GLOBAL RATCHETING BY ELASTIC-PLASTIC FEA ACCORDING TO ASME SECTION VIII RULES

Proceedings of the ASME 2011 Pressure Vessels & Piping Division Conference Proceedings of PVP2011 2011 ASME Pressure July Vessels 17-21, and 2011, Piping Baltimore, Division Maryland, Conference USA July 17-22, 2011, Baltimore, Maryland GLOBAL RATCHETING BY ELASTIC-PLASTIC FEA ACCORDING TO ASME SECTION VIII RULES PVP2011-57196 PVP2011-57196 Arturs Kalnins Lehigh University Bethlehem, PA, U.S.A. ak01@lehigh.edu Jürgen Rudolph AREVA NP GmbH Erlangen, Germany rudolph.juergen@areva.com ABSTRACT A framework of global ratcheting is developed for which the ratcheting measure is incremental permanent change of shape of a vessel or component. The changes of shape are such that they reduce the resistance to collapse, and, upon extended cycling, can lead to eventual failure by collapse. A ratcheting check is designed to prevent this failure. This framework fits the ratcheting rules that are currently in the 2010 Section VIII of the ASME B&PV Code. Tools are provided for ratcheting assessment based on this framework, and examples are presented that illustrate their application. 1 INTRODUCTION While the major design codes leave no doubt that a ratcheting check is needed, there is still not much consensus on exactly how the ratcheting of a vessel is to be measured and the failure that a ratcheting check is to address. In the early days, when the major analysis tool was based on elastic shell theory, there was not much dispute about that. For obvious reasons, the applications of today require more. It is now commonly agreed that doing more should be based on elastic-plastic FEA. Beyond that, two different kinds of events, both referred to as "ratcheting", are recognized. One is called local ratcheting and the other global ratcheting. For local ratcheting, the ratcheting measure is the accumulated strain magnitude within a small volume (theoretically a point). The mode of failure that is being prevented by the ratcheting check is of the same kind as that associated with a strain limit for static loading. Local ratcheting will not be considered in this paper but will be left to a separate paper, PVP2011-57229, which is also scheduled to appear in the Proceedings of the PVP2011 conference. The following is a typical scenario of the global ratcheting that is considered in this paper. A vessel or component is subjected first to static mechanical loading, which is kept steady in time. The applied magnitude of the loading is set below the collapse load, so that the design requirements for static loading are met. Then cyclic loading that does not influence collapse, such as a temperature field, is superimposed on the stress state of the steady mechanical loading. Failure by ratcheting can occur if this combination of steady and cyclic loading produces incremental permanent changes of shape of the vessel that reduce its resistance to collapse. Upon extended cycling, the resistance to collapse is lowered to the point at which the steady mechanical loading can produce collapse. This is the failure mode that the ratcheting check is designed to prevent. Since collapse is a global event, such cyclic behavior of a vessel will be called global ratcheting. The scope of this paper is limited to this kind of global ratcheting. The first objective of this paper is to provide the technical basis for global ratcheting according to the above scenario. The second objective is to provide the tools for its assessment and to present examples that illustrate their application. 2 BACKGROUND Numerous papers have been published on ratcheting in conference proceedings and journals. Among those that use direct cycle-by-cycle, elastic-plastic FEA, the following are cited. Garud (1993) 1 reported experimental results on ratcheting of an elbow-pipe assembly and compared them with 1 Copyright 2011 by ASME

theoretical results; Hübel (1996) 2 considered the concepts of material and structural ratcheting; Kalnins (2001) 3 developed the concept of the elastic core; Okamoto, Nishiguchi, and Aoki (2001) 4 proposed a limit on the equivalent plastic strain; Kalnins (2002) 5 proposed the permanent displacement as the measure to address the failure mode of incremental growth; Bhagwagar and Gurdal (2003) 6 considered the ratcheting of a straight pipe and Yang and Gurdal (2003) 7 a piping elbow; and Rahman, Hassan, and Corona (2008) 8 presented experimental results for a pressurized pipe subjected to cyclic bending and compared them with theoretical simulations. An excellent summary of the basic issues involved in ratcheting was given by Reinhardt (2003) 9. In addition, non-cyclic methods that identify the boundary between the shakedown and ratcheting domains directly, without the need for cycle-by-cycle analysis, have also been developed, pioneered by Reinhardt 10 and Ponter 11. 3 HOW GLOBAL RATCHETING LEADS TO COLLAPSE The key part of the scenario for global ratcheting in section 1 above is that the incremental permanent changes of shape of the vessel reduce its resistance to collapse, and that it is the steady mechanical loading that produces that collapse. The objective of this section is to identify the parameters that are involved in reducing the resistance to collapse. This is achieved by considering the Bree 12 problem. Its geometry consists of a cylindrical shell, sketched in Figure 1. The crosshatched object on top is a rigid, circular plate that keeps the upper edge of the shell from rotating. For the purposes of this paper, the mean shell radius is assumed 95 mm and the thickness 10 mm. A material with yield strength of 262 MPa (38 ksi) is assumed. The shell is subjected to internal pressure. Temperature on the inside surface is cycled between 371 C (700 F) and 21 C (70 F), and the outside kept at 21 C. According to the theory of limit analysis, collapse of the shell for a steady internal pressure is governed by the hoop membrane stress defined by equation(1), in which S y is the yield strength. If the hoop membrane stress is equal to the yield strength, then collapse occurs. R σ hoop membrane = Pressure S y (1) t If the pressure is set at 2/3 of the limit pressure, then collapse does not occur provided that the inside radius, R, and thickness, t, are the dimensions of the undeformed shell. The key is that it could occur if the R / t ratio is increased in the event of global ratcheting. This can be seen to happen from the results of heat transfer and stress analysis problems that are obtained from the Abaqus 13 finite element program. In the stress analysis problem, the elastic-perfectly plastic (EPP) material model and the non-linear geometry option are used (the basis for using the EPP model is in section 5.1 below). From these results, the contour of the displacement magnitude, U, is plotted in Figure 2. The Legend shows that, after 15 cycles, the internal radius, initially at 95 mm, has increased permanently by 2.281 mm as indicated by the maximum U (red line of elements), and that the thickness, initially at 10 mm, has decreased by 0.168 mm, which is obtained by subtracting the minimum U (blue) from the maximum U (red). According to equation(1), the incremental change of the shape of increased radius and decreased thickness has reduced the resistance to collapse and moved the shell closer to eventual collapse by 2/3 of the limit pressure. TEMP= 371 21 C TEMP=21 C Steady Pressure Figure 1: Bree problem Figure 2: Contour of U after 15 thermal cycles, plotted on undeformed (left) and permanently deformed shell geometry 2 Copyright 2011 by ASME

To demonstrate how collapse could occur, the cycle-bycycle changes of the radius and the thickness observed from Figure 2 are shown in Figure 3 and Figure 4. If the straightline behavior in these figures remained after 15 cycles, the hoop membrane stress of equation(1) could be extrapolated until it reached the yield strength of 262 MPa, which would indicate collapse. Figure 5 shows that to happen at 196 cycles. This is not to be interpreted as a prediction of the actual collapse at 196 cycles. The purpose of Figure 5 is only to show how the parameters that are involved in reducing the resistance to collapse could eventually lead to collapse. The prediction of the number of cycles for the actual collapse plays no role in the ratcheting assessment of this paper. Mean Radius Thickness 98.0 97.5 97.0 96.5 96.0 95.5 95.0 94.5 94.0 Figure 3: Cyclic increase of radius 10.1 10.0 9.9 9.8 9.7 Figure 4: Cyclic decrease of thickness Hoop Membrane Stress 300 280 260 240 220 200 180 160 Collapse 0 40 80 120 160 200 Figure 5: Extrapolation of hoop membrane stress to yield This completes the illustration of how global ratcheting can lead to collapse. It fits the rules of Section VIII of the 2010 ASME B&PV Code 14, from which excerpts are cited next. 4 EXCERPTS FROM ASME SECTION VIII The rules for ratcheting assessment by elastic-plastic analysis appear in Section VIII-Division 2 14, paragraph 5.5.7.2 and in Division 3, paragraph KD-234.1. Additional commentaries are given by Osage 15, in paragraph 5.5.7. A sample problem was provided by Sowinski, Osage, and Brown 16. The relevant parts for this paper are listed below: 5.5.7.2c): STEP 3 An elastic-perfectly plastic [EPP] material model shall be used in the analysis. The von Mises yield function and associated flow rule should be utilized. The yield strength defining the plastic limit shall be the minimum specified yield strength at temperature from Annex 3.D. The effects of non-linear geometry shall be considered in the analysis. 5.5.7.2e): STEP 5 The ratcheting criteria below shall be evaluated after application of a minimum of three complete repetitions of the cycle. Additional cycles may need to be applied to demonstrate convergence. If any one of the following conditions is met, the ratcheting criteria are satisfied. If the criteria shown below are not satisfied, the component configuration (i.e. thickness) shall be modified or applied loads reduced and the analysis repeated. 1) There is no plastic action (i.e. zero plastic strains incurred) in the component. 2) There is an elastic core in the primary-load-bearing boundary of the component. 3) There is not a permanent change in the overall dimensions of the component. This can be demonstrated by developing a plot of relevant component dimensions versus time between the last and the next to the last cycles. 3 Copyright 2011 by ASME

5 EXAMPLES 5.1 Description of Examples In the examples, the models shown are subjected to a steady mechanical load and a cycled temperature field. The mechanical load is set at 2/3 of the limit load that is obtained from paragraph 5.2.3 of ASME Section VIII, Division 2 14. After the pressure is applied, the temperature on the inside (or top) surface is then cycled slowly between a specified value for each example and 21 C (70 F). The outside surface (or bottom) is kept at 21 C. The material is SA-516 Gr. 70 steel, with a yield strength of 262 MPa (38 ksi) at 21 C. In the FEA, the bold-faced parts of Section VIII paragraph 5.5.7.2c) in section 4 above, are used. The use of the EPP model in this paper is justified on the following grounds. Test data (e.g., Fig. 2.16 in Ellyin 17, and Lefebvre and Ellyin 18 ) have shown strain amplitude-stress amplitude curves for SA- 516 Gr.70 steel that indicate softening with cycles up to strain amplitude of 0.3 % and hardening above that. Material models that can handle such cyclic action are still in research stage. An EPP model with zero cyclic softening and hardening with cycles provides a middle-ground for design purposes. The only material parameters that are needed are the elasticity modulus, yield strength, and thermal expansion coefficient. Their temperature dependent values are used in the FEA. As per Section VIII paragraph 5.5.7.2e)3), the ratcheting measure is the permanent change in the overall dimensions of the component. In this paper, this measure is indicated by the permanent displacement magnitude at the location at which it is greatest. That location is determined from the contour plots of the displacement magnitude (U, in Abaqus). 5.2 Shell with Hemispherical Head We begin with the vessel shown in Figure 6. The vessel consists of a cylindrical shell with a hemispherical head. The mean radius of the shell is 250 mm and the thickness of the shell and head are both 16 mm. Limit analysis for internal pressure is performed and the limit pressure of 19.40 MPa (2.81 ksi) is obtained. The limit state indicates that the critical pressure boundary for collapse is the shell. The vessel is subjected to a steady pressure of 12.93 MPa (1.88 ksi), which is 2/3 of the limit pressure. A cyclic temperature field is superimposed to the steady pressure. 5.2.1 Cycling between 200 C and 21 C The temperature on the inside surface is cycled slowly between 200 C (392 F) and 21 C (70 F) and the outside surface is kept at a steady 21 C (70 F). After 15 cycles, the contour of the permanently deformed vessel is shown in Figure 7. Its Legend indicates the critical location of the ratcheting measure that is to be used in the ratcheting check. It shows that, after the 15 th cycle (load Step 31), the maximum U is 0.330, which occurs at node 331. For the global ratcheting check, the change of U per cycle at that node is recorded in Table 1. As seen from Table 1, the change in U is already zero at the end of the 3 rd cycle, within four significant digits. U is also plotted at the ends of 15 cycles in Figure 8. No further change in U is detected. The conclusion is that, according to paragraph 5.5.7.2e)3) of section 4 above, the Section VIII ratcheting check for this case passes. Table 1: U and its change per cycle for cycling at 200 C Cycle Node U Change 0 331 0.2269 1 331 0.3300 0.1030 2 331 0. 3300 0.0000 3 331 0. 3300 0.0000 Figure 6: One-half of model of undeformed vessel Figure 7: Contour of U plotted on undeformed (left) and permanently deformed shell geometry after 15 temperature cycles between 200 C and 21 C 4 Copyright 2011 by ASME

Displacement Magnitude 0.35 0.30 0.25 0.20 0.15 0.10 0.05 As seen from Table 2, the change in U indicates a slight increase from the 13 th to the 15 th cycle. U is also plotted at the ends of 15 cycles in Figure 10. An almost linear change in U is detected. The conclusion is that, according to paragraph 5.5.7.2e)3) cited in section 4 above, the Section VIII ratcheting check for this case does not pass. 0.00 Figure 8: Permanent displacements at node 332 at ends of 15 temperature cycles between 200 C and 21 C 5.2.2 Cycling between 250 C and 21 C Next, the cycling of the temperature between 250 C (464 F) and 21 C (70 F) is performed. After 15 cycles, the contour of the permanently deformed vessel is shown in Figure 9. Its Legend indicates that the critical location of the ratcheting measure is at node 251. For the global ratcheting check, the change of U per cycle at that node is recorded in Table 2. Table 2: U and its change per cycle for cycling at 250 C Cycle Node U Change 0 251 0.222 1 251 0.461 0.2392 2 251 0.601 0.1401 3 251 0.740 0.1393 4 251 0.879 0.1383 5 251 1.016 0.1377 6 251 1.154 0.1378 7 251 1.291 0.1370 8 251 1.429 0.1380 9 251 1.567 0.1381 10 251 1.707 0.1392 11 251 1.846 0.1393 12 251 1.985 0.1394 13 251 2.127 0.1414 14 251 2.268 0.1415 15 251 2.410 0.1416 Figure 9: Contour of U plotted on undeformed and deformed shell geometry after 15 temperature cycles between 250 C and 21 C Displacement Magnitude 3.00 2.50 2.00 1.50 1.00 0.50 0.00 Figure 10: Permanent displacements at node 252 at ends of 15 temperature cycles between 250 C and 21 C This completes the ratcheting assessment of the vessel. The conclusion is that the cycling between 200 C and 21 C passes the ratcheting check but that between 250 C and 21 C does not. 5 Copyright 2011 by ASME

5.3 Shell with Flat Plate Bottom We consider now a vessel that consists of a cylindrical shell with a welded-on flat circular bottom. One-half of the cross section of its undeformed model is shown in Figure 11. The mean radius of the shell is 250 mm, its thickness is 4 mm, and the plate thickness is 10 mm. the outside surface is kept at 21 C. The 371 C (700 F) is meant to be the highest temperature at which the effects of creep can be neglected and the highest at which this ratcheting assessment is regarded as appropriate. Figure 13 shows the contour plot needed for identifying the location of the ratcheting measure, U. Its Legend shows that the maximum permanent U occurs at node 9001, which is at the center of the bottom plate. This means that U at node 9001 is plotted for the ratcheting check, as shown in Figure 14. Figure 11: One-half of model of undeformed vessel In order to identify the critical pressure boundary for collapse, limit analysis for the applied pressure is performed first. The limit pressure of 0.954 MPa (0.138 ksi) is obtained. The limit state is shown in Figure 12. It identifies the bottom plate as the critical pressure boundary. Figure 13: Contour of U plotted on undeformed and deformed shell geometry after 15 temperature cycles between 371 C and 21 C 25 Displacement Magnitude 20 15 10 5 0 Figure 12: Limit state of vessel at limit pressure at Plate Center node 9601 The vessel is subjected to a steady internal pressure of 0.636 MPa (0.092 ksi), which is 2/3 of the limit pressure. The temperature on the inside surface of the vessel is cycled between 371 C (700 F) and 21 C (70 F) for 15 cycles, and Figure 14: Permanent displacements of plate center at ends of 15 temperature cycles between 371 C and 21 C 6 Copyright 2011 by ASME

Table 3 provides the details for performing the ratcheting check of ASME Section VIII, which is cited in paragraph 5.5.7.2.e)3) of section 4 above. It consists of the comparison of the component dimensions (represented by displacements) between the last and the next to the last cycles. Its last row shows that the difference of the displacement at the ends of the 15 th and 14 th cycle is 0.08 mm, which amounts to 0.36% of that of the 15 th cycle. Section VIII leaves it up to the analyst to decide whether the check passes. It is recognized that this difference only appears to approach zero but will never reach it. At some point, it will lose accuracy of subtracting all significant digits used in the analysis, which renders any results of additional cycling meaningless. In the judgment of the authors, the results indicate that the vessel passes the ratcheting check. Table 3: U and its change per cycle for cycling at 371 C Cycles Node U Change % of 15 th cycle 1 9001 15.660 10.804 48.318 2 9001 18.350 2.690 12.030 3 9001 19.600 1.250 5.590 4 9001 20.330 0.730 3.265 5 9001 20.810 0.480 2.147 6 9001 21.150 0.340 1.521 7 9001 21.410 0.260 1.163 8 9001 21.610 0.200 0.894 9 9001 21.780 0.170 0.760 10 9001 21.910 0.130 0.581 11 9001 22.030 0.120 0.537 12 9001 22.120 0.090 0.403 13 9001 22.210 0.090 0.403 14 9001 22.290 0.080 0.358 15 9001 22.360 0.070 0.313 6 CONCLUSIONS For the global ratcheting considered in this paper: 1. Loading consists of steady mechanical loading and cyclic loading that does not influence collapse, such as a temperature field. 2. Magnitude of the mechanical loading must meet design requirements for static loading. 3. Ratcheting measure is permanent change of shape at end of a cycle, which can be represented by overall dimensions or displacements. 4. Ratcheting is a threat to serviceability of a vessel if the changing shape reduces the resistance to collapse. 5. Eventual failure mode is collapse produced by the steady mechanical loading after extended cycling. 6. After a reasonable number of cycles, the deformed shape of the vessel should be consistent with that of the limit state for the steady mechanical loading. 7. Location of the critical change of shape can be determined from the Legend of the contour plot of the displacement magnitude, which is provided by the postprocessors of commonly used finite element programs. REFERENCES 1 Garud Y.S., (1993), Analysis and Prediction of Fatigue- Ratcheting: Comparison with Tests and Code Rules, Proceedings of ASME Pressure Vessel & Piping Division Conference, PVP vol. 266, Creep, Fatigue Evaluation, and Leak-Before-Break Assessment, pp. 23-32. 2 Hübel, H., (1996), Basic conditions for material and structural ratcheting, Nuclear Engineering and Design, vol. 162, pp. 55-65. 3 Kalnins, A., (2001), Shakedown Check for Pressure Vessels Using Plastic FEA, ASME Bound Vol. 419, pp. 9-16, Pressure Vessel and Piping Codes Standards, edited by M. D. Rana. 4 Okamoto, A., Nishiguchi, I., and Aoki, M., (2001), Recent Advancement on the Draft of Alternate Stress Evaluation Criteria in Japan Based on Partial Inelastic Analyses, ASME PVP vol. 419, pp. 17-24, Pressure Vessel and Piping Codes Standards, edited by M. D. Rana. 5 Kalnins, A., (2002), Shakedown and Ratcheting Directives of ASME B&PV Code and their Execution, ASME PVP vol. No. 439, pp. 47-55, Pressure Vessel and Piping Codes and Standards, edited by R.D. Rana. 6 Bhagwagar, T., and Gurdal, R., (2003), Straight Pipe Cyclic Analyses for Shakedown Verification Code Criteria, ASME PVP vol. No. 453, pp. 19-30, Pressure Vessel and Piping Codes and Standards, edited by G.S. Chakrabarti. 7 Yang, J., and Gurdal, R., (2003), Piping Elbow Cyclic Analyses for Shakedown Verification, ASME PVP vol. No. 453, pp. 49-59, Pressure Vessel and Piping Codes and Standards, edited by G.S. Chakrabarti. 8 Rahman, S.M., Hassan, T., and Corona, E., (2008), "Evaluation of Cyclic Plasticity Models in Ratcheting Simulation of Straight Pipes Under Cyclic Bending and Steady Internal Pressure", International Journal of Plasticity, vol. 24, pp. 1756 1791. 9 Reinhardt, W., (2003), Distinguishing Ratcheting and Shakedown Conditions in Pressure Vessels, Proceedings of ASME Pressure Vessel & Piping Division Conference, PVP2003-1885, PVP vol. 458, Computer Technology and Applications, pp. 13-26. 7 Copyright 2011 by ASME

10 Reinhardt, W., (2008), A Non-Cyclic Method for Plastic Shakedown Analysis, ASME Journal of Pressure Vessel Technology, vol. 130, No. 3, paper No. 031209. 11 Chen, H., and Ponter, A.R.S., (2001), A Method for the Evaluation of a Ratchet Limit and the Amplitude of Plastic Strain for Bodies Subjected to Cyclic Loading, European Journal of Mechanics-A/Solids, vol. 20, pp. 555-571. 12 Bree, J., (1967), Elastic-Plastic Behaviour of Thin Tubes Subjected to Internal Pressure and Intermittent High Heat Fluxes with Application to Fast-Nuclear-Reactor Fuel Elements, Journal of Strain Analysis, vol. 2, No. 3, pp. 226-238. 13 ABAQUS Finite Element Program, Abaqus/Standard 6.9-1, Hibbitt, Karlsson, and Sorensen, Inc., Pawtucket, R.I., by Educational License to Lehigh University. 14 ASME Boiler and Pressure Vessel Code, (2010), American Society of Mechanical Engineers, New York. 15 Osage, D.A., (2009), "ASME Section VIII-Division 2 Criteria and Commentary", ASME PTB-1-2009, American Society of Mechanical Engineers, New York. 16 Sowinski, J.C., Osage, D.A., and Brown, R.G., (2010), "ASME Section VIII-Division 2 Example Problem Manual", ASME PTB-3-2010, American Society of Mechanical Engineers, New York. 17 Ellyin, F., (1997), Fatigue Damage, Crack Growth and Life Prediction, Chapman & Hall, New York, 1 st Edition. 18 Lefebvre, D. and Ellyin, F., (1984), "Cyclic Response and Inelastic Strain Energy in Low Cycle Fatigue", International Journal of Fatigue, vol. 6, Elsevier Science Ltd., Oxford, pp. 9-15. 8 Copyright 2011 by ASME