Random Fatigue. ! Module 3. ! Lecture 23 :Random Vibrations & Failure Analysis

Transcription

1 !! Module 3! Lecture 23 :Random Vibrations & Failure Analysis Random Fatigue!! Sayan Gupta Department of Applied Mechanics Indian Institute of Technology Madras

2 Random fatigue As has been mentioned earlier, the two primary modes of failure in randomly vibrating structures are (a) due to overloading: this is technically termed as first passage type of failures and has already been discussed in the earlier slides, and (b) due to gradual deterioration, through fatigue. Fatigue damage occurs in a structure due to stress reversals that take place, within the structure material. In vibrating structures, the stress reversals occur often and hence fatigue damage is a primary mode of failure. In deterministic systems, fatigue life is typically characterised in terms of a S-N curve, which is available for di erent materials. 2

3 Random fatigue A S-N curve is basically a plot of the number of cycles required for a specimen made of a specified material and subjected to a uniaxial tensile loading for various stress levels, for failure. The dots represent experimental observations for number of cycles required for failure for a given stress level. The empirically obtained best-fit curve through these points is termed as the S-N curve. 3

4 Random fatigue In metals, usually the S-N curve is bi-linear and is typically of the form as shown. Note that the x-axis denoting the number of cycles is plotted in the log-scale. This implies that the relationship between the stress levels and the number of cycles is through a power law of the form N = as b,..(1) where, N is the number of cycles to failure, s is the load amplitude and a and b are material constants. 4

5 Random fatigue For loads with variable (but regular) amplitude levels, the fatigue life is computed by computing the fractional damage in the material due to one cycle of loading with amplitude level s i.thus,ifn i cycles are required for failure with loading of amplitude s i, the fractional damage due to one such cycle is 1 N i. Now, if there are n i such cycles, the fractional damage is computed as D i = n i N i. Thus, the total fractional damage due to a loading with k di erent amplitude levels is given by kx n i D =...(2) N i This damage accumulation rule is known as the Palmgren-Miner s rule. i=1 5

6 Random fatigue Substituting for N i from Eq.(1) in Eq.(2), we get D = kx i=1 n i a 1 s b = X i s b....(3) The summation in Eq.(3) assumes that the loading comprises of large number of cycles with di erent amplitude levels, but there exists only one cycle corresponding to a level s i. This is not such a restrictive assumption for an irregular but continuous load. Clearly, when the loading is a random process, s i is random which in turn, implies that D - the fatigue damage after time duration T - is a random variable which can be completely characterised in terms of its pdf. 6

7 Random fatigue The primary di culties associated with applying this damage accumulation rule in characterizing random fatigue damage due to random loadings are as follows: (a) expressing the time history into an equivalent number of cycles, (b) estimating the incremental fatigue damage due to each such cycle, and (c) applying a suitable damage accumulation rule to compute the total fatigue damage. Clearly, the crux lies in developing an appropriate cycle counting method. In this lecture, we discuss three four methods of cycle counting schemes: 1. peak counting method 2. range counting method 3. rain-flow counting method 4. level crossing counting method. 7

8 Random fatigue: peak counting method In the peak counting method, all local maxima above zero are counted. A local maximum is paired with a local minimum of the same size (that can occur anywhere in the time history) and an equivalent loading history is obtained. It is thus implicitly assumed here that the probability distribution function of the local maxima, F max ( ) and the local minima F min ( ) satisfy the relation F min ( x) =1 F max (x). Additionally, the local maxima and the local minima are paired to form load cycles independent of their relative location in the actual load time history. 8

9 Random fatigue Now computing the probability density function of random fatigue is a very di cult problem. A simpler problem would be to compute the mean fatigue damage, denoted as E[D(T )]. Now assuming that the amplitude of the cycles are continuously distributed, and that the loadings are stationary which implies that one can state that E[D(T )] = dt, where, d is the expected fatigue damage rate and T is time duration. Further, assuming that a linear damage accumulation rule is used and the amplitudes are continuously distributed, the summation sign in Eq.(3) can be replaced with an integral sign. 9

10 Random fatigue: peak counting method This leads to obtaining an expression for the expected value of fatigue damage per unit time, neglecting any mean stress e ects, as d = a Z 1 0 s b p a (s)ds,...(4) Here, p a (s) is the pdf of the stress amplitudes, a is the mean occurrence rate of cycles, and and b have the same meaning as in Eq.(3). It is to be noted here that counted cycles are treated as random events and have a probability distribution p am (s, m) which is a function of two random variables - the amplitude and the mean value. The expression developed in Eq.(4) shows that the fatigue damage is related to the constant amplitude loading strength, cycle distribution and the expected rate of occurrence of cycles. 10

11 Random fatigue: peak counting method Here, we first develop some expressions that will be useful in understanding the expressions developed in the previous slide. Let h be the probability density of counted cycles as a function of maximum and minimum. Thus, the probability distribution function can be expressed as H(u, v) = Z u 1 Z v 1 h(x, y)dydx. Here, H(u, v) represents the probability of a cycle with peak lower than or equal to u and a valley lower than or equal to v. The probability density of counted cycles as a function of amplitude and mean values is related to h through the transformation function p am (s, m) = 1 h(m + s, m 2 s). 11

12 Random fatigue: peak counting method The marginal probability density function for amplitude can be obtained by integrating out the mean parameter and can be expressed as p a (s) = Z 1 1 p am (s, m)dm. Thus, for every complete counting of the cycles, on an average, it is expected that the expected rate of occurrence of cycles is equal to the expected rate of occurrence of peaks. In other words, this implies, that a = p. This explains the expressions that were developed a few slides earlier. 12

13 Random fatigue: peak counting method In the peak counting approach, the peaks below zero are neglected. Thus, the expected number of cycles is lower than the expected number of peaks. However, Eq.(4) may be modified by relating the damage to the peak distribution because its positive part agrees with the amplitude distribution. This enables rewriting Eq(4) as d = p Z 1 0 s b p p (s)ds, where, p is the expected rate of occurrence of peaks and p p (s) is the peak density function. 13

14 Random fatigue: peak counting method It can be shown that for a Gaussian processes, the expected fatigue damage rate can be expressed as Z 1 p 1 d = p [ d = s b 2 + p p [ x Z 1 0 s 2 s 2 p 1 s b 2 exp[ p 2 exp[ 2 2 x(1 2 x 2 x(1 2 2 ) 2 ] 2 ) 2 ] + Z 1 0 s b 2s 2 x exp[ s x ] ( x 2 s p )ds]. Here, i = p i 0 2i i = p i 0 2i (x) = 2 Z x 1 e t2 dt i = Z 1 1! i S XX (!)d!. 14

15 Random fatigue: level crossing approach Peak counting usually overestimates the total damage. It has been mention din the literature that it is not appropriate to pair every positive peak with an opposite valley. Instead, it is more reasonable to pair only the number of peaks minus the number of valleys above zero, because the valleys above zero should not be rejected. Thus, the number of counted cycles with amplitudes greater than a value S is equal to the number of peaks minus the number of valleys greater than S, i.e., the number of up-crossings of level S. This approach is defined as the level crossing approach. The expected fatigue damage rate for a Gaussian process is given by d = p Z 1 0 Z 1 = p 2 s b s exp[ 2 x 0 s b [p p (s) p v (s)]ds (1) s x ]ds (2) Here, = p 2 ( p 2 x ) b (1 + b ). (3) 2...(5) [ ] is the gamma function. 15

16 Random fatigue: range counting method Note that the fatigue damage given by Eq.(5) is equal to the damage of a narrow band process having Rayleigh distribution with the same standard deviation and the expected number of peaks is equal to the number of up crossings of the mean level of the actual loading. This is the so called narrow-band approximation of fatigue damage of broad banded loadings, and has been frequently used in engineering literature. In range counting algorithm, every peak is paired with an appropriate valley so that cycles and ranges coincide. The expected damage due to this approach can be studied by investigating the crossings of a double envelope process and approximating the range counting by means of the amplitude envelope. For more details, see the following: 1 Madsen HO, Krenk S, Lind NC. Methods of structural safety, Prentice- Hall, Tovo R. Cycle distribution and fatigue damage under broad-band random loading. International Journal of Fatigue 24 (2002)

17 Random fatigue: rain-flow cycle counting Of all the counting cycles, the rain flow cycle counting algorithm is observed to give predictions which are at least variance with experimental observations. The rain flow cycle counting algorithm is based on the principle of hysteresis due to the random variations in the loadings. In the rain flow cycle counting method, the stress history is first converted into a series of peaks and valleys, with the peaks evenly numbered. The time axis is oriented vertically downwards with the positive direction downward. The time series is then viewed as a sequence of roofs with rain falling on them. The path of the rain-flow are defined according to the rules given in the next slide. 17

18 Random fatigue: rain-flow cycle counting 1. A rain-flow is started at each peak and at each valley. 2. When a rain flow path started at a valley comes to the tip of the roof, then the flow stops if the opposite valley is more negative than at the start of the path under consideration. Similarly, a path started at the peak is stopped at a peak which is more positive than that at the start of the path. 3. If the rain flowing down a roof intercepts flow from a previous path, the present path is stopped. 4. A new path is not started until the path under consideration is stopped. In this method, small reversals are treated as interruptions of the larger ranges. Also, the method identifies a mean stress for each stress cycle. The algorithm given in the previous slide is di cult to model mathematically. Instead, a mathematically more elegant but equivalent rain-flow cycle counting algorithm is provided by Rychlik [ see Rychlik I (1987). A new definition of rain flow cycle counting method. International Journal of Fatigue, 9, ] This is presented in the next slide. 18

19 Random fatigue: rain-flow cycle counting Let x(t), t 2 [0,T] be a variable load function having a finite number of local maxima. We assume that a local maximum v i = x(t i )inx(t) is paired with one particular local minimum u k, determined as follows: from the i-th local maximum (value v i ) one determines the lowest values in forward and backward directions between t i and the nearest points at which x(t) exceedsv i. The larger (less negative) of those two values, denoted by u rfc i, is the rain-flow minimum paired with v i,i.e. u rfc i is the least drop before reaching the value v i again on either side. Thus the ith rain-flow pair is (u rfc i,v i ). N S(u,v) = 1 f(u, v) = 1 (v u) b, The total damage D rfc (x) defined using the rain-flow method of Endo and linear Palmgren Miner damage accumulation rule is equal to D rfc (x(t)) = X f(u rfc i,v i ). (1) 19

20 Random fatigue: rain-flow cycle counting We next provide an alternative definition for rain flow cycle counting algorithm. For a smooth load x(t), the rain-flow damage is given by D rfc (x(t)) = Z +1 Z v 1 where, f 2 (u, v 1 f 12 (u, v)n + (u, v)du dv + and f 12 (u, v) u@ v. Z +1 1 f 2 (v, v)n + (u)du, (1) Note that the above equation is valid for deterministic loading. For random loads, clear;y the fatigue damage is a random variable, and one would need to take expectations on both sides of the equation, with the left hand side denoting the expected fatigue damage in time duration [0,T]. The meaning of the terms in the right hand side of the equation is explained in the next slide. 20

21 Random fatigue: rain-flow cycle counting Here, for a smooth loading function x(t), N + (u) is the number of up-crossings of level u by x(t), i.e., the number of solutions to equation x(t) =u, such that, ẋ(t) > 0. N (u) is the number of down-crossings of level u by x(t), i.e. the number of solutions to equation x(t) =u, such that ẋ(t) < 0. N + (u, v) is the number of up-crossings of an interval [u, v] byx(t), the number of solutions to equation system x(t) = u, x(s) = v, t<s, such that ẋ(t) > 0, ẋ(s) > 0 and for all z, t<z<s, u<x(z) <v. (Note that N + (u, u) =N + (u).) N (u, v) is the number of down-crossings of an interval [u, v] byx(t), i.e. the number of solutions to equation system x(t) = v, x(s) = u, t<s, such that ẋ(t) < 0, ẋ(s) < 0 and for all z, t<z<s, u<x(z) <v. (Note that N (u, u) =N (u).) Here, ẋ(t) denotes derivative of x(t) withrespecttotimet. It is obvious that N + (u, v) and N (u, v) can di er by at most one and that N + (u, v) is equal to the number of rain-flow pair (u rfc i,v i ), such that v i >vand u rfc i <u. 21

22 Random fatigue: rain-flow cycle counting Now, when the loads are random, the application of this algorithm would imply the need to compute the expected values of N + (u), mathematically referred to as E[N + (u)]. This is the expected up-crossing rate of a process X(t) across level u. We now know that we can compute this from Rice s formula. Additionally, we need to compute E[N + (u, v)] which denotes the expected range crossings of levels u and v. Estimating this quantity mathematically is however not a trivial issue and remains an open problem. Instead, we recognise that E[N(u, v)] apple min{e[n(u)],e[n(v)]}. 22

23 Random fatigue: rain-flow cycle counting For stationary processes, E[N(u, v)] = (u, v)t, where, (u, v) is the mean interval crossing rate. We also know that E[N(u)] = (u)t. Here, (u) is the mean level crossing rate. It follows that we can get a bound (u, v) apple min[ (u), (v)] = ˆ (u, v) Consequently, we can get a bound for the expected fatigue damage as well. 23

24 Random fatigue: rain-flow cycle counting It is worth noting that adopting this approach enables us to estimate the expected fatigue damage simply from calculating the mean level crossing rates associated with a random process. Thus, computing the mean level crossing rate using Rice s integral allows us to estimate failure probabilities against (a) overloading - first passage failures: as the mean level crossing rate has an important bearing on calculating the extreme value distributions, and (b) gradual structural deterioration - due to accumulation of damage due to fatigue. 24