Structural fatigue reliability analysis method based on cross-threshold time of random process

Abstract

The invention relates to the technical field of structural fatigue analysis, in particular to a structural fatigue reliability analysis method based on random process threshold-crossing time, which comprises the following steps: introducing total time delta Ts exceeding a damage threshold into a fatigue accumulated damage index D, and then linking delta Ts with a time interval delta T1 and crossing times X1 between first-time exceeding time and second-time exceeding time; secondly, popularizing a solution method of a discrete random variable to a mean value in any range to a continuous random variable situation, and replacing the delta T1 with a mean value M (delta T1) of the delta T1 in the time q required by one-time stress circulation, so that the delta T1 is numeralized; and finally, according to the critical fatigue accumulated damage index, analyzing a fatigue reliability calculation method in consideration of threshold crossing, and solving the fatigue reliability only by observing the probability distribution of the intersection point between the Wiener process and the damage threshold b in the time period.

Description

Technical Field

[0001] The present invention relates to the technical field of structural fatigue analysis, and particularly to a method for analyzing the structural fatigue reliability based on the cross-threshold holding time of a random process. Background Art

[0002] Traditional structural fatigue reliability analysis is mostly based on relatively idealized models, and the factors affecting fatigue cumulative damage are not comprehensively considered. Most of them only mainly consider the influence of the stress cycle times. Therefore, there is still a large gap between the existing theory and the actual situation. In fact, the structural fatigue cumulative damage is related to the duration of the structural stress process exceeding the safety threshold. Since it involves the analysis of multiple exceedances of the safety threshold by a random process, the theoretical analysis of this problem is quite difficult. Therefore, the influence of the cross-threshold holding time on the random fatigue cumulative damage has not been considered. Summary of the Invention

[0003] The purpose of the present invention is to provide a method for analyzing the structural fatigue reliability based on the cross-threshold holding time of a random process. The technical problem solved by the present invention is that the influence of the cross-threshold holding time on the random fatigue cumulative damage has not been considered.

[0004] The purpose of the present invention can be achieved by the following technical solutions:

[0005] A method for analyzing the structural fatigue reliability based on the cross-threshold holding time of a random process includes the following steps:

[0006] Step 1: Divide the stress development process into i classical short-time Wiener segments;

[0007] Step 2: Calculate the fatigue cumulative damage index including the cross-threshold holding time:

[0008] For the single-boundary case, the fatigue damage index D(T) at the T-th moment is

[0009]

[0010] where ΔT s is the total cross-threshold time, N e is the number of stress cycles per unit time, C is the material fatigue test constant, and E(S m ) is the m-th origin moment of the stress amplitude S;

[0011] Step 3: Calculate the total time for the random response to exceed the single-sided threshold:

[0012] ΔT 1 The mean value M(ΔT 1 ) within the time [0, q] for one stress cycle is:

[0013]

[0014] where ΔT 1 is the threshold crossing duration, Γ(x, y) is the incomplete gamma function, and b is the damage threshold;

[0015] Step 4: Fatigue reliability calculation considering the threshold crossing duration:

[0016] The structural fatigue reliability P within [0, T] r is given by:

[0017]

[0018] After obtaining the value on the right side of the inequality sign in the above formula, the probability value expressed by this formula is then determined by the Monte Carlo method.

[0019] As a further solution of the present invention: In step 2, if the number of stress cycles per unit time is N e , then according to the Miner's rule, the cumulative damage degree of the structure at the T-th moment is

[0020]

[0021] As a further solution of the present invention: In step 2, if at the moment of T 0 it just reaches the threshold b, and there is no intersection between the stress process and the unilateral limit b before T 0 , based on the assumption that fatigue damage to the structure occurs only when the tensile stress exceeds the threshold b, then at the moment of T 0 +ΔT, the fatigue damage index of the structure is:

[0022]

[0023] As a further solution of the present invention: In step 3, the probability distribution of the time interval between two adjacent crossing points of the Wiener process and any non-zero threshold is:

[0024]

[0025] where Γ(x, y) is the incomplete gamma function and b is the damage threshold.

[0026] As a further solution of the present invention: In step 3, if the mean value M(Y) within the range of any y i ~y j (i < j < n) is to be obtained, according to the definition of the mean value of a discrete random variable, we have:

[0027]

[0028] As a further solution of the present invention: In step 3, when extended to a continuous random variable Y from y i ~y jThe mean value M(Y) within the range (i < j < n) can be expressed as:

[0029]

[0030] As a further solution of the present invention: In step 4, the fatigue cumulative damage index within [0, T]

[0031]

[0032] As a further solution of the present invention: When the fatigue cumulative damage index D of the structure or component is greater than or equal to 1, the structure undergoes fatigue failure.

[0033] As a further solution of the present invention: When the fatigue cumulative damage index D of the structure or component is less than 1, the structure has not reached the fatigue limit and is in a safe state.

[0034] Advantages of the present invention:

[0035] Based on the analytical solution of the probability distribution of the time interval (duration exceeding the threshold) between two adjacent crossing points of the existing Wiener process, the present invention explores the probability distribution of the number of crossings of the Wiener process and a non-zero threshold within a certain time period, and compares the crossing probabilities calculated by the Monte Carlo method and the proposed numerical method under different threshold conditions, providing a basis for the subsequent analysis of the fatigue reliability of the Wiener-type random stress process. By considering the duration of the random stress process exceeding the damage threshold, the present invention analyzes the influence of the duration of crossing the threshold on the fatigue cumulative damage of the structure, derives the fatigue cumulative damage index considering the total duration of exceeding the damage threshold, and establishes a fatigue reliability analysis method considering the duration of crossing the threshold.

[0036] Compared with the traditional method, the structure fatigue reliability analysis method based on the duration of crossing the threshold in this paper starts from the random stress process itself, realizes the influence of the duration of the stress process exceeding the damage threshold on the fatigue cumulative damage, and makes the calculation results closer to the actual situation. Description of the drawings

[0037] In order to more clearly illustrate the technical solutions in the embodiments of the present invention, the following will briefly introduce the drawings required for the description of the embodiments. Obviously, the following drawings are only some embodiments of the present invention. For those of ordinary skill in the art, other drawings can be obtained based on these drawings without creative efforts.

[0038] Figure 1 is the flowchart of the structure fatigue reliability analysis method based on the duration of crossing the threshold of the random process of the present invention;

[0039] Figure 2It is the curve graph of the measured stress-time history of a certain subway bogie during the operation period of the present invention;

[0040] Figure 3 It is the curve graph of the sample curves of the Wiener process for 1400s and 200s of the present invention;

[0041] Figure 4 It is the curve graph of the cross-threshold duration of the stress process and the number of cross-threshold time intervals within [0, T] of the present invention. Detailed implementation manners

[0042] In order to enable those skilled in the art to better understand the solution of the present invention, the technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings in the embodiments of the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those of ordinary skill in the art without making creative efforts shall fall within the protection scope of the present invention.

[0043] The structural fatigue reliability analysis method based on the cross-threshold duration of a random process provided by the embodiments of the present invention includes the following steps:

[0044] Step 1: Divide the stress development process into i classical short-time Wiener processes:

[0045] The stress response process of the structure under fatigue loads should be regarded as a random process; the dynamic stress cycle process of some structures has the characteristics of a Wiener process. Figure 2 is the measured stress-time history of a certain subway bogie during the operation period, Figure 3 The sample curves of the Wiener process for 1400s and 200s are given. It can be seen by comparison:

[0046] Generally, the dynamic stress cycle process of the actual structure fluctuates up and down around the mean value of 0, and the random stress-time change history has non-stationary characteristics.

[0047] Due to the extremely fast stress cycle speed under variable amplitude loads, the entire stress development process of the subway bogie has similarity. Based on the idea of the cyclic rain flow counting method, the random stress-time spectrum is regarded as composed of a series of repeated stress-time histories based on typical stress-time spectrum blocks, that is, it is considered that after a certain period of the cycle, the subsequent stress development process is roughly the same as the stress change process in the previous period.

[0048] If the entire observation period of the component (such as 1400 s) is regarded as a standard Wiener process, it is found that the development path of the Wiener process does not quite match the stress-time change process of the component. However, if the observation period is short (such as 200 s), the stress change process is more similar to the standard Wiener process.

[0049] Therefore, based on the similarity between the short-period Wiener process and the stress cycle process of the component, assuming that the short-period Wiener process can be regarded as a typical stress-time spectral block, the random stress response process of the component can be regarded as a stress-time history repeated based on the short-period Wiener process. That is, it is assumed that the structural random stress response process X(t) within a certain period conforms to a non-stationary Wiener process. The short period t short Take values according to the actual stress-time change process.

[0050] Step 2: Calculate the fatigue cumulative damage index including the cross-threshold holding time:

[0051] The relationship between the stress amplitude and the number of stress cycles is expressed by the common Basquin equation as follows:

[0052] NS m =C (4.7)

[0053] In the formula, m and C represent the material fatigue test constants, which are obtained from fatigue tests. S represents the stress amplitude, and N represents the limit cycle number (fatigue life). In the random case, S and N are random variables. According to Miner's rule, the cumulative damage of the structure can be expressed by the fatigue cumulative damage index D as follows:

[0054]

[0055] In the formula, ΔD i is the fatigue damage caused to the structure under the action of the stress amplitude S i , n i and N i are the actual number of stress cycles and the limit cycle number with the amplitude of S i . For the case where the cyclic stress amplitude S changes continuously, Equation (4.8) can be written in the following integral form:

[0056]

[0057] Considering the random stress cycle situation, denote the total number of cycles of the stress response λ(t) within [0, T] as n(T), and the probability density function of the stress amplitude S as f s (s), then the number of cycles n(s, T) per unit amplitude with the amplitude of S can be expressed as

[0058] n(s, T)=n(T)fs (s) (4.10)

[0059] Substitute Eqs. (4.10) and (4.7) into Eq. (4.9), and the random cumulative damage index of the component within [0, T] can be obtained

[0060]

[0061] where E(S m ) is the m-th origin moment of the stress amplitude S

[0062] If the number of stress cycles per unit time is N e , then according to Miner's rule, the cumulative damage degree of the structure at the T-th moment is

[0063]

[0064] If at the moment T 0 it just reaches the threshold b, and there is no intersection between the stress process and the unilateral boundary b before T 0 , based on the assumption that fatigue damage to the structure will only occur when the tensile stress exceeds the threshold b, then at the moment T 0 +ΔT, the fatigue damage index of the structure is

[0065]

[0066] For the single-boundary case, the fatigue damage index D(T) at the T-th moment is

[0067]

[0068] where ΔT s is the total time of crossing the threshold

[0069] Step 3: Calculation of the total time for random response to exceed the unilateral threshold

[0070] For the case where the stress response of the structure changes rapidly under fatigue loads, but the change of the stress envelope is relatively gentle ( Figure 4 ). When the threshold b is small, the crossing events in a typical stress-time load spectrum block tend to occur in clusters. Based on the previous description, the entire random stress spectrum can be regarded as a stress-time history repeated based on the typical stress-time load spectrum block. Therefore, in the next spectrum block cycle, the crossing events still tend to occur in clusters. The above crossing characteristics and the change of the stress envelope are similar to those of a narrow-band process. Theoretically, when the number of crossings tends to infinity, the durations of each threshold crossing tend to be equal. Therefore, it can be approximately assumed that each threshold crossing duration is basically equal (see Figure 4 ). Denote the threshold crossing duration as ΔT 1 , then Eq. (4.14) can be written as

[0071]

[0072] where n is the number of crossing intervals within [0, T];

[0073] Obviously, the number of crossing intervals n within [0, T] has the following relationship with the intersection points of the random stress response process and the damage threshold b:

[0074]

[0075] where X 1 is the number of crossings of the Wiener process with the damage threshold within [0, T].

[0076] Substituting Equation (4.16) into Equation (4.15) gives:

[0077]

[0078] Equation (4.17) has related the value of the total crossing duration ΔT s to the time interval ΔT 1 between the first passage time and the second passage time and the number of crossings X 1 , greatly reducing the analysis difficulty. Next, the value of ΔT 1 is analyzed.

[0079] Based on the characteristic that the stress of the structure changes rapidly under fatigue loads, and in the evaluation of sample performance measures, the most commonly used point estimate is the sample mean. Therefore, the mean of ΔT 1 within the time q for one stress cycle is denoted as M(ΔT 1 ) to approximately represent ΔT 1 , that is, ΔT 1 ≈M(ΔT 1 ).

[0080] It is known that the distribution law of the discrete random variable Y is P{Y = y k} = p k , k = 1, 2... (Table 4.1). If the series converges, then the mean of the discrete random variable Y is Generalizing it to the case of a continuous random variable, the mean of the continuous random variable Y is

[0081] Table 4.1 Probability distribution of Y

[0082]

[0083] From the above analysis, it can be seen that if any y i ~y jThe mean value M(Y) within the range (i < j < n) can be obtained according to the definition of the mean value of a discrete random variable as follows:

[0084]

[0085] Generalizing to the continuous random variable Y from y i ~y j The mean value M(Y) within the range (i < j < n) can be expressed as:

[0086]

[0087] According to Equation (2.35), the probability distribution of the time interval between two adjacent crossing points of the Wiener process and any non-zero threshold is:

[0088]

[0089] Where Γ(x, y) is the incomplete gamma function and b is the damage threshold.

[0090] Combining Equations (4.19) and (2.35), we can obtain ΔT 1 The mean value M(ΔT 1 ) within the time [0, q] for one stress cycle is:

[0091]

[0092] Step 4: Calculation of fatigue reliability considering the duration of crossing the threshold:

[0093] Substituting Equation (4.20) into Equation (4.17), we can obtain the fatigue cumulative damage index within [0, T], that is

[0094]

[0095] Equation (4.21) is the expression of the random cumulative damage index considering the duration of crossing the threshold, which relates the duration of crossing the threshold of the Wiener process, the number of crossings of the Wiener process and the damage threshold, and the m-th origin moment of the stress amplitude S. Among them, m and C can be determined by fatigue tests.

[0096] According to Equation (4.4), when the fatigue cumulative damage index D of a structure or component is greater than or equal to 1, the structure undergoes fatigue failure; when the fatigue cumulative damage index D of a structure or component is less than 1, the structure has not reached the fatigue limit and is in a safe state. Then the fatigue reliability P of the structure within [0, T] r is:

[0097]

[0098] The number of crossings X 1The statistical law can be obtained by using the Monte Carlo method based on MATLAB. When analyzing practical problems, it is not necessary to count the number of crossovers X in the entire observation period [0, T]. 1 By analyzing the actual stress cycle history, the random stress-time spectrum can be regarded as a repeated stress-time history composed of a series of typical stress-time spectrum blocks. Therefore, it is only necessary to statistically select the typical short period t short The number of crossovers in the entire observation period [0, T] can be used to predict the number of crossovers in the entire observation period [0, T].

[0099] Working principle of the present invention: The present invention first takes the random stress response process with Wiener-type characteristics as the research object, based on the Miner criterion and the idea of ​​fatigue cutoff limit in the "Steel Structure Design Standard", the total time ΔT exceeding the damage threshold is s Introducing the fatigue cumulative damage index D, it is obvious that the longer the duration of crossing the damage threshold, the greater the fatigue damage caused. Then, ΔT s The time interval ΔT between the first surpassing time and the second surpassing time 1 and the number of crossovers X 1 Secondly, the method of solving the mean value of discrete random variables in any range is extended to the case of continuous random variables, and then ΔT is used to calculate the mean value of discrete random variables. 1 The mean value M(ΔT 1 ) instead of ΔT 1 , so that ΔT 1 Finally, according to the critical fatigue cumulative damage index, the fatigue reliability calculation method considering the cross-threshold duration is analyzed, and the fatigue reliability can be solved by only observing the probability distribution of the intersection between the Wiener process and the damage threshold b during the period.

[0100] The above is a detailed description of an embodiment of the present invention, but the content is only a preferred embodiment of the present invention and cannot be considered to limit the scope of implementation of the present invention. All equivalent changes and improvements made within the scope of the present invention should still fall within the scope of the patent coverage of the present invention.

Claims

1. A structural fatigue reliability analysis method based on the cross-threshold duration of a random process, characterized by: The following steps are involved: Step 1: The stress development process is divided into i classic short-term Wiener; Step 2: Calculate the fatigue cumulative damage index including the cross-threshold duration: For the single-limit case, the fatigue damage index D(T) at time T is Where ΔT s is the total time of crossing the threshold, N e is the number of stress cycles per unit time, C is the material fatigue test constant, E(S m ) is the m-order origin moment of the stress amplitude S; Step 3: Calculation of the total time that the random response exceeds the unilateral threshold: The mean value M(ΔT1) of ΔT1 in the stress cycle time [0,q] is: Where ΔT1 is the cross-threshold duration, Γ(x,y) is the incomplete gamma function, and b is the damage threshold; Step 4: Fatigue reliability calculation considering the cross-threshold duration: [0,T] Internal structure fatigue reliability P r for: After finding the value on the right side of the inequality sign in the above equation, the Monte Carlo method is used to determine the probability value expressed by the equation.

2. The structural fatigue reliability analysis method based on random process cross-threshold duration according to claim 1 is characterized in that: In step 2, if the number of stress cycles per unit time is N e , then according to Miner's law, the cumulative damage degree of the structure at the Tth moment is 3. The structural fatigue reliability analysis method based on random process cross-threshold duration according to claim 1 is characterized in that: In step 2, if the threshold value b is reached at time T0, and there is no intersection between the stress process and the unilateral limit b before T0, based on the assumption that fatigue damage will only occur to the structure when the tensile stress exceeds the threshold value b, the fatigue damage index of the structure at time T0+ΔT is:

4. The structural fatigue reliability analysis method based on random process cross-threshold duration according to claim 1 is characterized in that: In step 3, the probability distribution of the time interval between two adjacent intersections of the Wiener process and any non-zero threshold is: Where Γ(x, y) is the incomplete gamma function and b is the damage threshold.

5. The structural fatigue reliability analysis method based on random process cross-threshold duration according to claim 1 is characterized in that: In step 3, if we want any y i ~y j The mean value M(Y) in the range (i<j<n) can be obtained according to the definition of the mean of discrete random variables:

6. The structural fatigue reliability analysis method based on random process cross-threshold duration according to claim 1 is characterized in that: In step 3, generalize to the continuous random variable Y from y i ~y j The mean value M(Y) in the range (i<j<n) can be expressed as:

7. The structural fatigue reliability analysis method based on random process cross-threshold duration according to claim 1 is characterized in that: In step 4, the fatigue cumulative damage index in [0,T] 8. The structural fatigue reliability analysis method based on random process cross-threshold duration according to claim 1 is characterized in that: When the fatigue cumulative damage index D of a structure or component is greater than or equal to 1, fatigue failure occurs in the structure.

9. The structural fatigue reliability analysis method based on random process cross-threshold duration according to claim 1 is characterized in that: When the fatigue cumulative damage index D of a structure or component is less than 1, the structure has not reached the fatigue limit and is in a safe state.

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