Table 1 Calculation equation of failure pressure
From: Limit state equation and failure pressure prediction model of pipeline with complex loading
case | additional load | principal stress | judging condition | Calculation equation |
|---|---|---|---|---|
Case 1 | positive | \({\sigma }_{1}\) = \({\sigma }_{L}\), \({\sigma }_{2}\) = \({\sigma }_{h}\), \({\sigma }_{3}\) = 0. | \({\sigma }_{2}\le \frac{{\sigma }_{1}+{\sigma }_{3}}{2}\) | \(\left(\frac{1}{2}-\frac{b}{1+b}\right)\frac{p}{{p}_{0}}+\frac{\sum ({\sigma }_{L})}{{\sigma }_{u}}=1\) |
Case 2 | \({\sigma }_{2} > \frac{{\sigma }_{1}+{\sigma }_{3}}{2}\) | \(\left(\frac{1}{2(1+b)}+\frac{b}{b+1}\right)\frac{p}{{p}_{0}}+\frac{1}{1+b}\frac{\sum ({\sigma }_{L})}{{\sigma }_{u}}=1\) | ||
Case 3 | \({\sigma }_{1}\) = \({\sigma }_{h}\), \({\sigma }_{2}\) = \({\sigma }_{L}\), \({\sigma }_{3}\) = 0 | - | \(\frac{1}{1+b}\left(1+\frac{b}{2}\right)\frac{p}{{p}_{0}}+\frac{b}{1+b}\frac{\sum ({\sigma }_{L})}{{\sigma }_{u}}=1\) | |
Case 4 | negative | \({\sigma }_{1}\) = \({\sigma }_{h}\), \({\sigma }_{2}\)=0, \({\sigma }_{3}\) = \({\sigma }_{L}\) | \({\sigma }_{2}\le \frac{{\sigma }_{1}+{\sigma }_{3}}{2}\) | \(\left(1-\frac{1}{2(1+b)}\right)\frac{p}{{p}_{0}}+\frac{1}{1+b}\frac{\sum ({\sigma }_{L})}{{\sigma }_{u}}=1\) |
Case 5 | \({\sigma }_{2} > \frac{{\sigma }_{1}+{\sigma }_{3}}{2}\) | \(\left(\frac{1}{b+1}-\frac{1}{2}\right)\frac{p}{{p}_{0}}+\frac{\sum ({\sigma }_{L})}{{\sigma }_{u}}=1\) | ||
Case 6 | \({\sigma }_{1}\) = \({\sigma }_{h}\), \({\sigma }_{2}\) = \({\sigma }_{L}\), \({\sigma }_{3}\) = 0 | - | \(\left(1-\frac{b}{2(1+b)}\right)\frac{p}{{p}_{0}}+\frac{b}{1+b}\frac{\sum ({\sigma }_{L})}{{\sigma }_{u}}=1\) |
