1. LIMIT LOAD CALCULATION MODEL OF DUCTILE FAILURE OF DEFECTIVE PIPE AND PRESSURE VESSEL Orynyak I.V. G.S. Pisarenko Institute for Problems of Strength, National Academy of Sciences of Ukraine, Kyiv, Ukraine IPS NASU 1th Hungarian-Ukrainian Joint Conference on Safety-Reliability and Risk Engineering Plants and Components 11,12 April 2006, Miskolc, Hungary

2. IPS NASU 2W 2a2a  Application of Limit Load in Fracture Mechanics Thus another restriction (criterion) is needed

3. IPS NASU Examples of Limit Load Calculation P P RoRo RoRo RiRi RiRi N N N N M M M M t

4. IPS NASU The Existing Formulas for Defected Pipes Battelle’s local 1 c – defect halflength l – defect halfwidth a – defect depth  = 1- a /t - dimensionless ligament thickness = c /  Rt - dimensionless length 1 = c /  R a - dimensionless length c – defect halflength l – defect halfwidth a – defect depth  = 1- a /t - dimensionless ligament thickness = c /  Rt - dimensionless length 1 = c /  R a - dimensionless length Battelle’s local 2 Nuclear Electric global  - strength reduction cofficient

5. IPS NASU Strength Characteristic of Material Illustrating the viability of the UTS as a failure criterion for plastic collapse in a steel line pipe

6. IPS NASU Analytical Modeling of Ductile Failure Static limit load theorem (lower bound) of the theory of plasticity: an elastic–plastic structure will not collapse under monotone loads if a)it is in static equilibrium and b)the yield function is nowhere violated Static limit load theorem (lower bound) of the theory of plasticity: an elastic–plastic structure will not collapse under monotone loads if a)it is in static equilibrium and b)the yield function is nowhere violated Corollaries of the theorem : 1.Accounting for any additional bond in the body can only increase the limit load 2.Decreasing the volume of the body cannot increase the limit load 3.The solution which gives maximal value of limit load is more appropriate Corollaries of the theorem : 1.Accounting for any additional bond in the body can only increase the limit load 2.Decreasing the volume of the body cannot increase the limit load 3.The solution which gives maximal value of limit load is more appropriate

7. IPS NASU AXIAL SURFACE CRACK IN A CYLINDER 1. Assumptions: a)shear force is equal to zero b)shearing moments is equal to zero c)axial force is proportional to the net-section 1. Assumptions: a)shear force is equal to zero b)shearing moments is equal to zero c)axial force is proportional to the net-section 2. Equations of equilibrium 3. The Gist of the models The presence of an axial flaw causes imbalance between the circumferential stress and the internal pressure which must be balanced by the increment in the transverse forces to maintain equilibrium. The transverse forces, in turn, induce bending moments. The cylinder passes into the limit state when the bending moments reach a critical value corresponding to the chosen limit condition

8. IPS NASU AXIAL SURFACE CRACK IN A CYLINDER. 4. Limit condition: 5. Second boundary of limit area, x = x 1 where for crack-like defect for rectangular form for semi-elliptical form for parabolic form

9. IPS NASU AXIAL SURFACE CRACK IN A CYLINDER 6. The maximum value of the applied moment : 7. The analytical formulas for rectangular form for semi-elliptical form for parabolic form

10. IPS NASU AXIAL SURFACE CRACK IN A CYLINDER Illustration of the stages of solution for the rectangular defect

11. IPS NASU AXIAL SURFACE CRACK IN A CYLINDER The crack shape influence on the dimensionless limit pressure: 1 - for a rectangular crack, 2 - for a semi-elliptical crack, 3 - for a parabolic crack The crack shape influence on the dimensionless limit pressure: 1 - for a rectangular crack, 2 - for a semi-elliptical crack, 3 - for a parabolic crack

12. IPS NASU AXIAL SURFACE CRACK IN A CYLINDER The limit pressures for rectangular axial cracks as given by the different formulas: (I) “local” formula (1a), (II) “local” formula (1b), (III) “global” formula (2), (IV) – our solution (13a).

13. IPS NASU AXIAL SURFACE CRACK IN A CYLINDER N 2c, mm R, mm t, mm t n, mm kg/mm 2 13603819.53.7958.6437.410.6380.6200.634 22123819.53.7859.3640.780.6870.6890.684 3733819.53.7858.6448.940.8350.8850.875 43583819.55.6957.4425.030.4360.4320.445 52073819.55.6957.4428.830.5020.5050.516 6713819.55.6954.1443.450.8030.7990.801 73573819.57.7256.4610.970.1940.2290.239 82103819.57.7257.1616.170.2820.3090.320 9723819.57.7256.4639.090.6920.7010.694 102093819.58.5157.448.580.1490.2360.243 Comparison with Battelle’s experimental results for defected pipes

14. t x Q x (x) L2 L2 k2 А В н t нс t z z t IPS NASU THE TREATMENT OF THE MULTIPLE DEFECTS The strength reduction coefficient is determined from The strength reduction coefficient vs. the distance between adjacent defects

15. IPS NASU INFINITE DENT IN A THIN-WALLED PIPE 1. The form considered W << R – dent depth  – jump in the angle of the tangent to the pipe surface 2. Equations of equilibrium

16. IPS NASU INFINITE DENT IN A THIN-WALLED PIPE 3. Applied moment determination 4. Limit condition where

17. IPS NASU INFINITE DENT IN A THIN-WALLED PIPE 5. Analytical results What is the characteristic dent dimension? - Elastic solution -“Elastic ”

18. IPS NASU FINITE DENT IN A THIN-WALLED PIPE The base for analysis is that Finite length dent finite length crack 1.Rectangular dent if if 2. “Parabolic” form if

19. IPS NASU AXIALLY SYMMETRIC DEFECT 1. Complete analogy with the “crack” solution 2. The only difference is the limit moments : 3. Limit condition: or 4. The solution for the limit pressure:

20. IPS NASU FINITE LENGTH FINITE WIDTH 3D DEFECT (SLOT) 1. Differential equation of equilibrium 2. “Slot” solution is between the “crack” and “axially sym.” solutions Mechanism of transferring the applied moments from sections to sections where

21. IPS NASU FINITE LENGTH FINITE WIDTH 3D DEFECT (SLOT) 3. Limit force and moments in circumferential direction and 4. Limit conditions in circumferential direction  ptimization parameter 5. Limit conditions in axial direction

22. IPS NASU FINITE LENGTH FINITE WIDTH 3D DEFECT (SLOT) Nt R (inch) (t-a)2c (inch) 2lFracture pressure lbf/inch 2 (inch) Exp TheorOur model 10.1252.740.0621.5020.375153012791305 20.1282.740.0621.498 191012801200 30.1272.740.0621.5020.124163012821350 40.1222.74----- ------23001914 50.1122.760.0921.5840.398215016561610 60.1122.700.0331.5040.3941050616800 70.1242.590.0290.4780.157210011261720 80.1232.590.0230.2410.380227519431470 90.1232.590.0342.3480.404950620720 100.2482.620.0372.1060.38717007011380 110.2461.620.0610.5420.385330032353530 Comparison of the calculated and experimental values of the fracture pressure for aluminum pipes with 3D defects

23. IPS NASU FINITE LENGTH FINITE WIDTH 3D DEFECT (SLOT)

24. IPS NASU THROUGH CRACK 1. Equilibrium differential equations 2. Boundary region On the boundaries

25. 3. Redistribution axial stresses IPS NASU THROUGH CRACK

26. IPS NASU Redistribution axial stresses (epures)

27. IPS NASU 4. Redistribution tangent stresses Determination THROUGH CRACK

28. Redistribution tangent stresses (epures) IPS NASU THROUGH CRACK

29. IPS NASU 5. Redistribution circumference stresses THROUGH CRACK

30. IPS NASU Redistribution circumference stresses (epures) THROUGH CRACK

31. IPS NASU 6. Limit state in the longitudinal direction Limit equilibrium equation of the moments for the determination of 7. Limit state in the circumferential direction THROUGH CRACK Limit equilibrium equation of the moments

32. IPS NASU 8. Comparison with the Battelle’s formula Battelle00.150.30.50.70.250.9360.9560.9550.9620.9620.959 0.50.8140.8460.8520.8590.8750.874 0.750.7020.7130.720.7290.7370.713 10.5950.5890.5890.6010.6110.596 1.50.4360.3730.4180.4190.420.397 20.3480.3030.3080.2990.2910.286 30.2240.1870.1920.1760.1660.155 THROUGH CRACK

33. IPS NASU THROUGH CRACK - experiment N R, мм t, мм 2l,мм  в,МПа Pexp R/ t МПа Pcal R/ t МПа13819,55222,25537,9191,7187,47 23819,63222,25537,9194,5188,23 33819,14222,25553,1190,3190,98 43819,14222,25553,1190,3190,98 53819,22114,3533,1322,8342,96 63819,3225,4533,1486,9522,06 73819,3725,4533,1481,4522,20 83819,5083,82560,7384,8432,56 93819,4083,82560,7387,6432,23 103819,96224,0563,4206,9202,86 113819,53177,8532,4216,6233,40 123819,53177,8534,4227,6234,28 133819,53177,8495,2226,2217,10 143819,53177,8495,2223,4217,10 153819,53137,2518,6293,8292,22

34. IPS NASU THROUGH CRACK - experiment N R, мм t, мм 2l,мм  в,МПа Pexp R/ t МПа Pcal R/ t МПа163819,53162,6551,7267,6262,56 173819,53162,6862,1309,0410,29 183819,53162,6583,4205,5277,65 193819,14137,7648,3297,2355,49 203819,14138,9648,3285,5352,28 213819,14139,7688,3304,1376,20 223819,17152,4589,0269,6298,93 233819,17152,4583,4272,3296,08 243819,17152,4580,0257,9294,36 253819,12152,4641,4266,1325,50 263818,33381587,6126,294,92 273818,33508,0587,6100,764,69 283818,33222,25587,6217,9184,54 293818,33177,8558,6222,0233,30 303818,33279,4557,2182,7120,66

35. IPS NASU Leak before break Precondition of the leak before break phenomenon - net thickness section The boundary between leak and break is obtained Then

36. IPS NASU Leak before break

37. IPS NASU Leak before break

38. IPS NASU CONCUSIONS 1.The limit load used in the criteria formulation of fracture mechanics originated from the theory of plasticity and should be treated by appropriate methods. 2.The theoretical models of ductile failure of defected bodies provides understanding of the ductile failure mechanisms, establishes the dimensionless parameters that have the most influence on the limit load. 3.Theoretical models can be used for choosing the analytical pattern in constructing empirical formulas and the checkpoints when performing the FEM calculations.