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International Journal of Pressure Vessels and Piping 116 (2014) 10e19

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International Journal of Pressure Vessels and Piping

journal homepage: www.elsevier .com/locate/ i jpvp

Performance study of the simplified theory of plastic zones and theTwice-Yield method for the fatigue check

Hartwig Hübel a, Adrian Willuweit b, Jürgen Rudolph b,*, Rainer Ziegler b, Hermann Lang b,Klemens Rother c, Simon Deller c

aBrandenburg University of Technology Cottbus-Senftenberg, Cottbus, GermanybAREVA GmbH, Erlangen, GermanycMunich University of Applied Sciences, Munich, Germany

a r t i c l e i n f o

Article history:Received 31 October 2013Received in revised form25 January 2014Accepted 30 January 2014

Keywords:Simplified elasticeplastic analysesSimplified theory of plastic zones (STPZ)Zarka’s methodTwice-Yield methodThermal cyclic loading

* Corresponding author.E-mail address: rudolph.juergen@areva.com (J. Ru

http://dx.doi.org/10.1016/j.ijpvp.2014.01.0030308-0161/� 2014 Elsevier Ltd. All rights reserved.

a b s t r a c t

As elasticeplastic fatigue analyses are still time consuming the simplified elasticeplastic analysis(e.g. ASME Section III, NB 3228.5, the French RCC-M code, paragraphs B 3234.3, B 3234.5 and B3234.6 andthe German KTA rule 3201.2, paragraph 7.8.4) is often applied. Besides linearly elastic analyses andfactorial plasticity correction (Ke factors) direct methods are an option. In fact, calculation effort andaccuracy of results are growing in the following graded scheme: a) linearly elastic analysis along with Ke

correction, b) direct methods for the determination of stabilized elasticeplastic strain ranges and c)incremental elasticeplastic methods for the determination of stabilized elasticeplastic strain ranges.

The paper concentrates on option b) by substantiating the practical applicability of the simplifiedtheory of plastic zones STPZ (based on Zarka’s method) and e for comparison e the established Twice-Yield method. The Twice-Yield method is explicitly addressed in ASME Code, Section VIII, Div. 2.Application relevant aspects are particularly addressed. Furthermore, the applicability of the STPZ forarbitrary load time histories in connection with an appropriate cycle counting method is discussed.

Note, that the STPZ is applicable both for the determination of (fatigue relevant) elasticeplastic strainranges and (ratcheting relevant) locally accumulated strains. This paper concentrates on the performanceof the method in terms of the determination of elasticeplastic strain ranges and fatigue usage factors.The additional performance in terms of locally accumulated strains and ratcheting will be discussed in afuture publication.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Incremental elasticeplastic code conforming fatigue analysesare the most accurate and least conservative way of considering thecyclic deformation behavior. The major drawback is time and spaceconsuming analyses for complicated 3D structures and complexloading conditions. That’s why the simplified elasticeplastic anal-ysis is often the method of choice. It relies on elastic finite elementanalyses and the application of an appropriate correction procedurefor plasticity effects. Nevertheless, the standard correction pro-posed by the codes is known to be overly conservative in manydesign situations. In this context, the application of direct methods,such as the STPZ and the established Twice-Yieldmethod, appear to

dolph).

be interesting alternatives. This paper covers both the theoreticalfundamentals of the STPZ and Twice-Yield methods with emphasison thermal cyclic loading conditions as well as practical applicationexamples.

2. Fundamentals of the simplified theory of plastic zones(STPZ)

An alternative to simplify the calculation of elasticeplastic stressand strain ranges is the simplified theory of plastic zones (STPZ) asdescribed in more detail in the following section.

2.1. General outline

The STPZ aims at capturing the elasticeplastic stress and strainranges due to cyclic loading between two states of loading in thecondition of plastic shakedown. The material is assumed to exhibit

H. Hübel et al. / International Journal of Pressure Vessels and Piping 116 (2014) 10e19 11

piecewise linear kinematic hardening. At present, the theory isworked out for a bilinear or tri-linear stressestrain curve. A vonMises yield surface is adopted along with an associated flow rule.Small deformation and additivity of elastic and plastic strains areassumed. Yield stress may depend on temperature, but the elasticproperties and the elasticeplastic tangent moduli must not.

In this theory the state of plastic shakedown is addresseddirectly, i.e. without going through the entire load history step bystep. The basic idea of the present simplification goes back to Zar-ka’s method (see Refs. [1e3]). A transformed internal variable (TIV)is introduced in this context. Its value can be estimated in the stateof plastic shakedown by local consideration. Sequentially, theelasticeplastic response of the structure can be gained by a series oflinear elastic analyses. As a consequence, however, the results areonly approximations compared to those of an incremental analysisthrough the entire load history, and no information is obtainedabout the evolution of stress and strain during cycling and thenumber of cycles required to achieve shakedown. There exist othermethods as well to determine elasticeplastic strains by iterativeelastic analyses, e.g. the GLOSSmethod [4]. However, they are basedon a different theoretical background. The present paper does notintend to outline these differences in methodology andperformance.

For illustration purpose monotonic loading with linear kine-matic hardening and temperature independent yield stress isconsidered first (Section 2.2), before cyclic loading with tempera-ture dependent yield stress (Section 2.3) and multi-linear kine-matic hardening (Section 2.4) is addressed.

2.2. Outline for monotonic loading with linear kinematic hardeningand temperature independent yield stress

Let sij be the stress tensor, sf :elij the tensor of stress obtained by afictitious elastic analysis, xij the tensor of the backstress due tolinear kinematic hardening, and rij the residual stress. The TIV gij

introduced by Zarka is then defined by

s0ij ¼ s0f :elij þ r0ij; i; j ¼ 1;2;3 (1)

gij ¼ xij � r0ij (2)

where ()0 indicates the deviatoric portion of a tensor. Note that r0ij isthe deviatoric portion of the residual stress. The fictitious elasticstresses are always considered to be known, because they can beeasily obtained for any state of loading by a linear analysis.

If sy is the uniaxial yield stress, the von Mises yield surface canbe considered as a (hyper)circle with radius sy centered in xij in thedeviatoric stress (hyper)space and can be expressed as

f�s0ij � xij

�� sy � 0 (3)

Reformulation leads to

f�s0f :elij � gij

�� sy � 0 (4)

representing a (hyper)circle with radius sy centered in s0f :elij in the

(hyper)space of the TIV. Active yielding requires the equal sign (¼0)in equations (3) and (4), s0ij and gij being on the edge of the

respective circle, as illustrated in Fig. 1. The negative residual stress�r0ij is then outside of the circle in the TIV-space. The volume V of a

structure can be split up into a portion remaining elastic (Ve) and aportion yielding actively (Vp). It should be noted that equation (3)contains two unknown tensors s0ij and xij equation (4) only one: gij.

The situation shown in Fig. 1 is characterized by a significantamount of directional redistribution of stress as visualized by thelarge angles between s0ij; s

0f :elij and xij. If directional redistribution is

not that large, a good approximation of gij can be obtained byprojecting the negative of the residual stress to the von Mises circlein the TIV-space:

gij ¼ s0f :elij � s0ij

�sysv

�(5)

where sv is the von Mises effective stress of the stress tensor sij:

sv ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi32s0ijs

0ij

r(6)

If directional redistribution is not possible, e.g. in case of uni-axial stresses, this projection provides exact results.

Once gij is known (either exactly or approximately), the elasticeplastic strains can be reformulated adopting the additivityassumption and Hooke’s law as well as the linear kinematic hard-ening rule

3el�plij ¼ 3elij þ 3

plij (7)

3elij ¼ E�1ijklskl (8)

3plij ¼ 3

2Cxij; C ¼ E$Et

E � Et(9)

where Eijkl is the elasticity matrix, E the Young’s modulus and Et theelasticeplastic tangent modulus, see Fig. 2.

By using an Operator Lijkl to convert a tensor into its deviatoricpart and introducing the abbreviations

3*ij ¼ 3el�plij � 3f :elij (10)

�E*ijkl

��1 ¼ E�1ijkl þ

32C

Lijkl (11)

3ij;0 ¼ 32C

gij (12)

we get

3*ij ¼8<:

�E*ijkl

��1rkl þ 3ij;0 in Vp

E�1ijklrkl in Ve

(13)

Equation (13) is a linear relation between the residual stress rij andthe residual strain 3*ij. In case of isotropic material, Eijkl consists only

of Young’s modulus E and Poisson’s ratio v, while E*ijkl consists of a

modified Young’s modulus E* and modified Poisson’s ratio v*:

E* ¼ Et ; v* ¼ 0:5� ð0:5� vÞ EtE

(14)

The residual stresses can thus be determined by a linear elasticanalysis of the structure with homogenous boundary conditionsloaded only by the initial strains 3ij,0 in Vp. This is termed “modifiedelastic analysis” (MEA) because the elastic parameters as well as theloading are modified. Note that in this step ANSYS� is used for thesolution of the linearly elastic boundary value problem includingthe initial strains 3ij,0. The level of 3ij,0 depends on the level of thereal loading while the values of E* and v* do not and are known in

Fig. 1. Yield surface in the deviatoric stress space (left) and in the TIV-space (right).

H. Hübel et al. / International Journal of Pressure Vessels and Piping 116 (2014) 10e1912

advance. However, the geometry of Ve and Vp changes with theload level. Once the modified elastic analysis is performed, i.e. r’ijand 3*ij are known, the stress can be determined via equation (1) andthe elasticeplastic strain from equation (10). This gives rise to aniterative procedure, since any initial guess about the geometry ofVp and the residual stress field may now be improved.

Thus we get the following workflow:

� Step 1: Perform a fictitious elastic analysis / sf :elij� Step 2: Initial estimation of geometry of Vp (all locations ofstructure where the von Mises equivalent stress exceeds theyield stress: sf :elv > sy)

� Step 3: In each location within Vp: projection of origin (¼initialnegative deviatoric residual stress) on yield surface in TIV-space / gij ¼ s0f :elij ð1� sy=sf :elv Þ

� Step 4: Modify the elastic parameters in Vp (E / E*, v / v*),delete all loading, apply initial strains, perform MEA / rij asdescribed above

� Step 5: Superposition of the fictitious elastic (Step 1) andmodifiedelastic analysis MEA results (Step 4) / sij ¼ sf :elij;Step 1 þ rij;Step 4

� Step 6: Improved estimation of the geometry of Vp (all locationsof structure where the von Mises equivalent stress exceeds theyield stress: sv > sy)

� Step 7: In each location within Vp: projection of negativedeviatoric residual stress on the yield surface in TIV-space / gij ¼ s0f :elij � s0ijðsy=svÞ

� Step 8: Introduce the modified elastic parameters in Vp (E*, v*),(re)establish the elastic parameters in Ve (E, v), delete allloading, apply initial strains, perform MEA / rij

� Step 9: Superposition of the fictitious elastic andmodified elasticanalysis results / sij

� Repeat from Step 6 until differences between two iterationsbecome satisfactorily small.

Fig. 2. Linear kinem

At last, the elasticeplastic strain is obtained via

3el�plij ¼ 3*ij þ 3f :elij (15)

Note that the associated stress state sij is already known fromStep 5.

2.3. Cyclic loading with temperature dependent yield stress

As mentioned above, the STPZ is a direct method to provide anapproximation to the elasticeplastic strain range between two loadcases without performing incremental analyses throughout a loadhistogram. Thus, the two points in time causing the maximumstrain range in a structure during a transient thermomechanicalload history must be identified in advance. This can mostly be donewith sufficient accuracy on the basis of a number of fictitious elasticanalyses for many points in time during the load history. Theloading is then assumed to vary linearly with time between thesetwo extremes, named “minimum” and “maximum” state of loading.The corresponding fictitious elastic stress states are sf :elij;min and

sf :elij;max.

Once the loading conditions constituting the two extremes of athermomechanical loading cycle are identified, the range of loadingis treated in the same way as a monotonic loading, so that, forexample,

Dsf :elij ¼ sf :elij;max � sf :elij;min (16)

is applied instead of sf :elij in Section 2.2. The range of the TIV ac-

cording to equation (5) is then different to those obtained by thelower and the upper estimates in Zarka’s method (see Refs. [1e3]).

A number of analyses showed that the effect of temperaturedependent elasticeplastic material parameters of a bilinear stresse

atic hardening.

Fig. 3. Tri-linear stressestrain curve.

H. Hübel et al. / International Journal of Pressure Vessels and Piping 116 (2014) 10e19 13

strain relationship is largely attributed to the temperature depen-dence of the yield stress, rather than to the temperature depen-dence of Young’s modulus, Poisson’s ratio and hardening modulus.This can be accounted for within the framework of the STPZ bysubstituting sy in the previous section by the sum of the two yieldstresses according to the temperatures associated with themaximum and minimum states of loading, sy,max þ sy,min.

Thus, for cyclic loading with temperature dependent yield stressthe following modifications are introduced in Section 2.2 related toa bilinear stressestrain curve:

sf :elij / sf :elij;max � sf :elij;min (17)

sy/sy;max þ sy;min (18)

which eventually leads to the elasticeplastic strain range

3el�plij / 3

el�plij;max � 3

el�plij;min (19)

2.4. Multi-linear stressestrain curve

A multi-linear stressestrain curve can be adopted within theframework of Zarka’s method and the STPZ based on the idea ofoverlay models. The basic assumption of overlay models is that astructural volume is considered to be made of a number of differentvolumes (which may be called layers) with different materialbehavior, all occupying the same volume and coupled by the samedeformations.

The most widely used overlay model is the Besseling model,where each layer exhibits a linear elastic e perfectly plasticbehavior. Thus, three layers are required to model a tri-linearstressestrain curve with a tangent modulus > 0 in the thirdsegment. In the present framework, each layer can be and must bemodeled by a linear kinematic hardening material so that only twolayers are required for a tri-linear stressestrain curve. In each ofthese layers, the theory as outlined in Sections 2.2 and 2.3 for abilinear stressestrain relation applies. In the following, the indi-vidual layers are addressed by roman numbers I and II, while thestates representing the entire second and the third segment of thestressestrain curve, i.e. the combined action of two layers, areaddressed by Arabic numbers 1 and 2 respectively. The firstsegment of the tri-linear stressestrain curve is thus defined byYoung’s modulus E, the second by the first vertex in sy1 and thetangent modulus Et1, and the third segment by the second vertex insy2 and the tangent modulus Et2. The tri-linear segmentation isshown in Fig. 3.

In the third segment of the stressestrain curve, both layers aregoing plastic simultaneously so that equation (13) becomes for theindividual layers I and II:

layer I:

3*ij;I ¼�E*ijkl

��1

Irkl;I þ 3ij;0I in VpI (20)

layer II:

3*ij;II ¼�E*ijkl

��1

IIrkl;II þ 3ij;0II in VpII (21)

and for the entire material

3*ij;2 ¼�E*ijkl

��1

2rkl þ 3ij;02 in Vp2 (22)

In the modified elasticity matrices ðE*ijklÞI; ðE*ijklÞII; ðE*ijklÞ2 themodified elastic parameters are given by

E*I ¼ EtI (23)

v*I ¼ 0:5� ð0:5� vÞ EtIEI

(24)

E*II ¼ EtII (25)

v*II ¼ 0:5� ð0:5� vÞ EtIIEII

(26)

E*2 ¼ Et2 (27)

v*2 ¼ 0:5� ð0:5� vÞ Et2E

(28)

where EI and EII are the Young’s moduli in layers I and II, and EtI andEtII the tangent moduli respectively.

The six parameters of the bilinear stressestrain curves in thetwo layers (EI, EtI, EII, EtII, syI, syII) can be identified from the fiveparameters of the tri-linear stressestrain curve (E, Et1, Et2, sy1, sy2)via the field equations (equilibrium and continuity conditions) foran elementary volume of material subjected to a uniaxial stressstate. However, these conditions are not sufficient to determine theparameters uniquely, so that one additional assumption can bechosen freely. In the present framework it is convenient to choosethat the modified Poisson’s ratio is identical in both layers:

v*II ¼ v*I (29)

This leads to

EtIEI

¼ EtIIEII

¼ Et2E

(30)

v*II ¼ v*I ¼ v*2 (31)

and to the following relations for the material parameters in thetwo layers

EII ¼ 1þ v32 �

�12 � v

�$Et1E

$Et1 � Et21� Et2

E

(32)

H. Hübel et al. / International Journal of Pressure Vessels and Piping 116 (2014) 10e1914

EI ¼ E � EII (33)

EtI ¼ Et2E$EI (34)

EtII ¼ Et2E$EII (35)

syI ¼ sy1$EIE

(36)

syII ¼ sy1$EIIE

þ �sy2 � sy1

�$1� Et2

Et1

1� Et2E

(37)

Consequently, we get

E*ijkl;I ¼ E*ijkl;2$EtIEt2

(38)

E*ijkl;II ¼ E*ijkl;2$EtIIEt2

(39)

Making use of the equilibrium and the continuity conditions inan elementary volume of material for the residual stresses andstrains

rij;2 ¼ rij;I þ rij;II (40)

3*ij;II ¼ 3*ij;I ¼ 3*ij;2 (41)

and the initial strains in the two layers with linear kinematichardening according to equation (12)

3ij;0I ¼ 32$1EtI

$

�1� EtI

EI

�$gij;I (42)

3ij;0II ¼ 32$1EtII

$

�1� EtII

EII

�$gij;II (43)

we get the initial strain for the third segment of the stressestraincurve:

3ij;02 ¼ 32$1Et2

$

�1� Et2

E

�$�gij;I þ gij;II

�(44)

gij,I and gij,II can be found by projecting the negative of the residualstresses �rij,I and �rij,II to the von Mises circle in the TIV-space ofeach of the two layers separately as described in Section 2.2. Themodified elastic analyses can then be performed with the modifiedelastic parameters E*2; v

*2 given in equations (27) and (28) and the

loading by initial strains 3ij,02 given by equation (44).

2.5. Limits

The following points explain the limits of the STPZ in thecurrently available development version:

� For every identified cycle the whole process of the STPZ has tobe carried out. This may lead to high computing times. E.g. for atransient with a lot of sub cycles, which are identified by aRainflow algorithm [5], the computing time can even be longerthan for the incremental elasticeplastic analysis of one cycle(including the sub cycles).

� In some cases the results of the STPZ are sensitive to the esti-mated strain range. This may yield overestimated stress ranges.

� Manageability: A lot of preparation has to be done by hand (e.g.application of the boundary conditions to the model).

2.6. Remarks

The STPZ is implemented via user subroutines into ANSYS�.Many practical applications show that usually two to five

modified elastic analyses are sufficient to get a close approximationto the elasticeplastic strain range in the state of plastic shakedownof thermomechanical cyclic loading problems compared with in-cremental analyses.

Within the same framework, assessment of strain accumulationdue to a ratcheting mechanism is also possible. This feature is notdiscussed by way of example in this paper. It requires a moredetailed discussion of the general performance of material modelsfor the simulation of local ratcheting.

3. Twice-Yield method

Another alternative for the determination of stabilized elasticeplastic strain ranges is the Twice-Yield method which is based onwork by Kalnins (see Refs. [6e8]) and is a proposed method in theASME Code, Section VIII, Div. 2 [9].

The basis of the method as described in the ASME Code is theestimation of stress and strain ranges of each cycle based on a singleelasticeplastic analysis in a quasi-monotonic fashion. Reducing theanalysis effort to a simple monotonic loading is a key of this simplemethod.

Themethod also requires a simplified approach for any transienttemperatures involved within a cycle by using a weighted value fortransient temperature changes within a cycle, i.e. by the followingequation:

T ¼ 0:75$Tmax þ 0:25$Tmin (45)

Stressestrain hysteresis loops of cycledmaterial can be obtainedby using the cyclic stabilized stressestrain curve based on anynonlinear stressestrain law, i.e. based on a RambergeOsgood typeof equation or any type of nonlinear or piecewise linearrepresentation.

Starting loading from zero stress and strain, the local stressestrain path will follow the cyclic stressestrain curve. At thereversal of the loading direction, the path follows a stressestraincurve that is doubled both in stress as well as in strain direction(see Fig. 4). The doubled stressestrain curve (twice-yield curve)is used to create any hysteresis branches according to Masing’shypothesis (both up- and downward branch for any load range)for any subsequent load ranges. (In case of variable amplitudeloading, a number of memory rules like the rules described byClormann and Seeger [5] could be applied to design the differentstressestrain hysteresis loops. This will not be considered in thispaper though.)

The Twice-Yield method simplifies this approach and does nottake into account any memory rules but only the doubled stressestrain curve to obtain stress and strain ranges for each load cycle.Load cycles for variable amplitude loading may be counted bymethods as described in the ASME code up front and stored as loadranges vs. number of occurrences. The application of the Twice-Yield method for a single load range therefore results in stressand strain ranges of the hysteresis loop but not in information onmean strain or (often more important) mean stress.

Fig. 4. Masing behavior for uniaxial loading, shown using a RambergeOsgood formulation.

H. Hübel et al. / International Journal of Pressure Vessels and Piping 116 (2014) 10e19 15

The Twice-Yield method was expanded to multiaxial behaviorby using equivalent stress and strain quantities based on von Misesformulations for multiaxial stress and multiaxial plastic strain.

After cycle counting, e.g. using the minemax method asdescribed in the ASME code, a first reversal point at time t1 and asecond reversal point at time t2 define each hysteresis loop. Thefirst point of the stressestrain loop at t1 is taken as a point ofreference to calculate ranges of tensor quantities of stress and strainfor each hysteresis loop. This means for any hanging hysteresisloops the upper point is taken as reference point and for standinghysteresis loops the lower point is taken as reference point.

The starting point of each hysteresis loop defines a referencepoint involving stress sðt1Þij and plastic strain pðt1Þij to calculate thecorresponding ranges for the hysteresis loop. Relative stress andplastic strain values are then calculated for each hysteresis loop:

Dsij ¼ sðt1Þij � sðt2Þij (46)

Dpij ¼ pðt1Þij � pðt2Þij (47)

For every hysteresis loop separately, the following equationssupply an equivalent elasticeplastic stress range and an equivalentplastic strain range by using

Dseq ¼ 1ffiffiffi2

p $

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðDs11 � Ds22Þ2 þ ðDs22 � Ds33Þ2 þ ðDs33 � Ds11Þ2 þ/þ 6$

�Ds212 þ Ds223 þ Ds213

�q(48)

D3peq ¼ffiffiffi2

p

3$

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðDp11 � Dp22Þ2 þ ðDp22 � Dp33Þ2 þ ðDp33 � Dp11Þ2 þ/þ 3

2$�Dp212 þ Dp223 þ Dp213

�r(49)

The equivalent stress range and the equivalent plastic strainrange is then combined to obtain a damage relevant equivalenttotal strain range of a cycle by Ref. [7] (see Fig. 5)

D3eff ¼ DseqE

þ D3peq (50)

The resulting value of total equivalent strain 3eff together withthe (uniaxial) strain-life curve will be used to calculate either apartial damage value or the cycles to failure.

The Twice-Yield approach offers some significant advantages:

� Calculation is much simpler and thus faster than incrementalelasticeplastic analyses through many load cycles.

� It requires solving a finite element model only for a singlemonotonic loading up to the largest value of stress differencesobtained from the load history. The elasticeplastic stressestraintensors as obtained for this monotonic loading and stored in theresults file then can be taken to efficiently assess fatigue damageof any loading cycle of arbitrary loading histories.

� The multiaxial equivalent values for stress and plastic strain areusually two numbers obtained by post processing a finiteelement analysis. In case of each single hysteresis loop theanalysis reduces to store those two values for further processing.

� No plasticity law capable of cyclic and kinematic hardening isrequired since the analysis uses just a quasi-static solutioninvolving a nonlinear stress strain law based on the stabilizedcyclic stressestrain curve. For example RambergeOsgoodformat can be used for a monotonic incremental plasticityanalysis.

The method though does not capture the following effects:

� Well-known memory effects from uniaxial strain based fatigueanalysis (memory rules to correctly obtain the stressestraincycles of subsequent cyclic variable amplitude loading).

Fig. 5. Ranges of equivalent quantities for a single load range applied to the structure.

H. Hübel et al. / International Journal of Pressure Vessels and Piping 116 (2014) 10e1916

� Mean stress levels of each computed stress cycle. In case oftransient loading for specific repeated load sequences meanstress might not relax and thus be important.

� Transient effects like non-proportional hardening or any othertransient memory effects of the material.

� Multiaxial plasticity effects (i.e. kinematic hardening) occurringespecially for non-proportional stress response.

� Non-Masing material behavior, since Masing behavior is a pre-requisite for the simplifications.

� Ratcheting or shakedown under cyclic loading (except the cy-clic hardening or softening which is captured in the cyclicmaterial data required for computing) which would requirespecific kinematic hardening rules and suitable plasticitymodels.

The described expansion to multiaxial behavior should beused with caution in case of non-proportional loading. Sincepressure vessels often face proportional loading, this restrictionusually might be of less importance but this should bementioned.

Fig. 6. Geometry and thermal loading for nozzle example 1.

4. Performance study by way of examples

4.1. General remarks

The typical examples relevant for thermomechanical powerplant applications are considered in the following for a discussionof the performance of the STPZ and the Twice-Yield method withregard of the determination of fatigue relevant strain ranges andfatigue assessment:

� Nozzle example 1� Stepped pipe� Disk� Nozzle example 2

The examples (see also Ref. [10]) are shown in Figs. 6 through 9.A detailed respective study has been carried out in Ref. [11].

4.2. Flow chart for using the STPZ

This section describes the necessary steps to apply the STPZ aspart of a fatigue analysis.

� An elastic fatigue analysis according to e.g. ASMENB 3200 [12] isdone. From this analysis the following information is obtained:� Identification of the two load steps which form the stresscycle.

� The alternating stress amplitude Salt, from which the esti-mated strain range is calculated.

� The average cycle temperature.

Fig. 7. Geometry and thermal loading for the stepped pipe.

Fig. 8. Geometry and thermal loading for the disk example.

H. Hübel et al. / International Journal of Pressure Vessels and Piping 116 (2014) 10e19 17

� The stressestrain curve for the specific given average cycletemperature is created by interpolating between the givenstressestrain curves.

� From this newly created stressestrain curve the tri-linearstressestrain curve is calculated from the estimated strain ac-cording to 3est ¼ min(Salt/E; 0.01).

� All this information is gathered and prepared in order to createan ANSYS� input file for the STPZ.

� Besides the ANSYS� input file, the model database and thestructural result file of the elastic calculation is needed.

� According to the STPZ the elasticeplastic strain range is calcu-lated. It is transferred to the stress range and in the end theusage factor is calculated.

4.3. Results

The results of both the STPZ and the Twice-Yield method incomparison with incremental elasticeplastic analysis as well asdifferent Ke procedures (see Refs. [10,12e14]) are shown in Fig. 10.

All results are within 30% deviation with respect to the fatigueusage factor in comparison with the incremental elasticeplasticresults. Regarding the STPZ, all of the results are conservative (withregard to the incremental elasticeplastic reference analysis) anddeliver a lower usage factor than the simplified elasticeplasticanalysis according to different applicable design codes. Theconclusion is drawn from Fig. 10 that both the STPZ and the Twice-Yield method outperform all comparable design code based plas-ticity correction methods (Ke factors) by reducing overly conser-vative results. Note that ASME III Ke values (Fig. 10) are from NB3200 and not from CC N-779.

Thus, the STPZ and the Twice-Yield method can be seen as anintermediate between simplified elasticeplastic analysis (Ke fac-tors) and incremental elasticeplastic analysis. For practical

Fig. 9. Geometry and thermal loading for nozzle example 2.

Fig. 10. Analysis results of different methods compared to in

Fig. 11. Staged approach to elasticeplastic analyses.

H. Hübel et al. / International Journal of Pressure Vessels and Piping 116 (2014) 10e1918

engineering application, a case dependent graded approach asshown in Fig. 11 is proposed.

Note, that the cladding example studied in Ref. [10] (as wellas one complex 3D nozzle example) has consciously beenexempted from this comparison. The application of both thesimplified elasticeplastic analysis (Ke factors) and the STPZ iscritical in this case. For the time being, incremental elasticeplastic analyses are the method of choice in the case ofcladding.

5. Application to arbitrary load time histories

Both the STPZ and the Twice-Yieldmethod can be applied toarbitrary load time histories. In this case e.g. a Rainflow algo-rithm (e.g. Ref. [5]) can be used to identify the cycles. For theidentified cycles the procedures according to Section 3respectively 4.2 are applied afterward in a straight forwardway.

In general, the counting algorithm respectively the loadtime history has only an influence on the identification of cy-cles. The STPZ is using this cycle information as essential input.

cremental elasticeplastic reference analyses.

H. Hübel et al. / International Journal of Pressure Vessels and Piping 116 (2014) 10e19 19

This means that the initial cycle may be the result of a designtransient, measured transients or arbitrary load (temperature) timehistories.

6. Conclusions

Low cycle fatigue analyses of power plant components requirean appropriate consideration of cyclic plasticity for the predomi-nant thermomechanical loading conditions. The applicable designcodes such as Refs. [12e14] primarily differentiate between asimplified elasticeplastic (Ke factors, see Ref. [10]) and an incre-mental elasticeplastic analysis (i.e. incremental elasticeplastic FEAof one loading cycle based on the cyclic stressestrain curve). In thecase of thermal cyclic loading conditions typical for power plantoperation the plasticity correction factors Ke typically show short-comings by considerably overestimating the real elasticeplasticstrain ranges and thus the corresponding fatigue usage factors. Thealternative of an incremental elasticeplastic analysis is often timeconsuming and exhaustive for complicated geometry and loadingconditions. The application of the STPZ as well as the Twice-Yieldmethod offers an intermediate option and a compromise be-tween accuracy, conservatism and calculation effort. Thus, thepresented methods become a valuable part of the toolbox in agraded and case dependent application scheme. The applicabilityfor typical power plant components has been shown by way ofexample in this paper. Note, that a special numerical imple-mentation for the STPZ is required. The method is not part ofstandard distributions of finite element software. The theoreticalbasis of this implementation is explained in the first part of thispaper. Furthermore, the fundamentals of the Twice-Yield methodand its performance by way of example in comparison to the in-cremental elasticeplastic analysis and the STPZ were shown. Theapplication of this method does not require a special numericalimplementation and the procedure is an integrated part of ASME

Code, Section VIII, Division 2 [9]. Since the results obtained for alimited number of test cases led usage factors close to the valuesobtained using an incremental plasticity model, further validationis recommended though.

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