the onset of corrosion. The arrival of Fick's second law under the above boundary conditions (i.e., a constant surface chloride ion concentration on the concrete surface). Substituting the analytical solution under the condition that the initial chloride ion concentration inside the concrete is zero into the chloride ion concentration on the surface of the reinforcing steel, we get: Given the given conditions, the implicit equation for the corrosion initiation time is obtained by solving the equation as follows:

[0048]

[0049] in, This indicates the thickness of the concrete cover, which is the shortest distance from the concrete surface to the surface of the outermost steel reinforcement, in mm. This represents the exposure environment factor, used to correct for differences between the actual bridge's exposure environment and standard test environments, such as the splash zone, tidal zone, and underwater zone. Different values; This represents the curing condition coefficient, which reflects the influence of the curing quality after concrete pouring on the diffusion rate. The more thorough the curing, the larger the curing condition coefficient. This represents the test method coefficient, used to convert the results of rapid chloride ion migration tests in the laboratory into the equivalent diffusion coefficient under long-term natural diffusion conditions; Let represent the inverse function of the Gaussian error function, and the expression for the Gaussian error function is: ; The critical chloride ion concentration is usually expressed as a percentage of the mass of the cementitious material. This indicates the chloride ion concentration on the concrete surface, determined by on-site sampling and testing.

[0050] because pass Depends on The implicit equations described above cannot be solved analytically directly; a fixed-point iteration method must be used for numerical solutions. The iteration process is as follows: First, the initial guess for the corrosion initiation time is set to... Typically, a small percentage of the bridge's design service life can be taken; Substitute the time-varying diffusion coefficient In The calculation yielded Substituting the diffusion coefficient value into the right-hand side of the implicit equation, a new time estimate is obtained. Repeat the above two steps until the... After the second iteration, we obtain When the difference between two adjacent iterations Less than the preset tolerance (e.g.) When (year), take Corrosion initiation time for convergence .

[0051] Phase Two: Corrosion Evolution Calculation. (During service life...) Exceeding the corrosion initiation time Subsequently, the steel reinforcement enters the activated corrosion stage, where iron atoms in the anodic region continuously dissolve and release electrons, while an oxygen reduction reaction occurs in the cathodic region. The severity of corrosion is determined by the corrosion current density. Characterization shows that the corrosion current density decreases with time according to a power law after the onset of corrosion. This decrease is due to the accumulation of corrosion product layers on the steel reinforcement surface, which increases the resistance to ion transport and slows down the rate of electrochemical reactions. The formula for the decrease in corrosion current density is: .in, The initial corrosion current density at the moment of corrosion initiation, in units of The initial corrosion current density is positively correlated with the ratio of concrete resistivity to chloride ion concentration on the steel surface, and can be calibrated by on-site linear polarization resistance method or AC impedance spectroscopy method; the exponent −0.29 is an attenuation index obtained by statistical analysis of a large amount of long-term natural exposure and accelerated corrosion test data, reflecting the physical phenomenon that the hindrance effect of corrosion product film thickening on ion transport gradually becomes significant over time but the rate of increase decreases; the coefficient 0.85 is an empirical constant related to the density of corrosion product film.

[0052] The corrosion current density is converted into the cross-sectional loss rate of the reinforcing steel using Faraday's law. In real-world environments, periodic variations in temperature, humidity, and chloride ion concentration significantly affect the corrosion rate. Therefore, three environmental correction factors are introduced to adjust the corrosion rate, resulting in the complete corrosion rate calculation formula: .in, This indicates the corrosion rate in the radial direction of the reinforcing bar, expressed in mm / year. The constant representing the electrochemical-mechanical conversion is derived from Faraday's law, and its value is... ,in The molar mass of iron, The density of iron, This represents the number of electrons transferred in the iron dissolution reaction. It is Faraday's constant; The temperature correction factor, based on the Arrhenius relation, is expressed as follows: ,in, The annual average temperature of the bridge's service environment is expressed in degrees Celsius. 284.15 is the Kelvin temperature scale value corresponding to the reference temperature, and 2283 is the ratio constant of the activation energy of the corrosion reaction to the universal gas constant. The humidity correction factor is expressed as follows: ,in, The annual average relative humidity of the bridge's service environment is expressed as a percentage, with 75% as the reference relative humidity. The chloride ion concentration correction factor is expressed as follows: ,in, This indicates the chloride ion concentration on the concrete surface. This indicates the critical chloride ion concentration.

[0053] Because corrosion of steel reinforcement in actual bridge structures is not merely uniform, pitting corrosion can also form in areas of localized chloride ion enrichment, resulting in uneven cross-sectional loss. To comprehensively describe the combined effects of uniform and pitting corrosion, the remaining steel reinforcement cross-sectional area... The calculation uses the formula .in, This represents the remaining cross-sectional area of ​​the steel bar considering only uniform corrosion, which is obtained by subtracting the uniform corrosion loss area obtained by integrating the corrosion rate over time from the initial cross-sectional area. It represents the reference area parameter in the pitting correction calculation, and also represents the initial total cross-sectional area of ​​the reinforcing steel. The average width of the pitting corrosion is determined based on the degree of localization of the corrosion current density. Indicates the initial nominal diameter of the reinforcing bar; This represents the local remaining cross-sectional area of ​​the section where the pitting pit is located. It is calculated using a simplified geometric model of a semi-ellipse or rectangle based on the pitting depth and pit width.

[0054] The degree of corrosion is defined based on the cross-sectional area of ​​the remaining reinforcing steel. This represents the percentage of the cross-sectional area lost relative to the initial area, used to uniformly quantify the corrosion severity of steel bars of different diameters; it defines the degree of corrosion. The expression is: .in, This represents the initial total cross-sectional area of ​​the reinforcing steel when it is not corroded. The remaining cross-sectional area of ​​the reinforcing steel after considering the combined effects of uniform corrosion and pitting corrosion is given above. The unit is percentage.

[0055] The third stage: Calculation of mechanical property degradation. Based on the cross-sectional loss caused by corrosion, the degradation effect of corrosion on the mechanical properties of the reinforcing steel and the concrete cover is further calculated. First, the yield strength of the reinforcing steel... The degradation rate linearly increases with the degree of corrosion, and the degradation relationship is based on statistical results from tensile tests on corroded steel bars; the yield strength of the steel bars... The expression is: .in, Indicates service time The yield strength of the steel reinforcement, expressed in MPa; This indicates the initial yield strength of the steel bar when it leaves the factory; the coefficient 0.005 indicates the rate of decrease in yield strength for every 1% increase in corrosion degree. The rate ratio is related to the corrosion morphology (uniform or pitting corrosion). For corrosion morphology dominated by pitting corrosion, the local stress concentration effect will lead to a greater rate of decrease in yield strength. This can be corrected based on on-site sampling and testing.

[0056] Secondly, the volume expansion of corrosion products (rust volume is approximately 2 to 6 times the original iron volume) exerts radial expansion pressure on the concrete cover surrounding the reinforcing steel. When this expansion pressure exceeds the tensile strength of the concrete, longitudinal cracks form within the cover and gradually extend to the surface as corrosion intensifies, leading to a significant degradation in the compressive strength of the concrete cover. The degradation relationship is as follows: .in, Indicates service time The compressive strength of the protective concrete; This indicates the initial compressive strength of the protective concrete layer that has not been affected by corrosion. This represents the peak compressive strain of the protective concrete under uniaxial compression. This represents the transverse equivalent tensile strain caused by corrosion cracks. The physical meaning of transverse equivalent tensile strain is the average tensile strain level generated in the circumferential direction of the concrete cover due to the volume expansion of corrosion products. The formula for calculating transverse equivalent tensile strain is: .in, This indicates the number of longitudinal reinforcing bars within the pier cross-section that have undergone activated corrosion. Indicates service time The average opening width of corrosion cracks on the concrete surface of the protective layer is calculated by a mechanical equilibrium model of corrosion product expansion or obtained by actual measurement of crack width. This represents the characteristic dimension of the pier cross-section perpendicular to the crack propagation direction (e.g., the diameter of a circular pier or the shorter side dimension of a rectangular pier). The physical meaning of this formula is: the expansion effect of corroded reinforcing steel is distributed within the characteristic dimension of the pier cross-section, and the ratio of the total crack width caused by the corrosion to the cross-sectional dimension is the equivalent tensile strain.

[0057] The fourth stage: XGBoost model mapping. The degradation parameters calculated in the previous three stages are discretely distributed at different time points and in different physical dimension spaces. In order to comprehensively map the dispersed degradation parameters into the overall mechanical response of the bridge structure, the XGBoost (Extreme Gradient Boosting Tree) model is used to establish a nonlinear mapping from multi-dimensional degradation parameters to structural mechanical performance indicators.

[0058] The input to the XGBoost model is a multi-dimensional feature vector, which can contain the following eight dimensions: the first dimension is the degree of erosion. The second dimension is the yield strength degradation ratio. The value ranges from (0,1); the third dimension is the concrete strength degradation ratio of the protective layer. The fourth dimension is the ratio of the cross-sectional area of ​​the remaining reinforcing steel. The fifth dimension reflects the degree of cross-sectional loss; the fifth dimension is the width of the corrosion crack. The sixth dimension is service time. The seventh dimension is the pier shear span ratio, defined as the ratio of the effective height of the pier to the depth of the cross-section. The pier shear span ratio affects the failure mode of the pier (bending failure or shear failure). The eighth dimension is the pier axial compression ratio, defined as the ratio of the axial pressure borne by the pier to the compressive bearing capacity of the cross-section. The pier axial compression ratio significantly affects the ductility and energy dissipation characteristics of the pier. These eight dimensions collectively characterize the degradation state of bridges from the material, component, geometric, and temporal levels.

[0059] The parameter structure of the XGBoost model is as follows: the base learner is a CART regression tree, and the total number of trees can be set to 300; the maximum depth of a single tree can be set to 6 to achieve a balance between model fitting ability and generalization performance; the learning rate can be set to 0.05, as a smaller learning rate combined with a larger number of trees helps improve the model's prediction accuracy and stability; the loss function is the mean squared error (MSE), i.e. To suppress overfitting, an explicit regularization term is added to the objective function. The regularization term is defined as follows: .in, The penalty coefficient for leaf node splitting is increased. This value will make the tree's growth more conservative in order to reduce the number of leaf nodes; Indicates the first The total number of leaf nodes in the regression tree; This represents the L2 regularization coefficient, which penalizes the sum of squares of the leaf node weights to prevent individual leaf node weights from becoming too large; Indicates the first The first of the trees The predicted weight values ​​on each leaf node.

[0060] In the The nth iteration (i.e., the training nth iteration) When there are 100 trees, the objective function of the XGBoost model is... The sum of losses from the training set is added together. The regularization terms of the trees constitute the complete expression of the objective function: .in, This represents the total number of training samples; Indicates the first The true mechanical response label value of each training sample; Indicates the preceding After accumulating the trees, the first Predicted values ​​for each training sample; Indicates the first tree to the first 8-dimensional input feature vector of each sample The output value; Let be the mean squared error loss function. In each training round, the objective function is adjusted regarding... The second-order Taylor expansion, using the first-order gradient of each sample. And the second-order gradient (Hessian). Aggregation occurs at the leaf node level, where... Represents the loss function Regarding the previous round of forecasts First-order partial derivative, sign subscript The variable that the partial derivative operation is performed on is ; The loss function is expressed as follows: The second-order partial derivative, sign superscript Indicates second-order differentiation and subscript The variable to be differentiated is denoted; the expanded objective function is approximately in the following form: A greedy algorithm is used to search for a splitting scheme that maximizes the gain of the objective function in the above expansion form among all possible splitting points of all features.

[0061] To prevent overtraining, an early stopping strategy is employed: after each iteration, the mean squared error on the validation set is calculated. If the mean squared error on the validation set no longer decreases over 10 consecutive iterations, the iteration stops and the model parameters are rolled back to the iteration with the smallest mean squared error on the validation set. The XGBoost model output expression after training is shown below. The weighted sum of the outputs of all 300 trees is expressed as: .in, This represents an 8-dimensional input feature vector. Indicates the first The predicted values ​​of a CART regression tree for an 8-dimensional input feature vector. Indicates the first The weights of each tree, for example, all trees with a learning rate of 0.05. All can be equal to 0.05.

[0062] The training sample set required for the XGBoost model is constructed as follows: Six representative time points with equal intervals are selected along the bridge's service time axis as nodes for structural degradation analysis. At each time point, 500 different combinations of degradation parameters are generated within a reasonable range of degradation parameter values ​​using the Latin hypercube sampling method. Each set of degradation parameters corresponds to a possible degradation scenario. For each set of degradation parameter combinations, a nonlinear static pushover analysis or nonlinear time history analysis is performed using a structural finite element analysis workstation to calculate the mechanical response indicators of the bridge structure under the degradation scenario (such as base shear coefficient, top displacement angle, or energy dissipation capacity, etc.) as sample label values. A total of 3000 samples are generated at each of the six time points, and are randomly divided into a training set (2100 sets) and a validation set (900 sets) at a ratio of 70% and 30%, respectively.

[0063] In batch nonlinear analyses performed on a structural finite element analysis workstation, the seismic demand relationship is described using a log-linear model to represent the median seismic capacity of the structure. Statistical relationship with peak ground acceleration (PGA): .in, Indicates service time The median seismic capacity of a bridge structure at a given PGA level, expressed, for example, in terms of spectral acceleration or base shear coefficient; The intercept parameter reflects the initial seismic resistance level of the structure. As degradation deepens, the absolute value of the intercept parameter increases. The slope parameter reflects the sensitivity of a structure's seismic resistance to earthquake intensity. Severe degradation can cause the slope parameter to become steeper, indicating an increased degree of nonlinearity in the structure. (Parameter) and The values ​​at each time point are determined by least squares linear regression in logarithmic coordinates based on the finite element analysis results.

[0064] S2. Based on the degradation parameter sequence, a damage state probability model driven by historical cumulative damage is constructed for the bridge under various degradation states, and the output depends on the historical seismic loads borne by the bridge structure.

[0065] Specifically, the implementation process of this step is based on the degradation parameter sequence and time-varying seismic demand coefficient output from step S1. and The implementation process includes three sequentially executed stages: first, establishing a probabilistic model of earthquake occurrence to describe the stochastic patterns of earthquake events affecting the bridge at any given time; second, under each earthquake event, based on the earthquake demand relationship output in step S1... The median seismic demand under structural degradation is calculated, and then the Park-Ang damage increment is calculated. The calculation of the damage increment depends on the historical damage state accumulated by the bridge before this earthquake. Finally, a damage state probabilistic evolution model based on an inverse Gaussian process is constructed, which accumulates the damage increments of several discrete seismic events into a damage state probability distribution on a continuous time axis.

[0066] Phase 1: Construction of the Earthquake Occurrence Probability Model. The occurrence pattern of earthquake events experienced by a bridge throughout its entire service life is considered a stochastic point process. A non-homogeneous Poisson process is used for modeling to accommodate the time-varying characteristics of seismic activity (e.g., long-period activity trends caused by plate tectonics). The intensity function of the non-homogeneous Poisson process... Defined service time The average number of earthquake events expected to occur in the vicinity per unit time is expressed by the intensity function as follows: .in, Indicates the start time of bridge service ( The annual average earthquake occurrence rate benchmark value is obtained by statistically analyzing the annual average number of earthquakes in the target area from the historical earthquake catalog. A coefficient representing the trend of seismic activity over time, when The time signature indicates a gradual increase in seismic activity during the service life. Time indicates a gradual decrease. It degenerates into a homogeneous Poisson process; This indicates the design service life of the bridge, used to make time dimensionless, thus improving the trend coefficient. They are comparable in a physical sense.

[0067] During the period It just happened inside The probability of a sub-earthquake event is given by the probability mass function of a non-homogeneous Poisson process: .in, Indicates time period The random variable counting earthquake events within the event; The cumulative intensity function represents the integral of the intensity function from the start of the service to the current time. The physical meaning of the cumulative intensity function is a time period. The total number of earthquake events expected to occur within the period; express factorial.

[0068] For each seismic event, the randomness of the ground motion intensity also needs to be described. The seismic exceedance probability curve is used to describe the peak ground acceleration (PGA) exceeding a certain level. The annual average probability is expressed as a function: .in, This indicates that the PGA exceeded The annual average exceedance probability is a dimensionless value; This represents the value of PGA, with units of gravitational acceleration g; coefficient This represents the site's seismic hazard coefficient. The value of the site's seismic hazard coefficient is directly related to the seismic intensity zoning type of the bridge's location—for high-intensity zones, This indicates that the annual exceedance probability curve has shifted upward overall, and the PGA level of the site-set earthquake is higher; for moderate intensity zones, For low-intensity areas, This indicates that the annual exceedance probability curve has shifted downwards overall, and the PGA level of the site-specified earthquake is low. The coefficient −15 in the exponential term is the attenuation slope parameter obtained from the regional seismic hazard probability analysis based on log-linear regression.

[0069] Phase Two: Park-Ang Damage Increment Calculation. For each seismic event occurring at the service time point, it is necessary to calculate the incremental damage to the bridge structure caused by the seismic event. This embodiment uses the Park-Ang two-parameter damage model, which comprehensively considers the contributions of two physical quantities—maximum deformation demand and cumulative hysteretic energy dissipation—to the structural damage. Damage increment under the influence of a secondary earthquake event The calculation formula is: .in, Indicates by the first The incremental damage to the bridge structure caused by the secondary earthquake event is a dimensionless scalar quantity. Indicates the first The maximum displacement angle reached by the bridge pier under the action of the secondary earthquake is output by the dynamic time history analysis of the structural finite element analysis workstation; Indicates before experiencing Following the earthquake, the cumulative damage reached The equivalent yield displacement angle of the bridge pier. For the front The cumulative damage value of the earthquake; This indicates the aforementioned cumulative damage. The ultimate displacement angle of a bridge pier under certain conditions is the displacement angle corresponding to the point when the bearing capacity of the component drops to a certain preset proportion (e.g., 80%) of the peak bearing capacity. The contribution weighting coefficient of hysteretic energy dissipation to damage is related to factors such as the stirrup ratio, axial compression ratio, and shear span ratio of the component, and can be determined by regression calibration from relevant test data. Indicates the first The total hysteretic energy consumed by the bridge pier components under the action of a second earthquake is obtained by summing the areas of all hysteresis loops in the force-displacement hysteresis curve, with units of _____. The yield moment in the denominator (Unit is) and limit displacement angle The product of (dimensionless radian values) has the same [force × length] dimension, therefore the hysteresis energy dissipation term as a whole is a dimensionless ratio, which is consistent with the dimension of the maximum displacement term in the first term; The yield moment of the bridge pier is represented by the moment-curvature analysis of the cross section during the elastic analysis stage.

[0070] Damage increment The physical structure of the calculation formula is divided into two parts: the first part The first part measures the extent to which the maximum deformation demand exceeds the yield plateau relative to the available ductility capacity; the second part... The first part measures the proportion of hysteretic energy dissipation relative to the maximum energy that the structure can dissipate, while the second part characterizes the cumulative fatigue damage caused by cyclic loading; the sum of the two parts constitutes the total incremental damage to the structure caused by this earthquake.

[0071] With cumulative damage As the cumulative damage increases, the mechanical properties of reinforced concrete members undergo irreversible degradation, manifested as an increase in the equivalent yield displacement angle (stiffness softening) and a decrease in the ultimate displacement angle (ductility degradation). The degradation relationships of these two key displacement angle parameters with cumulative damage are described by the following linear degradation functions: , .in, The initial equivalent yield displacement angle of the bridge pier when it is completely undamaged is determined by the section moment-curvature analysis combined with the plastic hinge length model. The initial limit displacement angle of the pier when it is completely undamaged is also determined by cross-sectional analysis and component parameters. The damage degradation coefficient represents the yield displacement angle. A positive damage degradation coefficient reflects the softening effect caused by cumulative damage, which leads to stiffness degradation and earlier yield point. This represents the damage degradation coefficient at the ultimate displacement angle. The damage degradation coefficient at the ultimate displacement angle is positive, reflecting the embrittlement effect caused by the reduction of ductility capacity due to cumulative damage. This represents the cumulative damage value of the bridge at the current moment. In the above formula, The value is always greater than or equal to 1 (softening effect), while The value is always less than or equal to 1 (embrittlement effect). The above degradation law is consistent with the trend of increased pinching of hysteresis curve and degradation of bearing capacity observed in a large number of quasi-static tests of reinforced concrete bridge piers.

[0072] The third stage: Inverse Gaussian process probabilistic evolution. In the previous two stages, the occurrence of earthquake events and the damage increment of each event were treated as random variables. Thus, the cumulative damage evolution process of the bridge during its service life constitutes a non-Markovian process dependent on the complete historical load path of the earthquakes. In order to transform the discrete earthquake event sequence into a continuous-time damage state probability distribution, an inverse Gaussian process is used to model the probabilistic law of damage evolution over time.

[0073] For any tiny time interval, the damage increment It follows an inverse Gaussian distribution, and the expression for the probability density function of the inverse Gaussian distribution is: .in, This represents the incremental damage to the bridge structure over the considered time interval; the incremental damage is a positive real number. The mean parameter represents the inverse Gaussian distribution, which is the expected value of the damage increment, also known as the drift parameter; This represents the shape parameter of the inverse Gaussian distribution; the shape parameter controls the degree of dispersion of the distribution. The larger the value, the smaller the variance of the distribution, and the lower the randomness of the damage increment; Pi is a constant. The support set of the inverse Gaussian distribution is (0, +∞), meaning that the inverse Gaussian distribution is defined only in the positive real number domain. This property of the support set ensures that the damage increment is always non-negative in any time interval, thus perfectly matching the monotonic and irreversible physical property of damage accumulation.

[0074] In a continuous-time frame, the two core characteristic functions of the inverse Gaussian process are the drift function. Sum of variance functions The drift function and variance function respectively characterize the bridge damage state at time t. The variance of the expected growth rate and the random growth rate. The drift function couples the seismic intensity function at each time point with the expected value of the damage increment caused by a single earthquake. The drift function is defined as: .in, This represents the earthquake-damage scaling factor, used to convert the product of earthquake frequency and expected damage increment into the rate of drift of damage state on the time axis. The dimension of the earthquake-damage scaling factor is the reciprocal of time. Let be the intensity function of the aforementioned non-homogeneous Poisson process, at time . The value at; Indicates service time The unconditional expected value of the damage increment caused by a single earthquake in the vicinity is calculated as follows: Note that in the Park-Ang damage increment formula, the maximum displacement angle... Earthquake demand relationship output by step S1 Damage increment depends on the peak ground acceleration (PGA) of the seismic event. In essence The function, denoted as (in (Indicates the value of PGA); The expectation is obtained by integrating over the earthquake exceedance probability distribution, i.e. At the same time, the statistical mean is also taken for other random parameters affecting the damage increment (such as hysteresis energy consumption); among them, and These are the upper and lower bounds of the integral, respectively.

[0075] There is a proportional relationship between the variance function and the drift function. The expression for the variance function is: .in, This represents the variance-mean proportionality coefficient, a positive constant calibrated through statistical analysis of historical earthquake damage increment data (i.e., calculating the ratio of the variance to the mean of the damage increment samples within each time window). The proportional relationship between the variance function and the drift function is an inherent characteristic of the inverse Gaussian process, reflecting the physical intuition that uncertainty in damage evolution is proportionally amplified with increasing damage level.

[0076] Based on the definitions of the drift function and variance function above, the bridge's service life... The probability distribution of the damage state at time is given by parameters, which are the drift parameters. and shape parameters The inverse Gaussian distribution. Thus far, the output of this step depends on the probability of damage state based on the historical seismic loads borne by the bridge structure. This damage state probability quantifies the likelihood of the bridge being in each level of damage at the current moment, providing damage state input for the subsequent real option pricing step. The probability of damage state based on the historical seismic loads borne by the bridge structure is denoted as... For ease of reference, the probability of non-Markovian damage states will be uniformly expressed as follows: Its meaning is that from the known time... The historical sequence of earthquake loads that have occurred so far Under the condition that the bridge structure is in the first Level of damage The probability of.

[0077] S3. Based on the parameters in the degradation parameter sequence, the continuous degradation process of the bridge is discretized into multiple degradation levels; by simulating the fluctuation characteristics of the expected returns of the reinforcement project under each degradation level, the asset volatility corresponding to each degradation level is obtained, and a mapping relationship between degradation level and asset volatility is established.

[0078] Specifically, corrosion degree indicators are extracted from the multidimensional degradation parameter vectors at each time point of the degradation parameter sequence output in step S1. and degraded yield strength component Combined with the initial yield strength of the uncorroded steel bars Calculate the remaining yield ratio The two are combined to form a two-dimensional degenerate feature vector. The K-means clustering algorithm is used to perform clustering on the set of two-dimensional degenerate feature vectors at all discrete time points. It is initialized with k-means++ and the convergence criterion is that the displacement of the cluster centers is less than... The maximum number of iterations is 300, and the number of clusters is determined by maximizing the silhouette coefficient. Discretize the continuous degradation process into The degradation levels are denoted as follows: Each degradation level From the cluster center coordinate vector Characterization, in which .

[0079] Before proceeding with the volatility calibration process, an economic loss assessment model must be defined. This model consists of three components: direct repair costs, detour operating costs, and delay time costs. The formula for calculating direct repair costs is as follows: ,in, Indicates the first The ratio of repair cost to reconstruction cost for a damage level increases non-linearly with increasing damage level—for minor injuries. A smaller value indicates that only local repair is needed, which is suitable for moderate damage. A moderate value indicates the need for extensive reinforcement, especially for severe damage. Taking the larger value indicates that the pier needs to be replaced; for complete failure... Approaching It indicates a near-reconstruction; This indicates the comprehensive unit price for rebuilding a bridge deck area; and These represent the effective width of the bridge deck and the total length of the bridge, respectively. Indicates the discount rate; This indicates the point in time when the cost occurs. The formula for calculating the cost of detour operation is: ,in, and These represent the average operating cost per unit mileage for passenger cars and freight vehicles, respectively. Indicates the proportion of freight traffic; ADT represents the average daily traffic volume per year. Indicates the additional driving distance by taking a detour; Indicates the first The duration of traffic disruption on a bridge under Level 1 damage conditions. The formula for calculating the cost of delay time is: ,in, and These represent the average hourly wage per passenger car passenger and truck driver, respectively. and These represent the average number of passengers carried by passenger cars and freight vehicles, respectively. This indicates the loss of value per unit of time due to goods delay; This represents the average speed of vehicles traveling along the detour route. The sum of the above three losses is the [number of losses]. Total economic loss under level 1 damage .

[0080] For each degradation level Based on cluster center vector Select the Euclidean distance from the degenerate evolution trajectory. At each representative time point, execution is performed. Monte Carlo simulation (simulation step size) (Using a 1-year timeframe to match the time resolution of annual earthquake risk assessments), each simulation utilizes the XGBoost surrogate model in step S1 and the non-Markovian damage state probabilities output in step S2. The difference in expected risk loss between the unreinforced and reinforced scenarios within the simulation step size is calculated to obtain the single-time realized value of the reinforcement simulation benefit. Collect logarithmic return series and calculate annualized volatility The expression is:

[0081]

[0082] in, Indicates the level of degradation Corresponding representative time points Simulated benefits of reinforcement at the location; For the sample standard deviation operator; This is an annualized factor. For all... Each degradation level is sequentially labeled, and a mapping table between degradation levels and volatility is established. .

[0083] S4. Construct a degenerate state coupled binary tree with the planning period as the expiration time and the decision point as the starting time; wherein, each node of the degenerate state coupled binary tree corresponds to a different decision point and stores the asset value representing the expected return of the reinforcement project.

[0084] Specifically, the cumulative revenue over the entire planning period calculated using the above steps is used as the initial target asset value of the binomial tree. , i.e., root node The present value of the reinforcement at the start of the planning period. As the initial execution price, where This represents the cumulative revenue over the entire planning period (the specific calculation will be explained in the subsequent description of step S6). For the number of bridge piers, The comprehensive unit price for single-pier FRP outsourced reinforcement construction. The expiration date of the binary tree is set. (That is, the expiration time of the binary tree is equal to the length of the bridge planning period), number of steps Make step length The node uses a dual index. ,in For time-level indexing, , This is an intra-layer position index. Each node stores triples. ,in Represents a node The value of the underlying assets at that location. Represents a node The degradation level index at which it is located. Represents a node The calculated value of the reinforced real option is used as the value of the nodal option.

[0085] Forward traversal starts from the root node, and the root node has a degradation level. according to The value of the root node asset is determined by looking up the two-dimensional degradation feature vector in a table. From a time perspective Towards During the process, the degradation level of child nodes is looked up in a table according to the real-time degradation feature vector, and an irreversible degradation constraint is applied. The irreversible degradation constraint means that for any parent node and its child nodes in the degenerate state coupled binary tree, the degradation level index of the child node must not be less than the degradation level index of the parent node. For each parent node... Obtain volatility from the mapping table Calculate three core parameters: , , .in, and They are nodes The rising and falling multipliers at the position, The risk-free rate of continuous compounding, This represents a risk-neutral probability. The asset value of the child node is: , The forward traversal is performed until all terminal nodes are filled. Due to the different degradation levels of each node, the volatility changes node by node, and the binary tree exhibits non-stationary characteristics.

[0086] S5. Based on the mapping relationship, obtain the asset volatility corresponding to the degradation level of each node in the degenerate state coupled binary tree; based on the asset volatility corresponding to each node, evolve the asset value of each node in the degenerate state coupled binary tree, and calculate the root node option value by backward recursion based on the asset value of each node to obtain the reinforced real option value.

[0087] Specifically, backward recursive pricing is initiated at the terminal layer. The time-varying formula for execution price is: Execution price upon expiration The terminal node option value is: .from The value of holding a node is obtained by recursively pushing from the root node to the next level.

[0088]

[0089] in, and These represent the option values ​​for the up and down sub-nodes, respectively. Node option value. Take the greater of the holding value and the immediate execution value, that is:

[0090]

[0091] in, .when Continue holding if necessary, otherwise immediately implement reinforcement. Recursively calculate the reinforcement real option value at the root node. For each candidate year of reinforcement The cumulative profits obtained from subsequent steps The above process is re-executed using the initial asset value as the root node, outputting the corresponding reinforced real option value. .

[0092] S6. Based on the probability of damage state, calculate the cumulative risk value for different candidate reinforcement years; determine the cumulative benefit for each candidate reinforcement year based on the cumulative risk value.

[0093] Specifically, the non-Markov damage state probability output by step S2 This characterizes the probability distribution of bridges in various damage states under given seismic load histories. Based on this, the expected value of economic losses caused by seismic action, i.e., seismic risk, can be further quantified. The cumulative risk throughout the entire planning period without reinforcement is also considered. Defined from to The cumulative risks of not reinforcing throughout the entire process: ,in Indicates at a point in time The unconditional annual earthquake risk at a location, i.e., the expected economic loss caused by an earthquake within that year, is related to the year of reinforcement. Irrelevant. For each discrete time point within the planning period. First, use the time-varying coefficients output in step S1. and Substitute earthquake demand relationship Calculate the median seismic demand under the degradation state, and then input the median seismic demand into the damage state probability calculation process in step S2 to obtain the value at time point. Given ground motion intensity Probability of damage state under certain conditions Then calculate the conditional earthquake risk. Its expression is:

[0094]

[0095] in, The conditions also include the history of seismic loads. and current seismic intensity Two dimensions. Unconditional annual risk is obtained through the integral of total probability: ,in, and These are the lower and upper bounds of the PGA integral, respectively, and we take the smaller and larger values ​​for the PGA level of interest. The earthquake exceedance probability function defined for step S2; For step S2 in time The previously generated earthquake load history sequence.

[0096] For each candidate year of reinforcement The total cumulative risk over the entire planning period after reinforcement consists of two periods: before and after reinforcement. The first part represents the cumulative period without reinforcement, and the second part represents the cumulative period after reinforcement and correction. To apply the reinforcement reduction correction to the annual seismic risk, the calculation method is to use the median of the original seismic demand. Replace with the median of the reinforced seismic demand. The same full probability integral process as in S2 and S3, with the same risk as without reinforcement, is then repeated to obtain the result. The median seismic demand after reinforcement is corrected using a piecewise function: when... hour, ;otherwise .in The reinforcement reduction factor is set at 0.485 for the carbon fiber composite reinforcement scheme and 0.411 for the glass fiber composite reinforcement scheme. The bridge degradation process continues after reinforcement is implemented, and degradation parameters are continued to be read from the degradation parameter sequence established in step S1. This assumes that the reinforcement measures improved the bridge's seismic resistance but did not stop the corrosion degradation process of the bridge materials themselves—a conservative assumption consistent with engineering reality.

[0097] Cumulative benefit is defined as the difference between the cumulative risk over the entire planning period without reinforcement and with reinforcement: This is the integral of the risk difference before and after reinforcement within the reinforced period. The benefit ratio is: The present value of the reinforcement cost is: .

[0098] S7. Using cumulative returns and the value of the real option for reinforcement as evaluation dimensions, a multi-criteria evaluation function is constructed to weight and score each candidate reinforcement year to obtain a comprehensive score; based on the comprehensive score, the optimal reinforcement time is determined from all candidate reinforcement years.

[0099] Specifically, for the candidate reinforcement year sequence Each of them Construct a four-element evaluation index group Reconstruct the degenerate state coupled binary tree starting from each candidate reinforcement year and execute the complete backward recursive pricing process, outputting each... corresponding Based on the four-tuple sequence, four corresponding optimal solutions are determined according to a single index: , , , .

[0100] The four optimal solutions mentioned above do not overlap in most cases because: the cumulative risk indicator focuses only on safety while ignoring economic costs; the cumulative return indicator considers both risk reduction and costs but neglects investment efficiency; the return ratio indicator emphasizes the return per unit cost but ignores the scale of absolute returns; and the option value indicator captures the value of utilizing future uncertainties during the delay period and may be biased towards later reinforcement years. The biases of each indicator mean that relying on any single indicator for decision-making may lead to suboptimal judgments.

[0101] To comprehensively consider the four indicators mentioned above, a deviation standardization method is used to normalize each indicator, eliminating differences in their dimensions and numerical ranges. For the maximization indicator ( Using forward standardization, the normalized general formula is: ,in and Representing indicators Minimum and maximum values ​​across all candidate reinforcement years; cumulative risk Negative standardization is adopted (the smaller the cumulative risk, the better): ,in and These represent the maximum and minimum cumulative risk across all candidate reinforcement years, respectively.

[0102] After normalizing the four indicators, a comprehensive optimization index is constructed, and the four normalized indicators are weighted and summed according to preset weights: .in, For candidate reinforcement years The comprehensive selection index, , , , These are the weighting coefficients for the four evaluation dimensions: cumulative risk, cumulative return, return ratio, and option value. Each weighting coefficient satisfies the normalization constraint. The specific values ​​of the weighting coefficients are determined by decision-makers based on factors such as the importance level of the transportation network in which the bridge is located, seismic design standards, available budget, and the risk preference of the management. Under normal circumstances, for high-importance bridges that serve as lifeline passages, A larger value can be chosen to emphasize safety as a priority; for situations with tight budget constraints, A larger value can be chosen to emphasize the priority of investment efficiency.

[0103] Finally, a comprehensive optimization index was selected from all candidate reinforcement years. The year in which the maximum value is reached is considered the optimal time for reinforcement. .in, The selected optimal time for reinforcement.

[0104] Under the aforementioned comprehensive evaluation framework, carbon fiber composite reinforcement schemes (CFC scheme, The value is 0.485) and the glass fiber reinforced polymer (GFRP) reinforcement scheme. The seismic demand reduction factor (with a value of 0.411) is used to repeat all steps S2 to S7 to obtain the optimal reinforcement timing and corresponding comprehensive evaluation index system for each of the two schemes. By comparing the results of the two schemes, decision-makers can not only determine the optimal year for reinforcement implementation, but also compare and select the best among different reinforcement material schemes, realizing a synergistic optimization closed loop between reinforcement timing decision and reinforcement material selection. This provides comprehensive decision support for the whole life-cycle seismic management of deteriorated bridges, combining scientific rigor, economic efficiency, and engineering feasibility.

[0105] The aforementioned method for determining the timing of seismic reinforcement of deteriorated bridges based on real options obtains a sequence of degradation parameters reflecting structural corrosion and mechanical performance degradation. It constructs a damage state probability model driven by historical cumulative damage, enabling the damage probability to reflect the path-dependent impact of the seismic load sequence on the structure. Simultaneously, based on the degradation parameters, the continuous degradation process is discretized into multiple degradation levels. By simulating the fluctuation characteristics of the expected returns of reinforcement projects under each degradation level, a dynamic mapping relationship between degradation levels and asset volatility is established. After constructing a coupled binary tree of degradation states with the planning period as the maturity date, the method further refines the model based on the specific degradation parameters. The asset volatility corresponding to the degradation level of a node drives the evolution of asset value. By calculating the option value of the root node through backward recursion, a reinforcement real option value that reflects the risk differences at different degradation stages is obtained. Finally, using the cumulative return calculated based on the probability of damage state and the reinforcement real option value as evaluation dimensions, a multi-criteria evaluation is used to weight and score each candidate reinforcement year to determine the optimal reinforcement timing. This collaboratively overcomes the problem of distortion in reinforcement timing judgment caused by traditional methods ignoring path dependence effects and adopting constant volatility assumptions, thereby improving the accuracy and reliability of seismic reinforcement timing decisions for deteriorated bridges.

[0106] Example 2:

[0107] Obtaining the time-varying degradation parameter sequence of a bridge during its service life includes the following steps:

[0108] S11. Obtain the surface chloride ion concentration, critical chloride ion concentration, protective layer thickness, and reference chloride ion diffusion coefficient of the bridge service environment.

[0109] S12. Input the surface chloride ion concentration, critical chloride ion concentration, protective layer thickness, and reference chloride ion diffusion coefficient into the corrosion initiation time calculation model based on Fick's second law, and solve for the corrosion initiation time using the fixed-point iteration method. The fixed-point iteration method takes the set initial corrosion initiation time as the starting point of the iteration. In each iteration, the corrosion initiation time obtained in the previous iteration is substituted into the chloride ion diffusion coefficient expression to calculate the chloride ion diffusion coefficient for each iteration. The corrosion initiation time is then recalculated using the chloride ion diffusion coefficient of each iteration. The recalculated corrosion initiation time is used as the input for the next iteration until the difference between the corrosion initiation times obtained in two adjacent iterations is less than the convergence threshold. The corrosion initiation time obtained in the last iteration is taken as the final solution.

[0110] S13. Based on the corrosion initiation time and combined with the principle of electrochemical corrosion, calculate the corrosion rate at each subsequent moment, and integrate the corrosion rate over time to obtain the cumulative corrosion amount at each moment; calculate the remaining steel cross-sectional area and yield strength degradation value of the bridge at each moment based on the cumulative corrosion amount, and integrate the remaining steel cross-sectional area and yield strength degradation value at each moment into a degradation parameter sequence.

[0111] Specifically, surface chloride ion concentration By analyzing chloride ion content data at different depths using Fick's second law, The result was obtained by performing least-squares fitting and extrapolating to the concrete surface, where... Indicates the depth from the concrete surface. Indicates the chloride ion diffusion coefficient. Indicates exposure time. Critical chloride ion concentration. The chloride ion concentration corresponding to the location where visible corrosion products first appear on the surface of the reinforcing steel is measured. Protective layer thickness. Obtained using a rebar cover thickness measuring instrument. Reference chloride ion diffusion coefficient. Obtained through a rapid chloride ion permeation test.

[0112] The basic equation for corrosion initiation time is: ,in Let be the corrosion initiation time to be determined. The time-varying chloride ion diffusion coefficient is expressed as the product of three correction functions: The first correction function adopts a power-law decay form: In the formula For reference age, The age factor characterizes the rate at which the diffusion coefficient decreases with age due to concrete hydration and densification. The second correction function adopts the Arrhenius equation form: In the formula The activation energy for chloride ion diffusion. This is the universal gas constant. For reference temperature, The ambient temperature is used. The third correction function adopts the adsorption isotherm model form: In the formula For ambient relative humidity, The critical relative humidity reflects the effect of pore water saturation on the connectivity of diffusion paths. Because... The independent variable in The corrosion initiation time is the unknown, forming an implicit nonlinear equation, which is solved using the fixed-point iteration method: initial values ​​are set... And index the iteration rounds. Set to zero; in the first... In the round of iterations Substitution Expression calculation Substitute into the fundamental equations and solve using inversion. ,in Represents the inverse function of the error function; the difference between the results of two consecutive iterations. Less than the convergence threshold Output the final solution .

[0113] Obtain corrosion start time Subsequently, the corrosion rate at each time point was calculated according to Faraday's law of electrolysis. The cumulative corrosion amount and the cross-sectional area of ​​the remaining steel reinforcement at each time point were obtained by integrating the corrosion rate over time. Then, the degradation value of the yield strength at each time point was obtained. The cross-sectional area of ​​the remaining steel reinforcement, the degradation value of the yield strength, the degradation value of the concrete strength of the protective layer, and the crack width at each time point were integrated into a degradation parameter sequence.

[0114] Example 3:

[0115] Based on the degradation parameter sequence, a damage state probability model driven by historical cumulative damage is constructed for the bridge under various degradation states. The model outputs the damage state probability dependent on the historical seismic loads borne by the bridge structure, including the following steps:

[0116] S21. Model the arrival process of seismic events during the service life of the bridge as a non-homogeneous Poisson process. Based on the non-homogeneous Poisson process, randomly generate the peak ground acceleration for each seismic event through the probability distribution of site seismic hazard.

[0117] S22. For each earthquake event, determine the bridge structure degradation state at the time of the earthquake event based on the degradation parameter sequence; using peak ground acceleration, bridge structure degradation state, and current cumulative damage as inputs, employ a damage increment calculation model dependent on the current cumulative damage to calculate the damage increment generated by the earthquake event in the bridge structure degradation state; add the damage increment to the current cumulative damage to obtain the updated cumulative damage; wherein, in the damage increment calculation model, the yield displacement angle of the bridge structure degrades and shifts in the direction of increasing as the cumulative damage increases, and the ultimate displacement angle of the bridge structure degrades and shifts in the direction of decreasing as the cumulative damage increases.

[0118] S23. The time-varying probability distribution of the updated cumulative damage is modeled using an inverse Gaussian process to obtain a time-varying cumulative damage probability distribution model; wherein, the drift function and variance parameter function of the inverse Gaussian process are jointly determined by the random sequence of earthquake events and the degradation parameter sequence.

[0119] S24. Based on the time-varying cumulative damage probability distribution model, multiple cumulative damage paths driven by historical earthquake load event sequences are generated through Monte Carlo simulation; the frequency of occurrence of each damage state in the cumulative damage path is counted to obtain the damage state probability.

[0120] Specifically, the core parameters of the non-homogeneous Poisson process are determined from the site's historical seismic catalog using the maximum likelihood method: the Gutenberg-Richter relation... Perform linear regression fitting, where Indicates a magnitude greater than or equal to The cumulative frequency of earthquake events, For seismic activity parameters, This is the ratio coefficient for earthquakes of different magnitudes, which is then used to calculate the baseline earthquake occurrence rate. ,in This represents the minimum magnitude threshold considered in engineering. For each seismic event, the corresponding peak ground acceleration is randomly generated based on the site's ground motion attenuation relationship, and its basic form is as follows: ,in Let be the seismic attenuation function. The magnitude is [missing information]. The epicentral distance is... The equivalent shear wave velocity within a 30-meter depth of the site. For a standard normally distributed random variable, is the logarithmic standard deviation.

[0121] Yield bending moment The yield moment was obtained as follows: A fiber section discretization method was used. The confined concrete fibers were modeled using the Mander confined concrete constitutive model to consider the strength and ductility-enhancing effects of stirrups. The steel reinforcement fibers were modeled using the Menegotto-Pinto constitutive model to describe the Bauschinger effect of the steel reinforcement. The yield moment was determined using the equivalent energy method through section moment-curvature analysis. Hysteretic energy dissipation... The extraction method is as follows: the force-displacement hysteresis curve is integrally summed ring by ring using the composite trapezoidal numerical integration formula. ,in For resilience, For displacement, Indicates along a single hysteresis loop The closed-loop integral. The acceleration factor of the load sequence effect in the drift function. The calibration method is as follows: Construct two independent Monte Carlo simulations. The first simulation ignores the influence of previous cumulative damage on subsequent damage increments, while the second simulation fully incorporates the cumulative damage dependency. The calibration is performed by regression analysis using the ratio of the cumulative damage time series obtained from the two simulations. The proportionality coefficient between the variance function and the drift function is the dispersion coefficient. Estimated by the ratio of variance to mean of incremental damage samples, i.e. ,in Damage increment samples within each time window variance This represents the corresponding mean. In S24, the Monte Carlo simulation uses the MersenneTwister pseudo-random number generator to generate seismic event sequences and PGA samples, and uses a relative rate of change of the probability of damage state between two adjacent simulation batches that is less than a preset convergence criterion as the simulation termination condition.

[0122] Example 4:

[0123] Using peak ground acceleration, bridge structural degradation state, and current accumulated damage as inputs, a damage increment calculation model dependent on current accumulated damage is employed to calculate the damage increment generated by a seismic event under bridge structural degradation state. The calculation includes the following steps:

[0124] S221. Using peak ground acceleration and bridge structural degradation state as inputs, calculate the maximum inter-story drift angle of the bridge structure under the structural degradation state caused by the earthquake event through the seismic demand proxy model.

[0125] S222. Based on the maximum inter-story drift angle and the current cumulative damage, the damage increment of the bridge structure caused by the earthquake event is calculated using the Park-Ang type damage increment model.

[0126] The expression for the Park-Ang type damage increment model consists of the superposition of a maximum displacement term and a hysteresis energy dissipation term, expressed as: .in, Indicates the first The incremental damage to the bridge structure caused by the secondary earthquake event; in the maximum displacement term middle, Indicates the first The maximum inter-story drift angle of a bridge structure in a state of structural degradation caused by a secondary earthquake event. Indicates preceding The cumulative damage to the bridge structure caused by this earthquake event is [value missing]. The yield displacement angle of the bridge structure under the given conditions Indicates preceding The cumulative damage to the bridge structure caused by this earthquake event is [value missing]. The ultimate displacement angle of the bridge structure under the given conditions Indicates the preceding The cumulative damage to the bridge structure caused by the earthquake event; in the hysteretic energy dissipation term. middle, The dimensionless weighting coefficient representing the contribution of hysteretic energy dissipation to bridge structural damage. Indicates the first The cumulative hysteresis energy dissipation extracted by the area integral of the force-displacement hysteresis loop of the bridge structure under the action of a secondary earthquake. This represents the yield moment, which is proportional to the degradation yield strength of the bridge structure.

[0127] Yield displacement angle of bridge structure and the ultimate displacement angle of the bridge structure With cumulative damage The degenerate expressions are as follows: , .in, and Each indicates the first The cumulative damage caused by this earthquake to the bridge structure The yield displacement angle and ultimate displacement angle of the bridge structure at that time. and These represent the initial yield displacement angle and the initial ultimate displacement angle when the bridge structure is undamaged, respectively. This represents the sensitivity coefficient to the degradation of the yield displacement angle of a bridge structure. This represents the sensitivity coefficient to the degradation of the ultimate displacement angle of a bridge structure.

[0128] Specifically, the earthquake demand surrogate model takes the peak ground acceleration (PGA) and the bridge structure degradation state as inputs, and outputs the maximum inter-story drift angle. Its construction method is consistent with the sample construction and analysis process of the XGBoost model in Example 1. In the overall expression of the Park-Ang type damage increment model, Indicates by the first The incremental damage to bridge structures caused by the secondary earthquake event is a dimensionless scalar quantity. Indicates the first The maximum inter-story drift angle reached by the bridge piers under the action of a secondary earthquake; Indicates before experiencing The cumulative damage from this earthquake reached The equivalent yield displacement angle of the bridge pier. For the front The cumulative damage value of the earthquake; Indicates the same cumulative damage The ultimate displacement angle of a bridge pier under certain conditions is the displacement angle corresponding to the preset proportion by which the bearing capacity of the component drops to the peak bearing capacity. This represents the weighting coefficient for the contribution of hysteretic energy dissipation to damage, and its value is related to the stirrup ratio, axial compression ratio, and shear span ratio of the component. Indicates the first Total hysteretic energy consumed by bridge pier components under the action of a secondary earthquake; This represents the yield bending moment of the bridge pier section. Yield displacement angle. and limit displacement angle With cumulative damage In the degeneracy formula, The equivalent yield displacement angle of the pier in the initial undamaged state is determined by the section moment-curvature analysis combined with the plastic hinge length model; This represents the ultimate displacement angle of the pier in its initial undamaged state; This represents the degradation coefficient of the yield displacement angle. A positive value reflects the softening effect of cumulative damage leading to stiffness degradation and forward shift of the yield point; This represents the degradation coefficient of the limiting displacement angle. A positive value reflects the embrittlement effect that leads to a reduction in ductile capacity due to cumulative damage; This represents the current cumulative damage value. Degradation coefficient. and Multiple identical reinforced concrete pier specimens were subjected to different levels of pre-damage before being monotonically pushed over until failure, along with intact control specimens. The equivalent yield displacement angle and equivalent ultimate displacement angle were extracted from the push-over curves to accumulate damage. The degenerate formula is fitted with least squares regression on the x-axis, and the slope of the fitted line is normalized to obtain the calibration value.

[0129] Example 5:

[0130] refer to Figure 2 Based on the asset volatility corresponding to each node, the asset value of each node in the degenerate state coupled binary tree is evolved. The root node option value is calculated backward recursively based on the asset value of each node to obtain the reinforced real option value, including the following steps:

[0131] S51. Starting from the root node of the degenerate coupled binary tree, traverse forward. For each node, calculate the up multiplier and down multiplier based on the asset volatility of the corresponding node, and use the risk-free rate, up multiplier and down multiplier to calculate the risk-neutral probability of the corresponding node.

[0132] S52. Multiply the asset value of the parent node by the upward multiplier to obtain the asset value of the upward child node, multiply the asset value of the parent node by the downward multiplier to obtain the asset value of the downward child node, and assign the risk-neutral probability to the child node of the corresponding node; wherein, the degradation level of the child node is subject to the degradation irreversibility constraint, so that the degradation level of the child node shall not be lower than the degradation level of the parent node.

[0133] S53. In the terminal layer of the degenerate coupled binary tree, the immediate execution value obtained by subtracting the current execution price from the asset value of each terminal node is used as the option value of the corresponding terminal node; where the terminal layer is the expiration time layer of the planning period.

[0134] S54. Starting from the terminal layer, proceed backwards. For each node, calculate the expected value of the option to continue holding based on the node's risk-neutral probability and the option value of the corresponding child node. Compare the expected value of the option with the immediate execution value obtained by subtracting the current execution price from the asset value of the corresponding node. Use the larger of the expected value of the option and the immediate execution value as the option value of the corresponding node. When the recursion reaches the root node, complete the backward recursion and use the option value of the root node as the value of the solidified real option.

[0135] The formula for calculating the ascending multiplier is: The formula for calculating the downward multiplier is: The formula for calculating the risk-neutral probability is: .in, Represents a node The ascending multiplier, Represents a node The downward multiplier, Indicates the first The asset volatility corresponding to each degradation level This represents the time step of the binary tree. Represents a node The risk-neutral probability, Represents the risk-free interest rate that is compounded continuously; node middle, For time-level indexing, This provides the position index of a node within the same time layer.

[0136] Specifically, For the first The asset volatility corresponding to each degradation level is obtained from the mapping relationship in step S3. The time step width of the binary tree, index For time-level indexing, This is the index of the position within the layer. During the forward traversal, the parent node's asset value... Multiplying by the up-row multiplier yields the asset value of the up-row child node. Multiplying the value by the downlink multiplier yields the asset value of the downlink child node. Meanwhile, the degradation level of child nodes is subject to the constraint of irreversible degradation. In the backward recursion, the formula for calculating the option value of terminal layer nodes is: ,in For terminal nodes The asset value, This is the execution price at the end of the planning period. The formula for calculating the holding value of internal nodes is: ,in and These are the option values ​​for the upward and downward child nodes, respectively. For single-step risk-free discounting factors, and These represent the risk-neutral probabilities for upward and downward sub-nodes, respectively. The node option value is the greater of the holding value and the immediate exercise value. ,in The current node's asset value, This represents the execution price corresponding to the current time frame. Asset volatility at each node. Degradation level labeling By reading the S3 step mapping relationship layer by layer, different degradation levels correspond to different volatility, which makes the backward recursive discounted expectation differentiated layer by layer, thus fully preserving the non-stationary impact of the degradation process on option value.

[0137] The aforementioned method for determining the timing of seismic reinforcement of deteriorated bridges based on real options obtains a sequence of degradation parameters reflecting structural corrosion and mechanical performance degradation. It constructs a damage state probability model driven by historical cumulative damage, enabling the damage probability to reflect the path-dependent impact of the seismic load sequence on the structure. Simultaneously, based on the degradation parameters, the continuous degradation process is discretized into multiple degradation levels. By simulating the fluctuation characteristics of the expected returns of reinforcement projects under each degradation level, a dynamic mapping relationship between degradation levels and asset volatility is established. After constructing a coupled binary tree of degradation states with the planning period as the maturity date, the method further refines the model based on the specific degradation parameters. The asset volatility corresponding to the degradation level of a node drives the evolution of asset value. By calculating the option value of the root node through backward recursion, a reinforcement real option value that reflects the risk differences at different degradation stages is obtained. Finally, using the cumulative return calculated based on the probability of damage state and the reinforcement real option value as evaluation dimensions, a multi-criteria evaluation is used to weight and score each candidate reinforcement year to determine the optimal reinforcement timing. This collaboratively overcomes the problem of distortion in reinforcement timing judgment caused by traditional methods ignoring path dependence effects and adopting constant volatility assumptions, thereby improving the accuracy and reliability of seismic reinforcement timing decisions for deteriorated bridges.

[0138] It should be understood that although the steps in the flowcharts of the embodiments described above are shown sequentially according to the arrows, these steps are not necessarily executed in the order indicated by the arrows. Unless explicitly stated herein, there is no strict order restriction on the execution of these steps, and they can be executed in other orders. Moreover, at least some steps in the flowcharts of the embodiments described above may include multiple steps or multiple stages. These steps or stages are not necessarily completed at the same time, but can be executed at different times. The execution order of these steps or stages is not necessarily sequential, but can be performed alternately or in turn with other steps or at least some of the steps or stages of other steps.

[0139] Based on the same inventive concept, this application also provides a system for implementing the above-mentioned method for determining the timing of seismic reinforcement of deteriorated bridges based on real options. The solution provided by this system is similar to the implementation scheme described in the above method. Therefore, the specific limitations of one or more embodiments of the system for determining the timing of seismic reinforcement of deteriorated bridges based on real options provided below can be found in the limitations of the method for determining the timing of seismic reinforcement of deteriorated bridges based on real options described above, and will not be repeated here.

[0140] In one exemplary embodiment, such as Figure 3 As shown, a real options-based seismic reinforcement timing decision system 30 for deteriorated bridges is provided to implement the methods in the above embodiments. The system includes:

[0141] The degradation parameter acquisition module 31 is used to acquire the degradation parameter sequence of the bridge over time during its service life; wherein, the degradation parameter sequence includes parameters reflecting the corrosion state and mechanical performance degradation of the structure.

[0142] The damage probability modeling module 32 is used to construct a damage state probability model of the bridge under various degradation states driven by historical cumulative damage based on the degradation parameter sequence, and output the damage state probability that depends on the historical seismic loads that the bridge structure has borne.

[0143] The volatility mapping construction module 33 is used to discretize the continuous degradation process of the bridge into multiple degradation levels based on the parameters in the degradation parameter sequence; by simulating the volatility characteristics of the expected return of the reinforcement project under each degradation level, the asset volatility corresponding to each degradation level is obtained, and a mapping relationship between degradation level and asset volatility is established.

[0144] The coupled binary tree construction module 34 is used to construct a degenerate state coupled binary tree with the planning period as the expiration time and the decision point as the starting time; wherein, each node of the degenerate state coupled binary tree corresponds to a different decision point and stores the asset value representing the expected return of the reinforcement project.

[0145] The option value calculation module 35 is used to obtain the asset volatility corresponding to the degradation level of each node in the degenerate state coupled binary tree based on the mapping relationship; based on the asset volatility corresponding to each node, the asset value of each node in the degenerate state coupled binary tree is evolved, and the root node option value is calculated backward recursively based on the asset value of each node to obtain the reinforced real option value.

[0146] The risk-return calculation module 36 is used to calculate the cumulative risk value for different candidate reinforcement years based on the probability of damage status; and to determine the cumulative return for each candidate reinforcement year based on the cumulative risk value.

[0147] The decision optimization module 37 is used to evaluate the cumulative revenue and the value of the real option for reinforcement by constructing a multi-criteria evaluation function, weighted scoring of each candidate reinforcement year to obtain a comprehensive score; and based on the comprehensive score, the optimal reinforcement time is determined from all candidate reinforcement years.

[0148] Embodiments of this application also provide a computer device, including a memory and a processor, wherein the memory stores a computer program, and the processor executes the computer program to implement the steps in the aforementioned method embodiments.

[0149] Embodiments of this application also provide a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the steps in the above-described method embodiments.

[0150] For the device embodiments, since they basically correspond to the method embodiments, the relevant parts can be referred to in the description of the method embodiments. The device embodiments described above are merely illustrative. The components described as separate parts may or may not be physically separate, and the components shown as units may or may not be physical units, that is, they may be located in one place or distributed across multiple network units. Some or all of the modules can be selected to achieve the purpose of this disclosure according to actual needs. Those skilled in the art can understand and implement this without creative effort.

[0151] The above-described embodiments are merely illustrative of several implementation methods of the embodiments of this application, and their descriptions are relatively specific and detailed. However, they should not be construed as limiting the scope of the patent application. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the embodiments of this application, and these modifications and improvements all fall within the protection scope of the embodiments of this application.

Claims

1. A method for determining the timing of seismic reinforcement of deteriorated bridges based on real options, characterized in that, The method includes: S1. Obtain the degradation parameter sequence of the bridge over time during its service life; wherein, the degradation parameter sequence includes parameters reflecting the corrosion state and mechanical performance degradation of the structure; S2. Based on the degradation parameter sequence, a damage state probability model driven by historical cumulative damage is constructed for the bridge under each degradation state, and the damage state probability depends on the historical seismic loads borne by the bridge structure. S3. Based on the parameters in the degradation parameter sequence, the continuous degradation process of the bridge is discretized into multiple degradation levels; by simulating the fluctuation characteristics of the expected returns of the reinforcement project under each degradation level, the asset volatility corresponding to each degradation level is obtained, and a mapping relationship between the degradation level and the asset volatility is established. S4. Construct a degraded state coupled binary tree with the planning period as the expiration time and the decision point as the start time; wherein, each node of the degraded state coupled binary tree corresponds to a different decision point and stores the asset value representing the expected return of the reinforcement project; S5. Based on the mapping relationship, obtain the asset volatility corresponding to the degradation level of each node in the degenerate state coupled binary tree; based on the asset volatility corresponding to each node, evolve the asset value of each node in the degenerate state coupled binary tree, and calculate the root node option value by backward recursion based on the asset value of each node to obtain the reinforced real option value. S6. Based on the probability of the damage state, calculate the cumulative risk value for different candidate reinforcement years; determine the cumulative benefit for each candidate reinforcement year based on the cumulative risk value; S7. Using the cumulative revenue and the value of the reinforcement real option as evaluation dimensions, a multi-criteria evaluation function is constructed to weight and score each candidate reinforcement year to obtain a comprehensive score; based on the comprehensive score, the optimal reinforcement time is determined from all the candidate reinforcement years.

2. The method according to claim 1, characterized in that, The acquisition of the bridge's degradation parameter sequence over time during its service life includes: S11. Obtain the surface chloride ion concentration, critical chloride ion concentration, protective layer thickness, and reference chloride ion diffusion coefficient of the bridge service environment. S12. Input the surface chloride ion concentration, the critical chloride ion concentration, the protective layer thickness, and the reference chloride ion diffusion coefficient into the corrosion initiation time calculation model based on Fick's second law, and solve the corrosion initiation time using the fixed-point iteration method; wherein, the fixed-point iteration method takes the set initial corrosion initiation time as the iteration starting point, substitutes the corrosion initiation time obtained in the previous iteration into the chloride ion diffusion coefficient expression in each iteration, calculates the chloride ion diffusion coefficient for each iteration, and uses the chloride ion diffusion coefficient of each iteration to resolve the corrosion initiation time, and uses the resolved corrosion initiation time as the input for the next iteration, until the difference between the corrosion initiation times obtained in two adjacent iterations is less than the convergence threshold, and uses the corrosion initiation time obtained in the last iteration as the final solution; S13. Based on the corrosion initiation time, calculate the corrosion rate at each subsequent moment using the principle of electrochemical corrosion, and integrate the corrosion rate over time to obtain the cumulative corrosion amount at each moment; calculate the remaining steel reinforcement cross-sectional area and yield strength degradation value of the bridge at each moment based on the cumulative corrosion amount, and integrate the remaining steel reinforcement cross-sectional area and yield strength degradation value at each moment into the degradation parameter sequence.

3. The method according to claim 1, characterized in that, Based on the degradation parameter sequence, a damage state probability model driven by historical cumulative damage is constructed for the bridge under each degradation state. The output of the damage state probability depends on the historical seismic loads borne by the bridge structure, including: S21. The arrival process of seismic events during the service life of the bridge is modeled as a non-homogeneous Poisson process. Based on the non-homogeneous Poisson process, the peak ground acceleration of each seismic event is randomly generated through the site seismic hazard probability distribution. S22. For each earthquake event, determine the bridge structure degradation state at the time of the earthquake event based on the degradation parameter sequence; using the peak ground acceleration, the bridge structure degradation state, and the current cumulative damage as inputs, calculate the damage increment generated by the earthquake event under the bridge structure degradation state using a damage increment calculation model dependent on the current cumulative damage; add the damage increment to the current cumulative damage to obtain the updated cumulative damage; wherein, in the damage increment calculation model, the yield displacement angle of the bridge structure degrades and shifts in the direction of increasing as the cumulative damage increases, and the ultimate displacement angle of the bridge structure degrades and shifts in the direction of decreasing as the cumulative damage increases; S23. The time-varying probability distribution of the updated cumulative damage is modeled using an inverse Gaussian process to obtain a time-varying cumulative damage probability distribution model; wherein, the drift function and variance parameter function of the inverse Gaussian process are jointly determined by the random sequence of the earthquake event and the degradation parameter sequence; S24. Based on the time-varying cumulative damage probability distribution model, multiple cumulative damage paths driven by historical earthquake load event sequences are generated through Monte Carlo simulation; the frequency of occurrence of each damage state in the cumulative damage path is counted to obtain the probability of the damage state.

4. The method according to claim 3, characterized in that, The method uses the peak ground acceleration, the bridge structure degradation state, and the current cumulative damage as inputs, and employs a damage increment calculation model dependent on the current cumulative damage to calculate the damage increment generated by the seismic event under the bridge structure degradation state, including: S221. Using the peak ground acceleration and the bridge structure degradation state as inputs, calculate the maximum inter-story drift angle of the bridge structure under the degradation state caused by the earthquake event through the earthquake demand proxy model. S222. Based on the maximum inter-story drift angle and the current cumulative damage, calculate the damage increment of the bridge structure caused by the earthquake event using the Park-Ang type damage increment model; The expression for the Park-Ang type damage increment model consists of the superposition of a maximum displacement term and a hysteresis energy dissipation term, and is expressed as follows: ； in, Indicates the first The damage increment to the bridge structure caused by the secondary earthquake event; in the maximum displacement term middle, Indicates the first The maximum inter-story drift angle of the bridge structure in the degraded state caused by the secondary earthquake event. Indicates preceding The cumulative damage caused to the bridge structure by the earthquake event is [value missing]. The yield displacement angle of the bridge structure under the given conditions Indicates preceding The cumulative damage caused to the bridge structure by the earthquake event is [value missing]. The ultimate displacement angle of the bridge structure under the given conditions Indicates the preceding The cumulative damage value caused to the bridge structure by the earthquake event; in the hysteresis energy dissipation item middle, The dimensionless weighting coefficient represents the contribution of hysteresis energy dissipation to the damage to the bridge structure. Indicates the first The cumulative hysteresis energy dissipation is extracted by the area integral of the force-displacement hysteresis loop of the bridge structure under the action of the secondary earthquake. This represents the yield moment, which is proportional to the degraded yield strength of the bridge structure. The yield displacement angle of the bridge structure and the ultimate displacement angle of the bridge structure With cumulative damage The degenerate expressions are as follows: ； ； in, and Each indicates the first The cumulative damage caused by the earthquake to the bridge structure is as follows. The yield displacement angle and ultimate displacement angle of the bridge structure at that time. and These represent the initial yield displacement angle and the initial ultimate displacement angle when the bridge structure is undamaged, respectively. This represents the sensitivity coefficient to the degradation of the yield displacement angle of the bridge structure. This represents the degradation sensitivity coefficient of the ultimate displacement angle of the bridge structure.

5. The method according to claim 1, characterized in that, The asset value of each node in the degenerate state coupled binary tree is evolved based on the asset volatility corresponding to each node. The root node option value is then calculated backwards based on the asset value of each node to obtain the hardened real option value, including: S51. Starting from the root node of the degenerate coupled binary tree, traverse forward. For each node, calculate the upward multiplier and downward multiplier of the corresponding node based on the asset volatility of the node, and calculate the risk-neutral probability of the corresponding node using the risk-free rate, the upward multiplier and the downward multiplier. S52. Multiply the asset value of the parent node by the uplink multiplier to obtain the asset value of the uplink child node, multiply the asset value of the parent node by the downlink multiplier to obtain the asset value of the downlink child node, and assign the risk-neutral probability to the child node of the corresponding node; wherein, the degradation level of the child node is subject to the degradation irreversibility constraint, so that the degradation level of the child node shall not be lower than the degradation level of the parent node. S53. In the terminal layer of the degenerate state coupled binary tree, the immediate execution value obtained by subtracting the current execution price from the asset value of each terminal node is used as the option value of the corresponding terminal node; wherein, the terminal layer is the expiration time layer of the planning period; S54. Starting from the terminal layer, backward recursion is performed. For each node, the expected value of the option to continue holding is calculated based on the node's risk-neutral probability and the option value of the corresponding child node. The expected value of the option is compared with the immediate execution value obtained by subtracting the current execution price from the asset value of the corresponding node. The larger value between the expected value of the option and the immediate execution value is taken as the option value of the corresponding node. When the backward recursion reaches the root node, the backward recursion is completed, and the option value of the root node is taken as the value of the reinforced real option. The formula for calculating the upward multiplier is: ； The formula for calculating the downward multiplier is: ； The formula for calculating the risk-neutral probability is as follows: ； in, Represents a node The aforementioned ascending multipliers, Represents a node The aforementioned downlink multiplier, Indicates the first The asset volatility corresponding to each of the aforementioned degradation levels. This represents the time step of the binary tree. Represents a node The aforementioned risk-neutral probability, The risk-free rate represents continuously compounded interest; node middle, For time-level indexing, This provides the position index of a node within the same time layer.

6. A decision-making system for seismic reinforcement of deteriorated bridges based on real options, used to implement the method of any one of claims 1 to 5, characterized in that, The system includes: The degradation parameter acquisition module is used to acquire the degradation parameter sequence of the bridge over time during its service life; wherein, the degradation parameter sequence includes parameters reflecting the corrosion state and mechanical performance degradation of the structure; The damage probability modeling module is used to construct a damage state probability model of the bridge under each degradation state driven by historical cumulative damage based on the degradation parameter sequence, and output the damage state probability that depends on the historical seismic loads borne by the bridge structure. The volatility mapping construction module is used to discretize the continuous degradation process of the bridge into multiple degradation levels based on the parameters in the degradation parameter sequence; by simulating the volatility characteristics of the expected returns of the reinforcement project under each degradation level, the asset volatility corresponding to each degradation level is obtained, and a mapping relationship between the degradation level and the asset volatility is established. A coupled binary tree construction module is used to construct a degenerate state coupled binary tree with the planning period as the expiration time and the decision point as the start time; wherein, each node of the degenerate state coupled binary tree corresponds to a different decision point and stores the asset value representing the expected return of the reinforcement project; The option value calculation module is used to obtain the asset volatility corresponding to the degradation level of each node in the degenerate state coupled binary tree based on the mapping relationship; based on the asset volatility corresponding to each node, the asset value of each node in the degenerate state coupled binary tree is evolved, and the option value of the root node is calculated backward recursively based on the asset value of each node to obtain the value of the reinforced real option. The risk-return calculation module is used to calculate the cumulative risk value for different candidate reinforcement years based on the probability of the damage state; and to determine the cumulative return for each candidate reinforcement year based on the cumulative risk value. The decision optimization module is used to construct a multi-criteria evaluation function to weight and score each candidate reinforcement year based on the cumulative revenue and the reinforcement real option value as evaluation dimensions, and obtain a comprehensive score; based on the comprehensive score, the optimal reinforcement time is determined from all the candidate reinforcement years.

7. A computer device comprising a memory and a processor, wherein the memory stores a computer program, characterized in that, When the processor executes the computer program, it implements the method of any one of claims 1 to 5.

8. A computer-readable storage medium having a computer program stored thereon, characterized in that, When the computer program is executed by a processor, it implements the method of any one of claims 1 to 5.

Images