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

elastic–plastic calculations for limit loads, ratcheting and shakedown. In this paper we use a commercial non-linear finite element code for

these calculations. Two fairly complex and realistic geometries with cyclic loads are analysed, namely a pipe elbow and a traveling thermal

shock in a pressurized pipe. The special case of start-up shut-down cycles is also discussed. Creep and rupture predictions may be made from

the results. When reference stresses can be economically calculated, their use for high temperature design has the following advantages.

†

030

doi

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 time

and 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 failure

boundaries.

†

Conservatism. The rupture and strain calculations reflect the limit of rapid cycle behaviour. Cycles with relaxation will be associated with

longer 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 high

temperature 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 creep

range. The rules giving stress, which must be compared with

an allowable, are clearly intended and applicable for steady

loading. ASME VIII Division 2 [4], BS5500 [5] and ASME

IIINH [6] refer to stress ranges and cyclic loading and make

use of elastic finite element analysis with a stress

classification scheme and allowable stresses, for low and

high temperatures, where applicable. As noted in Part 1,

8-0161/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.

:10.1016/j.ijpvp.2004.06.010

E-mail address: peter.carter@ses-oh.com.

de-coupling the analysis technique from the material

properties for structural failure modes is reliable if the

analysis can reflect real structural failures. This approach

implicitly relies on a reference stress argument. The analysis

essentially gives a reference stress, which must be compared

with an allowable stress to assess the design.

The methodology described in these papers is that for

cyclic loading, creep strain accumulation is conservatively

defined by the ratcheting reference stress, and creep damage

is conservatively defined by the shakedown reference stress.

(In part 1, it was noted that a ratcheting reference stress

which tends to zero for finite thermal stress cycles could be

misleading.) ASME IIINH [6] uses a creep-fatigue damage

calculation and avoids the problem of distinguishing

between creep effective stress (or reference stress) for strain

and for creep damage. Linking creep strain accumulation

and rupture or damage as in API 579 [7] is only possible for

steady loading.

International Journal of Pressure Vessels and Piping 82 (2005) 27–33

www.elsevier.com/locate/ijpvp

P. Carter / International Journal of Pressure Vessels and Piping 82 (2005) 27–3328

2. Estimation of shakedown and ratcheting boundaries

We will use conventional elastic–plastic analysis to

determine shakedown and ratcheting reference stresses and

boundaries. There are techniques to calculate shakedown

and ratcheting without resorting to cyclic elastic–plastic

analysis such as rapid cycle analysis [8] and the linear

matching technique [9]. Usually these techniques are

benchmarked against a few simple cases with algebraic

solutions such as the Bree problem [10], or a full plastic

cyclic analysis using a non-linear commercial finite element

code. It is our experience that the advantages of one of the

alternative techniques (the rapid cycle solution) over full

cyclic analysis are marginal. In this paper we will use the

Abaqus version 6.4 code and cyclic elastic–plastic analysis

to obtain shakedown and ratcheting boundaries.

Fig. 1 is 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 cyclically

loaded structure. The axes are normalised primary (press-

ure) stress, and secondary (thermal) stress range. The

normalising stress is the material yield stress. There are four

main regions of interest. Region E is elastic, where pressure

plus thermal stress is always less than yield. Region S is

shakedown. Region R is reversed plasticity, where yielding

occurs on every cycle, but no incremental or ratcheting

strain occurs. Region R indicates ratcheting, where finite

strain growth occurs on every cycle.

We seek to generate similar diagrams for other structures

and loading. We consider cases, where primary stress is

constant and secondary stress is cyclic, but in general any

cyclic loading may be considered. Information is

Fig. 1. Bree diagram, showing elastic, shakedown, reverse plasticity and

ratcheting regions. Axes show stress normalized by yield stress.

summarized on ratcheting interaction diagrams similar to

Fig. 1, using normalized axes. In this paper ‘primary’ refers

to loads and stresses which are defined in terms of pressure

or load, ‘secondary’ refers to loads and stresses which are

thermal or displacement controlled.

For a given structure with cyclic loading, we look for the

lowest values of yield stresses for which (a) the structure

shakes down, and (b) does not ratchet. This is a trial and

error procedure, which can be made more efficient by

techniques to estimate the shakedown and ratcheting

boundaries in a ratcheting interaction diagram. If there is

only one load combination and cycle of interest, then a

ratcheting diagram is not necessary, and the plastic cyclic

analysis described below is adequate to calculate shake-

down and ratcheting reference stresses. However, it may

help to know which region of the ratcheting diagram is

applicable. The suggested procedure is to construct a trial or

estimated ratcheting diagram, which is then confirmed or

modified with detailed analysis. First we estimate the

shakedown boundary. For steady primary loads and cyclic

secondary loads, this could be assumed to be of the form of

the shakedown region in the original Bree diagram, and in

ASME IIINH. If the secondary stress is dominantly a

membrane or uniform stress, then a second typical shake-

down limit would be applicable (Fig. 2). This is easy to

derive using a Tresca yield surface, and assuming the

primary and secondary stress components are perpendicular.

It is similar to an interaction diagram derived by Ponter and

Cocks [12,13] for severe thermal shock. There is no reverse

plasticity region, and no part of the boundary having the

normalised secondary stress, qZconstant. This is consistent

with a general theory by Goodall [14] which predicts that a

shakedown surface having qZconstant is the boundary

between shakedown and reverse plasticity, whereas if the

shakedown surface does not have qZconstant, then it is the

boundary between shakedown and ratcheting. For typical

Fig. 2. Shakedown and ratcheting for cyclic membrane stress.

P. Carter / International Journal of Pressure Vessels and Piping 82 (2005) 27–33 29

cases, where secondary bending stress exists the suggested

procedure is:

†

Calculate the maximum elastic Mises stress (pe) and the

limit load reference stress (p0) for the primary (load-

controlled) load, and the maximum elastic Mises stress

range (qe) for secondary loads. sy is the yield stress.

†

Let pZpe/sy be the primary load parameter, and let

qZqe/sy be the secondary load parameter. Also define

p 0Zp0/sy as a primary load parameter having the value

one at the limit load. The estimated shakedown

boundary is defined by p 0Cq/4Z1 if p 0O0.5 and

qZ2 if p 0!0.5.

†

For any combination of p 0 and q in or on the shakedown

region, the estimated shakedown reference stress is the

greater of p 0Cq/4 and q/2.

Secondly, we need to estimate the ratcheting boundary.

There are two versions of the basic idea, and which is the

best to use depends on the information available. We

consider the effect of section thickness change on the

parameters (p,q), and whether it affects ratcheting. From this

we may derive an estimated ratcheting boundary based

either on the sensitivity to thickness, or on a distinction

between membrane (average) and bending thermal stress.

The dependence of primary and secondary elastic stress

on shell thickness is easily determined with a shell finite

element model. For the Bree diagram in Fig. 1, the equations

are simple. Let t be the normalised shell thickness with a

value of one in the original case. In this case the primary or

pressure stress is p0, and secondary or thermal stress is q0,

Then

pðtÞ Z p0=t; qðtÞ Z q0t (1)

where p(t) and q(t) are the maximum thickness-dependent

primary and secondary stress, respectively. The equation for

q(t) is true for displacement-controlled bending stress, or for

thermal stress, where the throughwall temperature differen-

tial is also proportional to t.

The procedure is based on the following assumption:

†

Fig. 3. Three predictions of non-ratcheting contours based on elastic stress

thickness dependence.

Increasing the shell thickness over the whole structure by

a uniform ratio, with other factors unchanged, does not

alter the structural behaviour from non-ratcheting to

ratcheting.

Generally, a structure which does not ratchet has an

elastic core. The elastic core is sufficient to prevent a

mechanism in the structure, which would imply ratcheting

or incremental collapse. In the case of the Bree problem, the

elastic core (if it exists) is that central region of the shell that

remains elastic. The non-ratcheting assumption above

would hold if increasing the shell thickness did not decrease

the elastic core of the structure. This would happen if the

following conditions were met.

(i)

The maximum mechanical stress p(t)!p0 if tO1. That

is, increasing the shell thickness by a uniform ratio

reduces the maximum stress due to primary loading.

(ii)

The thermal or displacement-controlled stress within

the elastic core is not increased by the shell thickness

increase.

These conditions are met in the Bree problem with

Eq. (1). For more general shell problems the relationship

between stress, shell thickness, curvature and temperature

suggests that the assumption should be true for pressure and

mechanical loading, linear throughwall thermal gradients

and displacement-controlled loads.

From the equations for p and q for the Bree problem it

follows that moving along a trajectory defined by pqZ1 is

equivalent to changing the wall thickness. Increasing the

wall thickness implies pqZ1 with p decreasing and q

increasing. Therefore if we select a point on the shakedown

boundary and move along the contour pqZ1 with p

decreasing and q increasing, then by the basic assumption,

we will not move into a ratcheting region. This can be seen

to be true for the points on the two shakedown boundaries

pCq/4Z1 and qZ2. Along pCq/4Z1, moving along

pqZ1 with p decreasing and q increasing is to move inside

the shakedown region. On qZ2 moving along pqZ1 in the

same way is to move into the reverse plasticity region.

Consider the intersection of the two shakedown lines at

pZ0.5, qZ2. Moving along pqZ1 with p decreasing and q

increasing defines the boundary between reverse plasticity

and ratcheting (P/R boundary).

This procedure is shown in Fig. 3. This has three stress

trajectories, each of which shows how stresses change as

shell thickness is increased. With the assumption that none

of these lines will cross the ratcheting boundary, it is clear

how to select the point at the beginning of the trajectory, so

that it defines the ratcheting boundary.

We may apply this idea directly for the case of

displacement-controlled secondary stress in a pipe elbow.

P. Carter / International Journal of Pressure Vessels and Piping 82 (2005) 27–3330

For the case of a moving thermal shock, the displacement-

controlled analogy is more difficult to make, since

thickening the shell changes the position of maximum

bending relative to the thermal front. However, we know

that a bending thermal stress with constant pressure is likely

to give a Bree-type of behavior as in Fig. 1. If the secondary

stress were membrane, the stress would not depend on shell

thickness, and the result would be similar to Fig. 2. So for

the moving thermal front we will construct an estimated

ratcheting line based on a combination of secondary

bending to secondary membrane stress.

Consider a uniform section subject to displacement-

controlled bending and tension. Let t is shell thickness,

where original thickness is 1. From the above arguments we

may put secondary stress q(t)Zq0(aCbt), where a and b

are the membrane and bending fractions of thermal stress q0.

We know pZp0/t, where p0 is the pressure stress at original

thickness. Eliminating t gives q/q0Z(aCbp0/p). This

relation between p and q is one estimated ratcheting line.

As before, the equation q/q0Z(aCbp/p0) applies to any

(p0,q0), and the optimum values are likely to be p0Z0.5,

q0Z2.

In addition there will be another limit when the

membrane component fails to shakedown. This line lies

between (p,q)Z(0,2/a) and (p,q)Z(0.5,1/a). The predicted

interaction diagram for the combination of a and b will be

the inner or most conservative, of the cases of membrane

effect alone, and using a ratcheting line predicted by

q/q0Z(aCbp0/p).

Returning to the use of the thickness stress sensitivity

technique, assume we have found the relation between p and

q using thickness variations with a shell finite element

model. We may then infer a and b for this relation using a

least squares fit. The membrane ratcheting line as in Fig. 2

may then be calculated.

Fig. 4. Mesh and thermal stress contours for section of tube. Plotted on

exaggerated displaced shape. Stress units ksi.

3. Cyclic plastic analysis

Confirmation of the shakedown boundary is relatively

straightforward. Based on experience, shakedown does not

require many cycles to occur, and it is easily identified by

the absence of iterations, zero incremental displacements

and increments of a variable such asÐ_3p dt over the cycle,

where _3p is plastic strain rate.

Identification of ratcheting behaviour has been made

easier with the the Abaqus ‘direct cyclic’ technique [15].

Time-dependent cyclic displacements are represented by a

Fourier series. Ratcheting is indicated by the non-conver-

gence of the residual force associated with the constant term

in the series. Positive identification of reverse plasticity is

more difficult. For severe cyclic loading well beyond

shakedown it is characterized by direct cyclic solutions

which are very slow to converge and which do not show a

clear divergence of the constant residual force term. Use of

conventional elastic–plastic analysis for cycles to detect

shakedown is feasible, but for cases that do not shakedown,

convergence to the steady cyclic solution can be slow and

difficult to detect. It is often difficult to distinguish between

numerical noise and ratcheting for conventional cyclic

analysis. Unless there is an ability to calculate the plastic

cyclic solution directly, the distinction between reverse

plasticity and ratcheting is often unclear in all but the

simplest of structures.

3.1. Pressurised tube subject to thermal transient

We consider a tube subject to constant pressure and a

traveling thermal shock. The tube starts isothermally at

K5.5 8C, and then a moving temperature front with ambient

T, 5.5 8C, film coefficient, 2500 W/m2/K, travels up the bore

at 76 mm/s. The external ambient temperature is a constant

K5.5 8C with a film coefficient of 816 W/m2/K. The tube

OD is 1295 mm and thickness is 50 mm. With typical

properties of mild steel, this produces a maximum thermal

stress of 10.5 MPa (Fig. 4). The membrane fraction a is

0.36, and the bending fraction b is 0.64. The elastic–plastic

cyclic analyses were performed with the temperature cycle

from this transient. Thermal stress was varied by varying the

expansion coefficient.

The conventional shakedown plot in terms of secondary

stress range is inconvenient for such transient thermal

stresses. Obtaining the maximum stress range from the

analysis is difficult. So the maximum thermal stress is used

to characterize the problem. The estimated limits are now

Fig. 5. Ratcheting and shakedown limits for pressurized tube subject to

thermal shock.

P. Carter / International Journal of Pressure Vessels and Piping 82 (2005) 27–33 31

based on the assumption that stress rangeZ2!maximum

stress.

The estimated shakedown and ratcheting limits were

constructed based on the Bree shakedown limit, and on

Fig. 6. Pipe mesh showing typical ratc

the ratcheting limit calculated from a and b as described

above. Then shakedown and ratcheting analyses were

performed. First the shakedown point at (1.0,0.5) was

found to be well inside the real limit, which was found to be

close to (1.0,0.71). The shakedown limits were adusted for

this. Fig. 5 shows the results with a number of other analysis

points. The estimates and predictions are in good agreement.

Also shown are the limits for shakedown and ratcheting

reference stressZ0.5! yield stress. The inner (shakedown)

limit is the more conservative, and would be used to

calculate rupture life. The ratcheting limit would be used to

calculate accelerated creep strain accumulation.

3.2. Pipe elbow under pressure and in-plane cyclic

pipework loads

We consider a pipe bend subject to pressure and

displacement-controlled in-plane bending. Fig. 6 shows

the finite element mesh using first order reduced integration

shell elements and the direction of displacement loading.

Also shown is a typical distribution of ratcheting strain. The

geometry is a 600 mm schedule 24 (9.5 mm thick) long

radius (900 mm) elbow.

Fig. 7 shows the estimated and calculated shakedown and

ratcheting limits. The estimated ratcheting line was based on

the effect of shell thickness on elastic stress at the position of

highest stress in the model. The membrane fraction was

found to be 0.8. The cyclic analysis results agree reasonably

well for secondary stress range less than twice yield.

heting plastic strain distribution.

Fig. 7. Shakedown and ratcheting of pipe elbow: effect of start-up/shutdown

cycles.

P. Carter / International Journal of Pressure Vessels and Piping 82 (2005) 27–3332

For higher secondary stress, the structural behaviour of the

elbow is significantly different (less conservative) from the

estimate based on the highest stress.

3.3. Analysis of start up–shut down cycles

For start up–shut down cycles, there is a region within the

shakedown boundary in which the contours of creep strain

accumulation are unaffected by the secondary stress. In the

ASME IIINH solution in Fig. 3, this is the elastic region E.

This is a general feature of start up–shut down cycles. An

estimate of the steady state region is a straight line between

the limit load and the initial yield point on the secondary

stress axis. Confirmation for complex structures is possible

by performing a creep steady state analysis, followed by an

elastic secondary load step. The size of the secondary load

to cause initial yielding is readily determined.

Fig. 7 shows the limit of steady state behaviour for start

up/shutdown cycles, and the shakedown reference stress of

50% of yield. This is a composite of steady state and

shakedown reference stress solutions. The steady state

region has creep contours independent of displacement-

controlled stress. Outside this region the more conservative

shakedown reference stress is used, which is appropriate for

the prediction of creep damage. The physical difference

between the two regions is that outside the steady state

region, the residual stress gets re-set every cycle, increasing

deformation and damage rates rates. The ratcheting

reference stress is a conservative representation of this

effect. At the boundary between the two regions in Fig. 7

there is a discontinuity. This is not necessarily physically

realistic, but reflects the different character of the two

solutions. The use of the less conservative ratcheting

reference stress outside the steady state limit would be

appropriate for strain prediction. This would be best

calculated from a contour passing through the minimum

calculated ratcheting points.

4. Effect of through wall temperature variations on yield

stress

A further factor to be considered in applying the Bree

interaction diagrams to real load cycles, is the effect of

temperature on mechanical properties. To assume that the

highest temperature in the cycle defines the properties to be

used over the complete cycle is an apparently conservative

assumption, which can be misleading. The assumption of a

uniform yield stress, even though it is a minimum value, can

underestimate ratcheting. As noted in Part 1, for the Bree

problem with high thermal stresses and zero or very small

primary stress, ratcheting can occur due to imbalances

brought about by non-uniform yield stress. It can be shown

that a variation of yield stress Dsy through the thickness

affects the elastic core in a manner similar to an additional

cyclic membrane thermal stress of the same magnitude,

Dsy. This can produce ratcheting as described above.

5. Conclusions

(i)

The calculation of shakedown and ratcheting has been

demonstrated for two reasonably complex cyclic

problems.

(ii)

The use of estimated limits and the direct cyclic

analysis technique is important for the efficiency of the

procedure.

(iii)

The results may be used to calculate creep strain

accumulation and rupture for high temperature cyclic

problems.

References

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York; 2003.

[4] ASME Boiler and Pressure Vessel Code, Section VIII Division 2,

ASME, New York; 2003.

[5] BS 5500: British Standard Specification for Fusion Welded Pressure

Vessels, British Standards Institute, London; 1996.

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