ITER
G 74 MA 8 01-05-28 W0.2
ITER STRUCTURAL DESIGN CRITERIA
FOR IN-VESSEL COMPONENTS
(SDC-IC)
APPENDIX C
RATIONALE OR JUSTIFICATION OF THE RULES
ITER
G 74 MA 8 01-05-28 W0.2
TABLE OF CONTENTS
CÊ1000
GENERAL ............................................................................................................................ 1
C 1100
Introduction.......................................................................................................................................1
CÊ1101
Purpose....................................................................................................................................1
C 3000
Design rules for single-layer homogeneous structures .................................................... 1
C 3020
Methods of analysis .....................................................................................................................1
C 3021
Test to determine if nonlinear (finite deformation) analysis is needed ................................1
C 3022
Negligible irradiation-induced swelling test..........................................................................2
C 3024
Inelastic analysis .....................................................................................................................2
C 3024.1
Simplified inelastic analysis...........................................................................................2
C 3024.1.1
Elastic-irradiation-induced-creep analysis............................................................2
C 3024.1.2
Neuber's rule ..........................................................................................................2
C 3024.1.3
Elastic follow-up factor (r) ....................................................................................3
CÊ3050
Negligible thermal creep test.....................................................................................................15
CÊ3100
Low-temperature rules ...................................................................................................... 17
C 3200
Rules for the prevention of M type damage.................................................................... 17
C 3210
Level A Criteria .........................................................................................................................19
C 3211
Immediate plastic collapse and plastic instability ...............................................................19
CÊ3211.1
Elastic analysis (Immediate plastic collapse and plastic instability)..........................19
C 3211.1.1
Primary membrane plus bending stress limit (bending shape factor)................19
C 3212
Immediate plastic flow localization .....................................................................................29
C 3212.1
Elastic analysis (Immediate plastic flow localization)................................................29
C 3212.1.1
Necking ................................................................................................................30
C 3212.1.2
Plastic flow localization ......................................................................................31
C 3213
Immediate local fracture due to exhaustion of ductility......................................................34
CÊ3213.1
Elastic analysis .............................................................................................................35
CÊ3214
Fast fracture .........................................................................................................................36
C 3214.1
Elastic analysis .............................................................................................................36
C 3300
Rules for the prevention of C type damage (Levels A and C) ...................................... 36
CÊ3310
Progressive deformation or ratcheting ......................................................................................36
C 3311
Elastic analysis......................................................................................................................36
C 3311.1
3Sm rule .......................................................................................................................37
C 3311.2
Bree-diagram rule.........................................................................................................37
C 3311.3
Efficiency Diagram rule ...............................................................................................39
C 3312
Elasto-plastic analysis...........................................................................................................48
CÊ3320
Time-independent fatigue..........................................................................................................48
C 3320.1
Historical Perspective...................................................................................................48
C 3320.2
Fatigue damage (Room temperature) ..........................................................................49
C 3320.3
Empirical correlations for fatigue life..........................................................................50
C 3320.4
Irradiation Effects.........................................................................................................51
C 3320.5
Effects of a mean stress or mean strain .......................................................................53
C 3320.6
Multiaxial loading ........................................................................................................54
C 3320.7
Strain cycles and their combination.............................................................................54
C 3320.8
Treatment of defects or geometrical singularities .......................................................56
C 3320.9
Design fatigue curves ...................................................................................................56
C 3320.10
Welded joints .............................................................................................................57
C 3323
Calculation of equivalent strain range De :.........................................................................57
C 3323.1
Elastic Analysis ............................................................................................................57
C 3323.1.1
Use of the cyclic stress-strain curve....................................................................57
C 3323.1.2
Strain distribution .....................................................................................................59
C 3323.1.3
Triaxiality effect ..................................................................................................60
C 3323.1.4
Notches and stress concentrations.......................................................................60
C 3323.1 5
Global cyclic plasticity and elastic follow-up effect ...............................................60
C 3323.1.6
Poisson's ratio effect and the coefficient Kn ......................................................62
C 3323.1.7
C 3400
Simplified procedure and the coefficient Ke ......................................................64
buckling Rules (Levels A, C, and D) ................................................................................ 69
SDC-IC, Appendix C - Rationale or Justification of the rules
page ii
ITER
G 74 MA 8 01-05-28 W0.2
C3410
Introduction.....................................................................................................................................69
C 3411
General Approach ......................................................................................................................69
C 3420
Theoretical basis for the buckling diagrams ..................................................................................70
C 3421
Basis for the construction of the buckling diagrams ................................................................70
C 3422
Stable and unstable post- buckling behavior ............................................................................72
C 3423
Equations for the buckling diagrams ........................................................................................73
C 3424
Influence of temperature on the buckling diagram...................................................................74
C 3430
Validation of the method ................................................................................................................74
C 3500
high temperature rules ...................................................................................................... 76
C 3510
Rules for the prevention of m-type damage.................................................................... 76
C 3521.1
Elastic analysis .............................................................................................................77
C 3530
Rules for the prevention of c-type damage (Levels A and C).......................................................78
C 3531
Progressive deformation or ratcheting ......................................................................................78
C 3532
Summary ...............................................................................................................................78
C 3533
Limits for inelastic strains ....................................................................................................79
C 3534
Elastic analysis rules.............................................................................................................79
C 3534.1
ASME Code (T-1320)..................................................................................................79
C 3534.2
RCC-MR (RB 3262) ....................................................................................................81
C 3535
Simplified Inelastic Analysis Rules .....................................................................................81
C 3535.1
ASME Code (T-1330)..................................................................................................81
C 3535.2
RCC-MR (RB 3262.1.1) ..............................................................................................83
C 3536
Comparison between the ASME Code and RCC-MR ratcheting limits.............................85
C 3537
Conclusions...........................................................................................................................86
C 3600
Rules for welded joints....................................................................................................... 86
C 3700
rules for brazed joints........................................................................................................ 86
C 3800
rules for bolted joints ......................................................................................................... 86
C 3810
CÊ4000:
CÊ4100
Mean stress effects on fatigue ...................................................................................................86
Design Rules for multiLayer Heterogeneous Structures .......................................... 87
Low-temperature rules ...................................................................................................... 87
C 4200
Rules for the prevention of M type damage ..................................................................................87
C 4211
Immediate plastic collapse ........................................................................................................87
C 4211.1
Elastic Analysis (Immediate plastic collapse)..................................................................87
C 4212
Immediate plastic instability .....................................................................................................90
C 4212.1
Elastic Analysis (Immediate plastic instability)...............................................................90
C 4213
Immediate local fracture due to exhaustion of ductility...........................................................93
C 4300
Rules for the prevention of C type damage ...................................................................................93
CÊ4310
Progressive deformation or ratcheting ......................................................................................93
REFERENCES ...................................................................................................................................................94
SDC-IC, Appendix C - Rationale or Justification of the rules
page iii
ITER
G 74 MA 8 01-05-28 W0.2
CÊ1000
C 1100
GENERAL
INTRODUCTION
This document provides the rationale or justification for the design rules presented in SDC-IC
on In-Vessel Components. Some of the rules were borrowed from or based on rules of either
the RCC-MR or the ASME Code, Section III (including code case N47). Because of the long
history and success of these rules, the extent of their justification has been abbreviated. On
the other hand, detailed justifications are provided for rules that are new or are significantly
modified from existing rules.
CÊ1101
Purpose
The purpose for the rationale or justification of the rules is to document background
information, including references to literature, that provided the foundation for the rules
proposed in the SDC-IC.
C 3000
C 3020
C 3021
DESIGN RULES FOR SINGLE-LAYER
HOMOGENEOUS STRUCTURES
Methods of analysis
Test to determine if nonlinear (finite deformation) analysis
is needed
This test is provided in appendix B to warn the designer when large distortions of the
structure, due to creep and irradiation-induced swelling, may make it necessary for the stress
analysis to take into account large displacements (finite deformation) effects.
At elevated temperatures, thermal creep strains, which generally dominate over irradiationinduced creep strains, are limited to 1% (for levelÊA and levelÊC criteria). Therefore, they
usually do not cause large distortions by themselves although together with irradiationinduced swelling strains they still could.
At low temperatures where thermal creep is negligible, irradiation-induced creep and
swelling strains are not limited, because they are considered to be non-damaging. Therefore,
the combined irradiation-induced swelling and creep strains could become large enough to
cause large distortions of the structure. Constrained swelling stresses are limited by the limits
of IC 3211.
In tests B 3021(see SDC-IC, Appendix B), the total operating period is divided into N
intervals of time and the maximum effective strain at end of life is limited to 2% by a linear
summation approach. For each interval i, the effective stress is set equal to the allowable
primary membrane stress intensity Smi (from Table A.MAT.5.1, SDC-IC, Appendix A)
corresponding to the maximum temperature and mean neutron fluence for the interval. It is
assumed that the secondary stresses, being deformation-controlled, are either relaxed by
irradiation-induced creep or do not contribute significantly to the overall distortion of the
structure.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 1
ITER
G 74 MA 8 01-05-28 W0.2
C 3022
Negligible irradiation-induced swelling test
The test in B 3022 of Appendix B is provided to help the designer identify conditions under
which the effects of irradiation-induced swelling can be ignored in the analysis. The total
operating period is divided into N intervals of time and it is determined, by a linear
summation approach, whether the maximum linear swelling strain at end of life is less than
0.017%. Calculated values of the neutron fluences ftsi needed for the test in B 3022 of
Appendix B are given in A.MAT.4.2 of Appendix A.
Irradiation-induced swelling strain accumulations during faulted events (for criteria level D)
are probably negligible, because these events are likely to be fast transients or occur during
postulated accidents when the plasma is off. Even if they are significant, those due to faulted
conditions are not to be included in this test because the reactor will not be able to restart
without inspection after a faulted event.
C 3024
Inelastic analysis
C 3024.1
Simplified inelastic analysis
C 3024.1.1
Elastic-irradiation-induced-creep analysis
C 3024.1.2
Neuber's rule
Neuber analyzed [(Neuber, 1961), (Neuber, 1965)] mode III (anti-plane) shear deformation
of a sharp notch with localized plasticity [(Roche, Aug.Ê1987)] and concluded that the
product of the shear stress t and the corresponding strain g is independent of the material's
behaviuor. The assumption that the plasticity is localized, i.e., the plastic deformation is
embedded well inside an elastically deforming zone, means that the product tg can be
determined from the knowledge of the surrounding stress field and the stress concentration
factor for an elastic material. Under the same conditions, Neuber's rule can be extended to
cyclic loading by replacing shear stress and shear strain by their respective ranges.
Neuber's rule can be generalized to other modes of loading via the concept of the J integral
proposed by Rice [(Rice, 1967), (Rice,1968)]. Detailed analyses [(Hutchinson,1968), (Rice
and Rosengren,1968)] of the stress-strain field at the crack tip have shown that the product of
the stress and strain is proportional to J. Therefore, a generalized Neuber's rule can be
justified and can be expressed as follows:
KsKe = KT2
where
Ks =
actual stress concentration factor
Ke =
actual strain concentration factor
KT =
stress concentration factor for a linear elastic material
The conservativeness of this rule has been verified for geometries containing notches with
finite stress concentrations rather than cracks [(Roche, 1986, 1987, 1989)].
SDC-IC, Appendix C - Rationale or Justification of the rules
page 2
ITER
G 74 MA 8 01-05-28 W0.2
C 3024.1.3
Elastic follow-up factor (r)
Neuber's rule is applicable if the remote stress field away from the notch is elastic. If the
remote stress-strain field itself undergoes plastic deformation, then a further correction is
necessary, because the remote strain is greater than the elastically calculated strain.
This may be illustrated through a simple example. Consider a cylindrical bar of length L,
cross-sectional area A, which is subjected to an axial load such that the extension would be
uel if it behaved elastically. As far as the material's properties are concerned, it is only
necessary to know the Young's modulus E and the stress-strain curve giving e as a function of
s.
There are a number of ways of applying the specified load, the two simplest being a
displacement u = eel.L and a force F = EAeel. imposed at the end (Figure C 3024-1). As long
as the behaviour is linear elastic, a strain eel. is effectively obtained for both loads. When the
behaviour ceases to be linear elastic, the two loadings no longer cause the same strain. For
the imposed displacement loading u, the real strain remains the same as the elastically
calculated strain eel, which means that no correction is necessary and the elastically
calculated stress = E u/L is a pure secondary stress. For the imposed force load, the real
strain corresponds to the real stress = F/A on the stress-strain curve. This stress is a pure
primary stress that can be seen to cause real strain (see Figure C 3024-2) which is much
higher than the elastically calculated eel..
Bar
F
Spring
primary
u
secondary
reality
Figure C 3024-1:
Primary stress, Secondary stress, and reality Ð
Effect of elastic follow-up
SDC-IC, Appendix C - Rationale or Justification of the rules
page 3
ITER
G 74 MA 8 01-05-28 W0.2
stress s
elastic
primary
s el
sN
-arctan [E/(r-1)]
reality
secondary
strain e
eel
Figure C 3024-2:
r e el
eN
Definition of elastic follow up factor, r
These two loads represent special cases. A more general method is to apply an imposed
displacement u to one end of a spring of stiffness K the other end of which is attached to the
bar (this spring represents for example the elasticity of the loading device). This case is more
general since if K is infinite (very stiff spring), it first imparts the displacement load and if K
is very low (extremely flexible spring), it then applies the imposed force load. The
displacement u required to produce a strain eel if the bar were to remain elastic is the sum of
e el
, i.e.:
the extension of the bar Leel and the extension of the spring E A
K
u = L e el r
where
r = 1+
AE
KL
.
When the behaviour is no longer linear elastic (established by noting the bar extension Leel
and the corresponding stress), this extension u should be written L er which is equal to
L e el r since the extension is imposed regardless of the material's behaviour. It follows that:
æ
ö r -1 æ
ö
- e N - e el =
s N - s el
è
ø
ø
E è
The point (eN, sN) representative of the real stress-strain state is the intersection of this
straight line with the stress-strain curve (Figure C 3024-2). It can hence be seen that the
multiplicative factor, which has to be applied to eel in order to obtain the actual strain eN,
depends on r, the so-called elastic follow-up factor.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 4
ITER
G 74 MA 8 01-05-28 W0.2
The stresses are neither purely primary (r = ¥) nor purely secondary (r = 1) but often
intermediary. This is the spring effect quantified by the coefficient r.
r=
Ee N - s N
s el - s N
(1)
where
sel = Eeel = Q.
The above definition is used in the SDC-IC.
NB: The above definition of the elastic follow-up factor is often used in the Japanese
literature, where the terminology is usually q rather than r. The quantity (r-1), with r defined
as above, has been used by Roche. In that case the definition of the elastic follow-up factor
becomes:
r* = r - 1 =
E e N - s el
s el - s N
In the simple example above, the spring is external to the component, i.e., the bar, being
analyzed. In a real component, parts of the component can act as a spring on the rest of the
component. One example is a bar of variable cross-section A(x) subjected to an axial
extension u. Here also,
æ
ö r -1 æ
ö
- e - e el =
s - s el
è
ø
è
ø
E
in each straight section in the x coordinate. The local value of r(x) is r(x) = T / t(x) where
t(x) is a value related to the cross-section characterizing the significance of the non-linearity,
T is a value related to the component which is an average of 1/ t = e p / e el , ep is the nonlinear portion of e corresponding to s el Êin the stress-strain curve, and
1/ T =
ò (1 / t)sel2dx / ò sel2dx
It should be noted that r is positive for the smallest cross-sections in which plasticity
increases the strain, and negative for the largest cross-sections in which it decreases the
strain.
If certain assumptions are made, the preceding expressions (i.e., Eq. 1) can be applied to a
large number of components [(Roche, 1986, AprilÊ1987, AugustÊ1987, 1989)]. The elevated
temperature structural design guide for the demonstration fast breeder reactor of Japan
recommends a conservative default value of r=3 for use in creep-fatigue design. Recently
conducted detailed monotonic and cyclic finite-element inelastic analyses of an support
shroud, a reactor vessel gas-dam wall, and a tube sheet model to investigate the effects of
primary stress and stress-strain law on the elastic follow-up factor r. The analyses confirmed
that, as long as the stresses were within the design allowable limits, the value of r was always
significantly less than 3, generally varying between 1.48 and 2.67.
It is possible to show that in actual cases, 4 is a conservative value of r. For evaluating the
elastic follow-up effect on primary stress, note that Neuber's rule (C 3024.1.3) usually (not
SDC-IC, Appendix C - Rationale or Justification of the rules
page 5
ITER
G 74 MA 8 01-05-28 W0.2
always three-point bend loading being an exception) gives a conservative estimate of the real
strain. Therefore, a conservative value of r can be obtained by equating the peak strain by the
r-factor methodology with that by Neuber's rule (i.e., . s e = s e e e )
s
P+Q
s
e
(P+Q)/E
Figure C 3024-3:
r (P+Q)/E
e
Elastic follow up factor, r
If the stress-strain curve is approximated by a power law,
s = A en,
then solving the following equations, after denoting the elastically calculated primary plus
secondary stress by P+Q (Figure C 3024-3),
s = A en
s e = se ee
P+Q
P + Q ( r - 1) E
=
P+Q
s
-e
r
E
Rearranging the above, one obtains
r=
A
n
n
B +1
-E
A
n
n
B +1
- ( P + Q)
1
n
B +1
,
SDC-IC, Appendix C - Rationale or Justification of the rules
page 6
ITER
G 74 MA 8 01-05-28 W0.2
with
2
P + Q)
(
B=
AE
or
1- n
é (P + Q)
r =1+ ê
A
êë
E
1
n ù n +1
ú
úû
(1a)
Thus, the value of r depends on the loading (P+Q) as well as the stress-strain curve. As it can
be seen from Figure C 3024-4, a conservative (higher than actual) value of r is obtained by
considering an elastic-perfectly-plastic material, corresponding to nÊ=Ê0:
s
P+Q
s(e)
Stress-strain curve,
s
Neuber's Rule, s e = C
Sy
(r' > r )
e
(P+Q)/E
Figure C 3024-4:
e'
r (P+Q)/E r' (P+Q)/E
e
Rigid-perfectly-plastic model gives
conservative value (r' ) of r factor
With this approximation (i.e., n=0), equation (1a) leads to:
A 2 - ( P + Q)
P+Q
r= 2
=1+
A - A ( P + Q)
Sy
2
If we use an upper limit of (PÊ+ÊQ) as 3Sm (=Ê2.7ÊSy),
=>r = 3.7
SDC-IC, Appendix C - Rationale or Justification of the rules
page 7
ITER
G 74 MA 8 01-05-28 W0.2
Analytical solutions for elastic-plastic behaviour of a few simple geometries are currently
available. Among these is a cantilevered beam subjected to a bending load at its free end
(Figure C 3024-5). The interest in such a geometry is that first, it exhibits a marked elastic
follow-up effect, more so than in real components, and thus provides an upper bound to the
elastic follow-up factor r. A second advantage is in the expression for r when the stress-strain
curve is expressed by the Ramberg-Osgood relation:
æ 1ö
e = +ç ÷
E ç A÷
è ø
1
s
n
s
1
n
(2)
For the fitted section,
r = (1+2n)/3n,
where
1/n is the exponent of the stress, giving r = 2 for n = 1/4.
It is evident that the elastic follow-up effect is highly dependent on the material's strain
hardening behaviour, being low for highly strain hardening materials (large n) and high for
low strain hardening materials (small n). In particular, r=¥ if n=0 (i.e., elastic-perfectly
plastic). This is of concern to ITER because, annealed type 316 austenitic stainless steel
loses its strain hardening capability with neutron irradiation and displays an almost elasticperfectly plastic stress-strain curve at a relatively modest neutron fluence (Figure C 3024-6).
Figure C 3024-5:
Cantilevered beam
SDC-IC, Appendix C - Rationale or Justification of the rules
page 8
ITER
G 74 MA 8 01-05-28 W0.2
900
6.8 dpa/220 appm He
800
5.1 dpa
140 appm He
700
Stress (P/Ao), MPa
600
0 dpa
500
400
300
3.1 dpa
17 appm He
10.9 dpa
85 appm He
200
100
0
0
5
10
15
20
25
30
35
Strain (DL/Lo), %
Figure C 3024-6:
Typical variation of the uniaxial stress-strain curve of Type
316 L(N)-IG stainless steel with neutron fluence
(displacement dose).
A second example which also shows larger elastic follow-up than real structures and for
which analytical solutions exist is the three-point bending of a beam subjected to a given
transverse displacement at the center. It is of particular interest, because failure data for such
tests on irradiated type 304 stainless steel are available. It is also sufficiently simple so that a
closed form solution can be obtained for a bilinear stress-strain curve (Figure C 3024-7) with
tangent modulus ET, yield stress Sy and yield strain ey. A plot of the computed variation of
the elastic follow-up factor r with the peak plastic strain (assuming ey = 0.4% for a typical
irradiation-hardened stainless steel) is also shown in Figure C 3024-8.
s
ET
Sy
E
ey
Figure C 3024-7:
e
Bilinear stress-strain curve
SDC-IC, Appendix C - Rationale or Justification of the rules
page 9
ITER
G 74 MA 8 01-05-28 W0.2
Three-Point-Bend Loading
10
E /E=0
T
8
0.005
6
r
0.01
4
0.05
2
0
0
1
Figure C 3024-8:
2
3
Max eP (%)
4
5
Effect of strain hardening (tangent modulus) on the r-factor
variation with maximum plastic strain for 3-point bend
tests
Several points should be noted:
-
r is a function of both the strain-hardening and the load level
-
if ET/E > 0, r first increases with loading and then, following a maximum, it
decreases
-
the higher the value of ET/E, the lower the maximum value of r
-
if ET/E = 0, r increases indefinitely with loading
-
if the peak plastic bending strain were limited to 2%, the maximum value of r » 5
-
for unirradiated annealed stainless steel (ET/E » 0.05), maximum value of r = 3
A third example for which an analytical expression for r-factor can be established
approximately by using Neuber's rule (C 3024.1.3) is the tensile loading of a notched bar
with a stress concentration factor KT. Substituting the expressions for peak elastic strain
KTeo and peak plastic strain from Eq. 2(b) of Appendix B 3024.1.3 into the expression for r
(Eq. 3 of Appendix B 3024.1.4), it can be shown that
1- n
r = (K T )1+ n
(3)
Three-point bend tests
A number of three-point bend tests on type 304 stainless steel irradiated in EBR-II were
conducted by Garkisch, Fish, and Haglund (1977). The specimens were 0.5 in wide by 0.04
SDC-IC, Appendix C - Rationale or Justification of the rules
page 10
ITER
G 74 MA 8 01-05-28 W0.2
in thick with a span of 1.5 in. The irradiation temperature and fast neutron fluence varied
from 735¡F to 835¡F and 3.9x1022 to 8.6x1022 n/cm2 (E>0.1 MeV), respectively. The test
temperature and cross head speed varied from 900¡F to 1150¡F and 0.002 to 0.2 in/min,
respectively. A summary of the measured collapse loads (P) and displacements (d) together
with tensile properties data from companion specimens are shown in Table C 3024-1. The
elastic stresses were calculated from the applied displacements using beam formula and the
elastic follow-up factor was determined using Eq. 1 above after correcting the true strain at
rupture (etr) with a plane strain triaxiality factor of Ö3. Since the strain hardening is small,
the use of Su instead of Str (true stress at rupture) in Eq. 1 should underestimate the r values
slightly.
Notched tensile tests
A few tensile tests on notched specimens (displaying notch-weakening, i.e., Su(notched)/Su <
1) of type 304 stainless steel irradiated in EBR-II were conducted by Fish (1976). The 0.04
in thick specimens with a gauge section of 0.75 in long by 0.125 in wide had 0.03 in deep
symmetrical 30¡ single-edge V-notches with a root radius of 0.003 in. The theoretical elastic
stress concentration factor was estimated as KT = 4. The irradiation temperature and fast
neutron fluence varied from 700¡F to 735¡F and 5.1x1022 to 9.2x1022 n/cm2 (E>0.1 MeV),
respectively. The test temperature was 1100¡F and the strain rate varied from 0.0027 to
2.7/min. A summary of the notched tensile test data together with tensile properties data
from companion smooth specimens are shown in Table C 3024-2. Compared to the smooth
specimens, the notched specimens showed significantly lower yield strength, ultimate
strength and uniform elongation, and exhibited very little total elongation beyond uniform
elongation and practically no hardening beyond initial yield (Sun » Syn). The r-factors for the
notched specimens were calculated with the assumption that failure was initiated at the notch
root as soon as the peak load was reached and, for that purpose, Eq. 1 was specialized as
follows:
r=
e tr
- Su
TF
K T ( Ee un + Sun ) - Su
E
(4)
where
E
= 19000 ksi
KT
= 4
TF
= 2.7
Su
= Ultimate tensile strength of smooth specimen
etr
= True strain at rupture of smooth specimen
Sun = Ultimate tensile strength of notched specimen
eun
= Uniform elongation of notched specimen
For comparison, the r factors have also been calculated analytically using Eq. 3 with n=eu for
the smooth specimens.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 11
ITER
G 74 MA 8 01-05-28 W0.2
Table C 3024-1 Three-point bend tests
P (Lbs)
d (in)
Sy (ksi)
Su (ksi)
E (ksi)
eu (%)
etr (%)
sel (ksi)
r
32.0
0.058
73.1
73.5
18300
0.2
13
113
34.6
44.5
0.076
85.0
88.0
19000
0.4
12
154
19.9
57.5
0.161
87.6
90.2
17200
0.6
54
295
26.1
46.0
0.133
71.0
73.5
19200
0.7
17
272
9.5
64.0
0.125
76.4
86.2
17300
0.8
34
231
23.5
54.0
0.136
77.3
80.2
17900
0.9
22
260
12.7
40.5
0.111
55.3
58.4
16400
1.3
20
194
13.9
Table C 3024-2 Smooth and notched tensile tests
Test type
%RA
eu (%)
etr (%)
Sy (ksi)
Su (ksi)
Smooth
6
0.04
6.2
-
56
Notched
5
0.04
5.1
-
36
Smooth
27
0.5
31.5
62
76
Notched
10
0.2
10.5
69
69
Smooth
29
0.5
34.2
60
75
Notched
17
0.4
18.6
70
72
r
3.2
6.1
4.5
The r-factors for both types of tests are plotted against the uniform elongation eu of the
smooth tensile specimens in Figure C 3024-9, the uniform elongation being a measure of the
strain hardening capability of the material. Note that the r-factors calculated from the
notched tensile test data do not increase with decreasing uniform elongation and agree well
with those predicted by Eq. 3. Although the number of 3-point bend test data is rather
limited, they fall within a relatively wide scatter band and follow the trend (solid line) of the
analytical prediction of reduced r-factor with increasing strain hardening (Figure C 3024-8).
To compare the analytical predictions with experiments, the tangent modulus for each threepoint bend test was first determined by a correlation of the measured displacement at collapse
and the true strain at rupture with the calculated maximum displacement and peak plastic
bending strain. Except for a couple of tests, the calculated r-factors compare very well with
the experimentally determined r-factors (Figure C 3024-10). Although the calculated fracture
loads do not compare as well (Figure C 3024-11), in most cases they are within 50% of the
experimentally measured values .
SDC-IC, Appendix C - Rationale or Justification of the rules
page 12
ITER
G 74 MA 8 01-05-28 W0.2
The r = r2 value for the peak stress (PL+Pb+Q+F) limit of the Sd rule (IC3213.1) is
conservatively set equal to the larger of 4 and KT, the elastic stress concentration factor.
r2 = Max {KT and 4}
(5a)
To be conservative, the value of r to be used with the Se rule (IC 3212.1) and PL+Pb+Q limit
of the Sd rule (IC 3213.1) are set equal to ¥ whenever the uniform elongation of the material
drops below 2%. For higher uniform elongations, i.e., eu >2%, r is set equal to 4.
ì¥ for e u £ .02
ï
r1 or r3 = í
ïî 4 for e u > .02
[2%]
(5b)
Elastic Follow-up Factor, r
100
3-point bend tests
Notched tensile tests
10
Predicted (Eq. 3)
1
0
Figure C 3024-9:
0.5
1
e u (%)
1.5
2
Calculated r-factors from 3-point bend tests and notched
tensile tests on irradiated type 304 stainless steel
SDC-IC, Appendix C - Rationale or Justification of the rules
page 13
ITER
G 74 MA 8 01-05-28 W0.2
35
Experimental r
30
25
20
15
10
5
5
Fracture Load (Lbs)
Figure C 3024-10:
10
15
20
Calculated r
25
30
Comparison of r-factors calculated from experiments with
those calculated analytically for 3-point bend tests.
100
90
80
70
60
Experimental
Calculated
50
40
30
20
0.04
Figure C 3024-11:
0.06
0.08
0.1
0.12 0.14
Displacement at Fracture (in)
0.16
0.18
Comparison of experimentally measured fracture loads and
displacements s at fracture with those calculated
analytically
SDC-IC, Appendix C - Rationale or Justification of the rules
page 14
ITER
G 74 MA 8 01-05-28 W0.2
Thermal bending and peak stresses
Detailed inelastic analysis of a plate (constrained to remain flat) subjected to a linear
temperature gradient through the thickness shows that r=1, irrespective of the strain
hardening of the material.
Thermal peak stress plays a role in enhancing the elastic follow up factor r and has been
studied analytically on the assumption of a power law stress-strain curve (Kasahara , 1993).
Results indicate that the value of r due to thermal peak stress and the accompanying Poisson's
ratio effect is bounded by r £ 5/3.
These values of r are significantly less than the default value of r that has been adopted for
the SDC-IC.
CÊ3050
Negligible thermal creep test
The design rules of SDC-IC have been divided into two separate categories depending on
whether or not the effects of thermal creep are negligible. The justification for such a
separation is derived from a general consensus in the materials community that although
creep strain at elevated temperature, whether accumulated in-pile or out-of-pile, is damaging,
low-temperature irradiation-induced creep and swelling strains are generally non-damaging
(Nichols, 1970, Hesketh, 1972). Such a separation between high temperature and low
temperature design rules also exists in both RCC-MR and the ASME Code. Irradiationinduced creep at low temperature is non-damaging for the following reasons.
-
Because of the high strain-rate sensitivity of irradiation-induced creep, necking
and onset of tertiary creep are delayed. Thus, although the uniform elongation of
a typical irradiated stainless steel in a tensile test may be reduced to <1%, it can
exhibit >5% creep strain (without rupture) during in-pile creep testing.
-
At low temperatures, defects, such as He bubbles, produced by neutron irradiation
are immobile and cannot migrate to grain boundaries to cause embrittlement.
-
At high temperatures, He bubbles are mobile and can migrate to the grain
boundaries. Together with cavitation damage due to grain boundary sliding they
can cause embrittlement of stainless steels.
The negligible thermal creep test proposed in IC 3050 requires the specification of time tc as
a function of temperature where tc is the time necessary to accumulate a thermal creep strain
of 0.05% at a stress of 1.5Sm (which is the value limiting the primary membrane-plusbending stress intensity in ICÊ3211.1.1, level A criteria) at a given temperature. tc is easily
obtained for out-of-pile creep tests. However, it is not as straightforward for in-pile creep
tests, because although thermal creep tends to dominate at high temperature and high stress
and irradiation-induced creep tends to dominate at low temperature and low stress, there is an
intermediate stress-temperature regime where both creep strains are important. In order to
separate the thermal creep component from the total experimentally measured creep strain in
this intermediate regime, one needs to first develop a constitutive model for in-pile creep
strain (Jung and Ansari, 1986, Coghlan and Mansur, 1983, Ehrlich, 1981, and Hudson et al.,
1977). Most constitutive models for in-pile creep of austenitic stainless and ferritic steels are
in the following general form (Hesketh, 1972 and Laidler, 1978):
e = B(ft, T)f (s ) + A( t, T )g(s )
(1)
where
SDC-IC, Appendix C - Rationale or Justification of the rules
page 15
ITER
G 74 MA 8 01-05-28 W0.2
t, ft and T are time, neutron fluence, and temperature, respectively, s and e
are effective stress and effective creep strain, respectively. The functions f
and g can often be approximated by power functions as follows:
f (s) = s n and g(s ) = s m
In Eq. (1), the first term on the right hand side represents the creep strain component driven
by neutron fluence and temperature (irradiation-induced) and the second term represents the
component driven by time and temperature (thermal). As an example, the constitutive model
adopted here is similar to that adopted for the US primary candidate alloy (PCA), which is a
cold-worked Nb-stabilized type 316 austenitic stainless steel. The time required to
accumulate a total creep strain of 0.05% as functions of stress and temperature is plotted in
Figure C 3050-1. Note that, for any given temperature, there are three distinct regimes. At
high stress and short time, deformation due to thermal creep dominates while at low stress
and long time deformation due to irradiation-induced creep dominates. In a small transition
region, both mechanisms contribute. At £ 400¡C, thermal creep is negligible and irradiationinduced creep is relatively insensitive to a variation in temperature. It is also in this region
that irradiation-induced creep is claimed to be non-damaging. At higher temperatures (T ³
500¡C), irradiation-induced creep is damaging.
104
Stress (MPa)
Thermal Creep
Dominated
Irradiation-Induced
Creep Dominated
103
102
101
10-1
Figure C 3050-1:
700¡C
600¡C
500¡C
400¡C
100
101
102
Time (h)
103
104
Calculated time to accumulate 0.05% combined thermal
and irradiation-induced creep strain in the US primary
candidate alloy (PCA) as a function of stress and
temperature and at a displacement rate of 3.15x10-3 dpa/h.
In Figure C 3050-2 are plotted the times necessary to accumulate 0.05% creep strain at a
stress of 1.5Sm as a function of temperature when the creep strain is induced by irradiation
(fluence) alone (dashed line), by thermal creep alone (dashed line) and by combined thermal
and irradiation-induced creep (solid line). As a conservative estimate for tc, either the solid
bold line of Figure CÊ3050-2 or a suitable smoothed out lower bound curve should be used.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 16
ITER
G 74 MA 8 01-05-28 W0.2
Note that the transition from a thermal creep-dominated regime to an irradiation-creep
dominated regime is dependent on the neutron flux. Thus, tc may also depend on the neutron
flux.
800
Irradiation-induced creep
700
Thermal creep
T (¡C)
600
500
400
300
200
t
c
Combined thermal and
irradiation-induced
creep
10-1
Figure C 3050-2:
100
101
102
103
Time (h)
104
105
106
Calculated time (tc) to accumulate a creep strain of 0.05%
at a stress of 1.5Sm at various temperatures.
A curve similar to that for combined thermal and irradiation-induced creep of Figure C 30502 should be used in the negligible irradiation-induced creep test of B 3101 of Appendix B. In
the absence of combined thermal and irradiation-induced creep data, the curve should be
derived from unirradiated thermal creep data.
Curves for negligible thermal creep tests are provided in A.MAT.4.1 of Appendix A for the
materials, temperature and neutron flux of interest to ITER.
CÊ3100
LOW-TEMPERATURE RULES
C 3200
RULES FOR THE PREVENTION OF M TYPE
DAMAGE
The rules of this article are aimed at providing sufficient safety margins with regard to M
type damage (IC 2110), excluding buckling phenomenon (IC 2130) and excessive
deformation affecting functional adequacy (IC 2140). The rules to prevent buckling and
excessive deformations affecting functional adequacy are dealt with in IC 3400 and IC 3040,
SDC-IC, Appendix C - Rationale or Justification of the rules
page 17
ITER
G 74 MA 8 01-05-28 W0.2
respectively. The main differences between these rules and those given in RCC-MR and the
ASME Code are the followings:
-
Mechanical properties (e.g., Sm, Su, eu, KJc) are fluence dependent
-
Additional damage modes occur due to irradiation effects (e.g., loss of ductility,
reduction in uniform elongation, loss of strain hardening capability, and loss of
fracture toughness)
-
Stresses due to constrained swelling may have to be considered
The safety margins included in the different rules are, at least, the ones imposed on pressure
vessel components in RCC-M rules (Roche, 1991). For the cracking and fast fracture failure,
the safety factors are chosen to be higher for the membrane failure mode (where the driving
force for cracking may be undiminished by crack growth) than other failure modes in which
the driving force for crack growth is significantly reduced with crack extension. The safety
factors are as follows:
- Level A:
Plastic collapse 1.5
Plastic instability and rupture 2.5
Cracking and fast fracture
Global (membrane loading) 3.0
Local (total loading including peak) 1.5
- Level C:
Plastic collapse 1.2
Plastic instability and rupture 2.
Cracking and fast fracture
Global (membrane loading) 2.5
Local (total loading including peak) 1.25
- Level D:
Plastic instability and rupture 1.1
Cracking and fast fracture
Global (membrane loading) 1.5
Local (total loading including peak) 1.1
Note:
1. A margin of 2.5 against rupture requires a factor 3 on Su . Indeed, Langer (1964)
has shown that this value ensures a safety margin of 3.2 for carbon steel cylinders,
but only 2.5 for austenitic spheres.
2. For elastic analysis, limits in level C are those of level A multiplied by 1.2. This
value is lower than RCC-MR (1.35), or PNEAG-7-002-86 (1.4) but equal to that
in ASME Code Case N47 (1.2) or MONJU Guide (1.2).
SDC-IC, Appendix C - Rationale or Justification of the rules
page 18
ITER
G 74 MA 8 01-05-28 W0.2
C 3210
Level A Criteria
At level A, the following rules for protection against M-type damage are applicable:
-
immediate plastic collapse (IC 3211)
-
immediate plastic instability and strain localization (IC 3211)
-
local fracture due to exhaustion of ductility (IC 3212)
-
fast fracture (IC 3213)
C 3211
Immediate plastic collapse and plastic instability
CÊ3211.1
Elastic analysis (Immediate plastic collapse and plastic
instability)
C 3211.1.1
Primary membrane plus bending stress limit (bending
shape factor)
Concept of Effective Bending Shape Factor
The usual limit on the combination of primary membrane and bending stress, used in the
RCC-MR and the ASME Code is given by the following two curves in conjunction:
Pm < Sm
(1a)
PL + Pb < K Sm
(1b)
where K is the bending shape factor. The second equation has been replaced in the SDC-IC
by a more conservative limit to account for possible embrittling of the material with
irradiation, expressed as :
P L + Pb £ K eff Sm
(1c)
where
the bar denotes the stress intensity of the sum of the membrane and bending
stress tensors,
PL
is the primary local (or general) membrane stress, as appropriate,
and
Keff
is an effective bending shape factor, which accounts for the
increased maximum bending moment carrying capability of an
elastic-plastic material with limited ductility as compared to that of
an elastic-brittle material. For unirradiated materials with unlimited
ductility, Keff = K, where K is the usual bending shape factor used in
the RCC-MR and the ASME Code. For irradiated materials with
limited ductility, an expression for Keff is derived below, based on
maintaining a consistent safety factor.
Both the ASME Code and RCC-MR allow the maximum elastically calculated primary
bending stress to exceed the Sm limit by a bending shape factor K, which depends on the
SDC-IC, Appendix C - Rationale or Justification of the rules
page 19
ITER
G 74 MA 8 01-05-28 W0.2
cross-sectional shape. K is equal to 1.5 for a solid rectangular section. The basis for this
factor is the limit analysis in pure bending of a material that is rigid-perfectly-plastic. The
analysis shows that the maximum bending moment carrying capability of this material is
higher by a factor K compared to that of an elastic-brittle material whose strength is equal to
the yield strength (or an elastic-plastic material at the onset of yield) of the rigid-perfectlyplastic material. The same factor also applies to an elastic-perfectly plastic material provided
the material has a high ductility - a condition which is always satisfied by the materials
considered in these codes. Both codes use this factor K together with the allowable primary
membrane stress intensity, Sm, in limiting the combined primary membrane plus bending
stress intensity using Eq. (1b).
The safety factor (SF) associated with the Eq. (1b) has two components. First, there are the
explicit safety factors of 3 on ultimate tensile strength and 1.5 on yield strength in the
definition of Sm. Second, there is a component in the SF which arises implicitly because of
applying the bending shape factor K for a rigid-perfectly-plastic material to a material that is
work hardening. The existing ASME code or RCC-MR rule for Pm vs Pb interaction
implicitly assumes a SF = 3.0 for pure tension. The SF in tension is higher when Sm is
controlled by Sy and the material is highly work hardening. In a similar fashion, the SF in
pure bending is 3.0 for the case of a zero work-hardening material upon which the definition
of the bending shape factor "K" in the codes is based.
For a work hardening material, when the original K = 1.5 is used, the implicit safety factor in
pure bending may be less than or greater than 3, depending on the work hardening and, by
implication, on whether Sm is controlled by Su or Sy. As will be shown below, a minimum
safety factor of 2.50 occurs when Su = 2 Sy.
An implicit assumption in both the ASME Code and RCC-MR is that the material has a
sufficiently high ductility to achieve the full potential increase in its bending moment
carrying capability because of plasticity effects. Although this assumption is reasonable for
the out-of-core components of fission reactors for which these codes were intended, it may
not necessarily be so for SDC-IC, because neutron irradiation has a significant influence on
work hardening and ductility of the material. The approach taken in the SDC-IC is to adopt a
similar equation (Eq. 1c) for allowable primary membrane plus bending stress as in the
ASME Code and RCC-MR, but replace K by a Keff such that comparable SFs are maintained.
In SDC-IC, Keff is assumed to be dependent on neutron fluence and temperature in addition
to the cross-sectional shape and is determined on the basis of the following assumptions.
(1) Safety factors for bending of irradiated materials as a function of work hardening
are used that are higher than those that are implicit in the ASME Code and RCCMR for unirradiated materials and reduce to them in the limit of zero neutron
fluence. However, the safety factors for bending are comparable to those for the
membrane case.
(2) No advantage is taken of a higher allowable (Sm) due to irradiation effects. This
procedure results in an additional safety factor in Sm for materials that harden due
to irradiation. However, the allowable stresses are lowered if the materials soften
during irradiation.
(3) The safety factors are evaluated assuming a bilinear stress-strain law. It is
assumed that the additional primary bending moment capability due to
deformations beyond the uniform elongation at ultimate tensile strength is small.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 20
ITER
G 74 MA 8 01-05-28 W0.2
(4) The derivations are carried out for a solid rectangular section and generalized for
other cross-sections by analogy.
Safety factors implicit in the ASME Code and RCC-MR
The maximum bending moment carrying capability of any material for a solid rectangular
section of unit width can be obtained by noting that the stress distribution through the section
(Figure C 3211-1) at the point of maximum bending moment is as shown in Figure C3211-1.
Su
z
h
2
Sy
gh
2
s
Figure C 3211-1:
Idealized stress distribution through the thickness at the
point of maximum bending moment
The maximum bending moment is given by
M max =
M max
h/2
ò- h / 2 s(z)zdz
Sy ( T, ft )h 2 é g 2 b - 1
ù
=
+
(1 - g )(2 + g )ú
ê1 4
3
3
ë
û
(2a)
and the corresponding maximum elastic bending stress Pb,max is given by
Pb,max =
Sy ( T, ft )
2
[3 - g 2 + (b - 1)(1 - g )(2 + g )]
SDC-IC, Appendix C - Rationale or Justification of the rules
(2b)
page 21
ITER
G 74 MA 8 01-05-28 W0.2
where
g = g ( T, ft ) =
b = b( T, ft ) =
e y ( T, ft )
e max ( T, ft )
,
Su ( T, ft )
,
Sy ( T, ft )
(2c)
(2d)
T is the temperature, and
ft is the fluence.
Since the tangent modulus must be less than the Young's modulus,
0 £ g £ 1/b £ 1
(2e)
The uniform elongation is given by
eu = emax - Su/E
(2f)
where E is the Young's modulus.
Unirradiated materials
For an unirradiated material with high ductility (which is the case for the ASME Code and
RCC-MR), g=0. Denoting the unirradiated values of the parameters at temperature by a
subscript T and those at room temperature by a subscript 0, for a rectangular section Eq. (2a)
reduces to
M max =
(1 + 2b T )
SyT h 2 ,
12
(3a)
and Eq. (2b) gives the maximum elastic bending stress in a rectangular section corresponding
to the bending moment Mmax as
Pb,max =
(1 + 2b T )
SyT
2
(3b)
Since the allowable bending stress is KrectSm, the safety factor implied by the ASME Code
and RCC-MR is given by (emphasizing the fact these are under unirradiated condition)
SF =
(1 + 2b T )SyT
2 K rect SmT
(4)
with Krect=1.5
SF can be shown to be a function of the work hardening parameter b only, as follows. For
unirradiated materials, SmT is defined by
1
1
2
SmT = min éê SuT , aSyT , Su 0 , Sy 0 ùú
3
3
ë3
û
SDC-IC, Appendix C - Rationale or Justification of the rules
page 22
ITER
G 74 MA 8 01-05-28 W0.2
There are four distinct possibilities for SmT , depending on the strength ratios and amount of
work hardening. They are as follows:
SmT
ì
ï
ï
ï
ï
ï
=í
ï
ï
ï
ï
ïî
SuT
3
if b T £ 3a,
bT £ b0
aSyT
if 3a £ b T ,
3a £ b 0
Su 0
3
2
Sy 0
3
if b 0 £ b T
SyT
Sy 0
SyT
,
if 2 £ b T
Sy 0
, b 0 £ 3a
Sy 0
SyT
Sy 0
SyT
SyT
Sy 0
SyT
2 £ 3a
,
Sy 0
, and b T £
, and 3a £
2 Sy 0
SyT
2 Sy 0
SyT
(5a)
, and b0 £ 2
and 2 £ b 0
where
ì0.9 for austenitic stainless steels
ï
a = í2
ï for all other materials
î3
(5b)
The safety factors for a rectangular section are obtained by substituting Eq. (5a) in Eq. (4).
Typical safety factor values for unirradiated type 316L(N)-IG austenitic stainless steel at
100¡C and at 250¡C are shown in the following table for both pure membrane and pure
bending loadings. Note that the room temperature yield and the yield at temperature control
the values of Sm at 100¡C and 250¡C, respectively.
Case
1
2
S mT
SF Equation
for bending
Unirradiated
state
controlled by
Subst. (5a) into
(4)
parameters
1
SuT
3
1 + 2b T
bT
bT=3
aSyT
1 + 2b T
3a
bT = 3
3
1
Su 0
3
1 + 2b T SyT
b 0 Sy 0
4
2
Sy 0
3
1 + 2b T SyT
Sy 0
2
Table C 3211-1
a = 0.9
Calculated
SF for 316 type SS
Membrane
Bending
3.4
at 250¡C
2.6
at 250¡C
3
at 100¡C
2.4
at 100¡C
b0 = 2.4
bT = 2.5
SyT/Sy 0 = 0.8
bT = 2.5
SyT/Sy 0 = 0.8
Typical implicit safety factors for 316L(N)-IG
based on Equations (4) and (5a)
SDC-IC, Appendix C - Rationale or Justification of the rules
page 23
ITER
G 74 MA 8 01-05-28 W0.2
Thus, for 316L(N)-IG, the safety factors for pure bending and for pure membrane implicit in
the ASME Code and RCC-MR are comparable (~3). But they are not a constant because
they depend on the work hardening as well as the yield stregth ratios. For stainless steel, the
safety factor in bending varies from 2.4 at 100¡C to 2.6 at 250¡C. The corresponding
numbers in pure membrane are 3.0 and 3.4, respectively.
In contrast, the minimum theoretically possible safety factors in the four regimes (considering
the inequalities in Eq. (5a)), for a hypothetical material, are as shown in Table C 3211-2.
Case
1-4
S mT
SF Equation
for bending
Unirradiated
state
controlled by
Subst. (5a)
into (4)
parameters
Membrane
Bending
as above
as above
b0 = 2
bT=2
3
2.5
3
2.37
(inconsistent
with actual SS
props)
Calculated
Minimum Possible SF
a = (2/3)
SyT/Sy 0 = 1
2a
aSyT
1 + 2b T
3a
Table C 3211-2
bT = 2.7
a = 0.9
Theoretically possible minimum safety factors based on
Equations (4) and (5a)
In most cases, the minimum safety factor in tension is 3 and the minimum in bending is 2.5.
The minimum in bending occurs at bT = 2. This is the point of transition between the regime
where Su controls Sm to the regime where Sy controls Sm. When a = 0.9, the transition and
the minimum SF occurs at 2.7. In this case, however, the minimum safety factor is
inconsistent with actual steel properties, which are the basis for letting a = 0.9, and so it may
be disregarded.
Determination of safety factors for SDC-IC
To maintain a specified safety factor (SF) against the maximum bending moment carrying
capability of the cross-section, the calculated bending stress must be related to the limit (Eq.
2b) through
Pb £
Pb,max
(6a)
SF
or, substituting Eq.(2b) into Eq.(6a), for rectangular sections,
Pb £
Sy ( T, ft )
2SF
[3 - g 2 + (b - 1)(1 - g )(2 + g )]
SDC-IC, Appendix C - Rationale or Justification of the rules
(6b)
page 24
ITER
G 74 MA 8 01-05-28 W0.2
where b and g are evaluated at (T, ft).
At the same time, to maintain consistency with the terminology used in the ASME code and
RCC-MR for the limit for pure primary bending, we arbitrarily express the limit in the form
Pb £ K eff , rect Sm ( T, ft )
(6c)
Equations, (6b) and (6c) are exactly equivalent, with the same safety factor, if we define
K eff , rect º
Sy ( T, ft )
3 - g 2 + (b - 1)(1 - g )(2 + g )]
[
2 SF Sm ( T, ft )
(6d)
where b and g are evaluated at (T, ft).
It is clear from Eq. (6d) that the determination of Keff is intimately connected with the safety
factor (SF) used. The safety factor for the irradiated materials can be defined in various
ways, all reducing to Eqs. (4) or (5b) in the limit of zero fluence. A natural candidate for SF
is a generalization of Eq. (4), defined for a rectangular section as follows:
SF =
[1 + 2b( T, ft )]Sy ( T, ft )
2 K rect Sm ( T, ft )
(7a)
Determination of Keff
a. Keff for rectangular cross section
Substituting Eq. (7a) into Eq. (6d),
é 3 - g 2 + (b - 1)(1 - g )(2 + g ) ù
K eff , rect = K rect ê
ú
(1 + 2b)
ë
û
(7b)
where b and g are evaluated at (T, ft).
Since b ³ 1,
1 £ Keff,rect £ Krect.
(7c)
Equation (7a) is used to calculate Keff for both unirradiated and irradiated material. The
parameters b and g are based on the state (T, ft) at the time in question, regardless of what
conditions control Sm. The variation of Keff,rect as a function of b and g is shown in Figure
CÊ3211-2 below.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 25
ITER
G 74 MA 8 01-05-28 W0.2
1.6
1.5
b=4
1.3
K
eff,rect
1.4
b=3
1.2
b=1
b=2
1.1
1
0
0.2
Figure C 3211-2
0.4
0.6
g
0.8
1
Variation of Keff,rect with the ductility parameter g .
b. Keff for cross-sections other than solid rectangular
The effective bending shape factor (Keff ) for an arbitrary cross section, for a material with
work hardening and reduced ductility, may be obtained by the following equation:
(
)
K eff = 1 + 2 (K - 1) K eff , rect - 1
(8)
where
K
is the usual bending shape factor for the same cross section,
derived on the basis of an elastic-perfectly plastic material, and
Keff, rect is the effective bending shape factor for a rectangular cross section
for a material with work hardening and reduced ductility, given by
Eq.Ê(7b).
Eq. (8) was selected arbitrarily as one that maps Keff, rect continuously into Keff and which has
the correct limiting behaviour,
1 < Keff < K ,
and
Keff = Keff, rect
when K = Krect = 1.5
More precise derivations for specific shapes would be possible, but the additional algebraic
complexity does not appear to be warranted.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 26
ITER
G 74 MA 8 01-05-28 W0.2
Specific values of K to use for certain common cross sections are given in IC 3211.1.1.
Safety factors implied by Eq. (7a) for irradiated materials
For irradiated materials, Sm is defined by
1
1
2
1
2
Sm ( T, ft ) = min éê Su ( T, 0), aSy ( T, 0), Su ( RT, 0), Sy ( RT, 0), Su ( T, ft ), Sy ( T, ft )ùú
3
3
3
3
ë3
û
where a is defined in Eq. (5b). Denoting the values of the various unirradiated parameters at
temperature T and at room temprature(RT) by subscripts T and 0, respectively, and the values
of the various irradiated parameters at temperature T and fluence ft by the parameters
without any subscript, Sm can be rewritten as in Eq. 5a as follows:
ì
ï
ï
ï
ï
ï
ï
ïï
Sm = í
ï
ï
ï
ï
ï
ï
ï
ïî
SuT
3
if b T £ 3a,
bT £ b0
aSyT
if 3a £ b T ,
3a £ b 0
Su 0
3
2
Sy 0
3
Su
3
2
Sy
3
if b 0 £ b T
SyT
Sy 0
SyT
if 2 £ b T
,
Sy 0
SyT
if b £ b T
,
Sy
SyT
,
if 2 £ b T
Sy
, b 0 £ 3a
Sy 0
SyT
Sy 0
SyT
SyT
Sy 0
SyT
2 £ 3a
,
Sy 0
SyT
b £ 3a
,
Sy
SyT
,
2 £ 3a
Sy
, bT £
, 3a £
2 Sy 0
SyT
2 Sy 0
SyT
, b 0 £ 2,
, bT £ b
, 3a £ b
b0 £ b
Sy
SyT
Sy
SyT
Sy
, and b T £
, and 3a £
,
Sy 0
Sy
2 £ b0 ,
2£b
,
Sy 0
Sy 0
2 Sy 0
, b£
,
b £ b0
Sy
Sy
Sy 0
Sy 0
, 2 £ b0
,
2 £ b0
Sy
Sy
and b 0 £
and 2 £
2 Sy
SyT
2 Sy
SyT
2 Sy
Sy 0
2 Sy
(9)
Sy 0
and b £ 2
and 2 £ b
There are six distinct possibilities for Sm , depending on the strength ratios, amount of work
hardening and on whether irradiation increases or decreases Sy or Su. For each of these cases,
we can substitute the appropriate value of SmÊ from Eq. (9), as well as Krect = 1.5, into the
expression for the safety factor, Eq. (7a). Table C 3211-3 below shows the results of this
substitution in the six cases. Cases 1 through 4 apply to a material in which irradiation
increases the strength, and cases 5 and 6 apply to a material in which irradiation reduces the
strength.
Table C 3211-3 also shows some possible numerical values of the safety factors for irradiated
316 stainless steel at 3 dpa and at 100¡C and 250¡C.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 27
ITER
Case
1
2
G 74 MA 8 01-05-28 W0.2
S mT
SF Equation
for bending
Irradiated
state
controlled by Subst. (9) into
(7a)
parameters
1
SuT
3
1 + 2b Sy
b T SyT
b=1
bT=3
Sy/SyT = 5
aSyT
1 + 2b Sy
3a SyT
b= 1
3
1
Su 0
3
1 + 2b Sy
b 0 Sy 0
4
2
Sy 0
3
1 + 2b Sy
2 Sy 0
5
1
Su
3
1 + 2b
b
6
2
Sy
3
1 + 2b
2
Table C 3211-3
a = 0.9
Sy/SyT = 5
Calculated
SF for 316 SS
Membrane
Bending
5.6
at 250¡C
and 3 dpa
5.6
at 250¡C
and 3 dpa
4.8
at 100¡C
and 3 dpa
4.8
at 100¡C
and 3 dpa
b0 = 2.4
b=1
Sy/Sy0 = 3.2
b0 = 2.4
b=1
Sy/Sy0 = 3.2
Typical implicit safety factors for irradiated
316L(N)ÊIG based on Equations (9) and (7a)
Note that the safety factors of irradiated type 316 stainless steel for both membrane and
bending loadings appear to be significantly greater than those of the unirradiated material
(Table C3211-1). This is due to the fact that no advantage is taken of the irradiation
hardening in the definition of Sm. The allowable stress Sm is conservatively low, whereas the
calculated limit stress, e.g. Pb,max , is based on estimated irradiated properties, which results
in a greater apparent safety factor. However, in all cases the safety factors for membrane and
bending are equal.
These higher safety factors for the irradiated material are based on a theoretical model in
which it is assumed that increases in strengths due to irradiation can be relied upon. In fact,
there is some doubt as to whether the increased strengths measured out-of-reactor on
irradiated material will remain in service. It is possible that material which is under cyclic
stress in the reactor or subjected to higher temperatures may be softened or annealed. This is
the reason why it was decided not to take credit for the hardening in the calculation of Sm for
irradiated materials.
Therefore, it was decided to retain Eqs. (7a) and (7b) as the definition of the safety factor and
SDC-IC, Appendix C - Rationale or Justification of the rules
page 28
ITER
G 74 MA 8 01-05-28 W0.2
the bending shape factor, respectively, for both irradiated and unirradiated material. In
practice, the value of Keff thus calculated is not reduced from its unirradiated value until the
uniform elongation for the irradiated material drops below 0.5%.
In contrast to the theoretical safety factors for the irradiation hardening case, the safety
factors for the irradiation softening materials (cases 5 and 6 in Table C 3211-3) are
comparable to those of the unirradiated materials, as can be verified by substituting b=2,
which leads to a minimum safety factor of 2.5.
Limiting cases
Infinite Ductility
In the limit of infinite ductility, g(T, ft) = 0, and Eq. (7b) becomes
Keff,rect = Krect.
(10)
Zero Ductility after Irradiation
In the limit of zero ductility, g(T, ft) = 1.0, and by Eq. (2e), b=1. Then, Eq. (7b) becomes
Keff,rect = 1.
(11)
as would be expected for an elastic brittle material.
Irradiated type 316L(N)-IG stainless steel
This material hardens significantly due to irradiation. The yield stress ratio Sy(irr)/Sy(unirr)
reaches a value of 5 at a fluence of ~3 dpa at 250¡C. At the same time, its work hardening
capability reduces drastically, and b =1. However, the uniform elongation remains at 10%
(gÊ<<1) up to 7 dpa. Thus, from Eq. (7b), Keff remains at 1.5 up to 7 dpa. Beyond 7 dpa, eu
drops precipitously to 0.3% which corresponds to g ~ 0.5l.. Thus, from Figure C 3211-2,
KeffÊ=1.38.
C 3212
Immediate plastic flow localization
C 3212.1
Elastic analysis (Immediate plastic flow localization)
The following new (not in RCC-MR or the ASME Code) limit on combined primary and
secondary membrane stress has been adopted for the SDC-IC in order to prevent premature
failure by plastic flow localization (plastic flow localization and necking) resulting from loss
of uniform elongation and loss of strain hardening capability of the material with irradiation.
PL + Q L £ Se ( Tm , ft m )
(1)
where
Se = b1Su,min ( Tm , ft m )
if eu (Tm, ftm) < 0.02
and
SDC-IC, Appendix C - Rationale or Justification of the rules
page 29
ITER
G 74 MA 8 01-05-28 W0.2
é
ù
Ea1
Se = b1 êSu,min ( Tm , ft m ) +
(e u (Tm , ft m ) - 0.02)ú
r1
ë
û
if eu (Tm, ftm) ³ 0.02
The notations used above are described in IC 2724.
When a significant volume of any material is subjected to stress (e.g., membrane stresses in a
shell), it may fail prematurely by plastic instability. For irradiated materials this instability
can occur either by necking or by a sliding off mechanism which localizes the plastic strain to
a narrow plane. Both can be related to a loss of uniform elongation of the material and thus
can be covered by a single rule. Since secondary membrane stresses with significant elastic
follow up may have an adverse influence on plastic instability of an embrittled material, both
primary and secondary membrane stresses must be limited. The elastic follow up effects of
the secondary stresses has been taken into account using the r-factor methodology (see C
3024.1.4 , Eq. 1).
s el = s +
Ee pl
r1
where
sel = elastically calculated stress
s and epl = actual stress and plastic strain, respectively
Setting the upper limit of stress s=Su at epl = eu, and using a safety factor of b1, the maximum
elastically calculated stress must be limited by the following:
é
Ee ù
s el £ b1 êSu + u ú
r1 û
ë
C 3212.1.1
(2)
Necking
Necking is a form of instability that occurs in all ductile materials and can lead to early
failure in a material whose uniform elongation has been reduced significantly (e.g., by
irradiation). In a uniaxial tensile specimen, onset of necking occurs at the ultimate stress Su
and uniform elongation eu. However, in a general structure, it depends on the geometry and
the multiaxial state of stress.
The values of eu (in Appendix A of SDC-IC) are generally obtained from uniaxial tensile
tests. It can be shown (Kellog, 1956) that the condition of necking instability is the following
relation between the logarithmic (true) strain (e) and the true stress s:
ds 1
= s
de a
with:
a = 0.5 for a thin-walled closed tube under internal pressure,
a = 2/3 for a sphere under internal pressure,
a = 1 for a tensile specimen, and
SDC-IC, Appendix C - Rationale or Justification of the rules
page 30
ITER
G 74 MA 8 01-05-28 W0.2
a = 2 for a plate under uniform tensile forces.
For materials with low uniform elongation, the true stress and true strain can be approximated
by the engineering stress and strain. Assuming that the plasticity law is of Ramberg-Osgood
( )
type: s = A e pl
n
and ignoring the elastic strain compared to the plastic strain:
ds
= n A e pl
de pl
( )
n -1
=n
s
1
= s
e pl a
=> e pl = a n
Since for a tensile specimen, a = 1, therefore, n = eu and the critical plastic strain to be used
in Eq. (2) above is epl = a eu. To be conservative, the value of a is set equal to 0.5.
C 3212.1.2
Plastic flow localization
Plastic flow localization is also a form of non-linear instability which is known to occur in
materials whose strain hardening capability has been reduced significantly by neutron
irradiation. Typical stress-strain curves of type 316 austenitic stainless steel at various
neutron fluences are shown in Figure C 3024-6. Note that the strain hardening capability of
316 stainless steel is reduced significantly at very low neutron fluences and is almost zero by
5 dpa. Generally, strain hardening helps to homogenize the plastic strain in a material.
Consequently, the effects of small surface notches in a strain hardening material are entirely
local and do not affect the overall behaviour of the structure. On the other hand, the presence
of small surface notches in a material with low strain hardening capability can lead to failure
of the structure by plastic flow localization.
To demonstrate the flow localization phenomenon, consider a plane specimen of unit width
and length to width ratio of 4 containing a semicircular surface notch (radius to width
=0.025) at the centre (Figure C 3212-1a) and subjected to an uniform axial displacement at
the end. The nominal elastic stress concentration factor for such a loading is 2.6. The
material true stress-strain curve was assumed to be bilinear with a yield stress of 700 MPa.
Elasto-plastic analyses, including the effects of finite deformation, were carried out using the
finite element code ANSYS.
Figure C 3212-1b shows the deformed shape (plane stress) for a ratio of ET/Sy =1. Note that
the plastic strain in the specimen is localized on one slip plane (FC in Figure C 3212-1a)
originating at the notch root at an angle of 60¡ to the applied stress direction. A similar slip
plane, but at an angle of 45¡to the applied stress direction, occurs for the plane strain case.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 31
ITER
G 74 MA 8 01-05-28 W0.2
d
D
A
y
2d
C
E
r=0.025d
B
F
x
Y
Z X
(a)
Figure C 3212-1a: Geometry of notched
tensile test analyzed by finite element
analysis
(b)
Figure C 3212-1b: Displaced shape
indicating flow localization
phenomenon
A plot of the peak plastic strain at location C (Figure C 3212-1a) as compared to the average
plastic strain as a function of the tangent modulus normalized by the yield stress is shown in
Figure C 3212-2. A similar plot for the value of r* (= r-1) for the same location is shown in
Figure C 3212-3. The solutions for the low tangent moduli (ET/Sy<2) should not be
considered as accurate, because they can be improved by refining the finite element mesh at
the line of discontinuity. However, they do indicate the lower limit of the tangent modulus
below which such a discontinuous solution is possible. Note that the value of the peak plastic
strain as well as r* increase rapidly with decreasing tangent modulus for ET/Sy < 2, indicating
that plastic flow localization is occurring. For ET/Sy > 2, r*=0 indicating that there is no
elastic follow up and plastic flow localization at location C.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 32
ITER
G 74 MA 8 01-05-28 W0.2
Notched Tensile test
r/d=0.025
14
epmax/epavg
12
10
8
Plane Stress
6
Plane Strain
4
2
0
0
2
Figure C 3212-2:
4
ET/Sy
6
8
10
Variation of calculated peak plastic strain away from notch
root normalized by the average plastic strain with tangent
modulus for the notched tensile test.
20
r*
15
10
5
0
-5
0
1
2
3
4
5
ET / S y
Figure C 3212-3:
Variation of analytically calculated r* (=r-1) factor with
tangent modulus for the notched tensile test
To relate tangent modulus with uniform elongation, consider a material obeying the
Ramberg-Osgood stress-strain law
SDC-IC, Appendix C - Rationale or Justification of the rules
page 33
ITER
G 74 MA 8 01-05-28 W0.2
s æ 1ö
e = +ç ÷
E ç A÷
è ø
1
n
s
1
n
(3)
The tangent modulus for such a material can be shown to be approximately given by
ET
Sy
=
1
Sy
E
+S
(4)
eu
u
Sy
-1
Su æ e u ö e u
=
Sy è 0.002 ø
where
The relationship between tangent modulus and uniform elongation for irradiated stainless
steels (E=190 GPa, Sy = 600-800 MPa) is shown in Figure C 3212-4. Note that ET/Sy =2
corresponds to a uniform elongation (eu) of ~1.6%. Thus, plastic flow localization is not a
problem for eu>2%.
It is thus possible to design against both necking and plastic strain (flow) localization by
using Eq. (2) above and setting r1=¥ whenever eu < 2%. Further, a requirement that the
allowable stress limit be continuous at eu =2% leads to Eq. (1) above and the Se rule of IC
3212.1.1.
0.05
0.04
eu
0.03
0.02
0.01
0
-1
0
1
2
3
4
ET / Sy
Figure C 3212-4:
C 3213
Calculated variation of uniform elongation with tangent
modulus
Immediate local fracture due to exhaustion of ductility
This is also a new requirement, not to be found in either the RCC-MR or the ASME Code,
whose purpose is to prevent local fracture (e.g., at notch root or at extreme fibre in bending)
due to exhaustion of ductility. In a normally ductile material, a peak stress limit is not
necessary, because the material has sufficient ductility to accommodate (relax) locally high
elastic peak stresses by plastic flow. However, if the ductility of the material is severely
SDC-IC, Appendix C - Rationale or Justification of the rules
page 34
ITER
G 74 MA 8 01-05-28 W0.2
reduced by irradiation, such a relaxation of locally high peak stress may not be possible
without cracking. In a similar fashion, if the strain hardening capability of the material is
severely reduced, local cracking may occur away from notches but in regions with high
elastic follow up. The purpose of this rule is to prevent such cracks from occurring.
CÊ3213.1
Elastic analysis
Local fracture due to exhaustion of ductility can occur in two ways. First, it can occur at
zones of stress concentration where due to local plastic flow, the available ductility may
become depleted. Second, it can occur in areas of high elastic follow up. In the former case,
because of the constraining effect of the notch, the elastic follow up factor r = r2 is always
bounded by the stress concentration factor of the notch, irrespective of the strain hardening
capability of the material (C 3024.1.4). In the second case, the r-factor r=r3 can become very
large for materials with low strain hardening capability (C 3024.1.4).
The objective of the rule is to limit the local plastic strain by the true strain at rupture (etr) and
the peak stress by the true stress at rupture. The ultimate stress Su, which is a lower bound to
the true stress at rupture, is used in the rule because true stress at rupture is not easily
obtained. There is experimental evidence that indicates that the peak plastic strain under a
high hydrostatic tensile stress can be reduced significantly compared to that in a uniaxial test.
A conservative estimate for the available ductility can be conservatively estimated from
etr/TF, where TF is the triaxiality factor.
Using the r-factor methodology and setting the upper limit of stress s=Su at epl = etr/TF, and
using a safety factor of b2, the maximum elastically calculated peak stress must be limited by
the following:
E e tr ( T, ft ) ö
Sd ( T, ft, r ) = b2 æ Su,min ( T, ft ) +
è
r
TF ø
(1)
The limits for the total stress intensity with peak ( PL + Pb + Q + F ) and without peak
( PL + Pb + Q ) are derived from the same formula but with different elastic follow up factors
r.
The r-values for the two cases are as follows (see C 3024.1.4):
Including peak stress ( PL + Pb + Q + F )
r2 = Max {KT and 4}
(2)
where KT is the elastic stress concentration factor.
Excluding peak stress ( PL + Pb + Q )
ì¥ for e u £ .02
ï
r3 = í
ïî4 for e u > .02
(3)
In the above, the notations are the same as in IC 2725. Justifications for the use of the two
values of r-factors are given in C 3024.1.4 (Figure C 3024-9).
SDC-IC, Appendix C - Rationale or Justification of the rules
page 35
ITER
G 74 MA 8 01-05-28 W0.2
CÊ3214
Fast fracture
Since neutron irradiation can reduce the fracture toughness of most structural materials, the
fast fracture limits are provided to protect against fast fracture originating from postulated
flaws. In safety studies, a postulated crack depth of quarter the wall thickness is often
required. The postulated crack depth should also be a function of the NDE technique to be
used for ITER. To be conservative, the postulated flaw, with a length of at least 10 times its
depth, is chosen conservatively as follows:
a0
=
postulated crack depth = max. [4au , h/4]
au
=
largest undetectable crack length by NDE
h
=
wall thickness
C 3214.1
Elastic analysis
In all cases, a mode I stress intensity factor KI has to be calculated for the postulated surface
crack subjected to the prescribed loading. If the postulated crack is embedded inside a
yielded region, elastic analysis rules do not apply and the elasto-plastic limit of IC 3213.2 has
to be applied. The material parameter KJC has to be provided as a function of temperature
and fluence. Two types of fast fracture are recognized - global and local.
Global fast fracture denotes fast fracture due to all primary membrane and primary plus
secondary membrane stresses. The crack driving force due to these types of stresses is not
likely to be diminished with crack extension and so a relatively large safety factor is
warranted. Local fast fracture is a fast fracture due to the total stress (including peak) and
there is a greater likelihood that the crack driving force diminishes with crack extension and
causes crack arrest. This type of fast fracture, therefore, requires a smaller safety factor than
that for the global fast fracture.
C 3300
RULES FOR THE PREVENTION OF C TYPE
DAMAGE (LEVELS A AND C)
The following rules for protection against C-type damage are applicable:
-
progressive deformation or ratcheting (IC 3310)
-
time-independent fatigue (IC 3320)
CÊ3310
C 3311
Progressive deformation or ratcheting
Elastic analysis
To prevent the occurrence of progressive deformation on the basis of elastic analysis, either
of the following two methods may be used:
SDC-IC, Appendix C - Rationale or Justification of the rules
page 36
ITER
G 74 MA 8 01-05-28 W0.2
a) 3 Sm rule
b) Bree- diagram rule
C 3311.1
3Sm rule
In the "3 Sm rule", the limit is placed at a level which ensures shakedown to elastic action
after a few repetitions of the stress cycle, except in regions containing significant local
structural discontinuities or local thermal stresses. The last two factors are considered only in
the performance of fatigue evaluation.
C 3311.2
Bree-diagram rule
The "Bree diagram rule" is based on the ASME Code Section III, Division 1 (NB-3222.5,
Thermal Stress Ratchet) and Code Case N47 (T-1330, test B1) to ensure that the structure
behaves in the domain E, S1, S2 or P of the Bree diagram (Figure C 3310-1), corresponding
to elasticity, shakedown or cyclic plasticity, without ratcheting (R1 and R2). To be
conservative, the provision in the ASME Code Section III of using 1.5Sm instead of Sy
whenever 1.5Sm>Sy has been dropped and replaced by the Code Case N47 approach of using
Sy. The use of Sy in all cases limits the amount of ratcheting strain that can accumulate prior
to shakedown to a lower value than that allowed by the ASME Code Section III.
5
4
R2
3
Y
P
R1
2
S2
S1
1
E
0
0
Figure C 3310-1
0.2
0.4
0.6
X
0.8
1
1.2
Bree diagram for axisymmetric structures
The Bree diagram rule is based on a 2-dimensional ratcheting analysis of a thin circular
cylindrical shell subjected to a steady internal pressure and cyclic axisymmetric temperature
drop through the thickness. In test 1 of the Bree diagram rule, one must ensure that the
structure is not in one of the ratcheting domains R1 or R2. In the Bree diagram the limit of
SDC-IC, Appendix C - Rationale or Justification of the rules
page 37
ITER
G 74 MA 8 01-05-28 W0.2
these domains is given by Z = 1, where Z definition differs with X and Y values. The
ordinate chosen for the Bree diagram in all cases is
Y=
DQ
Sy
(1a)
where
DQ is the cyclic secondary stress intensity range.
The abscissa X varies depending on the type of primary loading.
General primary membrane loading
X=
Pm
Sy
(1b)
where
Pm is the steady general primary membrane stress intensity, and
Local primary membrane plus bending loading
X=
PL + Pb / K
Sy
(1c)
where
PL and Pb
are the steady local primary membrane and bending stress
intensities, respectively, and
K
is the bending shape factor.
The domain limit can be given as a function of the secondary ratio: SR =
If SR > 4, then:
Z=X´Y
Z <1
If SR < 4, then:
=>
(cf. CASE N47-29, T1330 test B-1)
Y<
1
X
Z = Y + 1 - 2 (1 - X )Y
Z <1
=>
Y
X
(cf. CASE N47-29, T1330 test B-1)
Y < 4(1 - X )
The above rule can also be represented by the function F(X) as follows:
ì 1
ï
Y £ F( X ) = í X for 0 < X £ 0.5
ï 4(1 - X ) for 0.5 £ X £ 1.0
î
SDC-IC, Appendix C - Rationale or Justification of the rules
(2)
page 38
ITER
G 74 MA 8 01-05-28 W0.2
C 3311.3
Efficiency Diagram rule
A third method "efficiency diagram rule", which is derived from RCC-MR (RB 3261.1.1.4:
efficiency diagram), was also considered. In this approach, progressive deformation is linked
to the amplification of primary stress effects by a cyclic deformation. The main effect is an
increase of the mean strain after a large number of cycles. The efficiency diagram approach
provides a means of determining an effective primary stress (Peff) that would give the same
immediate deformation as the actual cycling load combination. The value of Peff is estimated
by entering the stress-strain curve at the experimentally determined accumulated strain
(Figure C 3310-2).
s
Stress-strain curve at test temperature
Peff
ec
Figure C 3310-2
e
Stress-strain curve
The efficiency diagram has been developed on the basis of tests on a large variety of
specimen geometry and loading, including non-axisymmetrical ones:
-
Three bar assemblies, with constant load and different cyclically varying
temperatures between the central and the lateral bars: (Swaroop,1973),
(Uga,1978), (Uga,1984).
-
metal band with applied weight and cyclically varying curvature:
(Anderson,1971), (Anderson,1973), (Goodall,).
-
tension-torsion [(Inoue,1976), (Udoguchi,1977), (Lebey,1979), (Cousseran,1980),
(Clement,1984), (ClŽment,1985)]
-
tests on components under thermal shocks: piping [(Corum,1975), (Hyde,)],
cylindrical vessels [(Cousin,1981-1), (Cousin,1981-2)]
as well as different materials (ferritic steel, austenitic steel, copper, lead,. etc.) and tests
temperatures (from room to creep temperatures). Each test is characterized by the efficiency
index (v) and the secondary ratio (SR):
v=
P
Peff
SR =
DQ
P
SDC-IC, Appendix C - Rationale or Justification of the rules
page 39
ITER
G 74 MA 8 01-05-28 W0.2
For application to codes and standards, Peff is determined from a lower bound to all
experimental data leading to the following functional relationship between SR and v:
ì
ï1 for SR £ 0.46
ï
v(SR ) = í1.093 - 0.926SR 2 /(1 + SR )2 for 0.46 £ SR £ 4
ï1 / SR for 4 £ SR
ï
î
(3)
General primary membrane loading
The abscissa is chosen to be the stress ratio SR1 defined by
SR1 =
DQ
Pm
(4a)
and an effective primary stress P1 is defined in terms of a parameter v1(SR1)
P1 =
Pm
(4b)
v1
To prevent ratcheting, P1 has to be limited as a primary stress. Requiring P1 £ Sm would
ensure the same safety margin with and without progressive deformation. However, in RCCMR, the safety margin has been reduced for the case of progressive deformation by limiting
P1 to 1.2ÊSm (~Sy for austenitic stainless steels) which leads to the following
Pm
P
= v(SR1 ) ³ m
1.2Sm
P1
(4c)
where the functional relationship between v1 and SR1 is given by Eq. 3. It can be shown that,
for austenitic stainless steels, setting the limit of P1 to 1.2 Sm implies that, in the most
unfavourable case, a membrane strain of 0.45% is acceptable. This value is similar to the
weld strain limit of Code Case N47-29 (0.5%). This value is acceptable as long as the
cumulative strain is negligible compared to the material ductility. If a reasonable strain limit
for cumulative strain is 10% of the material ductility, a membrane strain of 0.45% is
acceptable as long as the material ductility is greater than 5%. Since the uniform elongation
for irradiated austenitic stainless steel can be <5%, a limit of P1 £ Sm (maximum membrane
ratcheting strain = 0.07%) would be justified for ITER.
Local primary membrane plus bending loading
The equations corresponding to Eqs 4a-c are given by
SR 2 =
DQ
( PL + Pb )
SDC-IC, Appendix C - Rationale or Justification of the rules
(5a)
page 40
ITER
G 74 MA 8 01-05-28 W0.2
P2 =
PL + Pb
v2
(5b)
PL + Pb
P + Pb
= v(SR 2 ) ³ L
1.2 KSm
P2
(5c)
where
P2
= the effective primary stress, and
the function v2 = v(SR2) is given by Eq. 3.
Comparison between the Efficiency diagram and the Bree diagram equations
The Bree diagram can be plotted in the Efficiency diagram coordinates by first transforming
the Bree diagram limit (Eq. 2) to give (after noting that Sy » 1.2Sm for stainless steels) the
following:
Primary membrane loading
Pm
1.2Sm
(6a)
ì 1
ï
for SR1 ³ 4
ï SR1
ï
G(SR1 ) = í
ï 4
ï 4 + SR for 4 ³ SR1 ³ 0
1
ïî
(6b)
G(SR1) ³ v1 =
where
Local primary membrane plus bending loading
For the local primary membrane plus bending load case, the results become dependent on the
factors K, PL , and Pb , because the primary variable for the Bree diagram is PL + Pb / K
whereas that for the Efficiency diagram is ( PL + Pb ) / K . Restricting to cases where PL + Pb
= PL + Pb , one can write
æ
ö
æ
ö
Pb
F( X ) Pb
ç
÷
ç
SR 2 = a
,K £
a
, K÷
ç
÷
ç
X ç PL ÷
X ç PL ÷÷
è
ø
è
ø
Y
where
SDC-IC, Appendix C - Rationale or Justification of the rules
page 41
ITER
G 74 MA 8 01-05-28 W0.2
Pb
æ
ö 1+
Pb
KPL
a ç , K÷ =
ç PL ÷
Pb
ç
÷
è
ø
1+
PL
The Bree diagram limit can be plotted in the Efficiency diagram coordinates (SR2 and v2)
with a and K as parameters:
PL + Pb
1.2 KSm
(7a)
ì
ï a
ï SR for SR 2 ³ 4a
2
1 ïï
H(SR 2 , a, K ) =
í
4
aK ï
for 4a ³ SR 2 ³ 0
SR 2
ï
ï4 + a
ïî
(7b)
H(SR2, a, K) ³ v2 =
where
A comparison between the efficiency diagram (Eq. 3) and the Bree diagram transformed to
the new coordinate system (Eqs. 6b-7b) with K=1.5 is shown in Figure C 3310.4 for several
loadings. Except at low values of SR1, the two plots coincide with each other for the case of
general membrane loading (Pm), which is to be expected from a comparison between Eqs 3
and 6b. The curve for pure bending and local primary membrane loading falls below the
curves for general primary membrane loading. The curve for local primary membrane plus
bending loading with PL = Pb falls between those for the pure bending and the pure
membrane loading cases, as expected. Note that all the transformed Bree diagrams fall below
the Efficiency diagram of RCC-MR and are generally also lower than the lower bound
(which is somewhat lower than the Efficiency diagram of RCC-MR, Autrusson, 1987) to the
cloud of data (Figure C 3310-3) used to justify the RCC-MR curve. Thus, the Bree diagrams,
transformed to the Efficiency diagram coordinate system, provide lower bounds not only to
the Efficiency diagram of RCC-MR but also to the test data used in the development of the
RCC-MR approach. The same general trends are also observed if the efficiency diagram is
transformed to the Bree diagram coordinate system and superimposed on the Bree diagram.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 42
G 74 MA 8 01-05-28 W0.2
v =P /1.2S or v = (P +P )/1.2KS
1
L
b
2
m
m
m
ITER
1
RCC-MR (Eq. 3)
0.1
Bree, P
m
Bree, PL=0
Bree, P =P
b
L
Bree, Pb=0
Lower Bound to
Cloud of Data
0.01
0.1
Figure C 3310-4:
1
10
100
DQ/P (SR1 or SR2)
1000
Comparison of the RCC-MR Efficiency diagram with those
calculated from Bree diagram analyses
The basis for the original Bree diagram was ratcheting of an axisymmetric cylindrical shell
subjected to steady internal pressure and cyclic temperature drop through thickness. To
check the applicability of the Bree diagram to non-axisymmetric structures under bending
stresses, a cross-section of a flat first wall with a drilled coolant channel (Figure C 3310-5)
and subjected to a steady bending moment and a cyclic thermal loading was analyzed both
analytically and numerically (Majumdar and Walters, 96). Results of that analysis are
presented in Figure CÊ3310-6 for the case of bth=0.25. T and M are temperature and bending
moment (positive if it causes compressive stress at the plasma edge) with the subscript y
denoting their respective values to cause first yielding. The regimes E, S, and P denote load
combinations resulting in cyclic elasticity, shakedown, and cyclic plasticity, respectively. A
single test data (triangle) for ratcheting conducted at Julich on a comparable structure made
of 316 stainless steel is also shown in the Figure C 3310-6. This test ultimately led to cyclic
plasticity (no steady ratcheting) in agreement with analytical prediction. The circular
symbols denote typical numerical analysis results, with the open symbols denoting
shakedown (or cyclic plasticity) and the filled symbols denoting steady ratcheting.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 43
ITER
G 74 MA 8 01-05-28 W0.2
Surface Heat Flux
T0 + T
y
d
bd
th
2d
x
d
T0
Figure C 3310-5:
First wall with drilled coolant channel
4
Analytical
Numerical
Experiment
3
b =0.25
th
P1
P
2
T/T
y
P2
2
S
1
1
S
E
S
2
2
0
-2
-1.5
-1
-0.5
0
M/M
0.5
1
1.5
2
y
Figure C 3310-6 Bree diagram for the first wall with drilled coolant channel
A second example, in which both the primary and secondary stresses are of the bending type,
is a double cantilever (Figure C 3310-7) beam subjected to steady lateral pressure and cyclic
SDC-IC, Appendix C - Rationale or Justification of the rules
page 44
ITER
G 74 MA 8 01-05-28 W0.2
linear temperature drop through thickness (Majumdar, 92). The results for both examples
together with the predictions from Bree diagram rule (Eq. 2) are plotted in Figure C 3310-8,
which shows that, except for the double cantilever (DC model) at low primary bending
stresses, the Bree diagram rule predictions are never unconservative and usually quite
conservative.
Surface Heat Flux
T+D T
T
Coolant Pressure
Figure C 3310-7:
Loading on a double cantilever beam
5
D Q/Sy
b th=0.5
4
b th=0.25
3
b th=0
DC Model
2
1
Bree
Diagram
(Eq. 2)
0
-1
-0.2
Figure C 3310-8:
0
0.2
0.4
0.6
P b/KS y
0.8
1
1.2
Comparison of the ASME Code Bree diagram with those
calculated for structures primarily loaded in bending
Although the Bree diagram equation and the Efficiency diagram equation are mathematically
almost equivalent, the Bree diagram (Eq. 2) is adopted for SDC-IC because of the following
considerations.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 45
ITER
G 74 MA 8 01-05-28 W0.2
(1) The Bree diagram is not only a lower bound to the RCC-MR Efficiency diagram
but is also a better lower bound to the test data used in the justification of the
RCC-MR approach.
(2) The Bree diagram is a lower bound to the ratcheting curves of the flat first wall
design (with drilled coolant channel) and, except at low primary stresses, is also a
lower bound to the curve for the double cantilever (DC model) beam.
(3) The Bree diagram is more intuitive than the Efficiency diagram and may be a
little easier to implement for ITER analysis where the analyst is often interested
in computing the maximum permissible surface heat flux on the first wall or
divertor plate for a given primary loading and the stress ratio is not known a
priori.
Plasma disruption effects
In all the examples considered so far, the primary stress is steady and the thermal stress is
cyclic. However, in ITER, plasma disruptions will induce cyclic primary loading on top of
cyclic thermal stress.
A series of elastic-plastic finite element analyses on ratcheting of a semi-circular and a flat
first wall were conducted to check the adequacy of the elastic analysis ratcheting rule (Bree
diagram) that have been proposed for the SDC-IC. To this end, two simple geometries of the
first wall were selected - a flat first wall and a semicircular first wall.
The geometry and boundary conditions used are shown in Figure C 3310-9. The first wall
thickness was 5 mm for the semicircular first wall and 15 mm for the flat first wall. The
radius of the semicircular first wall was 20 cm and the toroidal span of the flat first wall was
16 cm. In the case of the flat first wall, a section with additional coolant channels inside the
first wall (Figure C 3310-5) was assumed. Thus the through-thickness temperature
distribution in the semicircular first wall was linear whereas that in the flat first wall was
bilinear.
A constant coolant pressure of 2 MPa with a superimposed cyclic disruption-induced
pressure (additive to the coolant pressure) of 2 MPa occurring once every 15 plasma on-off
cycles were used. In the case of the flat first wall, only the bending stresses due to the
coolant present behind the first wall were considered. A surface heat flux of either 0.5
MW/m2 or 0.75 MW/m2 were used for the plasma on-off cycles and all transient effects on
temperature distribution were ignored. The temperature of the first wall at the coolant
interface was assumed to be the same as that of the coolant (100¡C). The effect of a
strongback on the first wall deformation behaviour was simulated by assuming plane strain
deformation in the poloidal direction so that no ratcheting could occur in the poloidal
direction.
The material was assumed to be type 316 stainless steel with the following mechanical
properties (assumed to be temperature-independent):
E=185 GPa, n=0.27, and a=16.5x10-6/¡C, k=16 W/m ¡C
A bilinear approximation for the stress-strain curve was used with the following values for
yield strength (Sy ), ultimate strength (Su), uniform elongation (eu), and tangent modulus ET.
Sy = 180 MPa Su = 480 MPa eu =30%
ET = 1000 MPa
The corresponding Sm value for the material was 160 MPa.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 46
ITER
G 74 MA 8 01-05-28 W0.2
2
q MW/ m
2
q MW/ m
p
c
p
c
q
N cycl es
N cycl es
t
p
d
Figure C 3310-9:
Thermal cycles and intermittent plasma disruption cycles
considered for finite element analysis
The main conclusions from these analyses are as follows.
-
Although in the absence of plasma disruptions (at a surface heat flux of 0.5
MW/m2) both first walls would eventually shakedown to cyclic plasticity without
further increase of ratcheting strain, the occurrence of periodic plasma disruptions
would prevent such a shakedown from occurring and promote ratcheting.
-
The 3Sm rule would be violated for both designs at the surface heat fluxes
considered, with or without disruption loading. As expected, it was the most
conservative rule.
-
At a surface heat flux of 0.75 MW/m2, when disruption cycles were included in
detailed inelastic analyses, ratcheting was predicted for both designs. In the
absence of disruption cycles, the analyses indicated that ratcheting would occur in
the membrane loaded semicircular first wall but the bending loaded flat first wall
would eventually shakedown. If elastic analysis were used, both designs would
exceed the Bree diagram limit at a surface heat flux of 0.75 MW/m2, with or
without disruption loading, and the combined surface heat flux and disruption
loading for this case would be correctly disallowed.
-
At a surface heat flux of 0.5 MW/m2 and excluding disruption cycles, detailed
inelastic analyses showed that shakedown would occur for both designs. If elastic
analysis were used, the Bree diagram limit would be exceeded (i.e., conservative)
for both designs.
-
At a surface heat flux of 0.5 MW/m2 and including disruption cycles, detailed
inelastic analyses indicated that ratcheting would occur for both designs. If elastic
analysis were used, the Bree diagram limit would be exceeded (i.e., conservative)
for both designs if DQ + DPdis (where DPdis is the elastic stress range due to
SDC-IC, Appendix C - Rationale or Justification of the rules
page 47
ITER
G 74 MA 8 01-05-28 W0.2
disruption loading) was used as the measure for stress intensity range. However,
if DQ + DPdis were used instead, the Bree diagram limit would be satisfied (i.e.,
unconservative) for the flat first wall.
-
To conservatively design against ratcheting using elastic analysis and for all
loadings expected in ITER, either the 3Sm rule or the Bree diagram rule (using
DQ + DPdis to define cyclic stress intensity range) should be used.
C 3312
Elasto-plastic analysis
In order to check the limits on these strains, an elasto-plastic analysis, giving either the exact
value or an upper bound to the strains resulting from all the cycles envisaged, should be
carried out.
The limit on the significant mean plastic strain (eÄm )pl (IC 2616) is related to the uniform
elongation of the material either as half (to account for multiaxial stress effects) of the
minimum value over the whole life of the component or, to take advantage of the higher
ductility at the beginning of life, as a limit on a sum over blocks.
The limit on significant local plastic strain (eÄ)pl (IC 2616) is the lesser of 5% and the
minimum true fracture ductility over the whole life of the component. As before, to take
advantage of the higher ductility at the beginning of life, a limit on sum over blocks approach
is acceptable.
CÊ3320
Time-independent fatigue
SDC-IC defines fatigue in IC 3320 as one of C-type damage "which can result from the
repeated application of a load". In other words, fatigue is a damage which can be produced
by a load varying over time, irrespective of whether this load is applied in the form of applied
forces or imposed displacements (in particular, incompatible thermal strains). If a component
is subjected to such a variable loading, cracks may eventually appear, propagate and even
cause the component to fail. For ITER, because of its pulsed mode of operation, reliable rules
are required which, if followed, will ensure that fatigue damage is still small at the end of the
component's life.
Fatigue rules in most design codes currently in use (RCC-MR, ASME Code Section III, and
Code Case N47) are the result of an evolution over a long period of time (thirty years or so)
and have demonstrated that their use provides adequate margin against fatigue failure.
Therefore, SDC-IC fatigue rules have been based on these rules with modifications to take
into account the effects of irradiation.
C 3320.1
Historical Perspective
From the middle of the 19th century to the middle of the 20th century, applied stress variation
was considered to be the main variable in evaluating fatigue strength. Usually, rotating
bending tests were performed under imposed load (which at the time was believed to be
imposed stress), with the number of cycles to failure N being plotted as a function of the
amplitude of the stress variation S (half of the total variation DS ) on a log-log basis. These
conventional S-N curves are sometimes referred to as Wšhler curves. Typically the S-N
curves showed a characteristic knee which for some materials (notably mild steel) resulted in
SDC-IC, Appendix C - Rationale or Justification of the rules
page 48
ITER
G 74 MA 8 01-05-28 W0.2
a so-called endurance limit value below which the specimens seemed to have infinite life (no
fatigue crack initiation).
The existence of such a characteristic fatigue curve with an endurance limit turned out to be
an over-simplification. First of all, experimental results show a scatter, indicating that fatigue
strength is a statistical quantity. Although the large scatter observed during early testing due
to poor control of test parameters was later reduced somewhat by better test techniques (e.g.,
closed-loop servo-controlled tests using uniaxial specimens), it could not be avoided and
remains a physical reality for fatigue. A distribution exists which can be characterized by
curves of equal probability of failure. The scatter in fatigue life has been attributed to many
factors not all of which are fully understood - specimen geometry and volume effects,
specimen surface preparation, environment of testing (e.g., humidity), heat-to-heat variation,
and metallurgical condition of the material such as grain size and inclusions.
The S-N curves are usually reported for tests with zero mean stress. If mean stress
(especially tensile mean stress) is present, fatigue strength may depend on the mean stress,
particularly if the number of cycles to failure is large.
Historically, fatigue life has been divided arbitrarily into two regimes - high-cycle fatigue and
low-cycle fatigue. High-cycle fatigue is a regime in which the material's behaviour remains
essentially elastic during cyclic loading (failure occurs after a large number of cycles, in
excess of several tens or hundreds of thousands of cycles) and were historically studied first.
Low-cycle fatigue, on the other hand, is characterized by significant cyclic plasticity and
lower fatigue lives (less than ten thousand or so). Operationally one could separate highcycle from low-cycle fatigue on the basis of relative contributions of the elastic strain range
and the plastic strain range to the total strain range. When the two components are equal, the
corresponding life has sometimes been defined as the transition fatigue life. The scatter in
fatigue life is characteristically larger in the high-cycle than in the low-cycle regime. Much
of the scatter in low-cycle fatigue life observed in earlier tests was mainly due to the use of
stress rather than strain as the loading variable.
And finally, fatigue curves are mainly established from tests on uniaxial test specimens.
Multiaxial fatigue test data are much more limited. Since most design applications involve a
multiaxial stress field, correlations between uniaxial and multiaxial fatigue are needed before
the design curves can be applied.
C 3320.2
Fatigue damage (Room temperature)
Accumulation of fatigue damage in a material at a microscopic level is a highly complex
strain-localization phenomenon. It is generally held that irreversibility of to-and-fro slip in
the material leads to intense slip band formation at the free surface from where a fatigue
crack is generally initiated. The details of this process are not yet fully understood.
Therefore, currently there is no truly theoretical model for fatigue crack initiation that can be
used in design. Macroscopically, fatigue damage accumulation in a typical fatigue specimen
progresses through the following phases:
-
an incubation period prior to the appearance of a measurable crack. During this
phase, it may be possible to reverse the fatigue damage through annealing,
-
crack initiation occurs along the directions of maximum shear (or maximum
distortion),
-
crack propagation normal (macroscopically) to the applied stress,
SDC-IC, Appendix C - Rationale or Justification of the rules
page 49
ITER
G 74 MA 8 01-05-28 W0.2
-
final failure when the crack propagates through the specimen.
From a practical standpoint, only two stages are generally considered: an initiation stage and
a propagation stage. The transition between these two stages is not easily identifiable. A
practical definition for the transition from initiation to propagation is when the crack reaches
an appreciable depth (several times a grain size), i.e. about a millimetre. The relative
durations of the crack initiation phase and the crack propagation phase vary with the cyclic
life. Typically for high-cycle fatigue, a large fraction (virtually the whole duration) of the life
of a specimen is spent in crack initiation. On the other hand, in low-cycle fatigue, crack
growth is generally the dominant phase.
C 3320.3
Empirical correlations for fatigue life
Although initially stress range (or amplitude) was successful as a correlating parameter for
(high-cycle) fatigue life, with the interest shifting towards low-cycle fatigue after the war,
tests [(MANSON, 1953), (MANSON, 1966), (COFFIN, 1954)] using machines that impose
an extension (rather than a force) have clearly shown that strain range De (particularly plastic
strain range, Dep) is a superior correlating parameter for low-cycle fatigue life (Nf) than stress
range. The correlation has come to be known as Coffin-Manson (or Manson-Coffin) law:
De = De e + De p
(1)
where
Dee is the elastic strain range,
De e = B N fb , and
De p = A N cf .
An important fact to remember is that the strain range above does not refer to the measurable
geometrical strain but to the stress-induced fraction of this strain. The measurable
geometrical strain may be the sum of thermal expansion and stress-induced strains. A typical
example is a straight bar of constant cross section, wedged between two rigid abutments.
When subjected to temperature variations, the bar's length does not change and its strain
variation is zero, but the bar may nonetheless show signs of fatigue, because the stress and
mechanical strain in the bar vary with the temperature.
A large number of tests performed on various materials led Manson to propose the "universal
slopes" equation for the number of cycles to failure:
De = 3.5(Su / E ) N f -0.2 + (e tr )
0.6
N f -0.6
(2)
where
Su
is the ultimate tensile strength,
E
is Young's modulus,
(e tr )
is the true fracture ductility, i.e. ln [100/(100-%RA)],
%RA
is the per cent reduction of area.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 50
ITER
G 74 MA 8 01-05-28 W0.2
On the basis of fatigue tests on stainless steels, Coffin proposed a fatigue plastic strain life
correlation
De p = e tr ( N f )
-0.5
(3)
and claimed that the coefficient, which was equal to the plastic strain amplitude at 1/4Êcycle,
was the same as the monotonic tensile ductility. However, with more fatigue tests conducted
by various researchers on other materials, it became apparent that neither the exponents nor
the coefficients were truly universal but varied with materials. Subsequently, a strain-life
correlation was proposed (Morrow,1962) using strain reversals (2 N f ) rather than cycles ( N f )
as follows:
De =
2s f
( 2 N f ) b + 2 e f ( 2 N f )c
E
(4)
The exponents and coefficients of the above strain-life equation could be related to the cyclic
stress-strain curve of the material given by:
æ Ds ö
Ds
De =
+ 2e f ç
÷
E
è 2s f ø
n
where
n = c/b
By considering a monotonic tensile test to be equivalent to 1 reversal, e f and s f could (not
always) be related to the tensile ductility and true fracture strength respectively.
To take into account an "endurance limit", Manson (Manson,1966) proposed a modified
relation:
De = AN a + Se / E
(5)
where
Se is the endurance limit.
Based on this work, (Langer,1971) proposed a simple model:
De = [0.25 e tr ]N -0.5 + Se / E
(6)
Tests in the high-cycle fatigue regime on a variety of materials have shown that, with the
exception of mild steel, no other material displays a true endurance limit. Even in mild steel,
the endurance limit can be eliminated by periodic application of overload cycles (Brose et al.,
1974 and Dowling, 1973). From a practical viewpoint, an endurance limit may be defined as
the strain range at 108 cycles. In any case, Eq. 6 can be considered as a special case of Eq. 1
with the exponent b=0.
C 3320.4
Irradiation Effects
Experimental fatigue data on irradiated materials of interest to ITER are scarce. For design
applications, correlations such as Eqs 1-3 provide a way of estimating the fatigue behaviour
of irradiated materials from tensile data (ductility and ultimate tensile strength) which are
SDC-IC, Appendix C - Rationale or Justification of the rules
page 51
ITER
G 74 MA 8 01-05-28 W0.2
more readily accessible. We assume that the high cycle fatigue life is controlled by the
tensile strength and the low-cycle fatigue life is controlled by the ductility of the material and
that the fatigue curves for the irradiated material can also be expressed as in Eq. 1 and the
exponents b and c are not affected by irradiation. For the unirradiated material,
De = BN fb + AN cf
(7)
Together with Eqs 2 and 3, an elastic strain life modification factor Fe (£1, i.e., no advantage
taken for irradiation hardening) and a plastic strain life modification factor Fp can be defined
as follows to modify the coefficients B and A:
fe =
(S u ) i
(S u ) u
and
é (e tr ) ù
u
fp = ê
ú
e
êë ( tr )i úû
c
where
the subscripts u and i refer to unirradiated and irradiated properties, and
the estimated fatigue equation for the irradiated material can be expressed as
[
]
[
De = f e BN fb + f p AN cf
]
(8)
Note that Eq. 8 reduces to Eq. 7 if irradiated properties are set equal to the unirradiated
properties.
Particular attention should be given to the following situations where the available ductility
etr of the material may be reduced:
-
plastic strain accumulation due to progressive deformation (ratchet) cycles,
-
the ductility of welds in welded joint is often much lower than that of the base
metal,
-
cold work during manufacturing.
Although Eq. 8 may be used to estimate the fatigue strength, fatigue data on irradiated
material will be ultimately needed to validate the fatigue design of ITER. Available limited
fatigue tests on irradiated type 316L(N) stainless steel conducted at Petten appear to be
reasonably close to those predicted by the procedure recommended here (Figure C 3320-1).
Since the effects of irradiation on fatigue life of type 316L(N) stainless steel is rather small at
temperatures less than 400¡C up to a displacement dose of 10 dpa, the design fatigue curve
for the irradiated material is recommended to be the same as that for the unirradiated material
in Appendix A (A.S1.5.5).
SDC-IC, Appendix C - Rationale or Justification of the rules
page 52
ITER
G 74 MA 8 01-05-28 W0.2
Type 316 LN Stainless Steel
0.1
De =3.5S u/E(Nf)-0.065+e tr0.6(Nf)-0.6
Total Strain Range
0 dpa (RCC-MR data base)
0.01
10 dpa
Calculated
Petten Data, 325¡C, 10 dpa
Petten Data, 225¡C, 10 dpa
0.001
10
Figure C 3320-1:
C 3320.5
100
1000
10 4
10 5
Cycles to Failure
10 6
10 7
Comparison of calculated and experimental faitigue lives
for irradiated specimens
Effects of a mean stress or mean strain
Mean stress correction factors are needed for high cycle fatigue lives (ASME criteria,1972).
Indeed, for a given life, the stress range decreases with the mean applied stress. This effect
was observed as early as 1874 (Gerber,1874) and a great deal of research work has been
carried out since, highlighting the complexity of this phenomenon. In fact, the endurance
limit depends on the types of loads involved [(Sines,1959), (Hakem,1987)]. For an applied
mean stress, the effect can be taken into account by subtracting this stress from Su in Eq. 2 or
by using a modified Goodman diagram.
For low, steady primary (load-controlled, e.g., pressure, self-weight, etc.) stresses and larger
cyclic thermal stresses, the induced mean stresses can be relaxed by cyclic plasticity and it is
more realistic to correct for mean strain rather than mean stress. The abundant literature on
this subject concludes that the effect of mean strain on endurance is negligible as long as it is
small compared to the ductility of the material. For a highly ductile material like annealed
unirradiated stainless steel, the mean strain is always negligible compared to the ductility and
does not need to be taken into account in defining the permissible limits. If the ductility is
reduced by irradiation significantly, the ductility term in the Manson-Coffin law (see C
3320.3) should be corrected by subtracting the mean strain from the ductility [(Weiss,1963)
and (Sachs,1960)].
SDC-IC, Appendix C - Rationale or Justification of the rules
page 53
ITER
G 74 MA 8 01-05-28 W0.2
C 3320.6
Multiaxial loading
The tests considered in the preceding sections are uniaxial tension and compression tests.
Since real structures are subjected to multiaxial state of stress, it is necessary generalize the
conclusions drawn from these tests.
The various stages (e.g., initiation, propagation, etc.) of fatigue damage imply that a single
multiaxial criterion may not be suitable for all stages. However, most of the design codes
such as RCC-MR, ASME Code, etc. have opted for a rule which has already been used
industrially (Langer, 1971) and which stands out quite favourably when compared with its
rivals: this is the octahedral (or von Mises) strain range defined as follows:
De eq = maximum during cycle of
(2 / 3)De ijDe ij
The expression for Deeq does not reduce to the axial strain range De11 for a uniaxial test
unless the value of Poisson's ratio is 0.5. Since the Poisson's ratio for elastic strains is less
than 0.5, the uniaxial fatigue curve plotted against De11 differs from the fatigue curve plotted
against Deeq. Therefore, in order to use the uniaxial fatigue curves, the definition of the
equivalent strain range has to be modified as follows:
For a uniaxial fatigue curve expressed in the Manson-Coffin form: De11 = Dee + Dep, the
plastic strain range component is not changed (since np = 0.5), but the elastic strain range
component is multiplied by (1 + n)/(1 + 0.5). Hence the definition of equivalent strain
becomes:
De eq =
C 3320.7
2
(1 + n) De e + De p
3
Strain cycles and their combination
In the discussions so far, we have considered purely cyclic loads during which the strain
varies in an approximately sinusoidal manner over a cycle and all the cycles considered are
the same. In reality, the operating conditions involve much more complex loading cycles.
At the design stage, it is not possible to fully determine the load conditions to which a
component will be subjected during operation. Identical installations, intended for the same
use, will experience different loads. At the least, the equipment specification should contain
a list of typical events and the number of these events that the component must withstand
without damage. Each event may consist of either steady states or transients connecting
various steady states. Such information constitutes an infinite number of possibilities for
which it would be futile to attempt a complete analysis.
In design (e.g., in RCC-MR) it is customary to combine these events (which are either steady
states or transients) into cycles, i.e., sequence of events which, beginning with an initial state,
join up with it at the end of the sequence.
When creep is not significant, the time scale is of no importance - the only thing that matters
is the sequence in which events take place, the time they last being of no consequence.
The problem facing the designer is how to estimate the fatigue damage due to cyclic loads
that vary in amplitude and number of occurrences. In the absence of a clear resolution to this
SDC-IC, Appendix C - Rationale or Justification of the rules
page 54
ITER
G 74 MA 8 01-05-28 W0.2
problem, the so-called Miner's linear damage accumulation rule has been adopted for SDCIC. This rule has been followed by various industries for almost thirty years with reasonable
success [ASME Code, RCC-MR]. This criterion also has the great advantage of being a
simple rule whose outcome does not depend on the sequence in which the cycles are applied,
information which is usually not known to the designer. Stated simply, the linear damage
rule assumes that if a cycle type i causes failure in Ni cycles, then ni cycles of this type use up
a fraction ni/Ni of the material's fatigue strength capabilities. Thus the sum of the ni/Ni
fractions over all cycle types must not exceed 1.
Good technical judgment is needed to apply this linear damage accumulation rule correctly.
The cycle considered may be complex and include several elementary sub-cycles. A cycle
may be a juxtaposition of cycle De1 with a variation of cycle De2 (Figure C 3320-2). It is, of
course, more complex to apply the rule in the case of a non-proportional multiaxial loading.
Two isolated cycles may cause more damage if they follow one another (Figure C 3320-3).
As their number of occurrences are probably not the same, a "recombination of cycles" is
necessary. To be conservative, from amongst all physically possible combinations of cycles,
the one leading to a maximum fatigue damage should be adopted.
e
De 1
De2
t
Figure C 3320-2:
Combination of strain cycles, Example 1
e
cycle 1
cycle 2
Figure C 3320-3:
t
Combination of strain cycles, Example 2
SDC-IC, Appendix C - Rationale or Justification of the rules
page 55
ITER
G 74 MA 8 01-05-28 W0.2
C 3320.8
Treatment of defects or geometrical singularities
In design codes, such as the ASME Code, Section III and RCC-MR, the aim of the fatigue
rule is the prevention of "crack initiation" (which includes both the incubation and early
growth of a crack) from an undefected region. In reality, despite nondestructive examination,
small manufacturing or material defects and/or geometrical singularities may exist in the
structure. Although there are rules available within the framework of fatigue crack initiation
methodology for the prevention of fatigue crack re-initiation from such singularities, such
rules have been developed empirically from tests on unirradiated ductile materials
[(D'escatha,1980), (Jack,1970), (Aus,1977), (Bernard,1987)].and may not be applicable to
irradiated materials. Since the ITER in-vessel components will experience irradiation which
may cause embrittlement and accelerated fatigue crack growth, the acceptability of such
defects or singularities should be based on fatigue crack growth analysis using standard linear
elastic fracture mechanics methodology.
C 3320.9
Design fatigue curves
The experimental curves (De-N to failure) cannot be used directly for design purposes for the
following reasons:
-
first, the usually large scatter in the experimental results implies that the average
or mean fatigue curves cannot provide adequate reliability and safety factors must
be applied,
-
second, the laboratory tests are usually conducted on specimens that are polished
and defect-free and are generally manufactured much more carefully than large
fabricated structures (for example, surface condition, small undetectable defects,
etc.),
-
the curves correspond to failure of the specimens and, for low-cycle fatigue, the
number of cycles to crack initiation can be much lower than the number of cycles
to failure.
Therefore, the average or mean fatigue curves obtained from laboratory test specimens are
replaced by significantly reduced design fatigue curves. In SDC-IC, the following procedure
for deriving the design fatigue curves from the experimentally determined mean fatigue
curves has been borrowed from that of RCC-MR and the ASME Code, Section III
(ASME,1989): the design curve is deduced from the mean experimental curve by dividing
the number of cycles by twenty or the strain range by two (the most stringent criterion is
taken at each point).
Tests sponsored by the PVRC [(ASME criteria,1972), (Langer,1971)] have shown that, for
pressure vessels subjected to cyclic loading, no crack appeared at numbers of cycles lower
than those corresponding to the design curves.
Design fatigue curves obtained in this way have been used by industry for thirty years. This
wide experience shows that they provide great reliability. While B.F. Langer notes that
"safety factors of two or twenty are not excessive and the structure should not be expected to
survive 20 times the number of cycles designed for", this experience nevertheless
demonstrates that these curves are fairly conservative and that structures, in reality, are able
to withstand much higher numbers of cycles than the design allowables.
Note: The safety factors quoted above do not account for corrosive environment (e.g.,
water, Li, etc.) or irradiation effects.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 56
ITER
G 74 MA 8 01-05-28 W0.2
C 3320.10
Welded joints
Design fatigue curves are generally derived from tests on forged batches of the material. In
other words, the design curves for fatigue analysis relate to products, but not to assembled
products such as welded joints.
Operational experience shows that most fatigue cracks appear in welded joints. Many
reasons have been ascribed to this observation - improper choice of weld metal, inadequate
qualification of the welding process and poor welding quality being the most notable ones.
More fundamentally, the reason may also lie in the metallurgical structure of the heataffected zone which may be different from that of the base metal (ferrite in austenitic steel
welds for example). This structure may also change during operation due to thermal aging.
Fatigue lives of welded joints generally show more scatter than those of the base metal. The
geometry of non-flush welds may also lead to local strain concentrations. Finally, irradiation
may have deleterious effects on weld properties.
Operational experience shows that the safety margins in the design fatigue curves generally
cover these effects for welded joints which are well made according to a well qualified
process. However, expected lives for welded joints will very likely be less than those of the
base metal.
For these reasons, design criteria generally give particular attention to the mechanical
properties of welded joints. Fatigue tests will have to be conducted on welded specimens as
part of the qualification process for the welding process adopted for ITER. The effects of
welds in fatigue design may be introduced either directly or through a welded joint
coefficient.
C 3323
Calculation of equivalent strain range De :
C 3323.1
Elastic Analysis
Although calculations using a linearly elastic material behaviour simple, reliable and require
very little material properties data, the results obtained (Ds or Dee) cannot be used directly
for fatigue analysis purposes. Low-cycle fatigue usually occurs in areas where the material's
behavior is no longer elastic. To take into account the effects of plasticity, the elastic
calculation result has to be corrected. The scope of this chapter is basically to analyze the
correction applied to the elastic calculation results to obtain a suitable approximate value
(conservative) of the strain range.
The method used to obtain an equivalent strain range from the elastic calculation results is
based on that of the references [(Petrequin,1983), (Roche,1986), (Roche, AprilÊ1987),
(Roche,1989)]:
C 3323.1.1
Use of the cyclic stress-strain curve
When a material is subjected to fully reversed strain cycles, the stress/strain curve varies with
the number of cycles and does not remain the same as that of the virgin material. For
relatively large strain ranges, this variation generally ceases after a few tens of cycles
(although it may begin again towards the end of the fatigue life, this resumption may be
neglected since it does not commence as long as the number of cycles is less than the
allowable cycles). Thus, after the initial cycles, the material becomes cyclically stable, i.e.,
SDC-IC, Appendix C - Rationale or Justification of the rules
page 57
ITER
G 74 MA 8 01-05-28 W0.2
the s - e curve becomes independent of the number of cycles, with the strain varying between
- De/2 to + De/2 and the corresponding stress varying between - Ds/2 to + Ds/2. The curve
relating De with Ds for the stabilized cycle is the cyclic stress-strain curve (Figure C 3323-1).
Strain
s
Ds
De
Time
Stress Amplitude, sa , MPa
De
Cyclic
s1
Monotonic
e1
Figure C 3323-1:
Total Strain Amplitude, eta , %
Cyclic stress strain curve
Unlike the tensile curve, the cyclic curve does not represent an actual variation of e versus s
(the actual variation is the hysteresis loop during a stabilized cycle), but is the locus of the
tips of the hysteresis loops for the stabilized cycles at different strain amplitudes.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 58
ITER
G 74 MA 8 01-05-28 W0.2
A direct application of these cyclic curves for calculating the strain variation De was
highlighted in a study based on a careful comparison between numerical calculation and tests
[(Autrusson et al., 1986), (Autrusson et al., 1987), (Autrusson et al., 1989)]. This strain is
approximately equal to the strain calculated for the monotonic loading of a fictitious elastoplastic material whose tensile curve corresponds to the cyclic curve of the real material and
which would comply with the von Mises plastic flow criterion with a normality law and
isotropic work hardening.
C 3323.1.2
Strain distribution
The strain distribution in any structure can be separated into a global and a local part. For
example, consider a bar with a slowly varying cross section, a small notch at the smallest
cross section (Figure C 3323-2), and subjected to a cyclic extension Du . The stress and strain
concentration due to the notch is fairly localized so that it does not affect the distribution of
the average strain De along the bar. This global (or nominal) distribution may be calculated
prior to calculating the local concentration effects due to the notch. In general, such a
separate treatment of the two parts is acceptable provided they have a fairly low interaction.
Du
Figure C 3323-2 A notch in a bar with slowly varying cross-section
The term global (or nominal) behaviour refers to the bar's behaviour ignoring the very
localized perturbation caused by the notch. As long as the global behaviour is elastic, the
results of the elastic calculation directly provide the true variation of the strain along the
length (nominal strain DeN). When the behaviour ceases to be elastic, this distribution is
modified and the smaller sections are strained more than indicated by the elastic calculation,
whereas the larger sections are strained less. This is due to the fact that the average stress in a
section (or nominal stress) is always distributed in the same way along the bar (it remains
inversely proportional to the cross-sectional area), whereas the extension De ceases to be
proportional to the stress, and increases faster than the latter. This phenomenon is referred to
as the elastic follow-up or spring effect as the larger sections, which are less stressed, retain
an almost elastic behaviour and act like a spring on the smaller sections in which the bulk of
the extension is concentrated. This elastic follow-up effect has been considered in detail in
section C 3024.1.4.
Local strain concentrations, such as those caused by a notch, can be handled by the
generalized Neuber rule which has been considered in detail in section C 3024.1.3 and C
3323.1.4. Other types of local stress concentrations exist which are related not to a
geometrical discontinuity but to a loading such as a non-linear thermal gradient through the
SDC-IC, Appendix C - Rationale or Justification of the rules
page 59
ITER
G 74 MA 8 01-05-28 W0.2
thickness. If it can be shown that the associated strains are independent of the material
properties, these strains are classified as kinematically determined.
C 3323.1.3
Triaxiality effect
Whereas elastic deformation takes place with a relative volume change of sH/K where sH is
the hydrostatic stress and K [=E/(1-2n)] is the bulk modulus, the plastic deformation takes
place with no volume change. Consequently, the real equivalent strain is not equal to the
calculated elastic equivalent strain. Codes such as (ASME,1989) recommend that the elastic
deformation be calculated using a fictitious Poisson's ratio which is different from the real
Poisson's ratio for elastic deformation. In SDC-IC, an alternative procedure has been adopted
from the RCC-MR in which the equivalent strain range is determined from the real strain
tensor. For this purpose, it is only necessary to multiply the elastic expression for the
equivalent strain range by a correction coefficient Kn.
C 3323.1.4
Notches and stress concentrations
Stress concentration regions are small regions in which the elastically calculated stress
notably varies and reaches much higher values than those of neighbouring regions (peak
stress). These regions generally correspond to the area surrounding a free concave surface
such as a hole or notch bottom. In a component subjected to a cyclic loading, those regions
having the highest strain range De are most prone to fatigue damage.
If the material behaviour is elastic, De is proportional to Ds and is directly given by the
elastic calculation. This is rarely the case, since in low-cycle fatigue, the damage is
significant only if De exceeds the limits of elastic behaviour. When these regions are
subjected to cyclic plasticity, the real De is always equal to or greater than the one calculated
elastically. Neuber's rule (see section C 3024.1.3) allows the real De to be calculated from
the elastic De [(Autrusson and al, 1986), (Autrusson and al, 1987), (Autrusson and al, 1989)].
Neuber's rule [C 3024.1.3] for the case of cyclic loading can be expressed as Ds.De =
Dsel.Deel..
The generalized Neuber rule has been adopted for the SDC-IC. This rule gives results which
agree well with experimental results as long as it is applied to the stress concentration effect
and not the effect related to other phenomena such as Poisson's ratio effect, the elastic followup effect and even the scale effect due to the material's heterogeneity [(Roche, JulyÊ1984)].
SDC-IC considers these effects separately, and the Neuber rule is used only to account for the
stress concentration effect. However, for reasons of convenience, this rule is used to evaluate
two of these effects [(Petrequin,1983)].
C 3323.1 5
Global cyclic plasticity and elastic follow-up effect
[(Roche-Farr,1989), (Roche,1986), (Roche,1989), (Roche,1987), (Huebel,1987), (Iida,1980)]
The previous section only considered a very localized effect (stress concentration). A stress
concentration is a local disturbance due to a small geometrical fault (notch, hole, etc.) located
in a so-called nominal stress variation DsN and strain variation DeN region. The generalized
SDC-IC, Appendix C - Rationale or Justification of the rules
page 60
ITER
G 74 MA 8 01-05-28 W0.2
Neuber rule states that, when the behaviour is no longer elastic, the stress concentration Ds
2
and the strain concentration De satisfy the equation Ds.De = KT .DsN.DeN where KT is the
theoretical stress concentration factor. This allows the value of De to be deduced from the
material's cyclic curve (Figure C 3323-3).
Ds
K t Ds N
Cyclic curve
Ds
DsN
Neuber's Rule
D s D e = K t2 DsNDeN
DeN
Kt DeN
Figure C 3323-3:
De
De
Neuber's rule
If the behaviour remains elastic in the region the concentration is located, DsN and DeN =
(DsN/E) are given by the elastic calculation, but if this region undergoes cyclic plasticity, the
elastic calculation results have to be corrected, e.g., by using the elastic follow-up factor as
discussed in C 3024.1.4.
Using the same example of the cylindrical bar as discussed in C 3024.1.4. but replacing the
stresses and strains by their respective ranges,
- ( De N - De el ) =
r -1
(Ds N - Ds el )
E
The point (DeN, DsN) representative of the real cyclic stress-strain state is the intersection of
this straight line with the cyclic stress-strain curve (Figure C 3323-4). It can hence be seen
that the multiplicative factor, which has to be applied to Deel in order to obtain the actual
strain DeN, depends on r, the so-called elastic follow-up factor.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 61
ITER
G 74 MA 8 01-05-28 W0.2
stress s
elastic
primary
s el
sN
-arctan [E/(r-1)]
reality
secondary
strain e
eel
Figure C 3323-4:
eN
r e el
Elastic follow up factor, r
The stresses are neither purely primary (r = ¥) nor purely secondary (r = 1) but often
intermediate. This is the elastic follow-up effect quantified by the coefficient r.
r=
EDe N - Ds N
Ds el - Ds N
where
Dsel = EDeel = DQ.
A detailed discussion about the value of r is given in section C 3024.1.4.
Note: For some materials like irradiated type 316 austenitic stainless steel, the strain
hardening exponent under monotonic loading reduces almost to zero (n » 0), which creates a
potential for either plastic flow localization (C 3212.1.2) or in certain cases such as
cantilevered bending or three-point bending to very large values of r (C 3024.1.4) under lowcycle fatigue loading they tend to soften.
C 3323.1.6
Poisson's ratio effect and the coefficient Kn
This problem has been fully addressed by R. Roche [(Roche, MayÊ1984), (Moulin,1985)] and
it suffices to condense these publications here. The strain variation produced during a cyclic
tension test of an isotropic material is De = Ds/ES. This strain variation is accompanied by a
variation in the test specimen's lateral dimensions (diameter) - n*De. In the elastic range, ES
is equal to Young's modulus (E) and n* is equal to Poisson's ratio (n). Beyond yield, ES is
the secant modulus ( Ds / De ) and n* is different from n.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 62
ITER
G 74 MA 8 01-05-28 W0.2
As the relative volume change,
(1 - 2 n* ) DEss
is only that due to the elastic deformation,
(1 - 2 n)
Ds
,
E
it follows that
n* = n (ES /E) + 0.5 (1 - ES /E)
(Nadai's formula)
It is this value n*, different from n, which will characterize the Poisson's effect.
As cracks always appear at the surface of the component, consideration is given to what
happens in an isotropic material near a free surface area with a normal n3 for which s3 = 0.
The load is of the radial, imposed strain type (i.e. the variations of the main components e1
and e2 are in phase).
In the simplest and most frequent case (thermal shock), the surface stresses are equi-biaxial
and the two in-plane surface strains are equal, i.e., De1 = De2 =De (for example, for a rapid
thermal shock, De= aDT). As Ds3 is zero:
Ds 3 =
ES
n*
De 3 +
(De1 + De 2 + De 3 ) = 0
1 - 2n *
1+ n*
which leads to: De3 = - 2n*De/(1 - n*) and, as a result, the equivalent variation is:
De eq =
2 1+ n*
De
3 1- n*
This result shows that this strain range is not the same as the calculated equivalent strain
range (for the same strain De) using Poisson's ratio n but is equal to Kn times the latter
where:
Ku =
1+ n* 1- n
.
1- n* 1+ n
This factor is equal to 1 when the behaviour is elastic (n* = n). It increases with plasticity to
a value 3(1 - n)/(1 + n) when n* = 0.5 (hence up to 1.61 when n = 0.3).
When De1 and De2 are different (due to a phase variation), the calculation is similar, leading
to a value of Kn which depends on the ratio d = d/e where 2e = De1 + De2 and 2d = De1 De2. This value decreases when d increases, being equal to one for d = ¥ (pure shear).
SDC-IC, Appendix C - Rationale or Justification of the rules
page 63
ITER
G 74 MA 8 01-05-28 W0.2
C 3323.1.7
Simplified procedure and the coefficient Ke
The discussions in the previous sections show that for elastic fatigue analysis, the elastic
analysis results have to be modified to account for two effects both of which tend to increase
the strain range. First, the elastic follow-up effect in the case of global cyclic plasticity and
second, stress concentration effects, if any.
The first modification involves the whole component, ignoring local disturbances such as
stress concentrations. As a result, it is necessary at this stage to exclude from the elastically
calculated stresses, the portion due to stress concentrations. Thus, the non-primary portion of
the elastic stress range has to be divided into a secondary component DQ and a peak
component DF, which is the one caused by the presence of the stress concentration. The first
stage involves DQ only and provides a means of determining the nominal stress and strain
ranges DeN and Ds N which set the boundary conditions at infinity for the zone of stress
concentration. These variations must satisfy the cyclic stress-strain equation and the elastic
follow-up equation
- ( E De N - DQ) = ( r - 1) ( Ds N - DQ) .
It is thus necessary not only to have to identify the intensity of the secondary component DQ
(by removing the peak stress range DF) but also to have to evaluate its elastic follow-up
factor r.
The second modification accounts for the local disturbance due to stress concentration in a
material subject to nominal stress and strain ranges Ds N and De N . It is then necessary to
determine the elastic stress distribution in the region, or in other words the stress
concentration factor KT . The co-ordinates of the starting point for the Neuber construction
are K t De N and K t Ds N .
Both modifications are schematically illustrated in Figure C 3323-5 (DQÊ=>ÊMÊNÊPÊRÊ=>ÊDe)
SDC-IC, Appendix C - Rationale or Justification of the rules
page 64
ITER
G 74 MA 8 01-05-28 W0.2
Ds
OP = K ON
t
P
cyclic curve
M
DQ
R
N
Ds.De = cst
De
O
De°
Figure C 3323-5:
K De
De
t
N
r De°
N
De
Detrmination of local strain range, Example 1
The above procedure, based on the difference between a global effect (elastic follow-up) and
a local effect (stress concentration factor) is not always easy. First of all, it artificially makes
a distinction between a global effect and a local effect, i.e. between the secondary stress DQ
and the peak stress DF, a distinction which may introduce a subjective judgment on the part
of the designer (particularly when using a finite-element solution). It also calls for an
evaluation of the elastic follow-up factor r relative to DQ, an evaluation which is always
approximate regardless of the method used. Given these difficulties, it was decided to adopt
a simplified procedure doing away with the need to distinguish between secondary stress and
peak stress and using a conservative value of r.
The key to the simplified procedure is to replace the straight line:
- ( E De N - DQ) = ( r - 1) ( Ds N - DQ)
by the hyperbola:
E DeN DsN = DQ2.
As shown in Figure C 3323-6, this directly provides the elastic stress range Dsel (including
2
stress concentration effect) by the relation DeDs = Dsel /E.
Neuber's rule for the stress concentration can then be written as:
2
2
DeDs = KT DsN DeN = KT DQ2/E
2
and since, by definition, Dsel = KT DQ, we can eliminate KT and obtain DeDs = Dsel /E.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 65
ITER
G 74 MA 8 01-05-28 W0.2
The practical advantage in formulating the problem this way arises from the fact that the
elastic calculation result Dsel can be used directly without having to distinguish between
peak and secondary and it is no longer necessary to estimate the elastic follow-up factor r.
Ds
K T DQ
1
DQ
5
2
cyclic curve
8
6
3
7
Ds.De = cst
De
4
O
DeN
De
r De°
1234=56784
O8 = O7.KT
Figure C 3323-6:
Determination of local strain range, Example 2
The degree of conservatism of this procedure can be assessed by the value of the elastic
follow-up factor r that is implicitly assumed. If, in the absence of elastic follow-up effect, the
elastically calculated strain range is DQ/E and if, in the presence of the elastic follow-up
effect, it is De N = K e DQ / E , it can be shown that r = Ke+1 as follows (Figure C 3323-7):
SDC-IC, Appendix C - Rationale or Justification of the rules
page 66
ITER
G 74 MA 8 01-05-28 W0.2
Ds
DQ
cyclic curve
Ds
N
Ds.De = cst
De
DQ/E DeN
O
Ds N =
DQ
DQ
, De N = K e
,
Ke
E
Figure C 3323-7
De*
De* = (1 + K e )
DQ
E
Relationship between r-factor and K e
The elastic stress and strain ranges are DQ and DQ/E, respectively. The actual nominal strain
range is De N = K e DQ / E and since De N Ds N = DQ 2 / E, the actual nominal stress range is
DQ
Ds N =
. Therefore, it follows that:
Ke
r=
EDe N - Ds N
DQ - Ds N
= Ke+1
The outcome of this is that the correction is negligible (since K e is not much greater than 1),
the minimum elastic follow-up factor is 2, which even for the conservative cantilevered beam
corresponds to a strain range exponent n = 1/4 of the cyclic stress-strain curve (see C
3024.1.4), which is not an unrealistic value for n. The value of r increases with the correction
K e thereby providing additional conservatism in the cases of very high cyclic plasticity,
which is highly desirable.
Note: Although Neuber's rule is generally conservative, it does not always provide a
conservative value of r. For example, consider the three-point bend loading of a material
with a bilinear stress-strain curve considered in Figure C 3024-4. An application of Neuber's
rule gives the following value of r:
SDC-IC, Appendix C - Rationale or Justification of the rules
page 67
ITER
G 74 MA 8 01-05-28 W0.2
r =
ep
é
ê
ê
êë
ET
ET
æS
öæ S
öù
e p ö æ Sy
y
y
E
E
ep÷ ç +
e p ÷ úú
çE +
÷ -ç E +
E
E
E
E
ç
1 - T E ÷ø çè
1 - T E ÷ø çè
1 - T E ÷ø ú
è
û
where the nomenclature for the various symbols is given with Figure C 3024-4. A plot
comparing the analytical values of r with those computed by using Neuber's rule is shown in
Figure C 3323-8. Note that although at small values of peak plastic strains Neuber's rule
gives a conservative estimate of the elastic follow-up factor r, at larger plastic strains it
provides gross underestimates of r if the tangent modulus is low. For a typical unirradiated
annealed stainless steel, ET/E=0.05 in which case Neuber's rule provides a conservative
estimate of r for all plastic strains. However, for an irradiated stainless steel ET/E can be very
low (see Figure C 3024-6) in which case Neuber's rule does not give a conservative estimate
for r at ep>0.6%. Further, Neuber's rule is inapplicable for plastic strain (flow) localization
(C 3212.1.2) which can occur in very low strain hardening materials. Fortunately, low-cycle
fatigue loading of post-irradiated 316 stainless steel has shown that the material tends to
soften and regain some of its strain hardening capability (Horsten et al., 1993), which would
tend to suppress plastic flow localization with continued cycling. However, actual in-pile
fatigue data simulating ITER loading will be needed to settle some of these questions
conclusively.
10
Neuber's Rule
Analytical
8
E /E=0
E /E=0.01
r-factor
T
T
6
4
2
E /E=0.05
T
0
0
Figure C 3323-8:
1
2
e p (%)
3
4
5
Comparison of analytically calculated r-factors with those
calculated by Neuber's rule
On the other hand, Neuber's rule can sometimes be overly conservative. For example, some
elastic thermal stresses (e.g., thermal peak stress) have no elastic follow-up effect. Therefore,
SDC-IC, Appendix C - Rationale or Justification of the rules
page 68
ITER
G 74 MA 8 01-05-28 W0.2
a thermal peaks stress should not be subjected to the above increase of strain range and
K e should be set to1.
However, primary stress ranges can lead to even higher increases in strain ranges than
indicated above if they cause cyclic plasticity and have to be accounted for.
C 3400
BUCKLING RULES (LEVELS A, C, AND D)
C3410
INTRODUCTION
Buckling effects in current light water reactors (LWR) are not important because the thickwalled LWR vessels are internally pressurized with high pressure water. However, they have
been cosidered for low-pressure fast breeder reactor structures as they involve thin parts.
Among the in-vessel structures of ITER, some may be particularly prone to buckling
problems because of their low thickness.
The rules presented in SDC-IC (F.Touboul, 1996) aim to evaluate the margins against elastic
or elasto-plastic instability. They take into account the geometric imperfections of large
structures which may be small in comparison with the diameter but significant relative to the
thicknesses, and the eventual growth of these imperfections by plasticity. The proposed rules
are based on a simplified analysis method which requires classical elastic calculations
together with buckling diagrams.
The detailed rules of SDC-IC have been based on RCC-MR Design Rules for buckling
(RB3270 and AppendixÊA7). Buckling analyses have been mainly based on the theoretical
works of A.ÊCombescure (A.Combescure., 1986). Principles of a buckling diagram were
proposed first by A.ÊHoffmann (Chavant, 1981) and simplified design rules were deduced
from these theoretical works (D. Moulin, 1984). The initial bucking diagrams in RCC-MR
were derived from these rules. With time, further improvements have been added, in
particular concerning unstable post buckling (B. Autrusson, 1985) behaviour.
C 3411
General Approach
Rigorous analysis of buckling of shells is complex. Simplified rules are necessary,
particularly during preliminary design to choose shape and thickness. The aim of the
proposed rules is to develop a method similar to those used for buckling of beams and
columns.
Application of the method involves two steps.
-
In IC 3400, a simplified analysis method is proposed which requires only classical
elastic calculations on the perfect structure in order to determine the bifurcation
load ( PE ) and the load at which the structure plastically yields ( PY).
-
Then the buckling diagram provides a reduction factor X for this load as a
function of a "deviation index" d , which is the ratio between the greatest expected
f
geometrical imperfection f0 of the actual structure to the thickness ( d = 0 ); also,
e
PE
X depends on the ratio z =
which is a parameter of the structure that accounts
PY
for plasticity.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 69
ITER
G 74 MA 8 01-05-28 W0.2
The critical buckling load is then equal to: Pc = X PE
Parameters that are needed to use the rules proposed in SDC-IC are the following:
-
loading (and pressure),
-
material hardening,
-
defect,
-
post- buckling: stable or unstable.
The buckling diagrams take into account the following effects:
·
amplification of initial defect by applied load,
·
reduction of bifurcation load due to plasticity,
·
reduction of bifurcation load due to unstable post-buckling,
·
global collapse criterion.
C 3420
THEORETICAL BASIS FOR THE BUCKLING DIAGRAMS
The buckling diagrams were first proposed in 1980 by J.ÊDevos (Chavant, 1981), in order to
predict buckling taking into account geometric imperfections and thermal gradient. Using a
very simple example (double pin-ended beam), it was shown that the collapse load could be
assessed by taking into account the imperfection shape and size and assuming elastic-plastic
behaviour.
The rationale for the construction of the buckling diagram came from the fact that when a
real structure buckles in the elastic-plastic regime, two collapse modes interact: the elastic
buckling, characterized by the Euler buckling load, and the plastic collapse, characterized by
P
P
the limit load. A diagram was then proposed in term of
as a function of
(PL: limit
PL
PE
load).
To develop the buckling diagram, the authors first established the non-linear momentcurvature relationship and considered the equilibrium equation for the deformed shape. They
presented three buckling diagrams corresponding to different levels of simplification in the
estimation of the beam collapse load.
These diagrams have been generalized for the case of shells.
C 3421
Basis for the construction of the buckling diagrams
The method is based on the case of a beam with a rectangular cross section of unit width
subjected to an axial load (cf. FigureÊC 3421-1).
SDC-IC, Appendix C - Rationale or Justification of the rules
page 70
ITER
G 74 MA 8 01-05-28 W0.2
x
P
f0
f
y
Figure C 3421-1
It is assumed that the structure has an initial deflection f0. If P is the applied load, it can be
shown that the elastic deflection of the structure is given by the following formula :
f=
f0
1-
P
PE
(1)
where PE is the Euler buckling load.
It is assumed that Eq. (1) also holds when the column undergoes plastic deformation. Thus,
the section with the highest load is subjected to an axial load P and a bending moment
M = Pf.
If the material is assumed to be elastic/perfectly plastic, the classical relationship between the
axial load and the bending moment when limit load is reached is given by:
2
M æ Pö
+ç ÷ =1
M L è PL ø
(2)
where
PL : limit load of the perfect structure (equal to the yield load)
ML: limit moment of the perfect structure.
PL e
, where
4
PL is the limit load. Eqs. (1) and (2) can be used to derive the following relation between P/PL
f
, P/PE and d = 0 :
e
It can be shown that the limit moment of a rectangular cross section is equal to
2
f P æ Pö æ
Pö æ
Pö
4 0
+ ç ÷ ç1 - ÷ - ç1 - ÷ = 0
e PL è PL ø è
PE ø è
PE ø
Using the coordinates X= P/PE and Y = P/PL, the results can be plotted parametrically as load
f
reduction diagrams with d = 0 as a parameter. The same approach can be used for elastoe
plastic material using the elastic load limit (PY) instead of the limit load PL. These
preliminary diagrams have been improved and generalized in order to take into account post
buckling behaviour (cf. 2.2) and elasto-plastic material behaviour of shell structures (cf.Ê2.4).
SDC-IC, Appendix C - Rationale or Justification of the rules
page 71
ITER
G 74 MA 8 01-05-28 W0.2
C 3422
Stable and unstable post- buckling behavior
The equation giving the final deflection f can be written with a second order approximation
(B. Autrusson, 1985) (equation (1) is a first order approximation):
P
f
f - f0
f f 2
= PE
+ mæ - 0 ö
èe e ø
e
e
(3)
where
e: thickness of the shape.
PE: Euler load
m determines the post-buckling behaviour:
m £ 0, the post-buckling is unstable
m > 0, the post-buckling is stable
Typical stable or unstable buckling behaviour is presented in Figure C 3421-2. From Eq. (3),
it is possible to derive an expression for the curvature variation c :
PE
m æ f f0 ö 2
c=
(f - f0 ) + è - ø
EI
EI e e
(4)
The following assumptions are made:
a)
although Eq.(4) is strictly true for elastic behaviour, it is assumed that it can also
be used in the plastic regime,
b) In the proposed rules, the following arbitrary conservative values were selected
(cf. Figure C 3421-2):
·
m = 0 to characterize stable post-buckling behaviour
·
m = -0.31 PE to characterize unstable post-buckling behaviour.
These values for m correspond to the results obtained by (Koiter, 1963) on a cylindrical shell
with an axisymmetric imperfection.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 72
ITER
G 74 MA 8 01-05-28 W0.2
PE
Pf/e = PE(f-fo)/e, fo = 1 mm
1.0
Pf/e = PE[(f-fo)/e]x[1-0.31(f-fo)/e], fo = 1 mm
0.5
0.0
1
2
3
4
5
6
7
f, mm
Figure C 3421-2
C 3423
Equations for the buckling diagrams
The structure is assumed to be a beam subjected to an axial load P with an initial deflection
f0. If f is the deflection under load, the load gives a moment M:
M = Pf
M can be related to the curvature c by means of the mechanical properties. It has been shown
that c is a function of P, f0 and f (cf. equation (4)). Hence:
M = g ( P, f 0 , f , material strength )
Buckling occurs when the line M = P f is tangent to the curve
M = g ( P, f 0 , f , material strength ) in the plane (M, f) (cf. Figure C 3421-3). Therefore:
P=
¶g
(P, f , f0 )
¶f
This equation can be solved by successive approximations.
The resulting diagrams are shown in Appendix B, Figure B 3421-3a (unstable behaviour) and
Figure B 3421-4a (stable behaviour).
SDC-IC, Appendix C - Rationale or Justification of the rules
page 73
ITER
G 74 MA 8 01-05-28 W0.2
M=Pf
M
Moment
M = g ( P, f, f
f0
f
0
)
Deflection
Figure C 3421-3
C 3424
Influence of temperature on the buckling diagram
In the elastic range, since both the Euler load and the critical buckling load depend on the
P
Young's modulus, the ratio Xc = c must be independent of temperature.
PE
In the plastic range, for materials such as 316L, it can be assumed that the tensile curve only
P
depends on the temperature through the yield strength and that the ratio c is independent of
PY
the temperature. Consequently, the load reduction diagrams plotted with P/PY as a function
of P/PE should not depend on temperature.
This result was confirmed by calculations made at 20, 400 and 700¡C (B. Autrusson, 1987).
C 3430
VALIDATION OF THE METHOD
The "knock down" factors contained in the buckling diagrams have been validated by
comparison with the following experiments on specimens with imperfections varying from 0
to 4 times the shell thickness (Waeckel, 1984), (Combescure, 1984), (Hutchinson, 1971),
(Dostal, 1981), (Roche, 1984), (Autrusson, 1987),
-
12 thin tubes under axial compression,
-
36 cylinders under axial pressure,
-
5 cylinders under external pressure,
-
2 hemispherical shells under external pressure,
-
3 torispherical shells under external pressure,
-
1 cylindrical shell under shear tensile stress and internal pressure,
SDC-IC, Appendix C - Rationale or Justification of the rules
page 74
ITER
G 74 MA 8 01-05-28 W0.2
-
3 torispherical heads under internal pressure.
Application of the rules to the different tests has shown (B. Autrusson, 1987) that the
experimental buckling loads were estimated conservatively in all cases except the cases of
two thin cylinders, with imperfections, under axial compression (Hutchinson, 1971).
A comparison has been made between the proposed rules, ASME rules (sectionÊIII NB 3130
and Code CaseÊNÊ284) and the above experimental data. Code Case N284 takes into account
effects of initial imperfections and plasticity. The critical stress s c is given by the following
formula:
s c = a . h . s E / FS
where
sE
elastic buckling load
a
reduction coefficient due to imperfections
h
reduction coefficient due to plasticity
FS
Safety Factor
For cylinders under external pressure, ASME rules predict lower buckling loads than
experimental ones. In general, for small imperfections, the proposed rules give reasonable
predictions of experimental buckling loads. For large imperfections, the relative performance
of the different codes varies depending on the structures.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 75
ITER
G 74 MA 8 01-05-28 W0.2
C 3500
HIGH TEMPERATURE RULES
C 3510
RULES FOR THE PREVENTION OF M-TYPE
DAMAGE
There are three creep usage fractions used in the high temperature sections of the SDC-IC.
The first one (IC 2764) is the creep usage fraction for primary stress, Ut . Ut provides a
design margin against excessive creep deformation or creep rupture damage by monotonic
primary loading. The second one (IC 2765) is the creep strain usage fraction, Ue. Ue
provides a design margin against excessive creep deformation by cyclic ratcheting. Ut is
based on a time-fraction rule, which is used in satisfying the primary stress limit. Ue is based
on a strain fraction rule, which is used to satisfy the strain limits for ratcheting analysis. The
third usage fraction (IC 2766) is the creep rupture usage fraction Wt. Wt is used to evaluate
creep rupture damage from time dependent loading using a time fraction rule. Although the
strain fraction rule is preferred by many metallurgists for evaluating creep damage because it
can be justified on the basis of a ductility exhaustion theory, the time fraction rule is easier to
use and has been adopted by both the ASME Code and RCC-MR for evaluating creep
damage.
All three usage fractions require the use of an effective stress when applied to multiaxial
loading. The effective stress se is defined in RCC-MR either by the highest stress intensity
s (as in the Monju guide) or by the following expression:
s e = 0.87 s + 0.133 tr
where: 0.87 = 2
and 0.133 =
(1)
1+ n
3 ,
1 - 2n
3 ,
tr - is the trace of the stress tensor, which is the sum of the diagonal
components or the sum of the principle stresses.
For pure tensile stress, Eq. (1) is equal to s . For biaxial tensile stress, it is 1.23 time this
stress but if the applied stress is a compressive one, then s e = 3 / 8 s , which accounts for
the fact that compressive creep is often less damaging than tensile creep.
ASME Code case N47 uses the following expression:
é æJ
öù
s e = s exp êC ç 1 - 1÷ ú
ø úû
êë è Ss
(2)
where:
J1 = å s i ,
Ss =
å s 2i ,
SDC-IC, Appendix C - Rationale or Justification of the rules
page 76
ITER
G 74 MA 8 01-05-28 W0.2
C = 0.24 for stainless steels
Eqs. (1) and (2) take into account the less damaging effects of compressive creep than in
tensile creep in stainless steels. Eq. (1) is derived from the works of Contesti (1985) and Eq.
(2) derived by Huddleston (1985 and 1993).
The choice of Eq. (2) for ASME Code case N47, is derived from a detailed study by
Huddleston (1993), comparing the predictive accuracy of various definitions of effective
stress.
Eqs. (1) and (2) have been compared by Cabrillat (1989). It was noted that the two criteria
were quantitatively similar, Eq. (1) being slightly more conservative.
A more extensive comparative study of the results given by these expressions and
experimental results is necessary to propose a final expression. Moreover, the C value has to
be determined for steels other than unirradiated stainless steels.
Nevertheless, it is advisable to limit the use of Eq. (1) or (2), to cases where the strain
distribution is well known, e.g., that obtained by elasto-plastic analyses.
C 3521.1
Elastic analysis
The proposed rule is similar to the one of ASME Code CaseÊN47Ê(1988).
The factor Kt accounts for the reduction in extreme fibre bending stress due to the effect of
creep. For thermal creep, this factor is given by:
Kt = (K + 1)/2
(Code Case N47 1988, -3223Êlevel A and B service limits)
K is the bending shape factor defined in ICÊ3251.1.1.c.
It can be shown (Fig. C 3521.1) that the Kt value depends on the exponent of Norton's law for
creep n. The above Kt value is conservative for n greater than 3, which is the usual range for
the material of interest in the ASME Code, and for thermal creep.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 77
ITER
G 74 MA 8 01-05-28 W0.2
Bending shape factor K t for creep
1.5
1.4
K/[1+1/2n]
Calculated Kt
1.3
1.2
(K+1)/2
1.1
1
2
4
6
8
Norton creep exponent n
10
12
Fig. C 3521.1
C 3530
RULES FOR THE PREVENTION OF C-TYPE DAMAGE (LEVELS A
AND C)
C 3531
C 3532
Progressive deformation or ratcheting
Summary
Both ASME Code, Section III, Subsection NH (previously known as Code Case N-47) and
RCC-MR contain high-temperature ratcheting rules which are to be used when significant
creep strain can occur in the structure. A brief summary of the rules is presented here to
show that, as in the low-temperature case, the two sets of rules also lead to similar allowable
stresses and strains for the high-temperature case.
There are more choices (tests) available in the ASME Code than in RCC-MR for meeting the
strain limits using either elastic analysis or simplified inelastic analysis. The simple elastic
analysis rule of RCC-MR is virtually identical to Test No. A-3 of the ASME Code. The main
difference between the two codes is in the simplified inelastic analysis rules, which are based
on the Bree diagram for the ASME Code and on the efficiency diagram for RCC-MR. Both
approaches define an effective stress on the basis of the applied steady primary stress and
cyclic secondary stress. It is shown that the general shape and variation of the constant
effective stress contour plots of the ASME Code and RCC-MR are similar when plotted in
the Bree diagram coordinates, i.e., steady primary stress and cyclic secondary stress
parameters.
The ASME Code allowables are based on limiting the accumulated creep strain to 1% (0.5%
for welds) whereas the RCC-MR allowables are based on limiting the usage fraction defined
in terms of accumulated creep strain as well as creep rupture time and onset of tertiary creep.
By imposing the low-temperature ratcheting limits in addition to the high-temperature
SDC-IC, Appendix C - Rationale or Justification of the rules
page 78
ITER
G 74 MA 8 01-05-28 W0.2
ratcheting limits, RCC-MR effectively does not allow structures to operate in the ratcheting
regime (R1 and R2 of the Bree diagram). Although the ASME Code allows structures to
operate in the ratcheting regimes and gives additional formulas for calculating the ratcheting
strains when they do so, these rules apply strictly to axisymmetric structures under
axisymmetric loading.
Based on the comparison and also because the Bree diagram approach was adopted for low
temperature ratcheting (C 3310), ASME code rules (Test No. A-1, A-2, A-3, and B-1) with
some modifications were adopted for high-temperature ratcheting rules in the SDC-IC.
C 3533
Limits for inelastic strains
The ASME Code (T-1310) sets the following limits for the maximum accumulated inelastic
(i.e., plastic plus creep) strains:
(a)
strains averaged through thickness, 1%;
(b)
strains at the surface due to an equivalent linear distribution of strain through
thickness, 2%;
(c)
local strains at any point, 5%.
The above limits apply to the maximum positive value of the three principal strains
(significant strain in SDC-IC terminology).
The inelastic strain limits in RCC-MR (RB 3261.2.1.2) are similar to (a) and (b) with the
exception that 1% and 2% are replaced by allowable material-dependent ductilities Dmax and
2Dmax , respectively. In RCC-MR there is no limit comparable to (c) above.
In both the ASME Code and RCC-MR, it is implicitly recognized that conducting detailed
inelastic analysis for structures operating in the creep range can be complex and may be
unwarranted in the face of uncertainties in constitutive equations, material properties, etc.
Therefore, both codes provide a number of alternative rules that can be satisfied using either
elastic analysis or simplified inelastic analysis and give the designers the option for detailed
inelastic analysis for the few cases where elastic (or simplified inelastic) analysis rules cannot
be satisfied.
C 3534
Elastic analysis rules
C 3534.1
ASME Code (T-1320)
The rules for the ASME Code are expressed in terms of the following nondimensional
primary and secondary stress parameters:
X=
Pm
Sy
(1a)
for primary membrane stress intensities, and
X=
PL + Pb / K t
Sy
SDC-IC, Appendix C - Rationale or Justification of the rules
(1b)
page 79
ITER
G 74 MA 8 01-05-28 W0.2
for primary membrane plus bending stress intensities, where
Sy = average of the Sy values at the maximum and minimum wall-averaged
temperatures during the cycle, and
Kt = (K+1)/2 where K is the bending shape factor, and
Y=
DQ
Sy
(2)
for cyclic secondary stress intensity range.
Ignoring the special requirements for piping, there are basically three elastic analysis rules,
the satisfaction of any one of which is deemed sufficient for meeting the inelastic strain
limits.
Test No. A-1
X + Y £ Sa/Sy
(3)
where
Sa = lesser of {1.25St evaluated at t = 104 h and highest wall-averaged
temperature during the cycle, and Sy as defined above}
ì 2 / 3rds minimum stress to rupture
ï
St = Min.í80% of minimum stress to onset of tertiary creep
ï min imum stress to cause a total strain of 1%
î
(4)
with all of the above three stresses evaluated at time t and temperature T.
Test No. A-2
X+Y£1
(5)
provided one extreme of the stress cycle occurs at a wall-averaged
temperature that is below the significant creep temperature.
Test No. A-3
This is a combination of the low temperature 3Sm rule (modified) plus some additional rules
to limit creep damage. In the modification of the 3Sm rule, 3Sm is replaced by 3Sm as
follows:
3Sm = (1.5Sm + SrH) when only one extreme of the stress cycle occurs at a
temperature above the significant creep temperature limit and
3Sm =(SrH + SrL) when both extremes occur at temperatures above the
significant creep temperature limit where
SrH and SrL are the relaxed stresses, associated with the high and low
temperature extremes of the cycle, obtained from uniaxial stress relaxation
analysis starting with an initial stress of 1.5Sm.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 80
ITER
G 74 MA 8 01-05-28 W0.2
There are two additional rules under Test No. A-3 to limit the creep damage.
t
åti
i
£ 0.1
(6a)
id
and
å e i £ 0.2%
(6b)
i
where
ti = hold time at temperature Ti
tid = minimum stress rupture time at temperature Ti and stress 1.5Sy at
temperature Ti
ei = creep strain at stress 1.25Sy and at temperature Ti
C 3534.2
RCC-MR (RB 3262)
In RCC-MR, there are no rules corresponding to Test No. A-1 or Test No. A-2 of the ASME
Code. Since they are generally conservative and easy to use, they have been included in the
SDC-IC as options. There is a rule very similar to Test A-3 of the ASME Code with some
minor modifications. Being less conservative than either Test No. A-1 or Test No. A-2 and
yet simpler to use than the simplified inelastic analysis rule, it has been included in the SDCIC as a third option.
C 3535
Simplified Inelastic Analysis Rules
The basic objective in the ASME Code and RCC-MR is the same. Both recognize that the
creep strain due to a steady primary stress can be significantly increased if cyclic secondary
stresses are superimposed on the steady primary stress. Both try to put an upper bound to this
enhanced creep by first defining effective stresses as functions of steady primary and cyclic
secondary stresses. The effective stresses are then used to compute and limit either the
accumulated creep strain (ASME Code) or the creep usage fraction (RCC-MR).
C 3535.1
ASME Code (T-1330)
As in the case of elastic analysis rules, there are three tests for the simplified inelastic
analysis rules which are based on the Bree diagram approach (Fig. C 3535-1). The first two
tests restrict the loading to the elastic (E), shakedown (S1 and S2), and cyclic plasticity (P)
regimes. The third test, which is the least restrictive of all, allows loading in the ratcheting
regimes (R1 and R2) but is applicable only to axisymmetric structures under axisymmetric
loading away from local structural discontinuities.
Test No. B-1
Test No. B-1 is limited to either axisymmetric structures under axisymmetric loading or
general structures with linear distribution of through-thickness thermal stress. As in Test No.
A-2, Test No. B-1 is applicable only if the average wall temperature at one extreme of the
SDC-IC, Appendix C - Rationale or Justification of the rules
page 81
ITER
G 74 MA 8 01-05-28 W0.2
cycle is below the significant creep temperature limit. An effective stress sc is determined
from the following relation:
sc = Z¥SyL
(7)
where SyL is the yield stress at the cold end of the cycle.
4
Bree Diagm.
Efficiency Diagm.
Secondary Stress Parameter, Y
3.5
R2
3
P
2.5
R1
2
S2
1.5
z=1
S1
1
Z=0.6
Z=0.4
0.5
E
Z=0.2
Z=0.8
0
0
0.2
0.4
0.6
0.8
1
Primary Stress Parameter, X
Fig. C 3535-1
SDC-IC, Appendix C - Rationale or Justification of the rules
page 82
ITER
G 74 MA 8 01-05-28 W0.2
The effective creep stress parameter Z is obtained from the Bree diagram (Fig. C3535-1)
using the following equations:
Z = X¥Y in regimes S2 and P,
(8a)
Z = Y + 1 - 2 (1 - X )Y in regime S1,
(8b)
and
Z = X in regime E.
(8c)
The creep ratcheting strain is determined using the isochronous stress-strain curves by
applying a constant stress of 1.25sc throughout the time-temperature history of the
component. The resulting value is limited to 1% for parent metal and 0.5% for welds.
Test No. B-2
Test No. B-2, which is more conservative than Test No. B-1, is applicable to any structure
and loading (i.e., not restricted to axisymmetric structures and loading). Like Test No. B-1,
Test No. B-2 is applicable only if the average wall temperature at one extreme of the cycle is
below the significant creep temperature limit. The basic approach in Test No. B-2 is the
same as in Test No. B-1 except that the effective creep stress parameter Z is obtained from a
different figure (Fig. T-1332-2).
Test No. B-3
Test No. B-3 is less conservative than either Test No. B-1 or Test No. B-2 and is applicable
to cycles in all the regimes of the Bree diagram including R1 and R2. However, being
restricted to axisymmetric structures with axisymmetric loading, it is probably of less
relevance to ITER in-vessel components. The total creep strain in this test is obtained from
the following summations:
å e i = å ni + å hi + å d i
i
i
i
(9)
i
where
ni = inelastic strain increments obtained as in Test No. B-1 but without the
use of the multiplying factor 1.25,
hi = plastic ratchet strain increments for cycles in regimes S1, S2, P, R1, and
R2 using formulas given in ASME, T-1333, and
di = enhanced creep strain increments due to relaxation of the core stresses
sc using formulas given in ASME, T-1333.
C 3535.2
RCC-MR (RB 3262.1.1)
The high-temperature ratcheting rule in RCC-MR uses the same effective stress based on the
efficiency diagram approach as that used in the low temperature ratcheting rule (C 3311.3).
In the efficiency diagram, the abscissa is chosen to be the stress ratio defined by
SDC-IC, Appendix C - Rationale or Justification of the rules
page 83
ITER
G 74 MA 8 01-05-28 W0.2
SR1 =
DQ
for primary membrane stresses
Pm
(10a)
and
SR 3 =
DQ
for primary membrane plus bending stresses
( PL + Pb / K t )
(10b)
In contrast to low temperature, the high temperature ratcheting rule for primary membrane
plus bending stresses uses the factor PL + Pb/Kt in the definition of the stress ratio instead of
PL +Pb. The effective primary stresses are defined in terms of parameters v1(SR1) and
v3(SR3)
P1 =
Pm
v1
(11a)
and
P3 =
PL + Pb / K t
v3
(11b)
where both v1 and v3 are obtained from the efficiency diagram
ì
ï1 for SR £ 0.46
ï
v(SR ) = í1.093 - 0.926SR 2 /(1 + SR )2 for 0.46 £ SR £ 4
ï1 / SR for 4 £ SR
ï
î
(11c)
To prevent ratcheting, the stresses are restricted such that
P1 £ 1.2Sm ,
(12a)
P3 £ 1.2 x 1.5Sm,
(12b)
UA(P1/1.2) £ 1
(12c)
and
UA(P3/1.2) £ 1
(12d)
where UA is the usage fraction based on time and is defined by
N
ti
i =1 Ti
UA = å
(12e)
Ti = allowable time corresponding to P1/1.2 or P3/1.2 from the time vs St
plot (see Eq. 4)
SDC-IC, Appendix C - Rationale or Justification of the rules
page 84
ITER
G 74 MA 8 01-05-28 W0.2
C 3536
Comparison between the ASME Code and RCC-MR
ratcheting limits
The simplified inelastic analysis rules of the two codes are different because one is based on
the Bree diagram and the other on the efficiency diagram approach. Before comparing the
limits imposed by the two sets of rules, two aspects of the comparison should be noted. The
first relates to the definitions of the effective stresses and the second relates to how the
effective stresses are used to set the limits.
To make a direct comparison between the effective stress parameters P1 (or P3) of RCC-MR
and Z of the ASME Code (Test No. B-1), obvious candidates for the definition of Z for RCCMR are the following:
Z ( RCC - MR ) =
P1
Pm
=
1.2Sy 1.2 v1Sy
(13a)
for primary membrane stresses (Eq. 12c) and
Z ( RCC - MR ) =
P3
P + Pb / K t
= L
1.2Sy
1.2 v3Sy
(13b)
for primary membrane plus bending stresses (Eq. 12d).
Effective stresses for both cases, i.e., primary membrane stress and primary membrane plus
bending stresses, as given by Eqs. 13a-b (RCC-MR) have been plotted along with those given
by Eqs. 8a-c (ASME Code) in Fig. C 3535-1. Note that for small values of Z (< 0.6), RCCMR is more conservative, whereas for large values of Z (> 0.6), the ASME Code is more
conservative. In Test No. B-1 of the ASME Code, the maximum value of Z is set to 1.0
which rules out the two ratcheting regimes R1 and R2. Although at first sight it might appear
that RCC-MR allows structures to operate in the ratcheting regimes without restrictions,
satisfaction of Eq. 12a with Sm = 2/3Sy implies that Z (RCC-MR) £ 0.8 which is very close
to Z = 1 for the ASME Code. Thus, for primary membrane stresses, the limiting values of
the effective stresses essentially rule out the ratcheting regimes in both codes. However, for
primary membrane plus bending stresses, Eq. 12b would imply that Z(RCC-MR) £ 1, which
would include some part of the ratcheting regimes. In a similar fashion, if Sm = 0.9Sy, then
the limiting values of Z (RCC-MR) for the two cases are 0.9 and 1.35, respectively, both of
which would be in the ratcheting regimes. In fact, RCC-MR acknowledges that the use of the
factor 1.2 in Eqs 12a-b may result in a ratcheting strain of 0.45% for primary membrane
stress and 0.8% for primary membrane plus bending stress.
In the second part of the comparison, Test No. B-1 of the ASME Code limits the accumulated
creep strain over the lifetime of the component due to a stress of 1.25sc to 1% (0.5% for
welds). The factor 1.25 on the effective stress buys an additional safety factor of 3-5 to the
computed lifetime. In RCC-MR, the usage fraction due to the effective stresses is limited to
1. The allowable time that is used in computing the usage factor is determined from
consideration of times to accumulate 1% creep strain as well as creep rupture and onset of
tertiary creep and is therefore more conservative than the ASME Code approach.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 85
ITER
G 74 MA 8 01-05-28 W0.2
C 3537
Conclusions
Since the elastic analysis rules of ASME Code include the one that is contained in RCC-MR,
we adopted Test Nos. A-1, A-2, and A-3 of the ASME Code into SDC-IC.
For the simplified inelastic analysis rules, we excluded the ratcheting regimes R1 and R2
from the allowable domain, because Test No. B-3 of ASME Code, which is the only rule that
is applicable in these regimes, is strictly valid for axisymmetric structures with axisymmetric
loading only. Test No. B-1 of ASME Code and the procedure of RCC-MR give
approximately similar effective stress values. Since we have already adopted the Bree
diagram for low temperature ratcheting, it was decided that we adopt Test No. B-1 of the
ASME Code with the following modification. In ITER divertor we are considering high
temperature alloys that may be brittle. Therefore, instead of limiting the accumulated creep
strain to 1% as in the ASME Code, we adopted the RCC-MR approach of limiting the usage
fraction (based on strain fraction rather than time fraction) allowing for the possibility of
variation of creep ductility during operation.
For the inelastic analysis ratcheting rules, we adopted the RCC-MR approach of using
material- and history-dependent ductilities (function of stress and temperature) for limiting
membrane and peak strains.
C 3600
RULES FOR WELDED JOINTS
The design rules for welded joints were adopted from those of RCC-MR (which are similar to
those of the ASME Code) with minor modifications. However, it was recognized that the invessel components of ITER are not pressure vessels, and they have unique design challenges
with regard to access for fabrication and inspection. Therefore, the relationship between
weld categories and weld types is reduced herein to a recommendation rather than a
requirement. This may be modified as specific welds are adopted and qualified by R&D.
C 3700
RULES FOR BRAZED JOINTS
(Will be issued at a later date)
C 3800
RULES FOR BOLTED JOINTS
The design rules for bolts and bolted joints were adopted from those of the ASME Code,
Section III, Subsection NB and RCC-MR, with some modifications to account for the loss of
prestress by irradiation induced creep and for the mean stress effects on fatigue. Design rules
of the ASME Code and RCC-MR are virtually identical, except that the minimum bolt area
requirements in the ASME Code are based only on the primary coolant pressure loading,
whereas those in RCC-MR also include externally applied forces and moments on the joint.
Since most of the bolted joints in ITER are not required to be leak tight, an option is provided
in the SDC-IC whereby the requirement of minimum bolt and thread cross-sectional areas
need not be satisfied if leak tightness of the joint is not required.
C 3810
Mean stress effects on fatigue
Bolts are normally prestressed to a high fraction of the yield strength. Therefore, mean stress
effects on fatigue have to be taken into account. Fig. C 3810-1 shows room temperature
fatigue data from tests conducted on IN 718 bolts at various mean stresses. The stress
SDC-IC, Appendix C - Rationale or Justification of the rules
page 86
ITER
G 74 MA 8 01-05-28 W0.2
Nominal Stress Range (MPa)
ordinate is the nominal stress without any stress concentration factor. The deleterious effect
of a mean stress is clearly evident.
R=0.1
Test Data (20¡C)
1-1/8 in. IN 718 bolt
1000
s mean=840 MPa
0.8 > R > 0.5
thread failure
head failure
runout
100
3
10
10
4
5
10
Cycles to Failure
10
6
10
7
Fig. C 3810-1
In the absence of such data at temperature, a fatigue strength reduction factor of Kf =4 (IC
2753) to be applied on smooth specimen fatigue data to account for notch effects and a mean
stress correction factor based on a modified Goodman diagram has been proposed in the
SDC-IC.
CÊ4000:
DESIGN RULES FOR MULTILAYER
HETEROGENEOUS STRUCTURES
CÊ4100
LOW-TEMPERATURE RULES
C 4200
RULES FOR THE PREVENTION OF M TYPE DAMAGE
C 4211
C 4211.1
Immediate plastic collapse
Elastic Analysis (Immediate plastic collapse)
SDC-IC, Appendix C - Rationale or Justification of the rules
page 87
ITER
G 74 MA 8 01-05-28 W0.2
x3
k=n
h/2
h
x2
h/2
k
k=1
x1
Fig. C 4201-1
Consider a typical multilayer shell element consisting of n layers with the coordinate system
shown in Fig. C 4201-1. Assuming that the ratio of the thickness to the principal radius of
curvature of the shell is small, the membrane and bending stress resultants are defined by (IC
2531-IC 2532)
h/2
N ab =
ò sabdx3 , a, b = 1, 2
-h / 2
h/2
M ab =
ò sabx3dx3 , a, b = 1, 2
-h / 2
The collapse load corresponding to a uniaxial membrane loading in the xa (a = 1, 2)
direction is
h/2
k=n
-h / 2
k =1
N L,aa = N L =
ò Sy, k dx3 =
å h k Sy , k
(1a)
with an accompanying bending moment about the midsurface
M L,aa = N L De
where De is given by
SDC-IC, Appendix C - Rationale or Justification of the rules
page 88
ITER
G 74 MA 8 01-05-28 W0.2
h/2
De =
ò sy, k x3dx3
-h / 2
(1b)
NL
For the general biaxial case, the yield condition for the kth layer is (assuming von Mises
criterion)
(s11k ) + (s22k ) - (s11k )(s22k ) + 3(s12k )
2
2
2
= S2y, k
(2)
Assuming radial loading in all the layers,
s12, k = a1s11, k and s 22, k = a 2 s11, k ,
(3)
which on substitution in Eq. (2) gives
s11, k 1 + a 22 - a 2 + 3a12 = ±Sy, k
(4)
For tensile membrane loading (i.e., plus sign on RHS of Eq. 4), integrating Eq. 4 through the
thickness and using Eq. 1a,
N11 1 + a 22 - a 2 + 3a12 = N L
which on using Eq. 3, integrated through the thickness, gives
(N11 )2 + (N 22 )
2
( ) ( )
- ( N11 ) N 22 + 3 N12
2
= NL
(5)
This also leads to a bending stress resultant about the midsurface given by
Mab = Nab De
(6)
where De is given by Eq. 1b.
For the pure bending case (Nab = 0), multiplying Eq. 4 by x3 and integrating through the
thickness and proceeding as in the case of membrane loading gives
(M11 )2 + (M 22 )
2
( ) ( )
- (M11 ) M 22 + 3 M12
2
= ML
(7a)
where ML is given by
ML = -
Vo
h/2
-h / 2
Vo
ò Sy, k x3dx3 + ò Sy, k x3dx3
(7b)
and the location x3 = zo of the plastic neutral axis is determined from
Vo
-
h/2
ò Sy, k dx3 + ò Sy, k dx3 = 0
-h / 2
(7c)
Vo
SDC-IC, Appendix C - Rationale or Justification of the rules
page 89
ITER
G 74 MA 8 01-05-28 W0.2
For combined membrane and bending loading, define the effective membrane and bending
stresses for the multilayer structure as follows:
Pm* , L =
1
2
2
N11
+ N 222 - N11N 22 + 3N12
h
(8a)
and
Pb* =
6
h2
(M11* )
2
(
*
- M11
M*22 + M*22
)
2
( )
*
+ 3 M12
2
(8b)
where
M*ab = M ab - N ab De ,
K ms =
S*y =
6 ML
and
h NL
NL
h
(8c)
(8d)
(8e)
In the above, S*y represents an effective yield strength for the multilayer composite and Kms
is an effective bending shape factor. In analogy with the single layer equation, the following
interaction equation for collapse under combined membrane and bending loading is
postulated for the multilayer structure:
Pm* , L +
Pb*
= S*y
K ms
(9)
For the pure membrane and the pure bending cases, the effective failure stresses given by Eq.
5 and 7a reduce to
Pm* , L = S*y and
(10a)
Pb* = K msS*y
(10b)
which are in agreement with Eq. 9. Further, Eqs. 8d, 8e, and 9 reduce to the corresponding
equations for a single layer structure when the number of layer n is set equal to 1.
C 4212
Immediate plastic instability
C 4212.1
Elastic Analysis (Immediate plastic instability)
As in the case of a single layer structure (C 3212.1.1), consider a multilayer structure of n
layers each following a Ramberg-Osgood uniaxial stress-plastic strain law (ignoring elastic
strains)
s = a k e n k for k = 1 to n
(1)
where s and e are the true stress and true plastic strain, respectively.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 90
ITER
G 74 MA 8 01-05-28 W0.2
Following the same procedure as was used to derive the Consid•re criterion for necking in a
uniaxial bar, the total uniaxial load in the multilayer structure is
N=
k=n
å skAk
(2a)
k =1
where sk and Ak are the true stress and current area of the kth layer.
By compatibility of deformation, the plastic strains in all the layers are equal,
ek = e for k = 1 to n
(2b)
If the original area for the kth layer is Ako, the true plastic strain is given by the constancy of
volume and
A k = A ko e -e
(3a)
and the increment of area is given by
dAk = -Ak de
(3b)
The maximum load is attained when dN = 0, which on using Eqs. (2a) and (3b) and after
dividing by the original total area Ao leads to the effective ultimate tensile strength
S*u =
1
Ao
k=n
å Ak
k =1
ds k
de
(4)
Using Eq. (1), the true strain (eu ) at maximum load can be obtained by solving
k=n
å A k a k e n (e - n k ) = 0
k
(5)
k =1
Eq. (5) reduces to the well-known Consid•re criterion for a single layer, i.e., n=1. In general,
it has to be solved numerically. An effective uniform elongation for the multilayer structure
is then given by
e*u = e e u - 1
(6)
Computed variations of the effective ultimate tensile strength and effective uniform
elongation of a bilayer structure made of type 316L(N)-IG stainless steel and CuCrZr-IG (sol.
ann., quenched and cold worked) with their thickness ratio are shown in Figs. C 4212-1 and
C 4212-2 for two sets of temperatures.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 91
ITER
G 74 MA 8 01-05-28 W0.2
Ultimate Tensile Strength (MPa)
520
316SS LN-IG
500
T (Cu) = 100¡C
T (SS) = 100¡C
480
Thickness
Averaged UTS
460
440
420
Composite UTS
400
Cu-Cr-Zr (SA-Q-Aged)
380
0
1
2
3
4
Area (Cu)/Area (SS)
5
6
Fig. C 4212-1
0.4
316SS LN-IG
Uniform Elongation
0.35
T (Cu) = 100¡C
T (SS) = 100¡C
0.3
Composite e
0.25
0.2
Thickness
Averaged e
0.15
u
u
Cu-Cr-Zr (SA-Q-Aged)
0.1
0
1
2
3
4
Area (Cu)/Area (SS)
5
6
Fig. C 4212-2
Also shown in the figures are quantities obtained by the following simple area-weighted
average formulae
S*u =
1
Ao
k =2
å Su, k A ko
(7)
k =1
where Su,k (k = 1, 2) are the UTS of the two materials
SDC-IC, Appendix C - Rationale or Justification of the rules
page 92
ITER
G 74 MA 8 01-05-28 W0.2
and
e*u
1
=
Ao
k =2
å e u, k A ko
(8)
k =1
where eu,k (k = 1, 2) are the uniform elongations of the two materials.
Note that the approximate quantities are quite close to the actual values. Thus for elastic
analysis rule, the plastic instability of a multilayer structure can be estimated by the following
equation
Pm* , L = S*u
(9)
where S*u is obtained from Eq. (7).
For inelastic analysis rules, the onset of necking can be estimated from the following
equation
Avg e pl = e*u
(10)
where e*u is obtained from Eq. (8).
C 4213
Immediate local fracture due to exhaustion of ductility
The purpose of this rule is to prevent local fracture (e.g., at notch root or at extreme fibre in
bending) due to exhaustion of ductility of any layer. Normally, in a multilayer structure
using ductile materials, a peak stress limit is not necessary, because each individual layer has
sufficient ductility to accommodate (relax) locally high elastic peak stresses by plastic flow.
However, if the ductility of any layer is severely reduced by irradiation, such a relaxation of
locally high peak stress may not be possible without cracking. The rules proposed here are
the same as those for single layer structures (C 3213).
C 4300
CÊ4310
RULES FOR THE PREVENTION OF C TYPE DAMAGE
Progressive deformation or ratcheting
The 3Sm rule has been proposed as the elastic analysis rule for preventing ratcheting. For the
inelastic analysis rule, limits on average composite membrane plastic strain are provided (C4202). Also, limits on local peak plastic strain are provided to prevent failure of individual
layers.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 93
ITER
G 74 MA 8 01-05-28 W0.2
REFERENCES
Abo-El-Ata, M. M., Simplified inelastic Analysis methods applied to Fast Breeder Reactor
Core Design. Simplified methods in pressure Vessel Analysis, ASME/CSME Montreal PVP
Conf, 1978.
Anderson, W. F., Creep ratchetting deformation and rupture damage by thermal transient
stress cycle, Design for elevated temperature, ASME, PP.Ê1-11, 1971.
Anderson, W. F., Ratchetting deformation as affected by relative variation of loading
sequences, Trans. of the 2nd. Int. Conf. on Pres. Ves. Tech., pp.Ê277-289, ASME, San
AntonioÊ1973.
ASME Boiler and Pressure Vessel Code. SectionÊIII, DivisionÊ1, SubsectionÊNB. Edition
JulyÊ1989.
ASME Boiler and Pressure Vessel Code, SectionÊIII, DivisionÊ1, SubsectionÊNB. Edition
JulyÊ1995.
ASME Code Case 1331-4., New-York,Ê1971.
ASME Code Case N47-29, Class 1 components in Elevated Temperature Service,
SectionÊIII, divisionÊI, New York, July 1990.
ASME Code Case N47-32, Class 1 components in Elevated Temperature Service,
SectionÊIII, divisionÊI, New York, July 1995.
ASME Criteria for Design of Elevated Temperature Class 1 Components, in SectionÊIII,
DivisionÊ1 of the ASME Boiler and Pressure Vessel Code. Edition May, New York,Ê1976.
ASME Criteria of the ASME for Boiler and Pressure Vessel Code for design by analysis in
sectionsÊIII and VIII, DivisionÊ2. Pressure vessels and piping design and analysis, vol.Ê1,
ASME publisher,ÊNew York, 1972.
Aus, A. B. and al, Correlation between the fatigue crack initiation at the root of the notch
and low cycle fatigue data. Flow growth and fracture, ASTM-STPÊ631, pp.Ê99-111,
Philadelphia, 1977
Autrusson, B., Comparison of ASME Section III and RCC-MR Rules for the Prevention of
Ratcheting in the Absence of Significant Creep, Report DEMT 87/427, 1987.
Autrusson, B., Acker, D., Barrachin, A., Simplified elasto-plastic analysis in the notches,
E7/8, SMIRTÊ9, Lausanne, AugustÊ1987
Autrusson, B., Acker, D., Hoffmann, A., Fatigue simplified elasto-plastic analysis, Proc.
2nd int. sem. on standards and structural analysis in elevated temperature, VeniceÊ1986
Autrusson, B., Acker, D., Hoffmann, A., Simplified elasto-plastic analysis, Int. J. pres. ves.
and piping, Vol.Ê37, pp.Ê157-169,Ê1989
Autrusson, B., Drubay, B., Rathjen, P., Bauermeister, M., Creep Cross Over Curves,
SMIRTÊ11, TOKYO, August 1991.
Autrusson, B., Acker, D., and Devos, J. , Design Rules Against Elasto-plastic buckling,
SMIRT 7, Vol.E7/4, 1983.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 94
ITER
G 74 MA 8 01-05-28 W0.2
Autrusson B., .Acker D., Devos J., Design Rules Against Elasto-plastic buckling, Smirt 8,
E7/4, Bruxelles, 1985
Autrusson B., Acker D., Hoffmann A., Discussion and validation of a simplified analysis
against buckling, Nuclear engineering and design 98, ppÊ379-393, 1987
Bernard, J., Vagner, J., Pelissier-Tanon, A., Fatigue behavior of underclad cracks in
reactor vessel nozzles. Trans. 9th SMIRT (Ed. F.ÊWittmann) , pp.Ê373-379, 1987
Billone, M. C., unpublished memorandum, Allowable Design Stresses and Design Safety
Factors for ITER Type 316L(N) Stainless Steel, Argonne National Laboratory, July 31, 1995
Brose, W.R., Dowling, N.E., and Morrow, J., Effect of Periodic Large Strain Cycles on the
Fatigue Behavior of Steels, Automotive Eng. Cong., Soc. Automotive Eng., Detroit, Mich.,
Feb. 25 - March 1, 1974.
Cabrillat, M. T., MARTIN, Ph., Validation of a New Multiaxial Criteria for Creep-Fatigue
Damage evaluation, SMIRT 10, Los Angeles, August 1989.
Chavant, C. , Devos, J. , and Hoffmann, A. , On the Influence of Geometric Imperfections
and Thermal Gradients on Elastic-Plastic Buckling of Shells, SMIRT 6, Vol. E5/6, Paris,
1981
Clement, G., Cousseran, P., Lebey, J., Moulin, D., Roche, R.L., Tremblais, A, Analyse
pratique de l'effet de rochet, Rapport CEA-R-5178, 1982.
Clement, G., Lebey, J., Roche, R. L., A design rule for thermal ratchetting, Trans. ASME,
J. of Pres. Ves. Tech., Vol.Ê108, pp.Ê188-196, MayÊ1985.
Clement, G. and al., A design rule for thermal ratchetting, 5th Int. Conf. on Press. Vess.
Tech., Vol.Ê1, Design and analysis, San Francisco, 1984.
Coffin, L. F., A study of the effects of cyclic thermal stresses on ductile metals, Trans. ASME
Vol.Ê76, pp. 931, 1954
Coghlan, W.A. and Mansur, L.K., Irradiation Creep in the Fusion Reactor First Wall, Res
Mechanica, Vol. 7, 1983.
Combescure A. et al., A review of ten years of theoretical and experimental work in
buckling. Recent advances in nuclear component testing and theoretical studies on buckling.
PVP, Vol. 89, ASME, 1984
Combescure A., Static and dynamic buckling of large thin shells. Nuclear engineering and
design 92 (1986), ppÊ339-354, winning paper of the Thomas Jaeger Prize Smirt 8, 1985
Constenti, E., Cailletaud, G., Levaillant, C., Creep Damage in 17.12.SPH Stainless Steel
Notched Specimens: Metallographical Study and Numerical Modelling. 5th Int. seminar on
inelastic analysis and life prediction under high temperature environment, Paris, July 1985.
Corum,J. L and al., Thermal ratchetting in pipes subjected to intermittent thermal
downshocks at elevated temperature, 2nd Int. Conf. on Pres. Ves. and piping, pp.Ê47-58,
ASME, San Francisco, June 1975
Cousin, M. and al, Etude des dŽformations progressives de tubes mŽtalliques par le
phŽnom•ne de rochet thermique, Rap. INSA n)1.194, FebruaryÊ1981
SDC-IC, Appendix C - Rationale or Justification of the rules
page 95
ITER
G 74 MA 8 01-05-28 W0.2
Cousin, M. and al, Rochet thermiqu - ExpŽrimentation, 6th Smirt Conf., Vol.ÊL7/6, Paris,
AugustÊ1981
Cousseran, P. and al, Ratchetting- experimental tests and practical method of analysis, Int.
conf. on Eng. Aspects of Creep, Vol.Ê2, pp.Ê143-151, Inst. of mech. Eng., Londres, 1980
D'escatha, Y., and al. A criteria for analyzing fatigue crack initiation in geometrical
singularities. 3rd int. conf. on pres. ves. tech., London,Ê1980
Devos J. , Gonthier C. , Hoffmann A. , Buckling induced by cyclic straining: analysis of
simple models, Smirt 7, E3/2, Chicago, 1983
Dostal M. , Proposal for analytical and experimental work on buckling of primary vessel
under transverse load to CEC, WGCS2, FRD/DM(81) 394, 1981
Dowling, N.E., Fatigue Life and Inelastic Strain Response Under Complex Histories For an
Alloy Steel, J. Testing and Evaluation, Vol. 1, A.S.T.M., Philadelphia, 1973.
Dupouy, J.M., Erler, J., Huillery, R. Post irradiation Mechanical Properties of Annealed
and cold worked 316 Stainless Steel after Irradiation to High Neutron Fluences. international
Conference on Radiation Effects in Breeder Reactor Structural materials. American Institut
of mining, Metallurgical and Petroleum Engineers. p83-93.1997
Ehrlich, K., Irradiation Creep in Austenitic Stainless Steels, Mech. Behavior and Nucl.
Appl. of Stainless Steels at Elev. Temp., Proc. Int. Conf. Sponsored jointly by Commission
of the EC Joint Research Center, ISPRA and the Metals Soc., London, at Villa Ponti, Italy,
May 20-22, 1981.
Fish, R.L., Notch Effect on the Tensile Properties of Fast-reactor Irradiated Type 304
Stainless Steel, Nucl. Tech., vol. 31, October, 1976.
Garkisch, H.D., Fish, R.L., and Haglund, D.R., Irradiated EBR-II Duct Crushing Test and
Analysis, WARD-D-0164, January, 1977.
Gerber , Bestimmunganzulassingen Spannunugen in Eisen Constructionen, Z.ÊBayer Arch.
Ing.ÊV. Vol.Ê6, 1874
Hakem, N., S., Etude de l'interaction des dommages de fatigue et de dŽformation
progressive. Effet d'une charge primaire (en traction) sur la rŽsistance ˆ la fatigue (en
torsion) de l'acier 304L ˆ tempŽrature ambiante". Th•se de doctorat ˆ l'univertsitŽ de
ParisÊVI, sept.Ê1987.
Hesketh, R.V., Irradiation Creep, Irradiation Embrittlement and Creep in Fuel Cladding and
Core Components, Proc. Conf. Organized by the British Nucl. Eng. Soc., London, November
9-10, 1972.
Horsten, M.G., Van Hoepen, J., and De Vries, M. I., Low-Cycle Fatigue Tests on Plate
and Electron-Beam Welded Type 316 L(N) Material, ECN-CX-93-113, 1993.
Huddleston, R., L., An improved Multiaxial Creep Rupture Strength criteria, Journal of
Pres. Ves. Tech, Transac. ASME, vol.Ê107, NovÊ1985
Huddleston, R., L., Assessment of an improved Multiaxial Strength Theory Based on Creep
Rupture Data fot Type 316 Stainless Steel, Journal of Pres. Ves. Tech, Transac. ASME,
vol.Ê115, pp.Ê177-184, mayÊ1993
SDC-IC, Appendix C - Rationale or Justification of the rules
page 96
ITER
G 74 MA 8 01-05-28 W0.2
Hudson, J.A., Nelson, R.S., and McElroy, R.J., The Irradiation Creep of Nickel and ISI
321 Stainless steel During 4 MeV Proton Bombardment, J. Nucl. Mat., Vol. 65, 1977.
Huebel, H., Plastic strain concentration in a cylindrical shell subjected to an axial or radial
temperature gradient, Trans. ASME, Vol.Ê109, pp.Ê184-187, mayÊ1987
Hutchinson, J. W., Singular behavior at the end of a crack in a hardening material, J.
mech. phys. solids, Vol.Ê16, pp.Ê13-31, 1968
Hutchinson J. W. et al., Effect of a local axisymmetric imperfection on the buckling
behavior of a circular cylindrical shell under axial compression. AIAA Journal, Vol.9,
january 1971
Hyde, T. H., Yayaoui, K.,, An experimental study of the ratchetting and creep of thick
flanged tubes subjected to steady axial mechanical loading and transient thermal loading, In.
Jour. mech. scien., Vol.Ê26, pp.Ê47-61, 1984
Iida, K., and al, safety margin of the simplified elasto-plastic fatigue analysis method of
ASME PV code sectionIII, 3rd int. conf. on pres. ves. tech., London,Ê1980
Iida, K., Asada, Y., Ikavayashi, K., Nagata, T., Simplified Analysis and Design for
Elevated Temperature Component of MANJU. Nuclear Eng. and Design, vol.Ê98, pp.Ê314317, 1987.
Inoue, K., and al, Accumulation of axial strain of a cyclindrical bar under combined cyclic
torsion and axial load, Tech. Rep. Osaka Univ. Japan, pp.Ê223-231, marchÊ1976
Jack, A. R., Price, A. T., The initiation of fatigue cracks from notches in mild steel plates.
Int. J. fracture, Vol.Ê6, pp.Ê401-409, 1970
Jung, P. and Ansari, M.I., A Correlation Between Irradiation Creep Strength and Yield
Stress of FCC Metals and Alloys, J. Nucl. Mat., Vol. 138, 1986.
Kellog company, Design of Piping Systems, second edition, Ed. by John Wiley & sons inc.,
1956
Kempl, E., Low-cycle fatigue and stress concentration at room temperature and 550¡C for
three low strength structural steels. GEAP Reports 5474, 5714, 5726, and 10170,
marchÊ1970
Koiter W. T. , The effect of axisymmetric imperfection on the buckling of cylindrical shells
under axial compression, Koninkl. Ned. Akad. Weterschap Proc. B. 66, pp 265-270, 1963
Laidler, J.J., Alloy Properties Databook, HEDL, Richland, Washington, 1978.
Langer, B. F., Design of vessels involving fatigue.. Pressure vessel engineering tehnology.
Applied science pub. Ltd., London 1971.
Langer, B. F., Design-Stress Basis for Pressure Vessels.. The William M.Murray Lecture,
Exp. Mech., pp.Ê1-11, 1970.
Langer, B. F., PVRC Interpretive report of Pressure Vessel research. SectionÊI, design
considerations. Welding Research Council Bulletin n¡95. April 1964.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 97
ITER
G 74 MA 8 01-05-28 W0.2
Lebey, J., Roche, R. L., Tests on mechanical behaviour of 304L stainless steel under
constant stress associated with cyclic strain, Fatigue of engineering materials and structures,
Vol.Ê3, MayÊ1979
Majumdar, S., Relationship between creep, creep-fatigue, and cavitation damage in type
304 austenitic stainless steel, J. Eng. Mat. Tech., Vol. 111, p.123, April, 1989
Majumdar, S., Ratcheting Problems of The US International Thermonuclear Experimental
Reactor, Fusion Tech., Vol. 21, January 1992.
Majumdar, S. and Wolters, J., Shakedown and Ratcheting Analysis of Fusion Reactor First
Wall, Fusion Tech., Vol. 29, May, 1996.
Manson, S. S., Behavior of materials under conditions of thermal stresses, NACA technical
report 2933, 1953
Manson, S. S., Thermal stress and low cycle fatigue, Mac Graw-Hill, NewÊYork, 1966
Monju guide, Structural Design Guide for Class 1 Component of Prototype Fast Breeder
Reactor for Elevated Temperature Service, PNCÊN241Ê84-08, Sept. 1984
Morrow, J., and Halford, G. R., Low cycle fatigue in torsion Proceeding, American Society
of testing materials, Vol.Ê62, 1962
Moulin D. , Appraisal of allowable loads by simplified rules, PVP conf., SanÊAntonio,
juneÊ17-21, 1984
Moulin, D., Roche, R. L., Correction of the poisson effect in the elastic analysis of low cycle
fatigue, Int. J. pres. ves. and piping, Vol.Ê19, pp.Ê213-233,Ê1985
Neuber, H., Notch stress theory, Tech. rep. AFML-TR-95-225- Wright-paterton force base
DTIC, JulyÊ1965
Neuber, H., Theory of stress cocentration for shear-strained prismatical bodies with
arbitrary non linear stress strain law. Trans. ASME, J. appl. mech., pp.Ê544-550,
decemberÊ1961
Nichols, F. A., Evidences for enhanced ductility during irradiation creep, Mater. Sci. Eng.,
Vol. 6, pp.167-175, 1970
O'Donnell, W. J. and Porowski, J. S., Upper bounds for accumulated strains due to creep
ratcheting, Trans. ASME, J. Pres. Vessels Tech., Vol. 97, No.3, August, 1975
O'Donnell, W. J. and Porowski, J. S., Bounding methods in elevated temperature design,
Trans. 7th SMIRT, Chicago, Vol. L 9/4, August, 1983
O'Donnell, W. J., Porowski, J. S., and Badlani, M. L., Simplified inelastic methods for
bounding fatigue and creep rupture damage, Trans. ASME J. Pres. Vessels Tech., Vol. 102,
p. 394, November, 1980
Petrequin, P., Life prediction in low cycle fatigue., Int. conf. on advance in life prediction
methods-ASME, pp.Ê151-156,Ê1983
Porowski, J. S. and O'Donnell, W. J. , Creep ratcheting bounds from extended elastic core
concept, Trans. 5th SMIRT, Berlin, Vol. L 10/3, August, 1979
SDC-IC, Appendix C - Rationale or Justification of the rules
page 98
ITER
G 74 MA 8 01-05-28 W0.2
RCC-MR. Design and Construction Rules for Mechanical Components of FBR Nuclear
Islands. SectionÊI, SubsectionÊB: ClassÊ1 components. edition 1985
RCCM. Design and Construction Rules for Mechanical Components of PWR Nuclear
Islands. SectionÊI, SubsectionÊB: ClassÊ1 components. edition 1985
RDT F9-7, Structural Design Guideline for FBR Core Components, Structural Design
Criteria, RDT Standard (Draft), US Department of Energy, August, 1978.
RDT F9-8, Structural Design Guideline for FBR Core Components, Guidelines for Analysis,
RDT Standard (Draft), US Department of Energy, August, 1978.
RDT F9-9, Structural Design Guideline for FBR Core Components, Rationale, RDT
Standard (Draft), US Department of Energy, August, 1978.
Rice, J. R., A path independant integral and the analysis of strain concentration by notches,
J. appl. mech., pp.Ê379-386, 1968
Rice, J. R., Rosengren, G F, Plane strain deformation near a crack tip in a power-low
hardening material, J. mech. phys. solids, Vol.Ê16, pp.Ê1-12, 1968
Rice, J. R., Stress due to a sharp notch in a work-hardening elastic plastic material loaded
by longitudinal shear. J. appl. mech., pp.Ê287-298, JulyÊ1967
Roche, R. L., Amor•age des fissures de fatigue dans les singularitŽs gŽomŽtriques. Note
CEA-N2408 S. Doc. CEN Saclay, JulyÊ1984
Roche, R. L., Analyse ˆ la fatigue: dŽtermination de la variation rŽelle de dŽformation ˆ
partir d'un calcul Žlastique. Note CEA-N2523, Doc. CEN Saclay, AprilÊ1987
Roche, R. L., Correction de l'effet de poisson dans une analyse Žlastique de la fatigue
oligocyclique, Note CEA-N2389 S. Doc. CEN Saclay, MayÊ1984
Roche, R. L., Farr, J., Design Codes and structural mechanics. Elsevier applied science Pb,
London, pp.Ê201-218,Ê1989
Roche, R. L., Fatigue analysis of crack-like geometrical discontinuities. Res. mechanica,
Vol.Ê29, pp.Ê79-94,Ê1990
Roche, R. L., Simplified elastic-plastic fatigue analysis using an elastic follow-up method.
Fatigue and fracture, ASME-PVPÊ103, pp.Ê95-99, ChicagoÊ1986
Roche, R. L., The use of elastic computations for analysing fatigue damage, Trans. 9th
SMIRT (Ed. F.ÊWittmann), Vol.ÊD, pp.Ê325-333,augustÊ1987
Roche, R. L., Use of elastic calculations in analysis of fatigue. Nuclear eng. and design,
Vol.Ê113, pp.343-355,Ê1989
Roche, R.L., Appraisal of Elastic Follow-up, SMIRT 6, EÊ5/4, Paris 1981.
Roche, R.L., French Administrative Practice and Design Codes for Nuclear Vessels. Nuclear
Eng. and Design, vol.Ê129, n¡2, pp.Ê231-238, 1991.
Roche, R.L., Simplified Elastic-Plastic Fatigue Analysis using an Elastic Folow-up Method,
PVP. Vol 103, July 1986.
SDC-IC, Appendix C - Rationale or Justification of the rules
page 99
ITER
G 74 MA 8 01-05-28 W0.2
Roche, R.L., The use of Elastic Computation for Analyzing Fatigue Damage, SMIRTÊ9,
Lausanne, August 1987
Roche R. L. and Autrusson B. , Experimental tests on buckling of torisherical heads.
Comparison with plastic bifurcation analysis. Recent advances in nuclear component testing
and theoretical studies on buckling. PVP, Vol. 89, ASME, 1984
Sachs, G., et al., Low cycle fatigue of pressure vessels materials, Proc. of ASTM, Vol.Ê60,
pp.Ê512-529,Ê1960
Sines, G., Behavior of metal under complex static and alternating stresses in metal fatigue,
McGraw Hill Book compant, NewÊYork, 1959
Singh, B. N., Foreman, A. J. E., and Trinkaus, H., Radiation hardening revisited: role of
intracascade clustering, J. Nucl. Mat., Vol. 249, pp. 103-115, October, 1997
Smith, P. D., unpublished memorandum, Effect of Work Hardening on Allowable Membrane
vs. Bending Stress Curves, ITER document S 74 LS 10 95-07-26 W 1.1, July 26, 1995
Smith, P. D., unpublished memorandum, Calculation of Effective Bending Shape Factor,
ITER document S 74 LS 11 95-08-10 W 1.1, August 10, 1995.
Swaroop, A., Mc Evily, A.,, Analytical and experimental study of thermal ratchetting, in
fatigue elvated temperature STP 525, Amer. Soc. for test and mat.of testing materials,
pp.Ê563-572, 1973
Tagart, S. W.,, Plastic fatigue analysis of pressure components, ASME publication 68-PVP3, pp.Ê1-17, 1968
Udoguchi, T. nd al, Accumulation of longitudinal strain under cyclic torsion , Fracture, 197,
Vol.Ê2, Waterloo Un. press, pp.Ê767-775, Canada, JuneÊ1977
Uga, T., An experimental study of progressive strain growth due to thermal stress
ratchetting, Trans. ASME, Vol.Ê100, pp.Ê150-156, AprilÊ1978
Uga, T., An experimental study of thermal stress ratchetting on austenitic steel by a three
bars specimen, Nucl. Eng. and Design, Vol.Ê26, pp.Ê326-335,Ê1984
Waeckel N. , et al., Experimental studies in the instabilitiy of cylindrical shells with initial
geometric imperfections. PVP, Vol. 89, ASME, 1984
Walters, L.C., McVay, G. L., Hudman, G. D., Irradiation-Induced creep in 316 and 304L
Stainless steels. International Conference on Radiation Effects in Breeder Reactor Structural
materials. American Institut of mining, Metallurgical and Petroleum Engineers. p83-93, 1967
Wei, B.C., and Nelson, D.V., Structural Design Criteria for Highly Irradiated Core
Components, PVP Design Tech. 1982 - A Decade of Progress, Ed. S.Y. Zamrik and D.
Dietrich, ASME, 1982.
Weiss, V., Sessler, J., Packmann, P., Effect of several parameters on low cycle fatigue
behavior", Acta Metallurgica, Vil.Ê11, JulyÊ1963
SDC-IC, Appendix C - Rationale or Justification of the rules
page 100