An RC beam damage analysis and bearing capacity evaluation method based on a nonlinear vibration model

Abstract

The application discloses a kind of RC beam damage analysis and bearing capacity evaluation method based on nonlinear vibration model, belong to civil engineering structural health monitoring field.Free attenuation vibration signal of RC beam is collected by knocking, instantaneous amplitude, frequency, damping coefficient are extracted using wavelet transform or adjacent peak value method, and dynamic fitting parameter related to amplitude is obtained by fitting;Displacement related nonlinear vibration equation is constructed, and normalized linear and nonlinear dynamic parameters are identified by combining harmonic balance method, and linear natural frequency and linear damping ratio that eliminate amplitude interference can be obtained by solving;Concrete crack damage is distinguished using linear damping ratio, nonlinear damping parameter and nonlinear stiffness parameter, steel bar corrosion and bearing capacity degradation are distinguished using linear frequency and linear stiffness, and linear frequency and limit bearing capacity quantitative conversion formula is established, to realize the quantitative evaluation of RC beam residual bearing capacity, test data has high stability, damage identification sensitivity is strong, and evaluation error is small.

Description

Technical Field

[0001] This invention relates to the field of structural health monitoring technology, and in particular to a method for damage analysis and load-bearing capacity assessment of RC beams based on a nonlinear vibration model. Background Technology

[0002] During long-term service, reinforced concrete beams are susceptible to repeated loads, environmental erosion, steel corrosion, and crack propagation, leading to structural stiffness degradation, changes in energy dissipation mechanisms, and a decrease in load-bearing capacity. Traditional analysis methods based on linear modal parameters typically treat frequency and damping as constant values, making it difficult to fully characterize the actual vibration behavior of reinforced concrete beams during damage development. After damage, both frequency and damping of RC beams exhibit significant amplitude correlation, demonstrating nonlinear vibration characteristics. Relying solely on linear dynamic models is insufficient to accurately identify their damage state.

[0003] Existing damage analysis methods still have the following shortcomings in engineering applications:

[0004] 1. Insufficient ability to extract and analyze instantaneous frequency, instantaneous amplitude, and instantaneous damping and their variation laws in nonlinear decaying vibration signals;

[0005] 2. Dynamic test results often remain at the level of comparing characteristic quantities, lacking a unified framework for identifying and analyzing dynamic parameters and their nonlinear parameters;

[0006] 3. There is a lack of effective characterization of parameter evolution laws at different damage stages, especially during the corrosion development process. Therefore, it is necessary to propose a technical method that can identify the linear and nonlinear dynamic parameters of reinforced concrete beams based on free decay vibration response, and further apply it to damage analysis and bearing capacity assessment, and establish an analytical method that can reflect the changes in dynamic parameters and their nonlinear characteristics after damage to RC beams. Summary of the Invention

[0007] The purpose of this invention is to provide a method for damage analysis and load-bearing capacity assessment of RC beams based on a nonlinear vibration model. This method overcomes the shortcomings of traditional linear modal testing, such as amplitude interference, inability to distinguish between crack and corrosion damage, and inability to quantify load-bearing capacity. By collecting the free decay vibration displacement response of the beam, extracting the relationship between frequency and damping parameters and amplitude, establishing a displacement-related nonlinear vibration model, and identifying dynamic parameters, this method enables damage state analysis and load-bearing capacity assessment of reinforced concrete beams.

[0008] The technical solution adopted by this invention to solve its technical problem is:

[0009] A method for damage analysis and bearing capacity assessment of RC beams based on a nonlinear vibration model, characterized by the following steps:

[0010] S1. Acquire the free decay vibration displacement signal of the reinforced concrete beam;

[0011] S2. Based on the free decay vibration displacement signal, extract the instantaneous amplitude, instantaneous frequency, and instantaneous equivalent damping coefficient, establish the correlation between the square of the frequency, the normalized equivalent damping coefficient and the instantaneous amplitude, and obtain the fitting parameters p1, p2, q1, and q2 by linear fitting.

[0012] S3. Construct a displacement-dependent nonlinear vibration model. Combining the harmonic balance method with the fitting parameters obtained in step S2, identify the dynamic parameters normalized to equivalent mass: k0 / m, k n / m、c0 / m、c n / m;

[0013] S4. Calculate the linear vibration frequency f0 and linear damping ratio ζ to eliminate amplitude interference based on normalized dynamic parameters;

[0014] S5. Based on the variation characteristics of damping ratio and normalized nonlinear dynamic parameters, concrete crack damage is determined; based on the degree of degradation of linear vibration frequency and normalized linear stiffness parameters, the corrosion stiffness degradation of steel bars is determined, and the remaining bearing capacity of RC beam during service life is quantitatively calculated through linear vibration frequency.

[0015] The displacement-related nonlinear vibration model equation described in step S3 is as follows:

[0016]

[0017] In the formula, m is the equivalent mass, x is the structural vibration displacement, c0 is the linear viscous damping coefficient, and c n k is the nonlinear damping coefficient, k0 is the linear stiffness coefficient, and k n This is a nonlinear stiffness coefficient;

[0018] The formula for calculating the remaining bearing capacity is:

[0019]

[0020] In the formula, M(t) is the bearing capacity at service period t, f(t) is the linear frequency measured at that time, f(0) is the linear frequency in the initial state or in the undamaged state, and M0 is the initial ultimate bearing capacity.

[0021] S6. Perform damage analysis based on the changing trends and amplitudes of the damping ratio and nonlinear parameters; then, assess the bearing capacity based on the degradation of the linear vibration frequency or linear stiffness parameters.

[0022] The present invention provides a method for damage analysis and load-bearing capacity assessment of RC beams based on a nonlinear vibration model. Step S1 specifically includes: applying a transient impact excitation to the RC beam to generate free decaying vibration; directly acquiring displacement signals using a displacement sensor, or acquiring acceleration signals using an accelerometer and converting them into displacement response through a second integral, and eliminating signal segments with excessively low signal-to-noise ratios and obvious boundary interference.

[0023] The present invention provides a method for damage analysis and bearing capacity assessment of RC beams based on a nonlinear vibration model. In step S2, instantaneous parameters are extracted using wavelet transform or adjacent peak calculation method; the displacement analytical signal is processed using complex Morlet continuous wavelet transform, and the instantaneous amplitude, instantaneous frequency, and instantaneous equivalent damping coefficient are solved by wavelet coefficient ridges; or the vibration peak value is extracted periodically along the displacement time history, and the single-cycle equivalent amplitude, instantaneous frequency, and equivalent damping coefficient are calculated.

[0024] This invention provides a method for damage analysis and bearing capacity assessment of RC beams based on a nonlinear vibration model, wherein in step 2 (f i 2 A i ) and (c eq,i / m, A i The line graph obtained by linear fitting can be seen as follows:

[0025] f i 2 =p1A i +p2

[0026] c eq,i / m=q1A i +q2

[0027] Where p1, p2, q1, and q2 are the squares of the frequencies f. 2 Normalized equivalent viscous damping coefficient c eq The slope and intercept of the fitted straight line between / m and amplitude A are calculated; then, based on the displacement-dependent nonlinear vibration model and the harmonic balance method, k0 / m = 4π is calculated. 2 p2, k n / m=-(3π 3 / 2)p1、c0 / m=q2、c n / m=(3π / 4)q1, yielding four nonlinear vibration parameters normalized with respect to the equivalent mass m.

[0028] The present invention provides a method for damage analysis and bearing capacity assessment of RC beams based on a nonlinear vibration model. The normalized dynamic parameter calculation formula in step S3 is as follows:

[0029] ; ; ; ;

[0030] Where k0 / m is the mass-normalized linear stiffness parameter, k n / m is the mass-normalized nonlinear stiffness parameter, c0 / m is the mass-normalized linear damping parameter, and c n / m is the mass-normalized nonlinear damping parameter; when the equivalent mass m is known, the above normalized parameter can be multiplied by m to obtain the actual parameters k0 and k. n c0 and c n When the equivalent mass m is difficult to determine accurately, the above-mentioned normalized parameters can be directly used as indicators for damage analysis and bearing capacity degradation evaluation.

[0031] The present invention provides a method for damage analysis and bearing capacity assessment of RC beams based on a nonlinear vibration model. The formula for calculating the linear vibration frequency in step S4 is as follows:

[0032] f0=[(k0 / m)] 1 / 2 / (2π);

[0033] The formula for calculating the linear damping ratio is:

[0034] ζ=(c0 / m) / (4πf0)

[0035] When the equivalent mass *m* is known, multiplying the normalized parameter by *m* yields the actual stiffness and damping parameters. If the equivalent mass *m* is unknown, the linear frequency *f0*, damping ratios *ζ*, *k0*, *c0*, and *k* can be obtained. n / m、c n / m.

[0036] This invention discloses a method for damage analysis and bearing capacity assessment of RC beams based on a nonlinear vibration model. The damage determination rule in step S5 is: linear damping ratio ζ, normalized nonlinear stiffness coefficient k. n / m, normalized nonlinear damping coefficient c n A sudden change in / m exceeding the threshold indicates the initiation or expansion of concrete cracks; a continuous decrease in linear vibration frequency f0 and normalized linear stiffness k0 / m indicates steel corrosion and degradation of component bearing capacity; structural damage warning is triggered when parameters abnormally exceed preset thresholds.

[0037] The present invention provides a method for damage analysis and load-bearing capacity assessment of RC beams based on a nonlinear vibration model. The method involves repeatedly executing steps S1-S4 at fixed time intervals, under conditions of overload, impact, post-disaster, and before and after maintenance throughout the entire service life, thereby establishing a time-series dynamic parameter database and realizing long-term continuous damage monitoring and dynamic assessment of load-bearing capacity.

[0038] Compared with the prior art, the beneficial effects of the present invention are:

[0039] (1) By constructing a nonlinear vibration model, the nonlinear vibration phenomenon of the RC beam is characterized. Compared with the linear model, it is more in line with the physical law of actual structural vibration.

[0040] (2) Through parameter identification technology, linear and nonlinear dynamic parameters were obtained simultaneously, eliminating the influence of amplitude on frequency and damping ratio test results and enhancing the reliability of the data;

[0041] A multi-parameter damage evaluation system covering linear and nonlinear characteristics was constructed, and based on the intrinsic relationship between first-order linear frequency and bearing capacity, the joint determination of damage degree and remaining bearing capacity of RC beam was realized. Attached Figure Description

[0042] The present invention will be further described below with reference to the accompanying drawings and embodiments. In the accompanying drawings:

[0043] Figure 1 This is the time history curve of the free attenuation signal displacement of the beam collected in this embodiment;

[0044] Figure 2 This is the wavelet time-frequency diagram in the wavelet analysis of this embodiment;

[0045] Figure 3 This example illustrates the relationship between the damping coefficient ζ and the amplitude of the test beam.

[0046] Figure 4 The linear frequency f of the experimental beam in this embodiment is... 2 Relationship with amplitude;

[0047] Figure 5 This embodiment describes the damage analysis and load-bearing capacity assessment process for RC beams in actual engineering projects. Detailed Implementation

[0048] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, 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, not all embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention. 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.

[0049] A method for damage analysis and bearing capacity assessment of reinforced concrete beams based on a nonlinear vibration model is proposed. This method acquires the free decaying vibration displacement response of the beam, extracts the relationship between frequency and damping parameters and amplitude, establishes a displacement-dependent nonlinear vibration model, and identifies dynamic parameters to achieve damage state analysis and bearing capacity assessment of the reinforced concrete beam.

[0050] The technical solution of the present invention includes the following steps:

[0051] Step 1: Acquisition of Free-Damaged Vibration Response and Displacement Response

[0052] A transient impact excitation is applied to the reinforced concrete beam under test to induce free decaying vibration. Free decaying signals are collected using instruments such as displacement sensors and accelerometers placed on the beam. Displacement signals are directly used as input for subsequent identification; velocity or acceleration signals are converted to displacement signals, and data segments with significant end boundary effects and low signal-to-noise ratios are discarded.

[0053] Step 2: Based on the free decay vibration signal, calculate and analyze the relationship between frequency, damping coefficient and amplitude.

[0054] This section provides two methods for calculating the relationship between frequency, damping coefficient, and amplitude: the wavelet transform method and the adjacent peak calculation method. In practical calculations, either method can be chosen. The calculation principles and processes are as follows:

[0055] (1) Apply wavelet transform to calculate the relationship between frequency, damping coefficient and amplitude.

[0056] The displacement response signal is analyzed using complex Morlet wavelet continuous wavelet transform. The instantaneous frequency at each moment is determined by the wavelet coefficient ridges, and the instantaneous amplitude is calculated from the wavelet coefficient modulus at the ridges. Based on the definitions of the attenuation coefficient and viscous damping coefficient and their relationship with the wavelet coefficients on the ridges, the instantaneous viscous damping coefficient is obtained. Furthermore, the relationship between frequency, viscous damping coefficient, and amplitude is derived.

[0057] For the analytic signal x(t) = A(t)exp[iφ(t)] = A(t)cos(φ(t)) + iA(t)sin(φ(t)), a continuous wavelet transform is performed on it using a complex wavelet with good frequency domain locality. The wavelet coefficients are:

[0058] (1)

[0059] Where a is the scale parameter and b is the translation parameter corresponding to the time position. Let W be the wavelet basis function and the wavelet coefficients. x The magnitude of (a,b) reflects the local feature intensity of the signal at time b and scale a.

[0060] For a real-valued signal x(t) = A(t)cos(φ(t)) whose amplitude and frequency vary with time, a continuous wavelet transform is performed using a complex Morlet wavelet, and the wavelet coefficients are...

[0061] (2)

[0062] (3)

[0063] The ridges of the wavelet coefficients are the locations where the derivative of the wavelet coefficient phase with respect to scale a is zero. On these ridges, the instantaneous signal frequency φ'(b) / (2π) and the wavelet center frequency F... c Since / a is equal, the modulus of the wavelet coefficients on the ridge line can be determined from the above formula as follows:

[0064] (4)

[0065] Therefore, the instantaneous amplitude A(t) can be obtained from the wavelet coefficients on the ridge:

[0066] (5)

[0067] The instantaneous frequency can be obtained based on the scale corresponding to the ridge and the correspondence between the scale and the frequency.

[0068] For a single-degree-of-freedom freely decaying vibration, the amplitude A(t) is:

[0069] (6)

[0070] In the formula, A0 is the initial amplitude, δ is the attenuation coefficient, and its relationship with the viscous damping coefficient c in the vibration equation is c = 2mδ, where m is the mass of a single degree of freedom. Taking the logarithm of both sides of the above equation, we get:

[0071] (7)

[0072] From the relationship between the amplitude of the vibration and the wavelet coefficients on the ridge line, we can obtain:

[0073] (8)

[0074] The instantaneous viscous damping coefficient c can be obtained from the above formula:

[0075] (9)

[0076] (2) The relationship between frequency, damping coefficient and amplitude is calculated using the adjacent peak method.

[0077] Based on the displacement decay vibration curve, the vibration time is extracted cycle by cycle and the corresponding damping coefficient is calculated; the arithmetic mean of the initial and final peak values ​​in each cycle is taken as the equivalent amplitude of that cycle, and then the relationship between the square of the frequency and the damping coefficient and the amplitude is obtained.

[0078] Step 3: Establish a displacement-dependent nonlinear vibration model and perform parameter identification.

[0079] Based on the nonlinear phenomenon that both the frequency and damping of reinforced concrete beams vary with amplitude, the following displacement-dependent nonlinear vibration equation is constructed:

[0080] (10)

[0081] In the formula, m is the equivalent mass, x is the structural vibration displacement, c0 is the linear viscous damping coefficient, and c n k is the nonlinear damping coefficient, k0 is the linear stiffness coefficient, and k n This is a nonlinear stiffness coefficient;

[0082] The amplitude A and equivalent stiffness k are derived using the harmonic balance method. eq and the equivalent damping coefficient c eq The relationship is as follows. The restoring force in the nonlinear equation can be written as the equivalent relationship of the following equation.

[0083] (11)

[0084] If the excitation of the system is Ycos(ωt-φ), let the trial solution be x=Acosθ, θ=ωt. Substitute these into the above nonlinear terms and expand them into a Fourier series.

[0085] (12)

[0086] In the formula

[0087] (13)

[0088] Substituting the trial solution Acosθ into the restoring force f k (x), i.e., f k (x)=(k0-k n |Acosθ|)Acosθ, this function is an even function. In the above formula, b n =0. Therefore, the restoring force can be written as

[0089] (14)

[0090] In the formula

[0091] (15)

[0092] After calculation, a0=0, taking only the first harmonic (fundamental) term, we have:

[0093] (16)

[0094] And thus obtain

[0095] (17)

[0096] The equivalent stiffness k is obtained by solving the problem. eq The relationship with amplitude A is

[0097] (18)

[0098] Equivalent stiffness k eq This represents the ratio of average restoring force to displacement over one period, which is the average stiffness of the system over one period. Given the equivalent mass m of the system (the equivalent mass of a simply supported beam is half the total mass), the relationship between the instantaneous frequency f (Hz) and the amplitude A can be obtained as follows.

[0099] (19)

[0100] The damping force in the nonlinear equation can be expressed as the equivalent relationship of the following equation.

[0101] (20)

[0102] Substitute the trial solutions x=Acosθ, θ=ωt, ẋ=-Aωsin(ωt) into the above nonlinear terms and expand them into a Fourier series.

[0103] (twenty one)

[0104] After substituting the trial solution, f d Since it is an odd function, a in the above formula n =0 (n=0,1,2,…),

[0105] (twenty two)

[0106] Taking only the first harmonic (fundamental frequency) term, we have

[0107] (twenty three)

[0108] And thus obtain

[0109] (twenty four)

[0110] The equivalent damping coefficient c is obtained by solving the problem. eq The relationship with amplitude A is

[0111] (25)

[0112] Write it in the form of mass m normalization

[0113] (26)

[0114] In practice, the data obtained in step 2 (A) are processed separately. i f i2 ) and (A i c eq,i Linear fitting was performed on / m) to obtain:

[0115] (27)

[0116] (28)

[0117] p1 and p2 are the squares of the frequencies f, respectively. 2 - The slope and intercept of the fitted straight line of amplitude A, where q1 and q2 are the mass-normalized damping coefficients c. eq The slope and intercept of the straight line fitted between / m and amplitude A.

[0118] Based on formulas (19)(26)(27)(28), the normalized linear stiffness parameter, nonlinear stiffness parameter, linear damping parameter, and nonlinear damping parameter with respect to mass m can be calculated:

[0119] (29)

[0120] (30)

[0121] (31)

[0122] (32)

[0123] Where k0 / m is the mass-normalized linear stiffness parameter, k n / m is the mass-normalized nonlinear stiffness parameter, c0 / m is the mass-normalized linear damping parameter, and c n / m is the mass-normalized nonlinear damping parameter. When the equivalent mass m is known, the above normalized parameter can be multiplied by m to obtain the actual parameters k0 and k. n c0 and c n When the equivalent mass m is difficult to determine accurately, the above-mentioned normalized parameters can be directly used as indicators for damage analysis and bearing capacity degradation evaluation.

[0124] Alternatively, the above nonlinear vibration equation (10) can be used to directly fit the displacement vibration data to obtain relevant parameters.

[0125] Step 4: Calculate the linear vibration frequency and damping ratio based on the parameters obtained in Step 3.

[0126] Based on the identified k0 / m and c0 / m, the small-amplitude linearized dynamic characteristics of the beam can be further determined. The linear vibration frequency can be calculated using the following formula:

[0127] (33)

[0128] The linear damping ratio can be calculated using the following formula:

[0129] (34)

[0130] Where f0 is the linear vibration frequency and ζ is the linear damping ratio. This step separates the amplitude-dependent nonlinear effects in the free decaying vibration signal, yielding a linear frequency and linear damping ratio that better reflect the basic dynamic performance of the beam.

[0131] Step 5: Perform damage analysis based on the damping ratio, the changing trend and amplitude of nonlinear parameters, and assess the bearing capacity based on the linear vibration frequency.

[0132] At different points in the bridge's service life, such as at preset time intervals or under critical conditions like special loads, earthquakes, floods, impacts, and before and after maintenance, steps 1 to 4 are systematically repeated to obtain the linear frequency f0, damping ratio ζ, k0 / m, and k corresponding to each monitoring moment. n / m、c0 / m、c n / m, to obtain the dynamic characteristics and the evolution of their nonlinear parameters over time.

[0133] Crack development and steel corrosion in reinforced concrete beams have a significant impact on their dynamic characteristics. Crack initiation and development primarily alter the linear damping ratio, damping nonlinearity coefficient, and stiffness nonlinearity coefficient; therefore, the evolution of these three parameters can serve as sensitive indicators of crack initiation and propagation. Steel corrosion, on the other hand, directly leads to a decrease in the linear stiffness coefficient and linear frequency, both of which can serve as important bases for assessing the degradation of the structural bearing capacity. In other words, if a shift in the linear damping ratio, damping nonlinearity coefficient, or stiffness nonlinearity coefficient is detected, it usually indicates the appearance of new cracks within the concrete or the further propagation of existing cracks. A continuous decrease in linear stiffness and linear frequency more directly reflects the reduction in the structural bearing capacity.

[0134] Quantitative analysis is performed on the evolution trend and fluctuation characteristics of each parameter during the service life. When any parameter exhibits a significant abnormal change that deviates from historical patterns, or when the change in its current value relative to the pre-established baseline exceeds the preset engineering early warning threshold, an early warning mechanism is triggered, and an in-depth damage tracing and residual performance assessment procedure is initiated. For each discrete time node, the bearing capacity of the RC beam is quantitatively calculated according to the following formula.

[0135] (35)

[0136] In the formula, M(t) is the bearing capacity at service period t, f(t) is the linear frequency f0 measured at that time, f(0) is the linear frequency f0 in the initial state or the undamaged state, and M0 is the initial ultimate bearing capacity.

[0137] Example of specific implementation process:

[0138] After a period of use, a reinforced concrete beam developed steel bar corrosion, and its performance needs to be evaluated.

[0139] S1. The test was conducted at two time points: initially after construction and after a period of service. Two levels of loading (corresponding to 40% and 60% of the ultimate bearing capacity, respectively) were applied. After each loading, the load was unloaded and a dynamic test was performed, and the mid-span response signal was recorded. One of the displacement signals is shown below. Figure 1 As shown.

[0140] S2. Wavelet transform is applied to the signal to extract the instantaneous frequency, instantaneous amplitude, and instantaneous damping coefficient. The resulting wavelet time-frequency plot is shown below. Figure 2 This leads to the square of the frequency, f. i 2 Regarding the mass-normalized equivalent viscous damping coefficient c eq，i The relationship between / m and amplitude A, with amplitude A as the independent variable, is expressed as follows: i 2 -A relation and c eq，i Linear fitting was performed on the / mA relationship, see Figures 3-4 .

[0141] S3. Based on the relationship between amplitude, frequency, and damping coefficient in step 2, construct the displacement-dependent nonlinear vibration model as follows:

[0142] (1)

[0143] By combining parameter identification methods, the parameters in the model are identified, and the normalized nonlinear vibration parameters k0 / m and k are obtained with respect to mass m. n / m、c0 / m、c n / m.

[0144] S4. Obtain k0 / m and k from the above relationships. n / m, c0 / m and c n / m, calculate its linear vibration frequency and damping ratio, and the parameters identified in the two stages are shown in Tables 1 and 2;

[0145] f0=[(k0 / m)] 1 / 2 / (2π) (2)

[0146] ζ=(c0 / m) / (4πf0) (3)

[0147] Table 1. Identification results of dynamic parameters under different loading levels during the initial stage of construction.

[0148]

[0149] Table 2. Identification results of dynamic parameters under different loading levels after service corrosion.

[0150]

[0151] Table 1 shows that the linear frequencies f0 and k0 / m of the components in the initial unloaded state both decreased after the first loading stage. This indicates that the first loading stage caused cracking in the tensile concrete, resulting in a significant reduction in equivalent stiffness. After the second loading stage, the changes in linear frequencies f0 and k0 / m were smaller, indicating that the main stiffness degradation of the structure had already occurred after the first loading stage, and further loading had no significant effect on the overall linear frequency and linear stiffness. At the same time, the damping ratios ζ and c0 / m both increased slightly, indicating that crack development enhanced interfacial friction, crack opening and closing, and local energy dissipation.

[0152] From initially never loading to after loading, k n / m and c n The significant changes in the linear frequency f0 indicate that even though the linear frequency tends to stabilize after the first-stage loading, the nonlinear response characteristics within the structure still change with crack morphology and contact state. Therefore, relying solely on the linear frequency f0 to determine damage may miss some subtle damage information, while joint analysis of damping parameters and nonlinear parameters can more fully reflect the damage evolution process of the component, with nonlinear parameters having a more sensitive indicative role in crack state.

[0153] Comparing Tables 1 and 2 reveals that the linear frequency f0 and equivalent linear stiffness k0 / m of the corroded component under the same loading level are both lower than those at the initial construction stage. This change is due to the weakening of the steel reinforcement section caused by corrosion, leading to a decrease in the overall stiffness of the RC beam. Furthermore, the damping ratio ζ and nonlinear parameters of the corroded component show significant changes, indicating that corrosion damage and loading cracks alter the energy dissipation mechanism of the component. Therefore, the damping ratio ζ and nonlinear parameter k0 / m... n / m and c n / m can reflect the initiation and development of cracks, while the first-order linear frequency f0 can reflect changes in bearing capacity. The method for assessing residual bearing capacity using the first-order linear frequency f0 is as follows:

[0154] (4)

[0155] In the undamaged state, the bending moment bearing capacity of the beam section is M0 = 22.5 kN·m. f(0) is taken as the average of the first-order linear frequencies f0 measured after two stages of loading at the initial construction, which is 68.435 Hz. Similarly, f(t) is taken as the average of the first-order linear frequencies f0 measured after two stages of loading after a period of service, which is 63.425 Hz. The remaining bending moment bearing capacity of the section after corrosion is calculated to be M(t) = 20.85 kN·m. A destructive loading test was conducted on the beam in the laboratory, and the measured bearing capacity was 19.96 kN·m, with an evaluation error of approximately 4.4%. The first-order linear frequency f0 given by this method overcomes the influence of amplitude and can provide an estimate of the remaining bearing capacity of the RC beam that is close to the measured result without destructive loading.

[0156] S5. In practical engineering applications, dynamic tests can be repeatedly conducted at different service stages or maintenance points, and a parameter time series database for the same component can be established. This can be achieved by comparing linear frequency f0, linear damping ratio ζ, k0 / m, c0 / m, and k... n / m and c n The statistical characteristics of the evolution trend of / m over time and load level, such as Figure 5 As shown, this can further improve the stability of corrosion damage identification and residual load capacity assessment.

[0157] In the description of this invention, it should be noted that the terms "center," "longitudinal," "lateral," "upper," "lower," "front," "rear," "left," "right," "vertical," "horizontal," "top," "bottom," "inner," and "outer," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are used only for the convenience of describing the invention and for simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on the invention. Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance.

[0158] In the description of this invention, it should be noted that, unless otherwise explicitly specified and limited, the terms "installation," "connection," and "linking" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal connection of two components. Those skilled in the art can understand the specific meaning of the above terms in this invention based on the specific circumstances. Furthermore, in the description of this invention, unless otherwise stated, "a plurality of" means two or more.

[0159] It should be understood that those skilled in the art can make improvements or modifications based on the above description, and all such improvements and modifications should fall within the protection scope of the appended claims.

Claims

1. A method for damage analysis and bearing capacity assessment of RC beams based on a nonlinear vibration model, characterized in that, Includes the following steps: S1. Acquire the free decay vibration displacement signal of the reinforced concrete beam; S2. Based on the free decay vibration displacement signal, extract the instantaneous amplitude, instantaneous frequency, and instantaneous equivalent damping coefficient, establish the correlation between the square of the frequency, the normalized equivalent damping coefficient and the instantaneous amplitude, and obtain the fitting parameters p1, p2, q1, and q2 by linear fitting. S3. Construct a displacement-dependent nonlinear vibration model. Combining the harmonic balance method with the fitting parameters obtained in step S2, identify the dynamic parameters normalized to equivalent mass: k0 / m, k n / m、c0 / m、c n / m; S4. Calculate the linear vibration frequency f0 and linear damping ratio ζ to eliminate amplitude interference based on normalized dynamic parameters; S5. Based on the variation characteristics of damping ratio and normalized nonlinear dynamic parameters, concrete crack damage is determined; based on the degree of degradation of linear vibration frequency and normalized linear stiffness parameters, the corrosion stiffness degradation of steel bars is determined, and the remaining bearing capacity of RC beam during service life is quantitatively calculated through linear vibration frequency. The displacement-related nonlinear vibration model equation described in step S3 is as follows: ； In the formula, m is the equivalent mass, x is the structural vibration displacement, c0 is the linear viscous damping coefficient, and c n k is the nonlinear damping coefficient, k0 is the linear stiffness coefficient, and k n This is a nonlinear stiffness coefficient; The formula for calculating the remaining bearing capacity is: ； In the formula, M(t) is the bearing capacity at service period t, f(t) is the linear frequency measured at that time, f(0) is the linear frequency in the initial state or in the undamaged state, and M0 is the initial ultimate bearing capacity. S6. Perform damage analysis based on the changing trends and amplitudes of the damping ratio and nonlinear parameters; then, assess the bearing capacity based on the degradation of the linear vibration frequency or linear stiffness parameters.

2. The method for damage analysis and bearing capacity assessment of RC beams based on a nonlinear vibration model according to claim 1, characterized in that: Step S1 specifically includes: applying transient impact excitation to the RC beam to generate free decay vibration; directly acquiring displacement signals using a displacement sensor, or acquiring acceleration signals using an accelerometer and converting them into displacement response through double integration, and eliminating signal segments with excessively low signal-to-noise ratio and obvious boundary interference.

3. The method for damage analysis and bearing capacity assessment of RC beams based on a nonlinear vibration model according to claim 1, characterized in that: Step S2 involves extracting instantaneous parameters using wavelet transform or adjacent peak calculation; processing the displacement analytical signal using complex Morlet continuous wavelet transform, and solving for instantaneous amplitude, instantaneous frequency, and instantaneous equivalent damping coefficient from wavelet coefficient ridges; or extracting vibration peak values ​​periodically along the displacement time history and calculating single-cycle equivalent amplitude, instantaneous frequency, and equivalent damping coefficient.

4. The method for damage analysis and bearing capacity assessment of RC beams based on a nonlinear vibration model according to claim 1, characterized in that: In step 2 (f) i 2 A i ) and (c eq,i / m, A i The line graph obtained by linear fitting can be obtained as follows: f i 2 =p1A i +p 2； c eq,i / m=q1A i +q 2；； Where p1, p2, q1, and q2 are the squares of the frequencies f. 2 Normalized equivalent viscous damping coefficient c eq The slope and intercept of the fitted straight line between / m and amplitude A are calculated; then, based on the displacement-dependent nonlinear vibration model and the harmonic balance method, k0 / m = 4π is calculated. 2 p2, k n / m=-(3π 3 / 2)p1、c0 / m=q2、c n / m=(3π / 4)q1, yielding four nonlinear vibration parameters normalized with respect to the equivalent mass m.

5. The method for damage analysis and bearing capacity assessment of RC beams based on a nonlinear vibration model according to claim 1, characterized in that: The formula for calculating the normalized dynamic parameters in step S3 is as follows: ； ； ； ； Where k0 / m is the mass-normalized linear stiffness parameter, k n / m is the mass-normalized nonlinear stiffness parameter, c0 / m is the mass-normalized linear damping parameter, and c n / m is the mass-normalized nonlinear damping parameter; when the equivalent mass m is known, the above normalized parameter can be multiplied by m to obtain the actual parameters k0 and k. n c0 and c n When the equivalent mass m is difficult to determine accurately, the above-mentioned normalized parameters can be directly used as indicators for damage analysis and bearing capacity degradation evaluation.

6. The method for damage analysis and bearing capacity assessment of RC beams based on a nonlinear vibration model according to claim 1, characterized in that: The formula for calculating the linear vibration frequency in step S4 is as follows: f0=[(k0 / m)] 1 / 2 / (2π); The formula for calculating the linear damping ratio is: ζ=(c0 / m) / (4πf0); When the equivalent mass *m* is known, multiplying the normalized parameter by *m* yields the actual stiffness and damping parameters. If the equivalent mass *m* is unknown, the linear frequency *f0*, linear damping ratios *ζ*, *k0*, *c0*, and *k* can be obtained. n / m、c n / m.

7. The method for damage analysis and bearing capacity assessment of RC beams based on a nonlinear vibration model according to claim 1, characterized in that: The damage determination rule in step S5 is: linear damping ratio ζ, normalized nonlinear stiffness coefficient k. n / m, normalized nonlinear damping coefficient c n A sudden change in the linear vibration frequency f0 and normalized linear stiffness k0 / m exceeding the threshold indicates the initiation or propagation of concrete cracks; a continuous decrease in the linear vibration frequency f0 and normalized linear stiffness k0 / m indicates steel corrosion and degradation of the component's bearing capacity. When parameters abnormally exceed the preset threshold, a structural damage warning is triggered.

8. The method for damage analysis and bearing capacity assessment of RC beams based on a nonlinear vibration model according to claim 1, characterized in that: Throughout the entire service life, steps S1-S4 are repeatedly executed at fixed time intervals, under conditions of overload, impact, post-disaster, and before and after maintenance, to establish a time-series dynamic parameter database and achieve long-term continuous damage monitoring and dynamic assessment of bearing capacity.

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