PSI - Issue 21

Gürzap İ. Demirel et al. / Procedia Structural Integrity 21 (2019) 101 – 111 Gürzap / Structural Integrity Procedia 00 (2019) 000 – 000

104

4

[0] = √ 2 0 [ ] = √ 4 2

(1)

(2)

= [ [ 0] ] = √ 22 0 ∗ 4

(3)

Mathematically, the most convenient way of storing the stress range histogram is in the form of PDF of the stress range, [10]. It is easy to transform from a stress range histogram to a PDF or backwards. In order to calculate the expected fatigue damage, E(D), firstly the PDF of the stress ranges, p(S), should be determined. Multiplying the probability of the stress range p(S)*dS with the total number of cycles (S t ) in the histogram, total number of cycles N(S), for a given stress level S, can be obtained as shown by Equation (4). ( ) = ( ) ∗ ∗ (4) 2.1. Stress cycle counting method from PSD in the frequency domain Among all the stress cycle counting methods from PSD, Dirlik's empirical formula for the cycle counting is the most superior in terms of accuracy, [5]. Dirlik has derived an empirical closed form expression for the determination of the PDF of the cycle counting of stress ranges, which was obtained using extensive computer simulations to model the signals using the Monte Carlo technique, [11]. The fatigue damage (D) is calculated after the cycle counting by the Dirlik’s method and given by Equation (5), where m and A are material properties which are the fatigue strength exponent obtained from the material S-N curve and the fatigue strength coefficient, respectively. = ( 2 1 ∗ ) ∗ ∫ ∗ ( ) ∗ 0 ∞ (5) In Equation (5), histogram formula N(S) for stress cycles range is given by Equation (6) , where ν p and τ are the rate of peaks (number of expected peaks per unit time) and the exposure time, respectively. ( ) = ν ∗ ∗ ( ) (6) The best correlation for p(S) proposed by Dirlik is given by Equation (7). ( ) = 1 ∗ − + 2 2 ∗ ∗ 2 − ∗ 2 2 + 3 ∗ ∗ − 2 2 2∗√ 0 (7) where, is the stress range and other parameters in Equation (6) and (7) are given in Table 1. Table 1 Parameters in Equation (6) and (7)