Method and apparatus for assessing seismic resilience of lower structure of double-column pier bridge

Abstract

A method and apparatus for assessing the seismic resilience of a lower structure of a double-column pier bridge. The method comprises: determining a damage index of a single component; determining a damage probability of the single component by means of the damage index of the component; on the basis of the damage probability of the single component, using a Copula function to calculate the system vulnerability of a lower structure of a double-column pier bridge; on the basis of the system vulnerability, calculating a post-earthquake function loss function of the lower structure of the double-column pier bridge; and using the post-earthquake function loss function and a function recovery function of the lower structure of the double-column pier bridge to calculate a seismic resilience index of the lower structure of the double-column pier bridge.

Description

Methods and apparatus for assessing the seismic toughness of substructures of double-column pier bridges Technical Field

[0001] The invention belongs to the field of bridge seismic toughness assessment technology, and in particular relates to a method and device for assessing the seismic toughness of the substructure of a double-column pier bridge. Background Technology

[0002] Traditional bridge structures primarily rely on ductile seismic design to prevent collapse under strong earthquakes, utilizing the hysteretic energy dissipation capacity of the main structure to absorb seismic energy. Damaged sections are then repaired after the earthquake to ensure continued normal operation. However, the substructure of a double-column pier bridge is statically indeterminate. When an earthquake occurs along the transverse direction, the stress on the piers becomes complex, making it very difficult to select appropriate pier damage indices. Furthermore, because structural toughness assessment methods are relatively new, the process often considers only a few influencing factors, and the results are frequently categorized into grades, leading to somewhat ambiguous conclusions that significantly affect the reliability of post-earthquake toughness assessments. Summary of the Invention

[0003] The technical problem to be solved by the present invention is to provide a method and device for assessing the seismic toughness of the substructure of a double-column pier bridge. This method can simultaneously consider the damage correlation between various components of the substructure of a double-column pier bridge and the impact of delay time on seismic toughness. It can select a more suitable seismic toughness assessment method for different bridges and accurately calculate the seismic toughness index.

[0004] To achieve the above objectives, the present invention adopts the following technical solution:

[0005] A method for assessing the seismic toughness of the substructure of a double-column pier bridge includes:

[0006] Step 1: Determine the damage index of individual components;

[0007] Step 2: Determine the damage probability of a single component using the component damage index;

[0008] Step 3: Calculate the vulnerability of the double-column pier substructure system using the Copula function based on the damage probability of individual components;

[0009] Step 4: Calculate the post-earthquake functional loss function of the substructure of the double-column pier bridge based on the system's vulnerability.

[0010] Step 5: Calculate the seismic toughness index of the substructure of the double-column pier bridge using the post-earthquake functional loss function and functional recovery function. Preferably, the calculation formula for the index evaluating the degree of pier damage is as follows:

[0011] Among them, K DXF is an indicator for evaluating the degree of damage to a bridge pier. j K represents the frequency value of a certain axial force occurring at the bottom of a bridge pier. j Let be the curvature of a characteristic point on the moment-curvature curve of the pier bottom section corresponding to a certain axial force.

[0012] Preferably, in step three, the Gumb Copula function is used to calculate the component correlation. The distribution function expression of the multivariate Gumb Copula function is as follows:

[0013] Where α is the relevant parameter, and α∈(0,1),

[0014] Introducing the Copula function, the vulnerability of multiple components to simultaneous damage can be expressed as:

[0015] P(X1X2...X k )=C(P1,P2,…,P k )

[0016] Among them, P k For the fragility of individual components; C(P1,P2,…,P) k To consider the multivariate Copula function related to multiple components;

[0017] Substituting these values ​​into the vulnerability expressions for series and parallel systems, the vulnerability P of the series system... fcl for:

[0018] Preferably, in step four, the system vulnerability considering component correlation is calculated using the Copula function, and the functional loss of the bridge is calculated based on the exceedance probability of this vulnerability. The calculation formula is as follows:

[0019] Where j represents the damage level of the bridge system; C s,j Repair costs for structural systems at damage level j; I s Costs for reconstruction after complete destruction; The cost-loss ratio; P j Let be the probability of failure of a bridge with damage state j under a certain PGA.

[0020] The probability P of bridge damage state j under PGA action is P j The calculation formula is:

[0021] in, Let represent the exceedance probability of each failure state under a certain PGA.

[0022] As preferred options, the functional recovery functions for post-earthquake double-column pier bridges include: linear functional recovery functions, exponential functional recovery functions, and trigonometric functional recovery functions.

[0023] This invention also provides a device for evaluating the seismic toughness of the substructure of a double-column pier bridge, comprising:

[0024] The first calculation module is used to determine the damage index of individual components;

[0025] The second calculation module is used to determine the probability of damage to a single component using the component damage index.

[0026] The third calculation module is used to calculate the vulnerability of the double-column pier substructure system based on the damage probability of individual components using the Copula function;

[0027] The fourth calculation module is used to calculate the post-earthquake functional loss function of the substructure of a double-column pier bridge based on the system's vulnerability.

[0028] The fifth calculation module is used to calculate the seismic toughness index of the substructure of a double-column pier bridge using the post-earthquake functional loss function and functional recovery function of the substructure.

[0029] As a preferred embodiment, the calculation formula for the indicators used by the first calculation module to evaluate the degree of damage to the pier is as follows:

[0030] Among them, K DX F is an indicator for evaluating the degree of damage to a bridge pier. j K represents the frequency value of a certain axial force occurring at the bottom of a bridge pier. j Let be the curvature of a characteristic point on the moment-curvature curve of the pier bottom section corresponding to a certain axial force.

[0031] Preferably, the third calculation module uses the Gumb Copula function to calculate component correlation. The distribution function expression of the multivariate Gumb Copula function is as follows:

[0032] Where α is the relevant parameter, and α∈(0,1),

[0033] Introducing the Copula function, the vulnerability of multiple components to simultaneous damage can be expressed as:

[0034] p(X1X2…X k )=C(P1,P2,…,P k )

[0035] Among them, P k For the fragility of individual components; C(P1,P2,…,P) k To consider the multivariate Copula function related to multiple components;

[0036] Substituting these values ​​into the vulnerability expressions for series and parallel systems, the vulnerability P of the series system... fcl for:

[0037] Preferably, the fourth calculation module uses the Copula function to calculate the system vulnerability considering component correlation, and calculates the functional loss of the bridge based on the exceedance probability of this vulnerability. The calculation formula is as follows:

[0038] Where j represents the damage level of the bridge system; C s,j Repair costs for structural systems at damage level j; I s Costs for reconstruction after complete destruction; The cost-loss ratio; P j Let be the probability of failure of a bridge with damage state j under a certain PGA.

[0039] The probability P of bridge damage state j under PGA action is P j The calculation formula is:

[0040] in, Let represent the exceedance probability of each failure state under a certain PGA.

[0041] As preferred options, the functional recovery functions for post-earthquake double-column pier bridges include: linear functional recovery functions, exponential functional recovery functions, and trigonometric functional recovery functions.

[0042] This invention can determine the damage probability of each component in actual engineering projects by fitting existing seismic ground motion data; it can perform system vulnerability calculations for complex substructures of double-column piers; and it can correct and verify the seismic toughness index by considering delay time and different functional recovery functions. By considering the influence of delay time and different functional recovery functions on the seismic toughness index, this invention provides a more accurate and practical assessment of seismic toughness, thereby improving the accuracy and reliability of seismic toughness assessment results. Attached Figure Description

[0043] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on the provided drawings without creative effort.

[0044] Figure 1 shows the function-time curve corresponding to the linear function recovery function;

[0045] Figure 2 shows the function-time curves corresponding to the exponential recovery function;

[0046] Figure 3 shows the function-time curves corresponding to the trigonometric function recovery function;

[0047] Figure 4 shows the full bridge model;

[0048] Figure 5 shows the force-deformation diagram of the plate rubber bearing;

[0049] Figure 6 shows the restoring force model of the PTFE sliding plate rubber support;

[0050] Figure 7 shows the double-broken line stress-strain relationship of the reinforcing steel;

[0051] Figure 8 shows the stress-strain relationship of ordinary confined concrete;

[0052] Figure 9 shows the fiber cross-section of the pier column;

[0053] Figure 10 shows the response spectra of the 10 selected seismic waves;

[0054] Figure 11 shows the bending moment-curvature diagram of the pier bottom section;

[0055] Figure 12 shows the probability of different damage levels occurring in the bridge structure; where (a) represents minor damage, (b) represents moderate damage, (c) represents severe damage, and (d) represents complete damage.

[0056] Figure 13 shows the repair time of the bridge structure under different seismic intensities; where (a) represents the longitudinal direction of the bridge and (b) represents the transverse direction of the bridge.

[0057] Figure 13 shows the post-earthquake functional-PGA-time curves of the bridge (linear recovery); where (a) represents the longitudinal direction of the bridge and (b) represents the transverse direction of the bridge.

[0058] Figure 14 shows the post-earthquake functional-PGA-time curves of the bridge (exponential recovery); where (a) represents the longitudinal direction of the bridge and (b) represents the transverse direction of the bridge.

[0059] Figure 15 shows the post-earthquake functional-PGA-time curves of the bridge (trigonometric function type recovery); where (a) represents the longitudinal direction of the bridge and (b) represents the transverse direction of the bridge.

[0060] Figure 16 shows the resilience index of different types of functional recovery functions; where (a) represents the longitudinal direction of the bridge and (b) represents the transverse direction of the bridge.

[0061] Figure 18 is a flowchart of the seismic toughness assessment method for the substructure of a double-column pier bridge according to an embodiment of the present invention. Detailed Implementation

[0062] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0063] To make the above-mentioned objects, features and advantages of the present invention more apparent and understandable, the present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments.

[0064] Example 1:

[0065] As shown in Figure 18, this embodiment of the invention provides a method for assessing the seismic toughness of the substructure of a double-column pier bridge, including:

[0066] Step 1: Preliminary determination of damage indicators for individual components

[0067] Damage indices for supports and tie beams are determined based on existing databases. For pier damage indices, the curvature of the respective cracking point, equivalent yield point, maximum bending moment point, and limit point of the moment-curvature curve at the pier base is multiplied by the corresponding frequency to obtain an index evaluating the degree of pier damage. The specific calculation formula is shown below.

[0068] Where: K DX F is an indicator for evaluating the degree of damage to a bridge pier. j K represents the frequency value of a certain axial force occurring at the bottom of a bridge pier. j The curvature of the characteristic points on the moment-curvature curve of the pier bottom section corresponding to a certain axial force includes the cracking point, the equivalent yield point, the maximum bending moment point, and the limit point.

[0069] Step 2: Determine the damage probability of a single component using the aforementioned component damage index.

[0070] First, time-history analysis was performed on multiple seismic waves of varying intensities to obtain a series of seismic response data. Then, the seismic response and seismic intensity indices were logarithmically converted, and a probabilistic demand model for the structure was derived through linear regression. The fitted regression formula is as follows:

[0071] ln(μ d )=a+b ln(IM) (2)

[0072] Based on reliability theory, the expression for component failure cracking is as follows:

[0073] When both the seismic demand and seismic resistance of the bridge structure follow a log-normal distribution, substituting the logarithmized seismic demand and seismic resistance into formula (2), the probability of damage to a single component can be obtained as follows:

[0074] In the formula: φ[ ] is the standard normal distribution function; β d For earthquake demand variance; β c For the variance of seismic capacity; μ d It is the result of the seismic response; μ c Damage index; when SA is used as the independent variable in the structural seismic vulnerability curve, When using PGA as the independent variable

[0075] Step 3: Use the Copula function to calculate the vulnerability of the substructure system of the double-column pier.

[0076] The component correlation is calculated using the Gumbel Copula function. The distribution function expression of the multivariate Gumbel Copula function is as follows:

[0077] In the formula: α is the relevant parameter, and α∈(0,1).

[0078] Introducing the Copula function and considering the correlation of seismic demand among components, the vulnerability of multiple components to simultaneous damage can be expressed as: P(X1X2…X…) k )=C(P1,P2,…,P k ), where 1<k≤n (6)

[0079] In the formula: P k For the fragility of individual components; C(P1,P2,…,P) k () is a multivariate Copula function that considers multiple components.

[0080] Substituting the above equation into the fragility expressions for series and parallel systems, we can obtain the fragility P of the series system. fcl for

[0081] Step 4: Calculate the post-earthquake functional loss function of the substructure of the double-column pier bridge based on the system's vulnerability.

[0082] The system vulnerability considering component correlation is calculated using the Copula function, and the functional loss of the bridge is calculated based on the exceedance probability of this vulnerability. The calculation formula is as follows.

[0083] In the formula: j represents the damage level of the bridge system; C s,jThe repair cost for a structural system at damage level j is related to the degree of damage to the structure; s Costs for reconstruction after complete destruction; P represents the cost-loss ratio. j Let be the probability of failure of a bridge with damage state j under a certain PGA.

[0084] The probability P of bridge damage state j under a certain PGA action is... j The calculation formula is as follows:

[0085] Let represent the exceedance probability of each failure state under a certain PGA.

[0086] Step 5: Calculate the seismic toughness index of the substructure of the double-column pier bridge using the post-earthquake functional loss function and functional recovery function. Calculate the repair time based on the damage probability of different bridge failure states. The calculation formula is as follows:

[0087] In the formula: T RE The repair time for a bridge damaged by an earthquake; T j The repair time for the structural system at damage level j.

[0088] The function-time function expression is: Q(t) = 1 - L E *{H(tt 0E )-H[t-(t 0E +T RE )]}*f Rec (t,t 0E ,T RE (11)

[0089] In the formula: L E f is the functional loss function; Rec (t,t 0E ,T RE H(x) is the function recovery function; H(x) is the Heaviside step function, and its expression is as follows.

[0090] Based on various recovery functions, the following are the functional recovery functions for post-earthquake double-column pier bridges.

[0091] (1) Linear function recovery function

[0092] The specific expression for the linear function recovery function is:

[0093] After a bridge is subjected to an earthquake, the structural system function-time function expression corresponding to the linear function recovery function is:

[0094] The function-time curve of the linear function recovery function is shown in Figure 1:

[0095] When considering the delayed repair time of the structure after an earthquake, the mathematical expression for the toughness index of the structural system will be as follows:

[0096] In the formula: T 0E The moment when repairs begin; T LC To control time, T LC =t 0E +T RE -t0.

[0097] Structural repair was performed using a mathematical model based on a linear functional recovery function. The toughness index of the double-column pier bridge structure is as follows:

[0098] We can obtain the following by calculating using definite integrals:

[0099] (2) Exponential function recovery function

[0100] The specific expression for the exponential function recovery function is:

[0101] After a bridge is subjected to an earthquake, the structural system function-time function expression corresponding to the exponential functional recovery function is:

[0102] The power-time curve of the exponential recovery function is shown in Figure 2:

[0103] Structural repair was performed using a mathematical model based on an exponential functional recovery function. The toughness index of the double-column pier bridge structure is as follows:

[0104] We can obtain the following by calculating using definite integrals:

[0105] (3) Trigonometric function type function recovery function

[0106] The specific expression for the trigonometric function-type function recovery function is:

[0107] The structural system function-time function expression corresponding to the trigonometric function-type functional recovery function of the bridge after an earthquake is:

[0108] The function-time curve of the trigonometric function-type function recovery function is shown in Figure 3:

[0109] Structural repair was performed using a mathematical model based on a trigonometric function-type functional recovery function. The toughness index of the double-column pier bridge structure is as follows:

[0110] We can obtain the following by calculating using definite integrals:

[0111] This invention, based on the Copula function, performs system damage probability calculation and analysis on each component of the substructure of a double-column pier bridge, and implements a seismic toughness assessment method that considers both system damage probability and delay time, resulting in clearer and more accurate seismic toughness assessment results. Seismic toughness indices are calculated using three different functional recovery functions, and the most suitable seismic toughness assessment method for the actual project can be selected by comparing the results calculated by different functional recovery functions. Based on the stress characteristics of the substructure of a double-column pier bridge, a method for calculating the damage index of the double-column pier columns is proposed, further enhancing the reliability of the damage probability and seismic toughness assessment results.

[0112] Application example:

[0113] Model Introduction

[0114] This invention focuses on a double-column pier bridge in a real engineering project. The bridge employs a construction method of first simple support and then continuous support. The superstructure consists of 3×30m prestressed concrete simply supported T-beams, with 5 precast T-beams used per span. The bridge deck has a clear width of 11.4m and a main beam height of 2.0m, constructed using C50 precast concrete. The piers are double-column, the abutments are pile-column abutments, and the pile foundations are single-column pile foundations, all with solid circular cross-sections and a center-to-center spacing of 7.2m. Specifically, pier #1 has a diameter of 1.5m, abutment #0 and pier #1 have pile foundation diameters of 1.8m, piers #2 and #3 have diameters of 1.8m and pile foundation diameters of 2.0m. The longitudinal reinforcement of the piers and pile foundations is HRB400, 28mm in diameter, evenly spaced around the perimeter. The stirrups are HPB300, 10mm in diameter, spirally arranged at 10cm intervals. The concrete cover thickness of the piers and pile foundations is 4cm and 6cm, respectively. The cap beam is 11.6m long with a cross-section of 2.2m × 1.8m, the tie beam has a cross-section of 1.1m × 1.3m, and the ground tie beam has a cross-section of 1.3m × 1.5m. The substructure is constructed entirely of C35 concrete. The abutments and pier #3 are supported by GJZF4350×450×101 type PTFE sliding rubber bearings, while other piers use GJZ350×450×99 type plate rubber bearings. The complete bridge model is shown in Figure 4.

[0115] Superstructure simulation

[0116] When designing the superstructure, the primary consideration is usually bearing vertical loads. However, past earthquake records show that most vertical seismic forces are not very strong. Furthermore, seismic damage to the superstructure is mainly caused by factors such as collisions between adjacent spans of the superstructure due to displacement, collisions between the superstructure and abutments, and beam collapses, while damage to the superstructure itself is relatively rare. Given that the calculation example selected in this embodiment of the invention is a ductile seismic design, the piers will become the main components absorbing seismic wave energy. Therefore, when studying seismic forces, the superstructure is generally considered to be in the elastic stage, not yet in the plastic stage. Therefore, this embodiment of the invention uses beam elements to simulate the main beams, with the material defined as a linear elastic material, while the transverse connections between the main beams are simulated using virtual crossbeams. In addition, bridge deck loads such as secondary pavement and crash barriers are applied to the beam elements as line loads.

[0117] Support simulation

[0118] The double-column pier bridge structure studied in this embodiment of the invention has GJZF4350×450×101 type bearings installed on piers #0 and #3, and GJZ350×450×99 type bearings installed on piers #1 and #2. Here, the GJZ350×450×99 type bearing refers to a rectangular plate rubber bearing with a cross-section of 35cm×45cm and a thickness of 9.9cm; while the GJZF4350×450×101 type bearing refers to a rectangular PTFE sliding rubber bearing with a cross-section of 35cm×45cm and a thickness of 10.1cm. Considering damage conditions, the stress and deformation relationship of the plate rubber bearing in the horizontal direction is shown in Figure 5, and the restoring force model of the PTFE sliding rubber bearing is shown in Figure 6.

[0119] Pier Simulation

[0120] For bridges designed with ductile seismic resistance, the piers are the most critical components in the bridge system, both from the perspective of strength and deformation criteria. As ductile components, plastic hinges are generated at the pier top, pier body, and connection points with the tie beam in a double-column pier with tie beams. These hinges play an energy dissipation role under seismic loading, thereby protecting the energy-saving components. To study the nonlinear behavior of the piers after entering the plastic stage, this embodiment of the invention uses nonlinear materials and fiber sections in the pier simulation. The reinforcing steel is modeled using a bilinear model, while both the confined and unconfined concrete of the piers are modeled using Mander constitutive models, as shown in Figures 7 and 8.

[0121] After defining the properties of the inelastic material, the cross-sections of the pier columns with diameters of 1.5m and 1.8m were divided into fiber sections. The confined concrete and unconfined concrete of the pier columns were divided into 20 parts along the x-axis and y-axis respectively. The fiber section division is shown in Figure 9.

[0122] Tie beam simulation

[0123] As a capacity-protecting component, the tie beam forms a plastic hinge at its connection with the pier during an earthquake. This dissipates the energy generated by the seismic force, thus reducing the damage to the bridge structure. However, this also makes the area near the connection between the tie beam and the pier more susceptible to damage. To study the damage extent of the tie beam under seismic loading, this invention uses the Takeda three-fold hysteresis characteristic to define the skeleton hinge, simulating the plastic hinge region of the tie beam, and obtaining the rotation angle of the plastic hinge region of the tie beam using this method.

[0124] Seismic input

[0125] Seismic motion parameter selection

[0126] Currently, peak ground acceleration (PGA) and spectral ground acceleration (SGA) are two of the most commonly used seismic motion intensity indicators in structural vulnerability analysis. PGA is an easily obtainable parameter in bridge seismic resistance analysis and is also a widely recognized and applied seismic intensity indicator, holding an important position in bridge structural vulnerability analysis by scholars both domestically and internationally. Therefore, this embodiment of the invention selects PGA as the indicator for evaluating the magnitude of seismic motion intensity.

[0127] Seismic wave selection method

[0128] To ensure the accuracy of the vulnerability analysis calculation results for double-column pier bridge structures, this embodiment of the invention selects measured seismic waves as input values ​​for structural seismic response analysis. Randomness and accuracy are considered in the selection of seismic waves, which are chosen based on factors such as seismic wave spectral characteristics, peak ground acceleration (PGA), duration, and number of seismic waves. A design response spectrum is plotted based on the bridge site conditions and the "Code for Seismic Design of Highway Bridges." Using this as the target spectrum, 10 ground motion records meeting the above conditions are selected from 100 natural waves in the PEER strong earthquake database of the Pacific Earthquake Engineering Research Center (PEER). Since the ground motion intensity needs to be modulated to a state where more than 90% of the components are damaged during vulnerability analysis, this embodiment of the invention uses amplitude modulation amplification (IDA) to modulate the amplitude of each seismic wave, gradually increasing the PGA from 0.1g to 2.5g with an amplification of 0.1g, forming 10 groups of 250 seismic waves. These waves are then input into the model along both the longitudinal and transverse directions of the bridge to calculate the structural response, used for calculating the damage probability of double-column pier bridge structure failure. In this embodiment of the invention, the longitudinal and transverse directions represent the occurrence of earthquakes along the longitudinal and transverse directions of a double-column pier bridge, respectively. The amplitude adjustment formula is as follows:

[0129] In the formula: A and a(t) are the peak acceleration and time history acceleration functions before amplitude modulation, respectively; A' and a'(t) are the peak acceleration and time history acceleration functions after amplitude modulation, respectively.

[0130] Ten seismic waves that simultaneously meet the requirements of spectral characteristics, effective peak ground acceleration, and duration were selected from the PEER strong motion database of the Pacific Earthquake Engineering Center in the United States. The acceleration response spectra of the selected ground motion records are shown in Figure 10.

[0131] Determination of damage parameters for double-column pier bridge components

[0132] (1) Support

[0133] In this embodiment of the invention, the supports for piers #0 and #3 are polytetrafluoroethylene (PTFE) sliding rubber bearings, while the supports for piers #1 and #2 are plate rubber bearings. Displacement was selected as the damage index for the bearings. The damage state parameters for the PTFE sliding rubber bearings were based on research data from Li Lifeng et al., while the damage state parameters for the plate rubber bearings were based on research data from Nielson et al. Specific damage state index parameters are shown in Table 1.

[0134] Table 1

[0135] (2) Pier

[0136] The pier base curvature is derived from moment-curvature analysis of the pier base section of a double-column pier bridge. Based on research by scholars such as Pan, Agrawal, and Liu Hang, and the seismic damage classification standards for lifeline engineering projects, four corresponding characteristic points on the moment-curvature curve of the pier base section are typically selected as the dividing points for the pier's damage state: the cracking point, the equivalent yield point, the maximum bending moment point, and the ultimate limit point. The pier base section curvature damage index values ​​are shown in the figure.

[0137] The cracking point refers to the intersection of the moment-curvature relationship line and the ideal line within the elastic range. This point marks the cracking of the protective concrete layer in reinforced concrete. The equivalent yield point is the intersection of the two bi-broken lines of the moment-curvature ideal line. This point is considered the turning point where the reinforced concrete section changes from linear to nonlinear. The maximum bending moment point is the point on the moment-curvature curve where the bending moment reaches its peak value. The limit point represents the point with the largest curvature among the intersections of the moment-curvature ideal line and the relationship line. The damage state classification of bridge piers in this embodiment of the invention is shown in Table 2.

[0138] Table 2 Note: φ is the curvature value of the bottom section of a double-column pier bridge.

[0139] The substructure of a double-column pier bridge is a frame structure, belonging to a statically indeterminate structure. When an earthquake occurs along the longitudinal direction of the bridge, the axial force at the bottom of the pier remains constant. A constant axial force value can be used for moment-curvature analysis of the pier bottom section to obtain the curvature of the unique cracking point, equivalent yield point, maximum bending moment point, and ultimate point describing the damage state of the pier column. However, when an earthquake occurs along the transverse direction of the bridge, the axial force at the bottom of the pier changes over time. Therefore, the curvature of the four characteristic points mentioned above for moment-curvature analysis of the pier bottom section cannot be described using a single axial force value. Analysis of the axial forces of the piers calculated from 250 seismic waves input along the transverse direction reveals that the maximum axial force value at the bottom of the double-column pier is concentrated within a certain range. Using this range of axial forces for moment-curvature analysis of the pier bottom yields a series of moment-curvature curves. From these curves, it can be seen that the curvatures of the cracking point, equivalent yield point, maximum bending moment point, and ultimate point on these curves are all similar. Therefore, this embodiment of the invention calculates the frequency of occurrence of 250 sets of axial forces at the pier base by statistical analysis. Then, within the intervals in which these axial forces occur, the axial forces used for moment-curvature analysis of the cross-section are determined with a gradient of 500 kN. The obtained axial forces are then used to perform moment-curvature analysis on the pier base cross-section. Finally, the curvature of the cracking point, equivalent yield point, maximum bending moment point, and limit point of each of these moment-curvature curves is multiplied by the corresponding frequency to obtain an index for evaluating the degree of damage to the pier column. The specific calculation formula is shown below.

[0140] Where: K DX F is an indicator for evaluating the degree of damage to a bridge pier. j K represents the frequency value of a certain axial force occurring at the bottom of a bridge pier. j The curvature of the characteristic points on the moment-curvature curve of the pier bottom section corresponding to a certain axial force includes the cracking point, the equivalent yield point, the maximum bending moment point, and the limit point.

[0141] The damage parameters of piers #1, #2, and #3 were calculated using the above method and are shown in Table 3.

[0142] Table 3

[0143] (3) Tie beam

[0144] The rotation angle was used as the damage index for the tie beam. Based on the research of Nie Hongxin et al. and the structural design code, the damage parameters of the tie beam were selected as shown in Table 4.

[0145] Table 4

[0146] Seismic toughness parameter analysis

[0147] Functional loss

[0148] The economic losses caused by bridge structural damage under earthquake loading are called direct economic losses. The direct economic ratio refers to the ratio of the structural loss during an earthquake to the original construction cost. To facilitate rapid resilience assessment, this embodiment of the invention treats the economic losses of a bridge during an earthquake as earthquake-induced losses to the bridge's function, and then proceeds with the bridge resilience assessment accordingly.

[0149] The earthquake loss ratio is determined according to Table A.5 of the standard "Earthquake Field Work Part 4: Direct Disaster Loss Assessment" (GB / T 18208.4—2011). For highway bridge structures in life systems engineering, the structural failure loss ratio is shown in Table 5.

[0150] Table 5

[0151] According to seismic design theory, a bridge structure is considered to be in a basically intact state under seismic loading, requiring no repair and incurring no economic loss; the damage ratio for this state is considered to be 0. When the bridge structure is in a destroyed state, no repair is needed, and it is directly demolished and rebuilt; the damage ratio for this state is considered to be 100%. Therefore, the loss ratios for the basically intact, slightly damaged, moderately damaged, severely damaged, and destroyed states described in this embodiment of the invention are 0%, 16%, 31%, 56%, and 100%, respectively.

[0152] The probability of different damage levels of a double-column pier bridge is calculated based on the binary Gumbel Copula function. Then, the probability of different damage levels of the bridge structure system considering component correlation is calculated using formula (9). The curve is shown in Figure 12.

[0153] According to the data in Figure 12, the probability curves for minor and moderate damage show a trend of first increasing and then decreasing. In the minor damage state, the highest probability occurs in the transverse direction within the range of 0.2g-0.4g, while in the longitudinal direction it is mainly concentrated in the range of 0.3g-0.7g. For the moderate damage state, the highest probability occurs in the transverse direction within the range of 0.8g-1.1g, while in the longitudinal direction it is mainly concentrated in the range of 0.9g-2.0g. The probability curve for severe damage shows a trend of first increasing and then decreasing in the transverse and longitudinal directions of pier #1, while it shows a continuous increase in the longitudinal direction of piers #2 and #3. The highest probability of severe damage occurs in the transverse direction within the range of 1.6g-1.9g, while in the longitudinal direction it is mainly concentrated above 2.0g. The probability curve for complete damage shows a continuous increasing trend. Furthermore, the highest probability of complete damage is greater than 2.5g. Overall, when the earthquake is relatively weak, the main damage to bridges is minor. As the earthquake intensity increases, the damage gradually intensifies and may eventually reach a state of complete destruction.

[0154] Repair time

[0155] The repair time for bridges varies depending on the degree of damage caused by earthquakes. This invention analyzes the repair time for bridge structures under different damage states, taking into account the 240-day construction period of this bridge. Therefore, the repair times for double-column pier bridges under different damage states are shown in Table 6.

[0156] Table 6

[0157] The repair time of the bridge structure under different ground motion intensities PGA was calculated using formula (10), as shown in Figure 25.

[0158] According to the data in Figure 25, when the ground motion intensity PGA is less than 0.3g, the repair time remains constant and extremely short, indicating that the bridge structure is not damaged and no repair is required. However, when the ground motion intensity exceeds 1.8g, the rate of increase in repair time gradually slows down, indicating that the structure is close to complete destruction and the repair time is close to the reconstruction period, after which demolition and reconstruction are required.

[0159] The impact of different seismic motion intensities on the post-earthquake functional recovery of bridges

[0160] The functional restoration of bridges after an earthquake is affected by many factors, including delay time, repair time, and repair plan. Delay time is further influenced by factors such as inspection duration and material delivery, making functional restoration a highly complex process. This embodiment of the invention assumes a delay time to repair time ratio of A, and takes a post-repair functional value of 1 for analysis.

[0161] As discussed earlier, the post-earthquake functional recovery process of bridges is influenced by the ground motion intensity (PGA), repair time, delay time, and type of functional recovery function. Both repair time and delay time are related to the degree of bridge damage; the higher the damage level, the longer the required repair and delay times. If A = 0.2, the post-earthquake functional recovery process can be described by a three-dimensional curve of ground motion intensity (PGA) and time. The three-dimensional relationship between bridge functional recovery and ground motion intensity (PGA) and repair time is shown in Figure 13-16. According to the data in Figure 14-16, the opening of the surface reflecting the function-PGA-time curve under different types of functional recovery reflects the characteristics of each recovery function. The function Q in the longitudinal direction is higher than that in the transverse direction for all three types of functional recovery functions, indicating that when the earthquake occurs along the transverse direction, the bridge structural performance is most severely damaged, and post-earthquake reinforcement work is most difficult. As can be seen from the contour lines at the bottom of the 3D image, the area enclosed by the exponential contour lines is the smallest, followed by the trigonometric contour lines, and the area enclosed by the linear contour lines is the smallest. This indicates that when the bridge's function is restored to "1", the exponential contour lines restore the best, followed by the trigonometric contour lines, and the linear contour lines restore the worst.

[0162] Seismic toughness analysis of the substructure of a double-column pier bridge

[0163] By conducting seismic toughness analysis considering the delay time, and taking A=0.2, the functional losses in the longitudinal and transverse directions of the double-column pier bridge under different seismic intensity are substituted into the corresponding toughness index formulas, and the toughness index values ​​of the double-column pier bridge structure under different seismic intensity are obtained as shown in Figure 16.

[0164] According to the data in Figure 16, the seismic toughness indices of bridges calculated by the three functional recovery functions, from largest to smallest, are exponential recovery, trigonometric recovery, and linear recovery. The seismic toughness indices in the longitudinal direction are all higher than those in the transverse direction. When the peak ground acceleration (PGA) is less than 0.5g, the seismic toughness index R0 is greater than 0.9, indicating excellent bridge structure toughness; when PGA is between 0.5 and 1.5g, the seismic toughness index R0 is between 0.75 and 0.9, indicating good bridge structure toughness; when PGA is between 1.5 and 2.0g, the seismic toughness index R0 is between 0.6 and 0.75, indicating moderate bridge structure toughness; when PGA is greater than 2.0g, the seismic toughness index R0 is less than 0.6, indicating poor bridge structure toughness.

[0165] Example 2:

[0166] This invention also provides a device for evaluating the seismic toughness of the substructure of a double-column pier bridge, comprising:

[0167] The first calculation module is used to determine the damage index of individual components;

[0168] The second calculation module is used to determine the probability of damage to a single component using the component damage index.

[0169] The third calculation module is used to calculate the vulnerability of the double-column pier substructure system based on the damage probability of individual components using the Copula function;

[0170] The fourth calculation module is used to calculate the post-earthquake functional loss function of the substructure of a double-column pier bridge based on the system's vulnerability.

[0171] The fifth calculation module is used to calculate the seismic toughness index of the substructure of a double-column pier bridge using the post-earthquake functional loss function and functional recovery function of the substructure.

[0172] As one embodiment of the present invention, the calculation formula for the index used by the first calculation module to evaluate the degree of damage to the pier is as follows:

[0173] Among them, K DX F is an indicator for evaluating the degree of damage to a bridge pier. jK represents the frequency value of a certain axial force occurring at the bottom of a bridge pier. j Let be the curvature of a characteristic point on the moment-curvature curve of the pier bottom section corresponding to a certain axial force.

[0174] In one embodiment of the present invention, the third calculation module uses the GumbCopula function to calculate component correlation. The distribution function expression of the multivariate GumbCopula function is as follows:

[0175] Where α is the relevant parameter, and α∈(0,1),

[0176] Introducing the Copula function, the vulnerability of multiple components to simultaneous damage can be expressed as:

[0177] P(X1X2…X k )=C(P1,P2P,…P k )

[0178] Among them, P k For the fragility of individual components; C(P1,P2,…,P) k To consider the multivariate Copula function related to multiple components;

[0179] Substituting these values ​​into the vulnerability expressions for series and parallel systems, the vulnerability P of the series system... fcl for:

[0180] In one embodiment of the present invention, the fourth calculation module uses the Copula function to calculate the system vulnerability considering component correlation, and calculates the functional loss of the bridge based on the exceedance probability of this vulnerability. The calculation formula is as follows:

[0181] Where j represents the damage level of the bridge system; C s,j Repair costs for structural systems at damage level j; I s Costs for reconstruction after complete destruction; The cost-loss ratio; P j Let be the probability of failure of a bridge with damage state j under a certain PGA.

[0182] The probability P of bridge damage state j under PGA action is P. j The calculation formula is:

[0183] in, Let be the exceedance probability of each failure state under a certain PGA.

[0184] As one embodiment of the present invention, the functional recovery function of a post-earthquake double-column pier bridge includes: a linear functional recovery function, an exponential functional recovery function, and a trigonometric functional recovery function.

[0185] The embodiments described above are merely preferred embodiments of the present invention and are not intended to limit the scope of the present invention. Various modifications and improvements made to the technical solutions of the present invention by those skilled in the art without departing from the spirit of the present invention should fall within the protection scope defined by the claims of the present invention.

Claims

1. A method for evaluating the seismic toughness of the substructure of a double-column pier bridge, characterized in that, include: Step 1: Determine the damage index of individual components; Step 2: Determine the probability of damage to a single component using component damage indicators; Step 3: Calculate the vulnerability of the double-column pier substructure system using the Copula function based on the damage probability of individual components; Step 4: Calculate the post-earthquake functional loss function of the substructure of the double-column pier bridge based on the system's vulnerability. Step 5: Calculate the seismic toughness index of the substructure of the double-column pier bridge using the post-earthquake functional loss function and functional recovery function of the substructure.

2. The method for evaluating the seismic toughness of the substructure of a double-column pier bridge as described in claim 1, characterized in that, In step one, the calculation formula for the indicators evaluating the degree of damage to the pier is as follows: Among them, K DX F is an indicator for evaluating the degree of damage to a bridge pier. j K represents the frequency value of a certain axial force occurring at the bottom of a bridge pier. j Let be the curvature of a characteristic point on the moment-curvature curve of the pier bottom section corresponding to a certain axial force.

3. The method for evaluating the seismic toughness of the substructure of a double-column pier bridge as described in claim 2, characterized in that, In step three, the GumbCopula function is used to calculate component correlation. The distribution function expression of the multivariate Gumbel Copula function is as follows: Where α is the relevant parameter, and α∈(0,1), Introducing the Copula function, the vulnerability of multiple components to simultaneous damage is expressed as: P(X1X2…X k )=C(P1,,P2,…,P k ) Among them, P k For the fragility of individual components; C(P1,P2,...,P) k This is a multivariate Copula function that considers multiple components. Substituting these values ​​into the vulnerability expressions for series and parallel systems, the vulnerability P of the series system... fcl for:

4. The method for evaluating the seismic toughness of the substructure of a double-column pier bridge as described in claim 3, characterized in that, In step four, the system vulnerability considering component correlation is calculated using the Copula function. Based on the exceedance probability of this vulnerability, the functional loss of the bridge is calculated. The calculation formula is as follows: Where j represents the damage level of the bridge system; C s,j Repair costs for structural systems at damage level j; I s Costs for reconstruction after complete destruction; The cost-loss ratio; P j Let be the probability of failure of a bridge with damage state j under a certain PGA. The probability P of bridge damage state j under PGA action is P j The calculation formula is: in, Let represent the exceedance probability of each failure state under a certain PGA.

5. The method for evaluating the seismic toughness of the substructure of a double-column pier bridge as described in claim 4, characterized in that, The functional recovery functions for post-earthquake double-column pier bridges include: linear functional recovery functions, exponential functional recovery functions, and trigonometric functional recovery functions.

6. A device for evaluating the seismic toughness of the substructure of a double-column pier bridge, characterized in that, include: The first calculation module is used to determine the damage index of individual components; The second calculation module is used to determine the probability of damage to a single component using the component damage index. The third calculation module is used to calculate the vulnerability of the double-column pier substructure system based on the damage probability of individual components using the Copula function; The fourth calculation module is used to calculate the post-earthquake functional loss function of the substructure of a double-column pier bridge based on the system's vulnerability. The fifth calculation module is used to calculate the seismic toughness index of the substructure of a double-column pier bridge using the post-earthquake functional loss function and functional recovery function of the substructure.

7. The seismic toughness assessment device for the substructure of a double-column pier bridge as described in claim 6, characterized in that, The calculation formulas for the indicators used in the first calculation module to evaluate the degree of damage to the pier are as follows: Among them, K DX F is an indicator for evaluating the degree of damage to a bridge pier. j K represents the frequency value of a certain axial force occurring at the bottom of a bridge pier. j Let be the curvature of a characteristic point on the moment-curvature curve of the pier bottom section corresponding to a certain axial force.

8. The seismic toughness assessment device for the substructure of a double-column pier bridge as described in claim 7, characterized in that, The third calculation module uses the Gumb Copula function to calculate component correlation. The distribution function expression of the multivariate Gumb Copula function is as follows: Where α is the relevant parameter, and α∈(0,1), Introducing the Copula function, the vulnerability of multiple components to simultaneous damage is expressed as: P(X1X2…X k )=C(P1,P2,…,P k ) Among them, P k For the fragility of individual components; C(P1,P2,…,P) k This is a multivariate Copula function that considers multiple components. Substituting these values ​​into the vulnerability expressions for series and parallel systems, the vulnerability P of the series system... fcl for:

9. The seismic toughness assessment device for the substructure of a double-column pier bridge as described in claim 8, characterized in that, The fourth calculation module uses the Copula function to calculate the system vulnerability considering component correlation, and calculates the functional loss of the bridge based on the exceedance probability of this vulnerability. The calculation formula is as follows: Where j represents the damage level of the bridge system; C s,j Repair costs for structural systems at damage level j; I s Costs for reconstruction after complete destruction; The cost-loss ratio; P j Let be the probability of failure of a bridge with damage state j under a certain PGA. The probability P of bridge damage state j under PGA action is P j The calculation formula is: in, Let represent the exceedance probability of each failure state under a certain PGA.

10. The seismic toughness assessment device for the substructure of a double-column pier bridge as described in claim 9, characterized in that, The functional recovery functions for post-earthquake double-column pier bridges include: linear functional recovery functions, exponential functional recovery functions, and trigonometric functional recovery functions.

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