PSI - Issue 14

Viswa Teja Vanapalli et al. / Procedia Structural Integrity 14 (2019) 521–528 Author name / Structural Integrity Procedia 00 (2018) 000 – 000

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Table 4. A 1 , A 2 , q 0 and dx values of Eq. (5) for different sigmoidal curve fits. A 1 A 2

q 0

dx

Upper bound Sigmoidal fit Lower bound

1710.74005 1781.9732 1853.20635

908.59687 979.83002 1051.06317

0.31655 0.31655 0.31655

0.01205 0.01205 0.01205

where A 1 , A 2 , are the lower and upper transition value, q 0 is the inflection point and dx is the transition slope coefficient in Boltzmann sigmoidal function. The sigmoidal curve fit parameters found out are tabulated in Table 4.

3.2. A proposed normal variation of T over upper to lower bounds.

It is observed that there is a range of variation of peak stress for a particular value of ‘q’. Such variation is attributed to the microstructural changes at various zones of the material and position of the crack tip while fabricating specimens with respect to such changes. A range of upper and lower bounds of peak stress with respect to mid value is shown in Fig. 6a. The maximum probability of having the value of ‘T’ for any cracked component is the mid value given by Eq. (5). Similarly, the minimum probability is at the upper and lower bound values. A normal variation of parameter ‘T’ is proposed here over maximum and minimum limits. By considering that the upper and lower bounds lie within 3 σ limit, the value of Z score is varied between -3 to +3. The Z score for a standard normal distribution and the peak stress values (T i ) of intermediate Z scores can be obtained by using the following equations:

(6)

(7)

(8)

(9) where φ Z is probability density function, μ is the mean value, σ sd is the standard deviation and X u is the upper limit of peak stress (T). The typical values of ‘T’ for a constant value of q=0.31878 for seven values of Z scores are shown in Fig. 6b. Table 5. Computed values of Peak stress (T i ) at different Z scores for all pipes. Pipe component Peak stress, T (MPa) Z = -3 (X l ) Z = -1.5 Z = -0.67 Z = 0 (μ) Z = +0.68 Z = +1.5 Z = +3 (X u ) Pipe 8-1 1272.66 1308.279 1327.987 1343.896 1360.042 1379.513 1415.129 Pipe 8-2 1122.23 1157.856 1177.563 1193.472 1209.618 1229.089 1264.705 Pipe 8-3 950.316 985.9333 1005.641 1021.55 1037.696 1057.167 1092.783 Pipe 16-1 1081.45 1117.072 1136.779 1152.688 1168.834 1188.305 1223.921 Pipe 16-2 1102.32 1137.942 1157.65 1173.559 1189.705 1209.176 1244.792 Pipe 16-3 956.936 992.5526 1012.26 1028.169 1044.315 1063.786 1099.402 4. Cohesive zone analyses of straight pipes with through wall crack using probabilistic variation of peak stress ‘T’ suggested above The values of triaxiality parameters ‘q’ for all the six straight pipes are shown in Table 3. These values are then used to calculate peak stress ‘T’ for each pipe for seven values of Z score corresponding to a probability of occurrence. Table 5 shows such values for all the six straight pipes. Analyses are then carried out using these values along with the cohesive energy (G) as 220 kJ/m 2 . Results are shown in Fig. 7 along with the experimental results. It may be seen that the experimental values lie reasonably well within the computed results for all pipes except for an