A structural probabilistic damage identification and risk analysis method based on mode shape curvature uncertainty quantification

Abstract

The application discloses a structure probability damage identification and risk analysis method based on mode shape curvature uncertainty quantification. Firstly, the structure is subjected to vibration test to measure the structure response under environmental excitation; mode shapes are identified according to the measured response and uncertainty quantification is performed on the mode shapes; the uncertainty of the mode shapes is transferred to the mode shape curvatures before and after damage by using the matrix perturbation theory, and the curvature difference and its uncertainty are further calculated; the damage occurrence probability of each measuring point is calculated according to the obtained mode shape curvature uncertainty, the curvature difference is subjected to curvature normalization, and a damage risk index is constructed according to the obtained damage occurrence probability and the expectation of the normalized curvature difference. The application further extends the uncertainty quantification of parameters to the uncertainty quantification of damage, and integrates the uncertainty into the damage identification of the structure to evaluate the possibility and risk of damage occurrence at each position. The application can be used in the vibration-based health monitoring and nondestructive testing of beam structures.

Description

TECHNICAL FIELD

[0001] The present application relates to the field of structural health monitoring, and is a structural probability damage identification and risk analysis method based on mode shape curvature uncertainty quantification. BACKGROUND

[0002] Bridge structures will inevitably be damaged due to harsh service environment during service. In order to ensure the safety of the structure during service and avoid causing great safety threat, it is necessary to regularly check the state of the structure, determine the position and severity of the structural damage, and provide a basis for the maintenance of the structure.

[0003] In the vibration-based structural damage identification, the most commonly used method is to use the vibration response obtained by contact or non-contact sensing technology to determine the structural dynamic parameters such as modal frequency, modal damping, modal shape, modal scaling coefficient and modal flexibility, and to identify the structural damage by comparing the undamaged dynamic parameters and the damaged dynamic parameters. This method has the characteristics of direct physical interpretation and easy-to-obtain modal parameters. However, there are many uncertain factors such as noise, measurement error and model error in the field measurement, which interfere with the obtained modal parameters, and the vibration-based structural damage identification method is difficult to be applied in actual structure diagnosis. Therefore, when using modal parameters for structural damage identification, the uncertainty in the modal parameters needs to be considered. At present, most of the damage identification considering uncertainty is parameter-based, and there is a lack of in-depth research on risk assessment considering damage occurrence probability and damage hazard. SUMMARY

[0004] In view of the deficiencies of the existing damage index considering uncertainty, the present application provides a structural probability damage identification and risk analysis method based on mode shape curvature uncertainty quantification. The structural damage expectation considering the comprehensive influence of the structural position damage probability, damage sensitivity and damage degree is used to perform risk analysis on the possible damage positions of the structure based on probability statistics.

[0005] To achieve the above-mentioned purpose, the technical measures adopted by the present application are as follows:

[0006] 1) Step one: mode shape curvature of the area to be evaluated, curvature difference uncertainty quantification. Measure the response data under environmental vibration, identify the mode shape curvature and its uncertainty before and after the structural damage, and further calculate the mode shape curvature difference and its uncertainty;

[0007] 2) Step two: define the damage hazard degree of each area. According to the obtained mode shape curvature and curvature difference, further calculate the damage hazard degree of the structure;

[0008] 3) Step three: damage probability calculation. According to the determined mode shape curvature and its uncertainty before and after the damage, calculate the damage occurrence probability;

[0009] 4) Step four: Damage risk calculation. Calculate the damage risk according to the damage severity and damage probability obtained from each order parameter.

[0010] In step 1, the structure to be evaluated is preliminarily determined, and sensors are arranged through contact or non-contact means to perform vibration testing on the reference and the structure to be diagnosed, and the structural response is measured.

[0011] In step 1, the response data collected is used to identify the i-th mode shape curvature φ i (x) and φ id (x), and the uncertainty is quantified to obtain the curvature confidence interval cov(φ i (x) and cov(φ id (x).

[0012] In step 2, the mode shape curvature of the non-zero region obtained is normalized to obtain a constant Λ i (x):

[0013]

[0014] where Λ i (x) is the i-th normalized curvature difference at position x (a constant containing only the structural damage degree), φ i (x) is the i-th mode shape curvature, and the curvature difference is calculated as follows:

[0015] MD i (x) = φ id (x) - φ i (x)

[0016] In step 3, φ i (x) and φ id (x) are taken as random variables, and it is assumed that they follow a normal distribution, then the mode shape curvature probability density function before and after damage is:

[0017]

[0018]

[0019] where μ and μ d are the mean values of the identified undamaged mode shape curvature φ i (x) and the damaged mode shape curvature φ id (x), and σ = cov(φ i (x), σ d = cov(φ id (x).

[0020] In step 3, the damage occurrence probability pf is further calculated according to the obtained mode shape curvature probability density function i (x):

[0021]

[0022] Wherein, L1 represents φ i (x) lower bound of the probability confidence interval, L1 represents φ i (x) upper bound of the probability confidence interval, φ i (x) confidence interval in the interval (L1, L2) is (μ-3σ, μ+3σ), pf i (x) can represent the damage probability.

[0023] In step 4, the damage expectation is further calculated according to the obtained damage occurrence probability pf i (x) and the damage degree:

[0024] EMD i (x)) = MD ni (x) * pf i (x)

[0025] Wherein, EMD i (x)) is the damage expectation calculated according to the corresponding modal parameters, the curvature difference can indicate the structure damage position, and the size is related to the damage degree and the curvature itself; when the parameter identification has a certain accuracy, the structure damage severity can be obtained by normalizing the curvature difference; according to the curvature and its uncertainty before and after the damage, the real occurrence probability of the damage in each region can be obtained; according to the obtained damage probability and damage severity, the structure damage risk is further evaluated, and the corresponding maintenance strategy is guided to be taken.

[0026] Beneficial effects: based on the relationship between the mode shape curvature and the curvature difference, the normalized curvature difference is obtained to indirectly evaluate the damage degree of the structure, and the damage occurrence probability of each region is evaluated according to the mode shape curvature and its uncertainty before and after the damage; the damage risk index is constructed by using the normalized curvature difference and the damage probability, and the structure state is evaluated. The risk index combining the damage degree and the damage occurrence probability can obtain good damage positioning effect under noise interference, evaluates the structure damage risk, and provides an effective new method for damage identification of beam structures. BRIEF DESCRIPTION OF DRAWINGS

[0027] Figure 1 Flowchart of the application

[0028] Figure 2 Schematic diagram of vibration test of simply supported beam bridge

[0029] Figure 3 Mode shape curvature and confidence interval of the first two modes of a simply supported beam

[0030] Figure 4 First two order normalized curvature difference

[0031] Figure 5 First two order curvature difference and damage probability map

[0032] Figure 6 First two order curvature difference damage expectation DETAILED DESCRIPTION

[0033] The application will be further described in detail below in conjunction with the accompanying drawings and examples.

[0034] Referring to Figure 1 , which is a structural damage identification and risk analysis method flowchart based on mode shape curvature uncertainty of the application, the specific steps are as follows:

[0035] Step one: mode shape curvature of the region to be evaluated, curvature difference uncertainty quantification. Measure the response data under environmental vibration, identify the mode shape curvature and its uncertainty before and after structural damage, and further calculate the mode shape curvature difference and its uncertainty;

[0036] Step two: define the damage severity of each region. According to the obtained mode shape curvature and curvature difference, further calculate the damage severity of the structure;

[0037] Step three: damage probability calculation. According to the determined mode shape curvature and its uncertainty before and after damage, calculate the damage probability;

[0038] Step four: damage risk calculation. According to the damage severity and damage probability of the structure obtained by each order parameter, calculate the damage risk.

[0039] In step 1, the structure to be evaluated region is preliminarily determined, and sensors are arranged by contact or non-contact mode to perform vibration test on the reference and structure to be diagnosed, and the structure response is measured.

[0040] In step 1, according to the collected response data, the discrete time random space state model of the region is:

[0041]

[0042] Where y t ∈R l×1 is the measurement vector at time point t, x t ∈R n×1 is the system state vector, and A and C are system matrices. In the random subspace, let w t and v t be white noise sequences with zero mean, representing process noise and measurement noise respectively. Assuming that the number of reference sensors is r0(r0≤l), the output covariance matrix of the query region can be displayed as:

[0043]

[0044] wherein denotes the respective measurement of the reference sensor.

[0045] In step 1, the block matrix corresponding to the region of interest is constructed according to the output covariance matrix of the region of interest and the eigenvectors are identified:

[0046]

[0047] wherein T is derived from the output covariance matrix R i by singular value decomposition of T, the extended observability matrix is obtained, and the system matrix A and C are identified.

[0048] In step 1, the modal frequencies f i and the damping ξ i of the structure can be identified by the eigen equation of the system matrix A: i (x) = λ i i The eigenvectors ψ i (x) in combination with the system matrix C allow to obtain the modal shapes of the region of interest:

[0049]

[0050] wherein is the i-th modal shape at position x (x ∈ [x0 x n ]) and the maximum normalized modal shape φ i (x) is obtained by identifying the maximum of the modal shape and the uncertainty of the maximum normalized modal shape is:

[0051]

[0052] wherein cov(φ i (x) is the covariance of the modal shape, is the covariance of the block matrix, is the sensitivity of φ i (x) with respect to the system matrix, and are the sensitivities of the system matrix A and C with respect to the block matrix, respectively.

[0053] As an example, a simply supported beam is considered, which is 6 meters long and divided into 20 elements, each element being 0.3 meters long. The simply supported beam vibration test model is shown in Figure 2 The damage of the structure is simulated by the stiffness reduction of the local elements. The damage position is the element shown in Figure 2 Gaussian white noise is applied to the measured response. ​

[0054] In step 1, the structures in the two states to be evaluated are tested for vibration, and the curvature difference and its covariance are calculated according to the mode curvature and its uncertainty in the two states:

[0055] MD i (x) = φ" id (x) - φ" i (x)

[0056] where MD i (x) represents the i-th order mode curvature difference, φ" id (x) represents the i-th order mode curvature of the damaged structure, and the first two order mode curvatures and their uncertainties can be seen in Figure 3 .

[0057] In step 2, the damage degree is related to the mode curvature and the curvature difference, and the constant Λ containing only the structural damage degree can be obtained by normalizing the curvature difference according to the mode curvature in the non-zero region i (x):

[0058]

[0059] where Λ i (x) is the i-th order normalized curvature difference at position x (a constant containing only the structural damage degree), φ" i (x) is the i-th order mode curvature (φ" i (x) ≠ 0), and the first two order normalized curvature differences can be seen in Figure 4 When the damage degree is fixed, the normalized two order curvature difference values are the same for the damage location.

[0060] In step 3, φ" i (x) and φ" id (x) are assumed to follow a normal distribution when they are taken as random variables, then the probability density function of the mode curvature before and after damage is:

[0061]

[0062]

[0063] where μ and μ d are the mean values of the identified undamaged mode curvature φ" i (x) and the damaged mode curvature φ" id (x), σ = cov(φ" i (x), σ d = cov(φ" id (x)).

[0064] In step 3, the damage occurrence probability pf is further calculated according to the obtained mode curvature probability density function.i (x) : L1

[0065]

[0066] wherein L1 represents φ i (x) : L1 i (x) : L2 i (x) : (L1, L2) i (x) : pf Figure 5 The first order curvature difference changes at node 5, which actually has only 5% probability to be identified as real damage.

[0067] The step 4, according to the obtained damage occurrence probability pf i (x) and damage degree further calculates damage expectation:

[0068] E(MD i (x)) = MD ni (x) × pf i (x)

[0069] wherein E(MD i (x)) is damage expectation calculated according to corresponding modal parameters. The first two order curvature difference expectations are seen Figure 6 According to the damage risk index, false damage is greatly filtered, and the damage hazard can also be evaluated.

[0070] The above is the structural damage identification and risk analysis method based on mode shape curvature uncertainty of the present application, the damage severity is obtained by normalizing the obtained curvature difference, the damage occurrence probability of each region is calculated according to the mode shape curvature before and after damage, the damage expectation is calculated by combining the damage probability and the normalized curvature difference to evaluate the damage risk, but the present application is not limited to the above-mentioned embodiments. Any equivalent changes or modifications made according to the spirit and principle of the present application shall be covered within the protection scope of the present application.

Claims

1. A structural probabilistic damage identification and risk analysis method based on mode shape curvature uncertainty quantification, characterized in that The method comprises the following steps: Step 1: evaluate the mode curvature of the region to be assessed, quantify the curvature difference uncertainty, measure the response data under environmental vibration, identify the mode curvature and its uncertainty before and after structural damage, and further calculate the mode curvature difference and its uncertainty; identify the mode curvature φ" (x) and its uncertainty cov(φ" (x)) before and after damage according to the measured structural response, and further calculate the curvature difference MD i (x) and its uncertainty cov(MD i (x)); Step 2: Define the damage degree of each region, and further calculate the damage degree of the structure according to the obtained mode curvature and curvature difference; the damage degree is related to the mode curvature and the curvature difference, and the curvature normalization is performed on the curvature difference according to the obtained mode curvature of the non-zero node, so that a constant Λ containing only the damage degree of the structure can be obtained i (x): where Λ i (x) is the i-th order normalized curvature difference at position x, which only contains the constant of structural damage degree, φ" i (x) is the i-th order mode shape curvature; Step 3: Damage probability calculation, according to the determined mode curvature before and after damage and its uncertainty, the damage occurrence probability is calculated; φ" i (x) and φ" id (x) as random variables, assuming that they follow a normal distribution, the mode curvature probability density function before and after damage is: Where μ and μ d The undamaged i-th order mode curvature φ″ is identified. i (x) and the curvature of the damaged i-th order mode shape φ″ id (x) mean, σ=cov(φ″) i (x)), σ d =cov(φ″ id (x)); Based on the obtained probability density function of the mode curvature before and after the damage, the probability of damage occurrence pf is further calculated. i (x): where L1 represents φ" i (x) lower bound of the probability confidence interval, L2 represents φ" i (x) upper bound of the probability confidence interval, φ" i (x) the confidence interval is (μ-3σ, μ+3σ) in the interval (L1, L2), pf i (x) represents the damage probability; Step 4: damage risk calculation, calculating the damage risk according to the structural damage hazard degree and the damage probability obtained according to the parameters of each order.

2. The method of claim 1, wherein the method is based on quantification of mode shape curvature uncertainty. In step 4, the expected damage is further calculated from the resulting damage occurrence probability pf i (x) and the extent of damage. E(MD i (x)) = MD i (x) x pf i (x) where E(MD i (x)) is the damage expectation computed according to the corresponding modal parameters.

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