Arturs Kalnins Lehigh University, Bethlehem, PA 18015-3085 e-mail:
[email protected]
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Fatigue Analysis in Pressure Vessel Design by Local Strain Approach: Methods and Software Requirements The purpose, methods for the analysis, software requirements, and meaning of the results of the local strain approach are discussed for fatigue evaluation of a pressure vessel or its component designed for cyclic service service.. Three methods that are consistent consistent with the approach are evaluated: the cycle-by-cycle method and two half-cycle methods, twice yield and Seeger’s. For the cycle-by-cycle method, the linear kinematic hardening model is identified as the cyclic plasticity model that produces results consistent with the local strain approach. A total equivalent strain range, which is entered on a material strain-life curve to read cycles, is defined for multiax multiaxial ial stress situations situations DOI: 10.1115/1.2137770
Intro Int roduc ductio tion n
Coffin 1 and Dowling et al. 2 explain the basic idea of the local strain approach. Detailed coverage is given in Chapter 14, “Strain-Based Approach to Fatigue” of Dowling’s book 3. It has been used in practice for fatigue assessment assessment of pressu pressure re vessel compon com ponents ents.. Its app appeal eal is the appl applicab icabilit ility y to any smo smooth oth loca locall geomet geo metry ry that can be defi defined. ned. The ASME Boi Boiler ler and Pres Pressur suree Vessel B&PV Code 4 uses the local strain approach for fatigue evaluation on plastic basis. The paper considers applications applications to cases in which cyclic action experiences experie nces alternating plasticity. plasticity. The main objectives are to identify the methods and software that are capable of calculating stress and strain strain ran ranges ges within the fra framew mework ork of the local strain approach and to identify a multiaxial total equivalent strain range that is consistent with the approach. This strain range is the counterpart to the uniaxial total strain range that is listed as ordinate of a material strain-life curve.
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Basicc Assu Basi Assumpti mptions ons and and Consequen Consequences ces
The local strain approach follows from the assumption that a sufficiently small crack of the same size is developed at about the same number of cycles on the surfaces of a smooth test specimen and a smooth location of a pressure vessel component, when both are cycled at the same surface strain range and made of the same material. It is assumed that this equality holds true from the very first crack appearance up to some crack size, which depends on the local geometry and the magnitude of the cycled strain range and is generally unknown. Alth Al thoug ough h th thee act actua uall cra crack ck si size ze up to wh which ich th thee sp speci ecime menncomponent equality can be relied on is of no importance in its design procedure, the local strain approach is justified for design purposes only on the condition that the number of cycles to develop a crack of a given size in the component is not less than that in the smooth fatigue test specimens, at least on a statistical basis. If that is true, then the allowable cycles for a component taken from a material design fatigue curve that is constructed from the smooth specimen data can be expected to have a positive margin with wit h res respect pect to fail failure ure how however ever defined, defined, but the same for the specimens specim ens and compon component ent. It is pos possib sible le that for som somee pres pressur suree ves vessel sel app applica lication tionss the Contributed by the Pressure Vessels and Piping Division of ASME for publication in the JOURN Manuscript cript received August 8, OURNAL AL OF PRESSURE VESSEL TECHNOLOGY. Manus 2005; final manuscript manuscript receiv received ed October 10, 2005. Review conducted conducted by G. E. Otto Widera.
above con above conditi dition on leads to a mar margin gin that may be judged overly overly generous. If that is unacceptable, a different design procedure has to be formulated and followed. If that is not an option, the generous margin has to be accepted as part of the price for the simplicity and wide applicability any modelable geometry, loading, and material of the local strain approach. As an illustration of a case for which the above condition is met, consider a component in which a crack enters a plastic zone with a decreasing strain range field that is surrounded by elastically call y cyc cycled led mat materia erial, l, whic which h is a com common mon situation situation in pres pressur suree vessels. vessel s. It is expected that, after a certain crack size is reached, the crack cr ack growth growth in thi thiss com compo ponen nentt wi will ll be sl slow ower er tha than n th that at in a small-d sma ll-diam iameter eter rou round nd bar fati fatigue gue spe specime cimen, n, in whic which h the cra crack ck enters an almost uniform strain range field. The paper by Kalnins and Dowling 5 supports this scenario. It uses test data cited in Figs. 10.8 and 14.9 of Dowling’s book 3 on blunt double-notch plate components and smooth, 6.35 mm dia 0.25 in. dia, round fatigue fat igue test spec specime imens, ns, both mad madee fro from m the same heat of AIS AISII 4340 steel. Both are cycled to two predefined conditions: the appearance of a 0.5 mm 0.02 in. crack and failure. Figures 1 and 2 show the number of cycles obtained from the tests for each of the two conditio conditions. ns. As seen from Fig. 1, the number of cycles to develop a 0.5 mm crack in the fatigue specimens and plate components is about the samee for the low sam lower er str strain ain ranges, ranges, but for str strain ain ranges above 0.014, the plate takes more cycles to reach a 0.5 mm crack. In Fig. 2, the difference in cycles to failure is far greater. For example, for a strain range of 0.0136, the specimen fails at 2400 cycles, while it takes 6027 cycles for the plate component to reach failure in the test. These results show that the above conditio condition n is met. If the strainlife curve is constructed from the specimen data, the positive margin for the plate components is apparent. Of course, for design purpos pur poses, es, factors factors will be app applied lied to the spe specim cimen en curv curve, e, whi which ch will increase the actual design margin.
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Irregu Irr egular lar Loa Loadin ding g
The local strain approach applies to constan constant-ampli t-amplitude tude cycling. If the loading histogram consists of an irregular loading pattern, it has to be res resolve olved d into loading loading rang ranges es that produce produce indi individu vidual al stress-strain cycles before the local-strain analysis can be begun. This means that if a repeated loading block is defined over a time interval for which all loading components are specified at a number of time points, the local strain approach requires requires that all stress stress-strain cycles that are produced by the loading block be identified. This can be achieved by appropriate cycle-counting methods see
Fig. 1 Cycles to reach 0.5 mm crack in plate and specimen by test
3. The accumulated usage factor is then calculated over all the individual stress-strain cycles of the loading block, following the Palmgren-Miner rule 3, Chap. 9. The same applies to cases in which more than one loading block may be applied in a random sequence, each repeated a specified number of times. In the remainder of the paper, the analyses will be assumed applied to one stress-strain cycle.
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Stabilized Cycle
As part of the design procedure, the local strain approach assumes that a single value of a strain range is used for assessing fatigue damage for the life of a pressure vessel component. Since hardening and softening with cycles accompany the initial phase of cycling, during which the strain and stress ranges may change, the question is: What strain range shall it be? Since for many metals the stress and strain ranges tend to stabilize, so that stabilized hysteresis loops are experienced for the major part of life, the obvious answer is to bypass the hardening and softening with cycles and to accept the strain range of the stabilized cycle as representative of whole life. In some cases, stabilization may be difficult to achieve even until failure. Such cases notwithstanding, the local strain approach assumes that the cyclic action from which the cyclic stress-strain curve of the material is derived has stabilized. That has to be accepted as part of the design procedure.
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Cyclic Material Curve
Having established that stabilized action is the target, it follows that the cyclic-stress-range–strain-range or amplitude curve of the material provides the information that is needed for the material model used in the analysis. Such cyclic test data are available for many materials. Typical curve fittings to these data can be obtained in terms of three parameters: cyclic elastic modulus E , a stress parameter, and an exponent.
Fig. 2 Cycles to reach failure in plate and specimen by test
Fig. 3 Monotonic „lines only… and cyclic „markers… curves for SA-516 Grade 70 steel. L denotes longitudinal and T denotes transverse orientation of specimens machined from a plate. „Reprinted from Fig. 2 of †6‡, Copyright 1984, with permission from Elsevier.…
Substitution of a monotonic curve for the cyclic curve may cause problems. For example, Lefebre and Ellyin 6 present curve fittings to test data on specimens made of SA-516 Grade 70 steel, shown in Fig. 3. Stress amplitude is plotted versus strain amplitude for cyclic loading and compared to plots for monotonic loading. The material shows softening with cycles up to strain amplitude of about 0.4% and hardening above that level. Problems may arise within strain amplitudes of 0.1–0.4%, where the use of the monotonic curve can predict strain ranges in a component that err on the unconservative side. Of course, an accurate curve fitting to the cyclic data of the material under consideration is preferable, if one is available. However, for design purposes, approximations could be agreed on for certain classes of materials, which would parallel those of the design fatigue curves now used in the ASME B&PV Code 4.
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Applicability
The local strain approach is applicable to cases in which all structural features that affect fatigue damage are defined and can be modeled with sufficient accuracy. It is not applicable to cases in which some structural detail is known to affect fatigue damage but cannot be modeled, either because its geometry is unknown e.g., flaws at the weld toe of an untreated weld or because its model is unreliable e.g., very sharp notch. Such cases require approaches that incorporate the unmodelable details in the test data, such as, for example, those described by Maddox 7 for weld joint classes, and more recently by Dong et al. 8. Limitations on loading are not so clear. Proportional loading presents no problems, but cases when the principal stress and strain axes rotate have been shown to pose a problem. Itoh et al. 9 present test data for shear and axial strains that are imposed nonproportionally to the test section of a thin cylindrical shell, forcing the principal axes to rotate. The data show unsatisfactory correlation with predictions using the principal and equivalent strain ranges that are in the current ASME B&PV Code 4. Kalnins 10 has shown that the hysteresis loops for some of the nonproportional cases e.g., case 10 in 9 exhibit no elastic unloading and the reversal points of different components do not coincide, which prevents the use of the methods of Sec. 7. It may
Fig. 5 Stabilized hysteresis loop Fig. 4 Cyclic curve and calculated ranges
be that such situations are rare in pressure vessel components. Whether a caveat for these cases is or is not needed in design standards is an open question. It is assumed for the remainder of the paper that stress and strain reversal points of all nontrivial components coincide so that a multiaxial equivalent stress-strain cycle can be defined Sec. 9.2. This is ensured for proportional loading. The conditions for which it may also be true for nonproportional loading require further investigation.
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Methods and Software
As per Sec. 3, the analysis is applied to each individual stressstrain cycle that is produced by the loading histogram. These cycles must be identified, and the two time points at their stress and strain reversals determined. The loading components at the reversal points can then be evaluated and the loading range for the cycle determined. At this point, it has been established that the cyclic curve of the material will be used to model the material and the loading will consist of the loading range. Now the question is: What methods and software will solve the problem in a way that is consistent with the local strain approach? Two basic methods are discussed: cycle-by-cycle and half-cycle methods. For the latter, the twiceyield and Seeger’s methods are included. The cycle-by-cycle method is discussed only because of its appeal for modeling cyclic action, but, when used for design purposes, it requires far more effort and is less generic to software than the half-cycle methods. The stress and strain ranges obtained by all three methods are the same. 7.1 Cycle-by-Cycle Method. Elastic-plastic finite element analysis FEA is performed over a sufficient number of repetitions of a selected cycle until the stress and strain values at the reversal points stabilize. The cyclic-stress-amplitude–strainamplitude curve of the material is used as input for the monotonic uniaxial material model. The loading can be specified as either between the loading at the reversal points of the cycle or between plus and minus of the loading amplitude. If in the former case the hysteresis loop indicates a mean stress, its effect is neglected as per Sec. 8. Since up- and downloading is performed, a cyclic plasticity model must be specified to model the unloading and reloading phases. Two cyclic plasticity models will be considered for the cycle-by-cycle method. One includes linear and the other nonlinear hardening. For details, see the ABAQUS 11 Standard User’s Manual II, 11.2.2-2, or Refs. 12,13. The question is which cyclic plasticity model is consistent with the local strain approach. To answer that question, a validity check is given next. 7.1.1 Cyclic Plasticity Validity Check . According to Sec. 4, the cyclic stress-range–strain-range or amplitude curve is used, which means that the hysteresis loops in the specimens stabilize. Since the material of the component is supposed to be the same as that of the test specimens, the loops in the components should also
stabilize in the same way. The software that is used for the cycleby-cycle method must reflect this behavior. This means that it must be able to replicate the cyclic curve for a uniaxial stress state in a component. In other words, the calculated stress and plastic strain ranges must lie on the cyclic-stress-range–plastic-strainrange curve that has been input. This requirement will be used as a validity check in evaluating the cyclic plasticity models of the software. To illustrate the validity check, consider a uniaxial stress state cycled in strain control using a cyclic plasticity model that is to be checked for consistency with the local strain approach. The cyclic curve shown in Fig. 4 rewritten in amplitudes is input for the monotonic material model. The calculated stabilized hysteresis loop is shown Fig. 5. It shows a plastic strain range of 0.033 and a stress range of 1330 MPa. Is the cyclic plasticity model consistent with the local strain approach? This is decided by plotting the coordinates for the calculated ranges, 0.033 and 1330, in Fig. 4. It is seen that this point lies on the cyclic curve. If the cycling were done at different strain ranges, the cyclic curve in Fig. 4 would be duplicated. Therefore, the cyclic plasticity model used in this analysis is consistent with the local strain approach. 7.1.2 Linear Hardening in Cyclic Plasticity. Cyclic plasticity models with linear hardening involve two separate components: isotropic and kinematic. The model with isotropic hardening expands the yield surface until purely elastic action remains, as shown in Fig. 6. This result does not meet the validity check of Sec. 7.1.1 and is not acceptable. The kinematic component is considered next. Cyclic plasticity models with linear kinematic hardening have been developed that assume Masing behavior Sec. 2.5 of 14, according to which magnifying the cyclic stress-strain amplitude curve by a factor of 2 approximates the two branches of a stabilized hysteresis loop. These models pass the validity check of Sec. 7.1.1. Among the popular finite element programs, ANSYS 15 linear kinematic hardening model KINH supports a multilinear
Fig. 6
Stabilized cycle using linear isotropic hardening model
Fig. 7 Stabilized cycle using linear kinematic hardening model
curved curve fitting to the cyclic data for the input of the monotonic material model, while that of ABAQUS 11 Version 6.3-1 supports only a bilinear curvefit to the cyclic data when the parameter “hardening= kinematic” is invoked. Note that ABAQUS permits the input of a curved curve fitting to the cyclic data for the input of the monotonic material model when no cyclic plasticity with kinematic hardening is specified, which is the case for the half-cycle methods. For an illustration, consider a single eight-noded brick element, cycled in uniaxial strain control between a strain of 0.03 and 1 −0.01. Figure 7 shows the stress-strain response for which the stress range of 1330 MPa is obtained. When cycled with the same strain range, but fully reversed between 0.02 and −0.02, exactly the same stress range is predicted. In both calculations, the cyclic curve in Fig. 4 rewritten in amplitudes is used. The square marker in Fig. 4 shows the point with the coordinates of the calculated stress range and plastic strain range. The fact that it lies on the cyclic curve indicates that the test of Sec. 7.1.1 has been met. 7.1.3 Nonlinear Cyclic Plasticity Models. Nonlinear cyclic plasticity models e.g., 11–13,15, which contain combined isotropic/kinematic components, are not designed to receive a generic cyclic stress-strain curve of the material as input and calculate the stress and strain ranges that represent a stabilized cycle of the same material. The problem is that the input is written for a specified strain range, which is the end product of the analysis and, therefore, unknown before the analysis. For this reason, the nonlinear hardening models used in Refs. 12,13 do not meet the validity check of Sec. 7.1.1. The following example illustrates the problem. Again a single eight-noded brick element is subjected to fully reversed, straincontrolled cycling in one direction, producing a uniaxial stress state. ABAQUS 11 “data-type=stabilized” parameter is selected, for which an approximation derived from the cyclic curve shown in Fig. 8 is used as input. No isotropic component is used. The model includes only the nonlinear kinematic NLK component. The response is shown by the curve marked NLK in Fig. 9. It shows clearly that the stabilized cycle of the cyclic curve that was input has not been replicated. The stress range given by the NLK model is 1240 MPa, while the corresponding value on the cyclic curve is 962 MPa, which is also plotted in Fig. 8. It is clear that the model does not replicate the cyclic curve that has been input and fails the validity check of Sec. 7.1.1.
Fig. 8 Cyclic curve and calculated ranges using NLK model
with no unloading and reloading, and do not require cyclic plasticity models. The advantage is simplicity no FEA over cycles and that they can be performed with any finite element program that has an incremental plasticity option for static loading. The half-cycle methods give strain and stress ranges that, for practical purposes, are the same as those obtained by the cycle-by-cycle method of Sec. 7.1. The two half-cycle methods are considered next. 7.2.1 Twice-Yield Method . Theoretical support of this method can be found in the work of Mroz 16. Dowling 17 and Dowling and Wilson 18 applied it to some special cases. More recently, Kalnins 19 proposed it as a general method for design and called it the twice-yield method . It is applicable to cyclic primary and nonprimary e.g., transient thermal loading; that is, its applicability is the same as that of the cycle-by-cycle method. The only limitations are stated in Sec. 6. From an FEA perspective, the twice-yield method is explained by the observation that if in the input the load is specified as the loading range and the cyclic stress-range–strain-range curve is used for the material model, then in the output the stress components are the stress component ranges and the strain components are the strain component ranges. Thus, in one FEA load step, for which the loading is specified from zero to that of the loading range, the output provides the stress and strain ranges that are needed in the local strain approach. When coupled with the multiaxial total strain range Sec. 9.2, the twice-yield method is far simpler than the cycle-by-cycle method. After the reversal points of the loading for the cycle and the loading range have been determined, the method is straightforward. The quantities that are taken from the output are the multiaxial equivalent stress range, eq, given by Eq. 4, and the
7.2 Half-Cycle Methods. These methods take advantage of the stabilized form of the hysteresis loop of the cycle. There is no need to perform the cycle-by-cycle method over a number of cycles if the two branches of the loop are geometrically similar, as shown in Fig. 7. FEA over just one branch of the loop gives the desired stress and strain ranges. That is the basis of the half-cycle methods. They require only one monotonic FEA of one load step, 1
Jürgen Rudolph of the University of Dortmund, Germany, performed the calculations for this figure using ANSYS KINH linear kinematic hardening model.
Fig. 9 Calculated cycle using NLK model
equivalent plastic strain range, peq, given by Eq. 5. Typical finite element programs calculate them automatically. For example, ABAQUS 11 calls eq MISES , and peq PEMAG. ANSYS 15 uses similar variable names in the output. A generic output file is scanned for the maximum value of peq, and eq is then recorded at the same location. No search of the solution database is required. The total strain range is then obtained from Eq. 6 using a hand calculator. 7.2.2 Seeger’s Method . Seeger gives the general background in 20. Rudolph and Weiss 21 describe the procedure and discuss its application to weld seams with postweld treatment. It is applicable to proportional loading; that is, to cases in which all loading components are multiplied by a single function of time, say, L. Seeger’s method performs only one FEA of the component from L =0 to the greatest magnitude of L on the histogram and records a selected stress and strain measure at a number of L values that is sufficient to permit a curve fitting by an equation e.g, Ramberg-Osgood. The curve fitting between L and is called the component yield curve and that between and is called the local - curve. After the two curvefits are derived, the unload branch of the hysteresis loop of each cycle is constructed by assuming Masing behavior see 14. The stress and strain ranges are determined from this branch. For details, see 20,21. Regarding the comparison between the two half-cycle methods, twice-yield method performs an FEA for each stress-strain cycle separately, but determines the stress and strain ranges with no postprocessing. The advantage of Seeger’s method is that the results of a single FEA can be used for a number of stress-strain cycles with different loading amplitudes of the same set of loading.
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Mean Stress
What is known about each stabilized stress-strain cycle is only its loading range, which is used to calculate the stress and strain ranges of the stabilized cycle. The loading may begin with asymmetric components at the reversal points, but once the stress-strain cycle has stabilized, no information is available regarding the mean stress of the cycle. Since the only description of cyclic behavior of the material is taken from the cyclic stress-range– strain-range curve Sec. 5, which contains no information on mean stress, the magnitude of mean stress, if one is present, is unknown. The lack of knowledge of the mean stress is not a problem when the design fatigue curves of the ASME B&PV Code 4 are used, in which mean stress is assumed zero when alternating plasticity is present. This is supported by Ellyin 14, who has shown that mean stress approaches negligible magnitudes when test specimens of SA-516 Grade 70 steel are cycled with various degrees of mean strain. A reasonable assumption is that the effect of any mean stress that may actually occur in an individual cycle with alternating plasticity in a real component can be neglected. This is made part of the design procedure considered in this paper.
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Multiaxial Stress and Strain Equivalents
The objective of this section is to obtain the multiaxial stress and strain equivalents that are appropriate for the local strain approach. This will be achieved based on one multiaxial equivalent hysteresis loop that represents the cycle as a whole. This loop is
1
eq = 2
1
needed here only to identify the consistent stress and strain parameters in the local strain approach. The user of any of the methods discussed in Sec. 7 does not have to construct one for an application. In multiaxial situations, hysteresis loops are commonly constructed for corresponding stress and strain components separately. This does not reveal which strain range is entered on the material strain-life curve. Kalnins et al. 22 developed the concept of a multiaxial equivalent hysteresis loop and showed how to construct one. This is discussed next. 9.1 Uniaxial Stress State. To obtain a template for a multiaxial stress case, a procedure is outlined first for a uniaxial stress state in a fatigue test specimen. The calculation is performed with the cycle-by-cycle method of Sec. 7.1 using the cyclic plasticity model with linear kinematic hardening. 1. Plot stress versus axial plastic strain over one stabilized cycle and obtain a hysteresis loop, which may look like that in Fig. 5. 2. Note that its height is the uniaxial stress range , and its width is the axial plastic strain range p. 3. Calculate the axial total strain range from t =
E
1
+ p
where E is the modulus of the elastic portion of the cyclic curve. 4. Enter t as ordinate on the strain-life curve to read cycles. 5. Note that and p lie on the cyclic stress range-plastic strain range curve of the material, just like the square marker in Fig. 4. 9.2 Multiaxial Stress State. The five steps in Sec. 9.1 are now retraced for the multiaxial stress case. 1. Use again cycle-by-cycle method to calculate all stress ij and plastic strain pij components at a number of output points over a stabilized cycle that would be sufficient to draw a graph. Then the following two quantities are calculated at each of the output points:
eq
=
1
2
1
− 2 2 + 2 − 3 2 + 3 − 1 2 + 6 122 + 232 + 312
2 2
= 3 peq
p11 − p222 + p22 − p332 + p33 − p112 + 2 p122 + p232 + p312
3
3
r
r
where ij = ij ij , p ij = pij pij , i , j =1,2,3. The superscript r , r =1,2, refers to the stress and plastic strain components at the reversal points, t 1 and t 2. The minus signs apply to the right-hand r r downward leg see Fig. 5 of the hysteresis loop, with ij , pij fixed at the upper extreme. The plus signs apply to the left-hand r r upward leg of the loop, with ij , pij fixed at the lower extreme. The resulting curve of eq versus peq is the multiaxial equivalent hysteresis loop of the cycle, which is the counterpart to the uniaxial hysteresis loop of the fatigue test specimen. It may look like that in Fig. 5. 2. Just as in the uniaxial case, this loop identifies the stress range and strain range that describe the size of the loop. Its height is the multiaxial equivalent stress range and its width is the multiaxial equivalent plastic strain range, which are now defined by
2
2
2
− 2 2 + 2 − 3 2 + 3 − 12 + 6 12 + 23 + 31
4
2
peq = 3
2
1
2
p11 − p222 + p22 − p332 + p33 − p112 + 32 p212 + p223 + p231
1
where ij = ij − ij , p ij = p ij − p ij , i , j =1 ,2 ,3. The superscripts denote the subscripts at reversal points t 1 and t 2. 3. In analogy to Eq. 1, the multiaxial total equivalent strain 2 range is defined by eq =
eq
E
+ peq
6
4. The strain range of Eq. 6 is the multiaxial counterpart to the uniaxial total strain range of Eq. 1 and is entered on the strainlife curve to read cycles. 5. According to the theory of plasticity used in typical FEA software, the calculated eq and eq lie on the cyclic stress range-plastic strain range curve that is input for the required monotonic material model, just like the square marker in Fig. 4, thus meeting the validity check of Sec. 7.1.1. 9.3 Discussion. The multiaxial total equivalent strain range of Eq. 6 is superior to the maximum principal total strain range, which is used in the ASME B&PV Code 4, both in Section 8-Div. 2, 4-136.2c, and in Section 3, NB-3228.4 c. A simple 3 example of equibiaxial, in-plane cycling of a plate refutes its general application in the local strain approach. The maximum principal strain is perpendicular to the plate while the stress component in that direction is zero. This produces a degenerate hysteresis loop of a straight line on the strain axis, which is not consistent with the hysteresis loop observed in the cycling of a fatigue test specimen. The multiaxial total equivalent strain range is also superior to the equivalent strain range that is defined in terms of total strain component ranges, which is used in 4, Section III-NH, Appendix T, T-1413 for elevated temperature service. Its problem is that it does not reduce to the correct strain range for a uniaxial stress state. The multiaxial total equivalent strain range of Eq. 6 is defined in terms of Mises stress and plastic strain components. It provides a smooth transition to purely elastic action per cycle if the Mises multiaxial elastic stress range is taken as the stress measure for fatigue analysis in the elastic case. This was assumed here because it has been recommended for the new Division 2 of Section 8 of the ASME B&PV Code 4. However, the current 2004 edition of the ASME B&PV Code 4 uses the Tresca stress components for purely elastic action. In that case, an effective combined strain range can be defined on the basis of the maximum shear strain as shown in a recent paper by Reinhardt 23.
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Conclusions
1. The local strain approach gives allowable cycles with a design margin that depends on the local geometry and the magnitude of the cycled strain range. 2. The cycle-by-cycle method must be used with linear kinematic cyclic plasticity model, not isotropic. Nonlinear isotropic/kinematic cyclic plasticity models do not give results consistent with the local strain approach and should not be used. 3. The cycle-by-cycle method is more labor intensive and requires software with a cyclic plasticity model but gives the same strain ranges as the twice-yield or Seeger’s method. 2 This strain range was introduced by Dowling 3 who called it the effective strain range 3 Professor Masao Sakane of Ritsumeikan University, Shiga, Japan, pointed this out to the author.
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4. The twice-yield and Seeger’s methods require software with only incremental plasticity model for monotonic loading. No cyclic plasticity models are used. 5. The multiaxial total equivalent strain range defined in the paper is the multiaxial counterpart to the strain range listed on the design fatigue curve. 6. Twice-yield is the simplest method for calculating the multiaxial total equivalent strain range for a selected stressstrain cycle.
Acknowledgment This research was supported in part by the Pressure Vessel Research Council through Grant No. 01-DIV2/PNV-23AS. The author also wishes to thank Dr. Wolf Reinhardt of Babcock & Wilcox Industries, Cambridge, Ontario, Canada, for many helpful discussions on the topic of this paper.
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