3 Fatigue Analysis in Pressure

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Arturs Kalnins

Lehigh University, Lehigh University, Bethlehem, PA 18015-3085 Bethlehem, PA 18015-3085 e-mail: [email protected] e-mail: [email protected]

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

its component designed for cyclic serviceservice. Three methods that . Three methods that are consisteare consistent with nt with thethe

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

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

curve to read cycles, is deﬁned for read cycles, is deﬁned for multiaxmultiaxial stress situatioial stress situationsns



DOI: 10.1115/1.2137770DOI: 10.1115/1.2137770

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1

1 Int

Int

ro

duc

tio

n

Cofﬁn



11

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and Dowling et al.and Dowling et al.

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22

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explain the basic idea of theexplain the basic idea of the

local strain approach. Detailed coverage is given in Chapter 14,



“Strain-Based Approach to Fatigue”“Strain-Based Approach to Fatigue”



of Dowling’s book of Dowling’s book



33



. . ItIt

has been used

has been used in practice for in practice for fatigue assessfatigue assessment of ment of pressupressure vesselre vessel

com

componponentsents. . Its Its appappeal eal is is the the applapplicabicabilitility y to to any any smosmooth oth localocall

geo

geometmetry ry that can that can be be deﬁdeﬁned. The ASME ned. The ASME BoiBoiler and ler and PresPressursuree

Vessel



B&PVB&PV



CodeCode



44



uses the local strain approach for fatigueuses the local strain approach for fatigue

evaluation on plastic basis.

The paper considers applicatio

The paper considers applications to ns to cases in which cases in which cyclic actioncyclic action

experie

experiences alternating plasticitynces alternating plasticity. The . The main objectives are main objectives are to to iden-

iden-tify the methods and software that are

tify the methods and software that are capable of calculating stresscapable of calculating stress

and strai

and strain n ranranges within the ges within the fraframewmework of ork of the local strain ap-the local strain

ap-proach and to identify a multiaxial total equivalent strain range

proach and to identify a multiaxial total equivalent strain range

that is consistent with the approach. This strain range is the

coun-terpart to the uniaxial total strain range that is listed as ordinate of

terpart to the uniaxial total strain range that is listed as ordinate of

a material strain-life curve.

2

2 Basi

Basi

c

Assu

mpti

ons an

d Cons

equen

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.

Al

Alththougough h ththe e actactuaual l cracrack ck sisize ze up up to to whwhich ich ththe e spspeciecimemen-

n-component equality can be relied on is of no importance in its

component 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

de-velop a crack of a given size in the component is not less than that

velop 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

wit

with h resrespect to pect to failfailureure



howhowever deﬁneever deﬁned, but d, but the same the same for thefor the

specim

specimens ens and and componcomponentent



..

It

It is is pospossibsible le that for that for somsome e prespressursure e vesvessel sel appapplicalicationtions s thethe

abo

above ve conconditidition leads to on leads to a a marmargin that may gin that may be judged overlbe judged overlyy

generous. If that is unacceptable, a different design procedure has

to be formulated and followed. If that is not an option, the

gener-ous margin has to be accepted as part of the price for the

ous margin has to be accepted as part of the price for the

simplic-ity and wide applicabilsimplic-ity

ity and wide applicability



any modelable geometry, loading, andany modelable geometry, loading, and

material



of the local strain approach.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

elasti-call

cally y cyccycled led matmateriaerial, l, whicwhich h is is a a comcommon situatimon situation on in in prespressursuree

vessel

vessels. It s. It is expected that, after a is expected that, after a certain crack size is certain crack size is reached, thereached, the

cr

crack growack growth th in in thithis s comcompoponennent t wiwill ll be be slslowower er thathan n ththat at in in aa

sma

small-dll-diamiameter eter rouround nd bar bar fatifatigue gue spespecimecimen, n, in in whicwhich h the the cracrack ck

enters an almost uniform strain range ﬁeld. The paper by Kalnins

and Dowling

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55



supports this scenario. It uses test data cited insupports this scenario. It uses test data cited in

Figs. 10.8 and 14.9 of Dowling’s book

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33

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on blunt double-notchon blunt double-notch

plate components and smooth, 6.35 mm dia



0.25 in. dia0.25 in. dia



, round, round

fat

fatigue test igue test specspecimeimens, both ns, both madmade e frofrom m the same the same heat of heat of AISAISII

4340 steel. Both are cycled to two predeﬁned conditions: the

ap-pearance of a 0.5 mm

pearance of a 0.5 mm



0.02 in.0.02 in.



crack and failure. Figures 1 and 2crack and failure. Figures 1 and 2

show the number of cycles obtained from the tests for each of the

two

two conditioconditions.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

sam

same e for the for the lowlower er strstrain rangesain ranges, , but for but for strstrain ranges aboveain ranges above

0.014, the plate takes more cycles to reach a 0.5 mm crack. In Fig.

2, the difference in cycles to

2, the difference in cycles to failure is far greater. For example, forfailure 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

These results show that the above conditiocondition is n is met. If the met. If the strain-

strain-life curve is constructed from the specimen data, the positive

life curve is constructed from the specimen data, the positive

mar-gin for the plate components is apparent. Of course, for design

gin for the plate components is apparent. Of course, for design

pur

purposposes, factores, factors s will be will be appapplied to lied to the the spespecimcimen en curvcurve, e, whiwhichch

will increase the actual design margin.

3

3 Irr

Irr

egu

lar

Loa

din

g

The local

The local strain approach applies to strain approach applies to constanconstant-amplit-amplitude cycling.tude cycling.

If the loading histogram consists of an irregular loading pattern, it

has to

has to be be resresolveolved d into loadininto loading g rangranges es that produthat produce ce indiindividuvidualal

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

num-ber of time

ber of time points, the local strain approach requirpoints, the local strain approach requires that all es that all stressstress-

-strain cycles that are produced by the loading block be identiﬁed.

strain cycles that are produced by the loading block be identiﬁed.

This can be achieved by appropriate cycle-counting methods



seesee

Contributed by the Pressure Vessels and Piping Division of ASME for publication

in the J

in the JOURNOURNAL AL OFOF PPRESSURERESSUREVVESSELESSELTTECHNOLOGYECHNOLOGY. . ManusManuscript received August 8,cript received August 8,

2005; ﬁnal manuscrip

2005; ﬁnal manuscript t receivreceived October 10, ed October 10, 2005. Review conduc2005. Review conducted by ted by G. E. G. E. OttoOtto

Widera.

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3

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. The accumulated usage factor is then calculated over all the individual stress-strain cycles of the loading block, following the Palmgren-Miner rule



3

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, 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 re-mainder of the paper, the analyses will be assumed applied to one stress-strain cycle.

4 Stabilized Cycle

As part of the design procedure, the local strain approach as-sumes 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 stabi-lized 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.

5 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 mate-rial 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.

Substitution of a monotonic curve for the cyclic curve may cause problems. For example, Lefebre and Ellyin

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6

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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. Prob-lems may arise within strain amplitudes of 0.1–0.4%, where the use of the monotonic curve can predict strain ranges in a compo-nent 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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.

6 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

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7

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for weld joint classes, and more recently by Dong et al.

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8

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.

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

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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

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4

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.

Kalnins

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10

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has shown that the hysteresis loops for some of the nonproportional cases



e.g., case 10 in

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9

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exhibit no elastic un-loading and the reversal points of different components do not coincide, which prevents the use of the methods of Sec. 7. It may Fig. 1 Cycles to reach 0.5 mm crack in plate and specimen by

test

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 andT denotes transverse orientation of specimens machined from a plate. „Reprinted from Fig. 2 of †6‡, Copyright 1984, with permission from Elsevier.…

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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.

7 Methods and Software

As per Sec. 3, the analysis is applied to each individual stress-strain 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 twice-yield 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 repeti-tions of a selected cycle until the stress and strain values at the reversal points stabilize. The cyclic-stress-amplitude–strain-amplitude 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

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11

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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 cycle-by-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-strain-range 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 consis-tent 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 ex-pands 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 con-sidered 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 stabi-lized hysteresis loop. These models pass the validity check of Sec. 7.1.1. Among the popular ﬁnite element programs, ANSYS

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15

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linear kinematic hardening model KINH supports a multilinear

Fig. 4 Cyclic curve and calculated ranges

Fig. 5 Stabilized hysteresis loop

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

curved



curve ﬁtting to the cyclic data for the input of the mono-tonic material model, while that of ABAQUS

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11

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Version 6.3-1

supports only a bilinear curveﬁt to the cyclic data when the pa-rameter “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 −0.01. Figure 7 shows1 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 cal-culated 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 ge-neric cyclic stress-strain curve of the material as input and calcu-late 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, strain-controlled cycling in one direction, producing a uniaxial stress state. ABAQUS



11

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“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.

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,

with no unloading and reloading, and do not require cyclic plas-ticity 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

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16

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. Dowling



17



and Dowl-ing and Wilson



18



applied it to some special cases. More re-cently, 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 compo-nents are the stress component ranges and the strain compocompo-nents 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 straight-forward. The quantities that are taken from the output are the multiaxial equivalent stress range, _eq, given by Eq.



4



, and the

1_{Jürgen Rudolph of the University of Dortmund, Germany, performed the}

calcu-lations for this ﬁgure usingANSYS KINHlinear kinematic hardening model.

Fig. 7 Stabilized cycle using linear kinematic hardening model

Fig. 8 Cyclic curve and calculated ranges using NLK model

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equivalent plastic strain range,_peq_{, given by Eq.}

_

₅

_

_{. Typical} ﬁnite element programs calculate them automatically. For ex-ample, 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} _eqis 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 dis-cuss 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 re-sults of a single FEA can be used for a number of stress-strain cycles with different loading amplitudes of the same set of load-ing.

8 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 asym-metric 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 plas-ticity 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 de-grees 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.

9 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 ap-proach. This will be achieved based on one multiaxial equivalent hysteresis loop that represents the cycle as a whole. This loop is

needed here only to identify the consistent stress and strain pa-rameters in the local strain approach. The user of any of the meth-ods discussed in Sec. 7 does not have to construct one for an application.

In multiaxial situations, hysteresis loops are commonly con-structed for corresponding stress and strain components sepa-rately. This does not reveal which strain range is entered on the material strain-life curve. Kalnins et al.



22



developed the con-cept 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 multi-axial stress case, a procedure is outlined ﬁrst for a unimulti-axial 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 + p



1



where E is the modulus of the elastic portion of the cyclic curve.

4. Enter_tas 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



p_ij



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+ ₂₃2+ ₃₁2





2



 peq = ₂ 3



 p11 −p₂₂2+ p₂₂ −p₃₃2+ p₃₃ −p₁₁2+ 3 2 p122+p₂₃2+p₃₁2



3 

where  ij = ij r _{ }  _ij, p_ij= p ij r _{ } p_ij, i , j =1,2,3. The superscript



r



, r =1,2, refers to the stress and plastic strain components at the reversal points, t ₁and t ₂. The minus signs apply to the right-hand downward leg



see Fig. 5



of the hysteresis loop, with 

ij

r _{, p}

ij

r  ﬁxed at the upper extreme. The plus signs apply to the left-hand upward leg of the loop, with

ij

r 

, p_ijr  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 mul-tiaxial equivalent plastic strain range, which are now deﬁned by

 _eq= 1 2





 1− 2



2+



 2− 3



2+



 3− 1



2+ 6



 12 2 ₊_  23 2 ₊_  31 2

_

_

₄

_

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_peq₌2 3





 p11− p22



2+



 p22− p33



2+



 p33− p11



2+ 3 2



 p12 2 ₊__p 23 2 ₊__p 31 2

_

_

₅

_

where  _ij= ij 2₋  ij 1_, __p ij= pij 2_− p ij 1_{, i , j =1 ,2 ,3. The} super-scripts denote the subsuper-scripts at reversal points t ₁and t ₂.

3. In analogy to Eq.



1



, the multiaxial total equivalent strain range2is 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 strain-life 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, 4-136.2



c



, and in Section 3, NB-3228.4



c



. A simple example of equibiaxial, in-plane cycling of a plate3 refutes its general application in the local strain approach. The maximum principal strain is perpendicular to the plate while the stress com-ponent in that direction is zero. This produces a degenerate hys-teresis 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, Appen-dix 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



.

10 Conclusions

1. The local strain approach gives allowable cycles with a de-sign margin that depends on the local geometry and the mag-nitude of the cycled strain range.

2. The cycle-by-cycle method must be used with linear kine-matic cyclic plasticity model, not isotropic. Nonlinear isotropic/kinematic cyclic plasticity models do not give re-sults consistent with the local strain approach and should not be used.

3. The cycle-by-cycle method is more labor intensive and re-quires software with a cyclic plasticity model but gives the same strain ranges as the twice-yield or Seeger’s method.

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 mul-tiaxial total equivalent strain range for a selected stress-strain cycle.

Acknowledgment

This research was supported in part by the Pressure Vessel Re-search Council through Grant No. 01-DIV2/PNV-23AS. The au-thor also wishes to thank Dr. Wolf Reinhardt of Babcock & Wil-cox Industries, Cambridge, Ontario, Canada, for many helpful discussions on the topic of this paper.

References

1 Cofﬁn, L. F., Jr., 1973, “Fatigue at High Temperature,” Fatigue at Elevated Temperatures, American Soc. for Testing and Materials, ASTM STP 520, pp. 5–34.

2 Dowling, N. E., Brose, W. R., and Wilson, W. K., 1977, “Notched Member Life Predictions by the Local Strain Approach,” Fatigue Under Complex Loading—Analysis and Experiments, Society of Automotive Engineers,

War-rendale, PA.

3 Dowling, N. E., 1999, Mechanical Behavior of Materials, 2nd ed., Prentice Hall, Englewood Cliffs, NJ.

4 ASME, 2004, ASME Boiler and Pressure Vessel Code, ASME, New York.

5 Kalnins, A., and Dowling, N. E., 2004, “Design Criterion of Fatigue Analysis on Plastic Basis by ASME Boiler and Pressure Vessel Code,” ASME J. Pres-sure Vessel Technol., 126, pp. 461–465.

6 Lefebre, D., and Ellyin, F., 1984, “Cyclic Response and Inelastic Strain Energy in Low Cycle Fatigue,” Int. J. Fatigue, 6, pp. 9–15.

7 Maddox, S. J., 1998, Fatigue Strength of Welded Structures, 2nd ed., Abington Publishing, Cambridge, England.

8 Dong, P., Hong, J. K., Osage, D. A., and Prager, M., 2003, “Master S-N Curve Method for Fatigue Evaluation of Welded Components,” Weld. Res. Counc. Bull., 474, pp. 1–50.

9 Itoh, T., Sakane, M., Ohnami, M., and Socie, D. F., 1995, “Nonproportional Low Cycle Fatigue Criterion for Type 304 Stainless Steel,” ASME J. Eng. Mater. Technol., 117, pp. 285–292.

10 Kalnins, A., 2003, “Fatigue Analysis of Pressure Vessels Based on Individual Stress-Strain Cycles,” J. L. Zeman, ed., Proc. ICPVT-10, Vienna, July 7–10, Oesterr. Ges. f. Schweisstechnik, Vienna, pp. 349–356.

11 ABAQUSFinite Element Program, Version 6.1, Hibbitt, Karlsson, and Sorensen, Inc., Pawtucket, RI, by Educational License to Lehigh University.

12 Armstrong, P. J., and Frederick, C. O., 1966, “A Mathematical Representation of the Multiaxial Bauschinger Effect,” Berkeley Nuclear Laboratories, R&D Department Report No. RD/B/N/731.

13 Chaboche, J.-L., “Time-independent Constitutive Theories for Cyclic Plastic-ity,” Int. J. Plast., 22, pp. 149–188.

14 Ellyin, F., 1997, Fatigue Damage, Crack Growth and Life Prediction, Chap-man and Hall, London.

15 ANSYSFinite Element Program, Southpointe, Canonsburg, PA.

16 Mroz, Z., 1973, “Boundary-Value Problems in Cyclic Plasticity,” Second Int. Conf. on Structural Mechanics in Reactor Technology, Berlin, Vol. 6B, Part L, Paper No. L7/6.

17 Dowling, N. E., 1978, “Stress-Strain Analysis of Cyclic Plastic Bending and Torsion,” ASME J. Eng. Mater. Technol., 100, pp. 157–163.

18 Dowling, N. E., and Wilson, W. K., 1979, “Analysis of Notch Strain for Cyclic Loading,” 5th Int. Conf. on Structural Mechanics in Reactor Technology, Vol. L, Berlin, Paper No. L13/4, pp. 1–8.

19 Kalnins, A., 2001, “Fatigue Analysis of Pressure Vessels With Twice-Yield Plastic FEA,” ASME Bound Vol. 419, pp. 43–52.

20 Seeger, T., 1996, “Grundlagen fu ˝ r Betriebsfestigkeits-nachweise,” Stahlbau-handbuch, Band 1, Teil B, Stahlbau-Verlagsgesellschaft, Ko ˝ ln.

21 Rudolph, J., and Weiss, E., 2003, “Service Durability of TIG Dressed and Ground Weld Seams Subjected to Proportional Loading Criteria,” Proc. ICPVT-10, J. L. Zeman, ed., Vienna, July 7–10, Oesterr. Ges. f.

Schweisstech-nik, Vienna, pp. 465–475.

22 Kalnins, A., Updike, D. P., and Park, I., 1999, “Evaluation of Fatigue in the Knuckle of a Torispherical Head in the Presence of a Nozzle,” Welding Re-search Council Progress Report, Sept./Oct., pp. 31–93.

23 Reinhardt, W., 2005, “Strain Measures for Fatigue Assessment Using Elastic-Plastic FEA,” ASME PVP Conference, July, Denver, PVP2005–71547.

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}

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