Evaluation method of seismic toughness of building structures based on triple integral

Abstract

The present invention provides a method for evaluating the seismic toughness of a building structure based on triple integrals, comprising: selecting appropriate earthquake motion records, performing finite element modeling on a reinforced concrete structure, performing nonlinear time history analysis on the finite element model of the structure under earthquake action, and obtaining damage data of the structure; establishing a probability demand model of the structure based on the obtained structural damage data, obtaining the posterior distribution of the model parameters based on the Bayesian updating principle, and then calculating the vulnerability band of the structure; calculating the direct economic loss band of the structure based on the obtained vulnerability band, calculating the repair time of the structure under continuous earthquake action based on the obtained structural damage data, and obtaining the probability distribution of the repair time; introducing the mathematical idea of ​​multiple integrals, considering the discreteness of vulnerability and repair time, expanding the traditional seismic toughness calculation formula to triple integrals, obtaining the seismic toughness calculation formula based on triple integrals, and calculating the median value and interval of the seismic toughness of the reinforced concrete structure.

Description

Technical Field

[0001] The invention belongs to the technical field of structural seismic toughness assessment, and in particular relates to a method for assessing the seismic toughness of a building structure based on triple integrals. Background Art

[0002] Reinforced concrete structures are widely used in various infrastructures such as buildings, bridges, and tunnels due to their excellent bearing capacity and durability. However, these structures can be severely damaged under dynamic loads such as earthquakes, resulting in huge economic losses and casualties. Therefore, accurately evaluating the seismic toughness of reinforced concrete structures is of great significance to ensuring structural safety and improving the level of seismic design.

[0003] Traditional seismic assessment methods include laboratory tests, numerical simulations, and empirical formulas. Although these methods can reflect the seismic performance of structures to a certain extent, they have some limitations. For example, numerical simulations rely on the accuracy of the model, require a large amount of calculation, and the results are affected by multiple parameters; empirical formulas are based on historical data and are difficult to adapt to new design standards and complex engineering conditions, and cannot consider the impact of various discreteness on the toughness assessment results during the calculation process.

[0004] In recent years, with the development of computational mechanics and numerical analysis methods, seismic resilience assessment methods based on mathematical models have gradually gained attention. As an important mathematical tool, triple integrals have unique advantages in describing complex physical phenomena and quantifying discreteness. Through triple integrals, the vulnerability of structures under earthquakes and the discreteness of repair time can be fully considered, thereby achieving a more comprehensive assessment of the seismic resilience of structures.

[0005] This patent proposes a method for evaluating the seismic toughness of building structures based on triple integrals. This method constructs a finite element model of reinforced concrete structures and uses triple integrals to fully consider the seismic vulnerability and discreteness of the repair time of the structure under earthquake action, thereby obtaining the seismic toughness index and toughness range of the structure. This method has the advantages of a wide range of calculation applications and clear physical meaning, and can provide a scientific basis for seismic design and evaluation. Summary of the invention

[0006] The purpose of the present invention is to solve the problem that the discreteness of earthquake vulnerability and repair time cannot be considered in the existing seismic toughness analysis, and to provide a method for evaluating the seismic toughness of reinforced concrete structures based on triple integration. The idea of ​​multiple integration is introduced into the toughness calculation formula, and a new seismic toughness calculation model is proposed, which provides a more comprehensive method for toughness analysis that considers the influence of earthquake vulnerability and discreteness of repair time in actual engineering, and the method is applied to the seismic toughness evaluation of actual engineering structures.

[0007] A method for evaluating the seismic toughness of a building structure based on triple integrals comprises the following steps:

[0008] Step 1: Select appropriate earthquake records and establish a finite element analysis model for reinforced concrete structures. Use the selected earthquake records as input to perform nonlinear time-history analysis on the reinforced concrete structure to obtain damage data of the structure under earthquake action;

[0009] Step 2: Establish a probabilistic demand model of the structure based on the structural damage data obtained in step 1, as shown in the following formula:

[0010] ln[DI MA (Θ MA )]=θ 0MA +θ 1MA ln(IM MS )+θ 2MA ln(IM AS )+θ 31MA [ln(IM MS )〗 2 +σ MA ε

[0011] Where, IM MS and IM AS Respectively represent the intensity parameters of the main shock and aftershock; DI MA represents the cumulative damage parameter of the structure; θ MA =(θ 0MA ,θ 1MA ,θ 2MA ,θ 31MA ,…) and Θ MA =(θ MA ,σ MA ) is the model parameter set; σ MA ε represents the model error.

[0012] Based on the Bayesian update principle, the posterior distribution of the model parameters is calculated as shown in the following formula:

[0013] f(Θ MA )=κL(Θ MA )p(Θ MA )

[0014] In the formula, p(Θ MA ) and f(Θ MA ) represent the prior distribution and posterior distribution of the model parameters respectively; κ is the normalization parameter; L(Θ MA ) is the parameter Θ MA The likelihood function of .

[0015] The obtained demand model is used to construct the limit state equation of the structure, as shown below:

[0016] g(IM,Θ MA )=ln(C)-ln[DI MA (Θ MA )]

[0017] Where IM is the general term for earthquake intensity parameters; C is the resistance parameter of the structure.

[0018] Then the median vulnerability value and its band are calculated as shown in the following formula:

[0019] Median value of vulnerability curve: F(LS|IM,Θ MA )=P[g(IM,Θ MA )≤0|IM,Θ MA ]

[0020] Vulnerability curve strip: {Φ[-β(IM)-σ β (IM)],Φ[-β(IM)+σ β (IM)]}

[0021] Where LS represents the limit state of the structure; β(IM) is the reliability index; σ β (IM) is the standard deviation of the reliability index β(IM).

[0022] Step 3: Based on the vulnerability bands obtained in step 2, calculate the direct economic loss of the structure L(·), as shown in the following formula:

[0023]

[0024]

[0025] Where, DS i and LS i represent the i-th damage state and i-th limit state of the structure respectively; R DS,i Indicates that the structure is in DS i t represents the time range.

[0026] Based on the structural damage data obtained in step 1, calculate the repair time (T R ), and obtain the probability distribution of repair time, as shown in the following formula;

[0027] T R (IM,θ T )=T R,MS +T R,AS

[0028] Where, T R,MS and T R,ASare the repair time caused by the main shock damage and the incremental repair time caused by the aftershock incremental damage, respectively, and Θ T is the set of model parameters. The two can be solved by the following formulas respectively.

[0029]

[0030]

[0031] where γ[·] is a function of IM, and θ R,MS , θ R,MA , and ε are all unknown parameters, and

[0032] Furthermore, the mean value and standard deviation of T R can be obtained as shown in the following formula:

[0033]

[0034] where μ and σ 2 represent the mean value and standard deviation respectively.

[0035] After obtaining the mean value and standard deviation of T R , the probability distribution of T R can be obtained.

[0036] Step 4: Introduce the mathematical idea of multiple integrals, consider the discreteness of vulnerability and repair time, and extend the traditional seismic resilience calculation formula to triple integrals to obtain the seismic resilience calculation formula based on triple integrals as shown in the following formula:

[0037]

[0038] where R(IM) is the index to measure the resilience level; represents the real number field; t0 represents the start time, defined as 0; T LC is the control time; Q(·) is the performance function, which can be expressed by the following formula:

[0039] Q(t|IM,Θ MA ) = Q(t < t0) - L(t|IM,Θ F )·[H(t - t0) - H(t - t0 + T R )]·f(t, t0, T R )

[0040] where Q(t < t0) represents the initial performance level of the structure; L(·) represents the loss function of the structure, H(·) is the Heaviside step function; f(·) represents the recovery function.

[0041] Substituting the calculation results of step 2 and step 3 into the formula of step 4, the median value and range of seismic toughness of reinforced concrete structure can be calculated.

[0042] Beneficial effects of the invention: The purpose of the invention is to solve the problem that the discreteness of earthquake vulnerability and repair time cannot be considered in the existing seismic toughness analysis, and to provide a method for evaluating the seismic toughness of reinforced concrete structures based on triple integration. The idea of ​​multiple integration is introduced into the toughness calculation formula, and a new seismic toughness calculation model is proposed, and then the median value and interval of the structural seismic toughness index are obtained, providing a more comprehensive method for toughness analysis that considers the influence of earthquake vulnerability and discreteness of repair time in actual engineering. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] Figure 1 It is a flow chart of a method for evaluating the seismic toughness of a building structure based on triple integrals of the present invention;

[0044] Figure 2 is a demand model diagram in one embodiment of the present invention;

[0045] Figure 3 is a vulnerability result diagram in one embodiment of the present invention;

[0046] In the figure, (a) is the fragility result diagram when the limit state is LS1, (b) is the fragility result diagram when the limit state is LS2, (c) is the fragility result diagram when the limit state is LS3, and (d) is the fragility result diagram when the limit state is LS4;

[0047] Figure 4 is the repair time T in one embodiment of the present invention R,MS and T R,AS The result graph of

[0048] In the figure, (a) is T R,MS The result diagram of T R,AS The result graph,

[0049] Figure 5 It is a diagram of structural seismic toughness results in one embodiment of the present invention. DETAILED DESCRIPTION

[0050] The specific implementation of the present invention is further described in detail below in conjunction with the accompanying drawings and examples. The following examples are used to illustrate the present invention, but are not intended to limit the scope of the present invention.

[0051] like Figure 1As shown, the triple integral-based seismic toughness assessment method for building structures provided in this embodiment expands the traditional seismic toughness calculation method, can further assess the uncertainty of the toughness results caused by the discreteness of vulnerability and repair time, and can more comprehensively evaluate the seismic toughness level of the structure under the action of main shocks and aftershocks.

[0052] A method for evaluating the seismic toughness of building structures based on triple integrals, the specific steps of the method are:

[0053] Step S1, selecting appropriate earthquake motion records, and performing finite element modeling on the reinforced concrete structure, performing nonlinear time history analysis on the finite element model of the structure under earthquake action, and obtaining damage data of the structure.

[0054] Specifically in this embodiment, the above step S1 also includes the following steps:

[0055] S11. Select appropriate earthquake records and establish a finite element analysis model of reinforced concrete structure;

[0056] S12. Use the selected earthquake records as input to perform nonlinear time-history analysis on the reinforced concrete structure to obtain damage data of the structure under earthquake action.

[0057] Step S2: establishing a probabilistic demand model of the structure based on the obtained structural damage data, obtaining the posterior distribution of the model parameters based on the Bayesian updating principle, and then calculating the vulnerability band of the structure.

[0058] Specifically in this embodiment, the above step S2 also includes the following steps:

[0059] S21. Based on the structural damage data obtained in step 1, a probabilistic demand model of the structure is established, and the spectral acceleration is selected as the main shock (Sa MS ) and aftershocks (Sa AS ) strength parameters, the model is shown as follows:

[0060] ln[DI MA (Θ MA )]=θ 0MA +θ 1MA ln(Sa MS )+θ 2MA ln(Sa AS )+θ 31MA [ln(Sa MS )] 2 +σ MA ε

[0061] Where, DI MA represents the cumulative damage parameter of the structure; θ MA =(θ 0MA ,θ1MA ,θ 2MA ,θ 31MA ,…) and Θ MA =(θ MA ,σ MA ) is the model parameter set; σ MA ε represents the model error.

[0062] S22. Based on the Bayesian updating principle, the posterior distribution of the model parameters is calculated as shown in the following formula:

[0063] f(Θ MA )=κL(Θ MA )p(Θ MA )

[0064] In the formula, p(Θ MA ) and f(Θ MA ) represent the prior distribution and posterior distribution of the model parameters respectively; κ is the normalization parameter; L(Θ MA ) is the parameter Θ MA The likelihood function of , the final demand model is as follows Figure 2 shown.

[0065] S23. Use the obtained demand model to construct the limit state equation of the structure, as shown in the following formula:

[0066] g(IM,Θ MA )=ln(C)-ln[DI MA (Θ MA )]

[0067] Where IM is the general term for earthquake intensity parameters; C is the resistance parameter of the structure.

[0068] Then the median vulnerability value and its band are calculated as shown in the following formula:

[0069] Median value of vulnerability curve: F(LS|IM,Θ MA )=P[g(IM,Θ MA )≤0|IM,Θ MA ]

[0070] Vulnerability curve strip: {Φ[-β(IM)-σ β (IM)],Φ[-β(IM)+σ β (IM)]}

[0071] Where LS represents the limit state of the structure; β(IM) is the reliability index; σ β (IM) is the standard deviation of the reliability index β(IM). The final vulnerability results are as follows: Figure 3 shown.

[0072] Step S3: Based on the hazard decomposition result of the project site, a main shock hazard curve of the project site is obtained, and based on the conditional probability formula and the Copula function, a main shock and aftershock hazard surface is obtained.

[0073] Specifically in this embodiment, the above step S3 also includes the following steps:

[0074] S31. Based on the vulnerability band obtained in step 2, the direct economic loss band of the structure is calculated as shown in the following formula:

[0075]

[0076] Where, DS i and LS i represent the i-th damage state and i-th limit state of the structure respectively; R DS,i Indicates that the structure is in DS i The loss coefficient at the time t represents the time range. The direct economic loss obtained.

[0077] S32, based on the structural damage data obtained in step 1, calculate the repair time (T R ), and obtain the probability distribution of repair time, as shown in the following formula;

[0078] T R (IM,θ T )=T R,MS +T R,AS

[0079] Where, T R,MS and T R,AS are the repair time caused by the main shock damage and the repair time increment caused by the aftershock incremental damage, Θ T is the set of model parameters. The two can be solved using the following formulas respectively.

[0080]

[0081] Where γ[·] is the function of IM, θ R,MS ,θ R,MA ,、 and ε are unknown parameters, and The obtained T R,MS and T R,AS like Figure 4 shown.

[0082] Furthermore, we can get T R The mean and standard deviation are shown below:

[0083]

[0084] In the formula, μ and σ2 respectively represent the mean value and the standard deviation.

[0085] Obtain T R After obtaining the mean value and the standard deviation of T, the probability distribution of T can be obtained. R of T.

[0086] Step S4: Introduce the mathematical idea of multiple integral, consider the discreteness of vulnerability and repair time, extend the traditional seismic resilience calculation formula to triple integral, obtain the seismic resilience calculation formula based on triple integral, and calculate the median value and interval of the seismic resilience of the reinforced concrete structure.

[0087] Specifically in this embodiment, the above step S4 further includes the following steps:

[0088] Introduce the mathematical idea of multiple integral, consider the discreteness of vulnerability and repair time, extend the traditional seismic resilience calculation formula to triple integral, obtain the seismic resilience calculation formula based on triple integral, as shown in the following formula:

[0089]

[0090] In the formula, R(IM) is an index to measure the resilience level; represents the real number field; t0 represents the start time, defined as 0; T LC is the control time; Q(·) is the performance function, which can be expressed by the following formula:

[0091] Q(t|IM,Θ MA ) = Q(t < t0) - L(t|IM,Θ F )·[H(t - t0) - H(t - t0 + T R )]·f(t, t0, T R )

[0092] In the formula, Q(t < t0) represents the initial performance level of the structure; L(·) represents the loss function of the structure, H(·) is the Heaviside step function; f(·) represents the recovery function. The obtained seismic resilience result of the structure is as Figure 5 shown.

[0093] The above is only the preferred embodiment of the present invention and is not used to limit the present invention. It should be noted that for those of ordinary skill in the art, without departing from the technical principle of the present invention, several improvements and modifications can be made, and these improvements and modifications should also be regarded as the protection scope of the present invention.

Claims

1. A method for evaluating the seismic toughness of building structures based on triple integrals, characterized in that: It includes the following steps: Step 1: Select ground motion records, perform finite element modeling on the reinforced concrete structure, conduct nonlinear time history analysis of the finite element model of the structure under seismic action, and obtain the structural damage data; Step 2: Based on the obtained structural damage data, establish a probabilistic demand model of the structure, obtain the posterior distribution of the model parameters based on the Bayesian updating principle, and then calculate the fragility strips of the structure; Step 3: Based on the obtained fragility strips, calculate the direct economic loss strips of the structure. Based on the obtained structural damage data, calculate the repair time of the structure under consecutive seismic actions, and obtain the probability distribution of the repair time; Step 4: Introduce the mathematical idea of multiple integrals, consider the discreteness of fragility and repair time, extend the traditional seismic resilience calculation formula to triple integrals, obtain the seismic resilience calculation formula based on triple integrals, and calculate the median value and interval of the seismic resilience of the reinforced concrete structure; The specific content of Step 2 includes: Based on the structural damage data obtained in Step 1, establish a probabilistic demand model of the structure, as shown in the following formula: ln[DI MA (I MA )]=θ 0MA +θ 1MA ln(IM MS )+θ 2MA ln(IM AS )+θ 31MA [ln(IM MS )] 2 +s MA e Where, IM MS and IM AS Respectively represent the intensity parameters of the main shock and aftershock; DI MA represents the cumulative damage parameter of the structure; θ MA =(θ 0MA ,θ 1MA ,θ 2MA ,θ 31MA ,…) and Θ MA =(θ MA ,σ MA ) is the model parameter set; σ MA ε represents the model error; Based on the Bayesian updating principle, calculate the posterior distribution of the model parameters, as shown in the following formula: f(Θ MA )=κL(Θ MA )p(Θ MA ) In the formula, p(Θ MA ) and f(Θ MA ) represent the prior distribution and posterior distribution of the model parameters respectively; κ is the normalization parameter; L(Θ MA ) is the parameter Θ MA The likelihood function of ; Use the obtained demand model to construct the limit state equation of the structure, as shown in the following formula: g(IM,Θ MA )=ln(C)-ln[DI MA (Θ MA )] In the formula, IM represents the general term of ground motion intensity parameters; C represents the resistance parameter of the structure; Furthermore, calculate the median value of fragility and its strips, as shown in the following formula: Median value of vulnerability curve: F(LS|IM,Θ MA )=P[g(IM,Θ MA )≤0|IM,Θ MA ] Vulnerability curve strip: {Φ[-β(IM)-σ β (IM)],Φ[-β(IM)+σ β (IM)]} Where LS represents the limit state of the structure; β(IM) is the reliability index; σ β (IM) is the standard deviation of the reliability index β(IM); The specific content of Step 3 includes: Based on the fragility strips obtained in Step 2, calculate the direct economic loss strips of the structure, as shown in the following formula: Where, DS i and LS i represent the i-th damage state and i-th limit state of the structure respectively; R DS,i Indicates that the structure is in DS i The loss coefficient at time t represents the time range; Based on the structural damage data obtained in step 1, calculate the repair time T of the structure under continuous earthquake action. R , and obtain the probability distribution of repair time, as shown in the following formula; T R (IM,Θ T )=T R,MS +T R,AS Where, T R,MS and T R,AS are the repair time caused by the main shock damage and the repair time increment caused by the aftershock incremental damage, Θ T is the set of model parameters; the two can be solved by the following formulas respectively; Where γ[·] is the function of IM, θ R,MS ,θ R,MA , and ε are unknown parameters, and Furthermore, we can get T R The mean and standard deviation are shown below: In the formula, μ and σ 2 denote the mean and standard deviation, respectively; Get T R After calculating the mean and standard deviation of T R The probability distribution of The specific content of Step 4 includes: Introduce the mathematical idea of multiple integrals, consider the discreteness of fragility and repair time, extend the traditional seismic resilience calculation formula to triple integrals, obtain the seismic resilience calculation formula based on triple integrals, as shown in the following formula: In the formula, R(IM) is an index to measure the level of resilience; represents the real number domain; t0 represents the start time, which is defined as 0; T LC is the control time; Q(·) is the performance function, which can be expressed by the following formula: Q(t|IM,Θ MA )=Q(t<t0)-L(t|IM,Θ F )·[H(t-t0)-H(t-t0+T R )]·f(t,t0,T R ) In the formula, Q(t<t0) represents the initial functional level of the structure; L(·) represents the loss function of the structure, H(·) is the Heaviside step function; f(·) represents the recovery function; Substitute the calculation results of Step 2 and Step 3 into the formula of Step 4, and then the median value and interval of the seismic resilience of the reinforced concrete structure can be calculated.

2. A method for evaluating seismic toughness of building structures based on triple integrals according to claim 1, characterized in that: The specific content of Step 1 includes: Select ground motion records, establish a finite element analysis model of the reinforced concrete structure, use the selected seismic records as input, conduct nonlinear time history analysis on the reinforced concrete structure, and obtain the damage data of the structure under seismic action.

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