Reliability Theory of Dynamic Loaded Structures (cont.) Calculation of Out-Crossing Frequencies Approximations to the Failure Probability.

Transcription

1 Outline of Reliability Theory of Dynamic Loaded Structures (cont.) Calculation of Out-Crossing Frequencies Approximations to the Failure Probability. Poisson Approximation. Upper Bound Solution. Approximation Based on Envelope Processes. Energy Envelope Process. Cramér and Leadbetter Envelope Process. 1

2 Reliability Theory of Dynamically Loaded Structures (cont.) Calculation of Out-Crossing Frequencies An out-crossing from the safe domain in the interval of the scalar process takes place, if Hence per definition, cf. Lecture 7, Eq. (36):. 2

3 The joint probability density function of and is presumed to be available. The probability on the right-hand side of (1) is equal to the probability of attains a sample value in the hatched area in Fig. 1b: (2) is known as Rice s formula, and specifies the out-crossing frequency through an upper time-varying barrier. 3

4 Double barrier problem: 4

5 Next, the out-crossing frequency for a two-barrier problem with time varying lower and upper barriers and is determined. The probability of a joint out-crossing through the lower and upper boundaries in is of the order of magnitude. Hence, the out-crossing events at the lower and upper boundaries are independent as, and the out-crossing probabilities can be added: The last term, representing the out-crossing probability through the lower limit state function, may be derived in the same way as the first term. 5

6 Out-crossing of a vector process: Given an -dimensional stochastic vector process. The safe domain is confined by a failure surface, which may expand or contract deterministically with time. 6

7 At the time an area element of is considered, defined by the position vector. The outward directed unit vector is denoted. The velocity vector relative to the expanding failure surface becomes, where. This is the velocity vector seen by an observer fixed to. Especially, the relative velocity in the direction of becomes: At first, the probability of an out-crossing through on condition of during the time interval is determined. Such out-crossings take place, if is placed in the hatched volume shown in Fig. 3, and if. Hence, the conditional out-crossing probability becomes: 7

8 : Out-crossing frequency through on condition of. : Joint pdf of on condition of. The unconditional out-crossing probability during with a relative velocity in the interval becomes: : Joint pdf of and. 8

9 The out-crossing events in disjoint intervals are independent events. Hence, the probability contributions may be added linearly. Then, the probability for an out-crossing at all relative velocities becomes: Out-crossing events through two disjoint area elements and during must be independent as. Again, the probability contributions can be added. It follows that the out-crossing frequency through the surface is obtained by adding the contributions (7) from all area elements: (8) is known as Belyaev s formula. 9

10 Example 1: Out-crossing frequency for a stationary Gaussian process with a constant barrier is a stationary Gaussian process, for which reason and become independent random variables, cf. Lecture 3, Eq. (34). Hence: : Pdf of standardized normal random variable. The barrier (limit state function) is constant: 10

11 Then, (2) becomes: Especially, the expected (mean) number of up-crossings per unit of time of the mean value level follows for : where and are spectral moments, cf. Lecture 2, Eq. (10). 11

12 Example 2: Expected number of local maxima per unit of time of a stationary Gaussian process 12

13 A local minima of is related with a zero down-crossing of the velocity process as shown on Fig. 4. Then, the expected (mean) number of local maxima per unit of time follows from (3) upon replacing with, with, and using the barriers, : 13

14 where: From (12) and (14) follows that the band-width parameter becomes, cf. Lecture 2, Eq. (11): 14

15 (15) only applies for Gaussian processes for which the acceleration process exists with a finite variance. In contrast, VanMarcke s bandwidth parameter also applies to stationary non-gaussian processes. Further, it is merely required that the velocity process exists with a finite variance. Example 3: Out-crossing frequency from a rectangular time-invariant safe domain 15

16 Given a 2-dimensional vector process,. The safe domain is given as: The out-crossing frequency follows from (8). Hence, we need to calculate the joint probability density along the 4 sides of the failure surface, see Fig. 5: Side 1: 16

17 Side 2: Side 3: Side 4: 17

18 where it has been used that: Then (8) attains the form: side 1 side 2 side 3 Side 4 18

19 Example 4: Out-crossing frequency of a stationary Gaussian vector process from a time-invariant safe domain and are - and -dimensional correlated normal vectors. Then, the conditional distribution of on condition of is normal with the mean value vector and the covariance matrix given as: where:

20 Next, this result is applied for and. Further,, since is time-invariant, so. Then, since the vector process is stationary it follows that: Hence: where: 20

21 Notice that is independent of time. Then, the out-crossing frequency becomes, cf. (8): The innermost integral becomes: 21

22 Then, (29) becomes: In most cases the surface integral in (31) must be evaluated numerically. Approximations to the Failure Probability Poisson Approximation Out-crossings during are assumed to be independent of previous out-crossing events. Then, the hazard function becomes, cf. Lecture 7, Eq. (44): 22

23 Then, cf. Lecture 7, Eq. (41): The approximation is valid for: High safety levels ( ). Broad-banded processes (correlation length ). where is the up-crossing frequency of the level. The approximation is not valid for narrow-banded response processes at low and medium barrier levels. 23

24 Upper Bound Solution An exact upper bound solution to the failure probability can be derived from the inequality, cf. Lecture 7, Eq. (35): : Number of first-passages in. : Number of out-crossings in. 24

25 Decisions based on an upper bound to the failure probability are conservative. The upper bound is only of interest, if the right hand side of (34) is significantly smaller than, i.e. if: 25

26 Approximations Based on Envelope Processes At low and medium barrier levels ( in Fig. 6) the out-crossings take place in clumps. This means the sequential out-crossing events are not independent. 26

27 At high levels ( in Fig. 6) at most a single out-crossing takes place in each clump. Hence, out-crossings at high barrier levels may be assumed to be independent events, leading to the approximation (33). An envelope process is a maximum process, which is a tangent to the underlying narrow-banded process at the local maxima with a smooth passing in between, as shown by the realization in Fig. 6. The clumps occur with a sufficiently large time separation that the maxima in adjacent clumps may be considered as stochastic independent. Then, the out-crossing events of the envelope process may be assumed to be independent events. Hence, the failure probability of the envelope process is given as, cf. (33): : Out-crossing frequency of the stationary envelope process through the constant barrier. 27

28 Since, the envelope process is a maximum process, (36) is an upper bound (hopefully a rather sharp upper bound) to the failure probability of the underlying process. This provides the approximation: (37) provides a solution valid for the failure probability of narrow-banded processes with low and medium level barrier. The approximation requires that the out-crossing frequency of the envelope process is available. Below, this will be determined for two definitions of the envelope process, valid for a stationary Gaussian process. 28

29 Energy Envelope Process Given a stationary Gaussian process with the mean value function, the standard deviation function and the auto-correlation coefficient function. The following normal distributed random vector with components normalized to zero mean and unit variance is introduced: where: 29

30 Since,. Then, the covariance matrix of becomes: denotes the correlation coefficient of and. For a harmonic process, we have. For a narrow-banded Gaussian process,. is determined as: 30

31 The joint probability density of the components of becomes: where is the marginal pdf of and is the pdf of on condition of : 31

32 Next, a coordinate transformation to is introduced, defined from: where: to is denoted the energy envelope process. This designation refers being proportional to the potential energy, and being proportional to the kinetic energy. and may be expressed as: 32

33 At the same time. Consistency requires that: The Jacobian of the transformation of as a function of is given as: 33

34 Then the joint pdf of becomes: 34

35 From (51), (52), (53): From (54) follows that and are independent. Further,. The marginal joint pdf of becomes: Out-crossings of through the barrier is equivalent to outcrossings of through the non-dimensional barrier 35

36 The out-crossing frequency of through the barrier becomes: 36

37 indicates the out-crossing frequency of the underlying process, as given by Eq. (2). is a reduction factor, taking the clumping of the out-crossings of the narrow-banded process in consideration in proportion to the single outcrossing of the envelope process. As follows from (58) the method requires that. 37

38 Cramér and Leadbetter Envelope Process The approximation based on the energy envelope only works, if the variance of the acceleration process is finite. This rules out the important special case of a SDOF oscillator exposed to Gaussian white noise. In order to deal with this case a modified envelope process due to Cramér and Leadbetter is introduced as: where is the Hilbert transform of, cf. Lecture 2, Eq. (39): A Hilbert transform merely changes a cosine into a sine, and visa versa. Consider a harmonic process. Then: 38

39 Hence, in this case, and the envelope definitions (46) and (60) become identical. For a narrow-banded stationary Gaussian process this is only approximately the case. Especially, realizations of is no longer tangent to the local maxima of. Further, the realizations of the Cramér and Leadbetter envelope are upper bounds (although close upper bounds) to those of the energy envelope. 39

40 The out-crossing frequency of the Cramér and Leadbetter envelope can be given on the same form as (57). However, in this case the reduction factor is given as: As seen, (63) can be evaluated, if merely the 2 nd order spectral moment exists. The derivation of (63) is left as an exercise. 40

41 Summary of Calculation of Out-crossing Frequencies. Out-crossing of scalar processes. Rice s formula: Out-crossings of vector processes. Belyaev s formula: Expected local maxima per unit of time: 41

42 Approximations to the Failure Probability. Poisson approximation: Upper bound solution: Approximations based on envelope processes: : Energy envelope. : Cramér and Leadbetter envelope. 42