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International Journal of Pressure Vessels and Piping 82 (2005) 27–33 www.elsevier.com/locate/ijpvp
Analysis of cyclic creep and rupture. Part 2: calculation of cyclic reference stresses and ratcheting interaction diagrams
P. Carter
Stress Engineering Services, Inc., Mason, OH 45040, USA Received 14 June 2003; revised 28 May 2004; accepted 17 June 2004
Abstract Part 1 gives the basis for the use of cyclic reference stresses for high temperature design and assessment. The methodology relies on e

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Analysis of cyclic creep and rupture. Part 2: calculation of cyclicreference stresses and ratcheting interaction diagrams
P. Carter
Stress Engineering Services, Inc., Mason, OH 45040, USA
Received 14 June 2003; revised 28 May 2004; accepted 17 June 2004
Abstract
Part 1 gives the basis for the use of cyclic reference stresses for high temperature design and assessment. The methodology relies onelastic–plastic calculations for limit loads, ratcheting and shakedown. In this paper we use a commercial non-linear ﬁnite element code forthese calculations. Two fairly complex and realistic geometries with cyclic loads are analysed, namely a pipe elbow and a traveling thermalshock in a pressurized pipe. The special case of start-up shut-down cycles is also discussed. Creep and rupture predictions may be made fromthe results. When reference stresses can be economically calculated, their use for high temperature design has the following advantages.
†
Accuracy
. Limit loads, shakedown and ratcheting limits are based on detailed analysis, and do not rely on rules or judgement.
†
Efﬁciency
. Use of shakedown and ratcheting reference stresses to predict rupture and creep strain, respectively, allowing details of timeand temperature to be dealt with as material data, not affecting the analysis.
†
Factors of safety
. For both low and high temperature problems, factors of safety can be determined or applied, based on the real failureboundaries.
†
Conservatism
. The rupture and strain calculations reﬂect the limit of rapid cycle behaviour. Cycles with relaxation will be associated withlonger lives.
q
2004 Elsevier Ltd. All rights reserved.
Keywords:
Cyclic loading; Pipe bend; Thermal shock; Shakedown; Ratcheting; Cyclic reference stress
1. Introduction
1.1. Design code approaches for creep and rupture
A number of different methodologies are evident in hightemperature codes. ASME Sections I[1]and VIII Division 1[2]have design rules, and ASME Section IID[3]has design
data, for materials and temperatures well into the creeprange. The rules giving stress, which must be compared withan allowable, are clearly intended and applicable for steadyloading. ASME VIII Division 2[4], BS5500[5]and ASME
IIINH[6]refer to stress ranges and cyclic loading and makeuse of elastic ﬁnite element analysis with a stressclassiﬁcation scheme and allowable stresses, for low andhigh temperatures, where applicable. As noted in Part 1,de-coupling the analysis technique from the materialproperties for structural failure modes is reliable if theanalysis can reﬂect real structural failures. This approachimplicitly relies on a reference stress argument. The analysisessentially gives a reference stress, which must be comparedwith an allowable stress to assess the design.The methodology described in these papers is that forcyclic loading, creep strain accumulation is conservativelydeﬁned by the ratcheting reference stress, and creep damageis conservatively deﬁned by the shakedown reference stress.(In part 1, it was noted that a ratcheting reference stresswhich tends to zero for ﬁnite thermal stress cycles could bemisleading.) ASME IIINH[6]uses a creep-fatigue damagecalculation and avoids the problem of distinguishingbetween creep effective stress (or reference stress) for strainand for creep damage. Linking creep strain accumulationand rupture or damage as in API 579[7]is only possible forsteady loading.
0308-0161/$ - see front matter
q
2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijpvp.2004.06.010International Journal of Pressure Vessels and Piping 82 (2005) 27–33www.elsevier.com/locate/ijpvp
E-mail address:
peter.carter@ses-oh.com.
2. Estimation of shakedown and ratcheting boundaries
We will use conventional elastic–plastic analysis todetermine shakedown and ratcheting reference stresses andboundaries. There are techniques to calculate shakedownand ratcheting without resorting to cyclic elastic–plasticanalysis such as rapid cycle analysis[8]and the linearmatching technique[9]. Usually these techniques arebenchmarked against a few simple cases with algebraicsolutions such as the Bree problem[10], or a full plasticcyclic analysis using a non-linear commercial ﬁnite elementcode. It is our experience that the advantages of one of thealternative techniques (the rapid cycle solution) over fullcyclic analysis are marginal. In this paper we will use theAbaqus version 6.4 code and cyclic elastic–plastic analysisto obtain shakedown and ratcheting boundaries.Fig. 1is the Bree[10]diagram, generalized in Ref.[11]
and used in ASME IIINH[6].It shows the different regions
of structural behaviour that are possible in a cyclicallyloaded structure. The axes are normalised primary (press-ure) stress, and secondary (thermal) stress range. Thenormalising stress is the material yield stress. There are fourmain regions of interest. Region E is elastic, where pressureplus thermal stress is always less than yield. Region S isshakedown. Region R is reversed plasticity, where yieldingoccurs on every cycle, but no incremental or ratchetingstrain occurs. Region R indicates ratcheting, where ﬁnitestrain growth occurs on every cycle.We seek to generate similar diagrams for other structuresand loading. We consider cases, where primary stress isconstant and secondary stress is cyclic, but in general anycyclic loading may be considered. Information issummarized on ratcheting interaction diagrams similar toFig. 1, using normalized axes. In this paper ‘primary’ refersto loads and stresses which are deﬁned in terms of pressureor load, ‘secondary’ refers to loads and stresses which arethermal or displacement controlled.For a given structure with cyclic loading, we look for thelowest values of yield stresses for which (a) the structureshakes down, and (b) does not ratchet. This is a trial anderror procedure, which can be made more efﬁcient bytechniques to estimate the shakedown and ratchetingboundaries in a ratcheting interaction diagram. If there isonly one load combination and cycle of interest, then aratcheting diagram is not necessary, and the plastic cyclicanalysis described below is adequate to calculate shake-down and ratcheting reference stresses. However, it mayhelp to know which region of the ratcheting diagram isapplicable. The suggested procedure is to construct a trial orestimated ratcheting diagram, which is then conﬁrmed ormodiﬁed with detailed analysis. First we estimate theshakedown boundary. For steady primary loads and cyclicsecondary loads, this could be assumed to be of the form of the shakedown region in the srcinal Bree diagram, and inASME IIINH. If the secondary stress is dominantly amembrane or uniform stress, then a second typical shake-down limit would be applicable (Fig. 2). This is easy toderive using a Tresca yield surface, and assuming theprimary and secondary stress components are perpendicular.It is similar to an interaction diagram derived by Ponter andCocks[12,13]for severe thermal shock. There is no reverseplasticity region, and no part of the boundary having thenormalised secondary stress,
q
Z
constant. This is consistentwith a general theory by Goodall[14]which predicts that ashakedown surface having
q
Z
constant is the boundarybetween shakedown and reverse plasticity, whereas if theshakedown surface does not have
q
Z
constant, then it is theboundary between shakedown and ratcheting. For typical
Fig. 1. Bree diagram, showing elastic, shakedown, reverse plasticity andratcheting regions. Axes show stress normalized by yield stress. Fig. 2. Shakedown and ratcheting for cyclic membrane stress.
P. Carter / International Journal of Pressure Vessels and Piping 82 (2005) 27–33
28
cases, where secondary bending stress exists the suggestedprocedure is:
†
Calculate the maximum elastic Mises stress (
p
e
) and thelimit load reference stress (
p
0
) for the primary (load-controlled) load, and the maximum elastic Mises stressrange (
q
e
) for secondary loads.
s
y
is the yield stress.
†
Let
p
Z
p
e
/
s
y
be the primary load parameter, and let
q
Z
q
e
/
s
y
be the secondary load parameter. Also deﬁne
p
0
Z
p
0
/
s
y
as a primary load parameter having the valueone at the limit load. The estimated shakedownboundary is deﬁned by
p
0
C
q
/4
Z
1 if
p
0
O
0.5 and
q
Z
2 if
p
0
!
0.5.
†
For any combination of
p
0
and
q
in or on the shakedownregion, the estimated shakedown reference stress is thegreater of
p
0
C
q
/4 and
q
/2.Secondly, we need to estimate the ratcheting boundary.There are two versions of the basic idea, and which is thebest to use depends on the information available. Weconsider the effect of section thickness change on theparameters (
p
,
q
), and whether it affects ratcheting. From thiswe may derive an estimated ratcheting boundary basedeither on the sensitivity to thickness, or on a distinctionbetween membrane (average) and bending thermal stress.The dependence of primary and secondary elastic stresson shell thickness is easily determined with a shell ﬁniteelement model. For the Bree diagram inFig. 1, the equationsare simple. Let
t
be the normalised shell thickness with avalue of one in the srcinal case. In this case the primary orpressure stress is
p
0
, and secondary or thermal stress is
q
0
,Then
p
ð
t
Þ
Z
p
0
=
t
;
q
ð
t
Þ
Z
q
0
t
(1)where
p
(
t
) and
q
(
t
) are the maximum thickness-dependentprimary and secondary stress, respectively. The equation for
q
(
t
) is true for displacement-controlled bending stress, or forthermal stress, where the throughwall temperature differen-tial is also proportional to
t
.The procedure is based on the following assumption:
†
Increasing the shell thickness over the whole structure bya uniform ratio, with other factors unchanged, does notalter the structural behaviour from non-ratcheting toratcheting.Generally, a structure which does not ratchet has anelastic core. The elastic core is sufﬁcient to prevent amechanism in the structure, which would imply ratchetingor incremental collapse. In the case of the Bree problem, theelastic core (if it exists) is that central region of the shell thatremains elastic. The non-ratcheting assumption abovewould hold if increasing the shell thickness did not decreasethe elastic core of the structure. This would happen if thefollowing conditions were met.(i) The maximum mechanical stress
p
(
t
)
!
p
0
if
t
O
1. Thatis, increasing the shell thickness by a uniform ratioreduces the maximum stress due to primary loading.(ii) The thermal or displacement-controlled stress withinthe elastic core is not increased by the shell thicknessincrease.These conditions are met in the Bree problem withEq. (1). For more general shell problems the relationshipbetween stress, shell thickness, curvature and temperaturesuggests that the assumption should be true for pressure andmechanical loading, linear throughwall thermal gradientsand displacement-controlled loads.From the equations for
p
and
q
for the Bree problem itfollows that moving along a trajectory deﬁned by
pq
Z
1 isequivalent to changing the wall thickness. Increasing thewall thickness implies
pq
Z
1 with
p
decreasing and
q
increasing. Therefore if we select a point on the shakedownboundary and move along the contour
pq
Z
1 with
p
decreasing and
q
increasing, then by the basic assumption,we will not move into a ratcheting region. This can be seento be true for the points on the two shakedown boundaries
p
C
q
/4
Z
1 and
q
Z
2. Along
p
C
q
/4
Z
1, moving along
pq
Z
1 with
p
decreasing and
q
increasing is to move insidethe shakedown region. On
q
Z
2 moving along
pq
Z
1 in thesame way is to move into the reverse plasticity region.Consider the intersection of the two shakedown lines at
p
Z
0.5,
q
Z
2. Moving along
pq
Z
1 with
p
decreasing and
q
increasing deﬁnes the boundary between reverse plasticityand ratcheting (P/R boundary).This procedure is shown inFig. 3. This has three stresstrajectories, each of which shows how stresses change asshell thickness is increased. With the assumption that noneof these lines will cross the ratcheting boundary, it is clearhow to select the point at the beginning of the trajectory, sothat it deﬁnes the ratcheting boundary.We may apply this idea directly for the case of displacement-controlled secondary stress in a pipe elbow.
Fig. 3. Three predictions of non-ratcheting contours based on elastic stressthickness dependence.
P. Carter / International Journal of Pressure Vessels and Piping 82 (2005) 27–33
29
For the case of a moving thermal shock, the displacement-controlled analogy is more difﬁcult to make, sincethickening the shell changes the position of maximumbending relative to the thermal front. However, we knowthat a bending thermal stress with constant pressure is likelyto give a Bree-type of behavior as inFig. 1. If the secondarystress were membrane, the stress would not depend on shellthickness, and the result would be similar toFig. 2.So for
the moving thermal front we will construct an estimatedratcheting line based on a combination of secondarybending to secondary membrane stress.Consider a uniform section subject to displacement-controlled bending and tension. Let
t
is shell thickness,where srcinal thickness is 1. From the above arguments wemay put secondary stress
q
(
t
)
Z
q
0
(
a
C
b
t
), where
a
and
b
are the membrane and bending fractions of thermal stress
q
0
.We know
p
Z
p
0
/
t
, where
p
0
is the pressure stress at srcinalthickness. Eliminating
t
gives
q
/
q
0
Z
(
a
C
b
p
0
/
p
). Thisrelation between
p
and
q
is one estimated ratcheting line.As before, the equation
q
/
q
0
Z
(
a
C
b
p
/
p
0
) applies to any(
p
0
,
q
0
), and the optimum values are likely to be
p
0
Z
0.5,
q
0
Z
2.In addition there will be another limit when themembrane component fails to shakedown. This line liesbetween (
p
,
q
)
Z
(0,2/
a
) and (
p
,
q
)
Z
(0.5,1/
a
). The predictedinteraction diagram for the combination of
a
and
b
will bethe inner or most conservative, of the cases of membraneeffect alone, and using a ratcheting line predicted by
q
/
q
0
Z
(
a
C
b
p
0
/
p
).Returning to the use of the thickness stress sensitivitytechnique, assume we have found the relation between
p
and
q
using thickness variations with a shell ﬁnite elementmodel. We may then infer
a
and
b
for this relation using aleast squares ﬁt. The membrane ratcheting line as inFig. 2may then be calculated.
3. Cyclic plastic analysis
Conﬁrmation of the shakedown boundary is relativelystraightforward. Based on experience, shakedown does notrequire many cycles to occur, and it is easily identiﬁed bythe absence of iterations, zero incremental displacementsand increments of a variable such as
Ð
_
3
p
d
t
over the cycle,where
_
3
p
is plastic strain rate.Identiﬁcation of ratcheting behaviour has been madeeasier with the the Abaqus ‘direct cyclic’ technique[15].Time-dependent cyclic displacements are represented by aFourier series. Ratcheting is indicated by the non-conver-gence of the residual force associated with the constant termin the series. Positive identiﬁcation of reverse plasticity ismore difﬁcult. For severe cyclic loading well beyondshakedown it is characterized by direct cyclic solutionswhich are very slow to converge and which do not show aclear divergence of the constant residual force term. Use of conventional elastic–plastic analysis for cycles to detectshakedown is feasible, but for cases that do not shakedown,convergence to the steady cyclic solution can be slow anddifﬁcult to detect. It is often difﬁcult to distinguish betweennumerical noise and ratcheting for conventional cyclicanalysis. Unless there is an ability to calculate the plasticcyclic solution directly, the distinction between reverseplasticity and ratcheting is often unclear in all but thesimplest of structures.
3.1. Pressurised tube subject to thermal transient
We consider a tube subject to constant pressure and atraveling thermal shock. The tube starts isothermally at
K
5.5
8
C, and then a moving temperature front with ambient
T
, 5.5
8
C, ﬁlm coefﬁcient, 2500 W/m
2
/K, travels up the boreat 76 mm/s. The external ambient temperature is a constant
K
5.5
8
C with a ﬁlm coefﬁcient of 816 W/m
2
/K. The tubeOD is 1295 mm and thickness is 50 mm. With typicalproperties of mild steel, this produces a maximum thermalstress of 10.5 MPa (Fig. 4). The membrane fraction
a
is0.36, and the bending fraction
b
is 0.64. The elastic–plasticcyclic analyses were performed with the temperature cyclefrom this transient. Thermal stress was varied by varying theexpansion coefﬁcient.The conventional shakedown plot in terms of secondarystress range is inconvenient for such transient thermalstresses. Obtaining the maximum stress range from theanalysis is difﬁcult. So the maximum thermal stress is usedto characterize the problem. The estimated limits are now
Fig. 4. Mesh and thermal stress contours for section of tube. Plotted onexaggerated displaced shape. Stress units ksi.
P. Carter / International Journal of Pressure Vessels and Piping 82 (2005) 27–33
30

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