Proceedings
of PVP2009
Proceedings of the ASME 2009 Pressure Vessels and Piping
Division Conference
2009 ASME Pressure Vessels and Piping Division Conference
PVP2009
July
26-30,
2009,
Prague,
Czech
Republic
July
26-30,
2009,
Prague,
Czech
Repulblic
PVP2009-77394
PVP2009-77394
INVESTIGATION OF THE ELASTIC-PLASTIC DESIGN METHOD
IN SECTION VIII DIV. 2
Wolf Reinhardt
Atomic Energy of Canada Ltd.
Mississauga, ON, Canada
ABSTRACT
Section VIII Div. 2 contains an elastic-plastic stress
analysis option that is based on the ultimate instability load.
The performance of the method is evaluated by using the
PVRC burst disk tests. The prediction of the failure load is
attempted using both a “best estimate” and the Section VIII
Div. 2 material model. The strain limit against local failure is
assessed as well.
Starting from the notion that Design Codes try to establish
safety margins against ultimate failure (burst) and against
excessive deformation from the as-designed shape, this paper
also examines the proposed method of using a factor of safety
against instability (burst) in conjunction with a strain limit
(protection against local failure). Implications for the
application of the method to design are discussed.
INTRODUCTION
Section VIII Div. 2 of the ASME Code [1] requires the
following failure modes to be evaluated in the design-byanalysis of pressure vessels (Paragraph 5.1.1.2):
a) Protection against plastic collapse.
b) Protection against local failure.
c) Protection against collapse from buckling.
d) Protection against failure from cyclic loading.
The present paper focuses on the first two evaluations.
From the 2007 Edition of Section VIII Div. 2 forward, the
evaluation of these first two failure modes, plastic collapse and
local failure, may be by elastic-plastic analysis methods.
The method to address a) has been called “ultimate load
analysis” elsewhere [2]. Appendix F of Section III of the
ASME Code [3] uses the term “plastic instability load” for this
type of analysis. The modes of failure a), and b) can be
addressed in a single analysis for each loading combination that
requires evaluation. The evaluation methods for a) and b) are
described in detail in Paragraph 5.2.4, Elastic-Plastic Stress
Analysis Method, and in 5.3.3, Elastic-Plastic Analysis, of
Section VIII Div. 2 [1]. The term “ultimate load analysis” refers
to a technique that predicts the highest load e.g. highest
pressure), or load combination, that can be supported by the
analyzed vessel or component [2]. At the ultimate load, there is
a balance between the material and geometrical strengthening
and weakening effects that influence the load carrying capacity.
Ultimate load analysis is more familiar from fitness-forservice evaluation rather than from design of pressure vessels.
The only other place where it has been used for design in the
ASME Code is to assess nuclear plants under Level D severe
accident conditions in Appendix F of Section III. Therefore, the
present paper discusses some of the background of ultimate
load analysis.
The paper illustrates an application of the Div. 2 rules to
rupture disks under pressure loading, where experimental
results on the ultimate load are available [4], Figure 1. The
prediction of the failure load and of the failure mode (plastic
instability versus “local failure” due to insufficient ductility of
the material) is evaluated
NOMENCLATURE
b
Exponent of Voce nonlinear stress-strain curve
e
Tensile elongation
Hardening modulus (slope of linear stress-strain
KH
curve)
Instability strain in Section VIII Div. 2 stress-strain
m2
curve
R
Ratio of minimum specified yield strength to
minimum specified ultimate tensile strength
Initial slope of Voce nonlinear stress-strain curve
R0
R∞
Factor on exponential term of nonlinear stress-strain
curve
RA
Tensile reduction in area
Tr
Stress triaxiality ratio (hydrostatic to equivalent stress)
1
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asl
Exponent of Section VIII Div. 2 local strain limit
ε
True (logarithmic) strain
εp,eq
True von Mises equivalent plastic strain
εph
Limit on equivalent plastic peak strain
εf
Equivalent plastic true strain at failure
εinst, tensile Uniaxial true strain at tensile instability
σ
True stress
σeq
True von Mises equivalent stress
σy
Yield strength
σu
Ultimate tensile strength (engineering stress)
BACKGROUND
Ultimate Load Analysis
Ultimate load analysis aims to predict the failure load of
the analyzed component. In its use for design, it is, however,
not necessary to determine the ultimate failure load. Generally,
it is sufficient to establish that the component is capable of
carrying the required loads or load combinations safely without
reaching the ultimate load. To arrive at a safe design, structural
factors are applied to the specified design loadings, as defined
in Table 5.5 of [1]. If loads are applied in combination, all loads
would be increased proportionally up to the target level.
The material undergoes extensive plasticity at the analyzed
load level that includes structural factors. When loaded to the
allowable Design level, structures and components with nearneutral geometric behavior, such as cylindrical or spherical
shells, exhibit no significant permanent deformation on
unloading. As this paper will show, the same is not true for
components with strong geometric strengthening. Geometric
strengthening occurs if the deformation of a component
increases its load carrying capacity.
The ultimate load analysis strives for a realistic prediction
of the plastic behavior of a component up to the ultimate load,
i.e. to the highest load or load combination that the component
can support. The analysis requires a realistic stress-strain curve
(which might be a lower bound to the stress-strain curves of the
actual material), and any effects of geometric nonlinearity need
to be taken into account. Logarithmic strain, also called true
strain, and true stress, i.e. stress based to the deformed cross
section as opposed to the initial undeformed area, must be used.
In a static finite element analysis (FEA) that attempts to
establish force and moment equilibrium, an instability occurs at
the ultimate load. The analysis cannot be continued beyond the
instability because any attempt to increase the load(s) would
not result in a valid equilibrium state. To the user, this will be
reported as a loss of convergence at the ultimate load. The loss
of convergence makes for an easy way to decide whether the
factored loads can be carried or not.
The instability at the ultimate load is a physical reality. A
tension specimen will exhibit an instability by not allowing an
increase in load at this point. At the same time, the deformation
starts to localize and the test specimen will begin to form a
neck. Since the deformation becomes non-uniform, knowing
the gross deformation and load on the specimen does not allow
the local stress and strain to be calculated beyond the ultimate
load.
Stress-Strain Curve
A true stress - true strain curve is needed to perform an
ultimate load analysis. In a plot of the engineering stress-strain
curve, the ultimate stress can be determined directly by finding
the maximum of the stress-strain curve. The true stress- true
strain curve does not reveal the ultimate stress directly. The
ultimate true stress at the point of tensile instability is obtained
from equating the slope of the true stress - true strain curve
with the true stress.
dσ
(1)
= σ Ultimate
dε Ultimate
This shows that the slope of the stress - strain curve is an
important characteristic that determines the ultimate tensile
failure. When the true stress – true strain material model for the
analysis is developed, preparing a plot of the slope of the
stress - strain curve is recommended to ensure that the slope
decreases monotonically, a non-monotonic slope of the stress strain curve near the point of instability could affect the
convergence of the analysis.
The Section VIII Div. 2 true stress - true strain curves are
interpolated between two power-law relationships, one near
yield and one near the ultimate strength. The curves are given
by the equation
1
1
⎡
⎤
⎛ σ eq ⎞ m2
1 ⎢⎛⎜ σ eq ⎞⎟ m1
(1 − tanh H ) + ⎜⎜ ⎟⎟ (1 + tanh H )⎥⎥
ε p, eq = ⎢⎜
⎟
2 ⎢⎝ A1 ⎠
⎝ A2 ⎠
⎥
⎣
⎦ (2)
⎛ 1 σ eq − σ y
⎞
− 1⎟
H = 2⎜
⎜ K σ u −σ y
⎟
⎝
⎠
where the parameters m1, m2, A1, A2 and K are defined in Annex
3.D of Section VIII Div. 2 [1]. These parameters depend on the
true stress and strain at yield, on the ultimate tensile true stress
and true strain, and on the yield to ultimate strength ratio, R.
The curve describes the relationship between the von Mises
true stress, σeq, and the von Mises true plastic strain, εeq. The
effective (von Mises) true plastic strain is defined in terms of
the principal plastic strains as
2
ε p ,eq =
(ε p ,1 − ε p , 2 ) 2 + (ε p , 2 − ε p ,3 ) 2 + (ε p ,3 − ε p ,1 ) 2 (3)
3
At the transition of the two power-law curves, a slight
“kink” can occur. When translating the appropriate stress-strain
curve in Section VIII Div. 2 into the curve that the FE model
will use, it is recommended to identify any such kinks and
eliminate them by a suitable interpolation of the curve.
The tensile instability of the Section VIII Div. 2 stress strain curve occurs at a true strain of approximately m2. Beyond
this strain, the Code requires the stress- strain curve used for
modeling to have zero slope, i.e. to be perfectly plastic. This is
a conservatism of the Code.
2
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Plastic Strain Based Failure Criterion
Various strain based failure criteria have been proposed
outside Section VIII Div. 2. For example, the ASME NUPACK
committee has proposed stress and strain limiting criteria to be
applied to the results of a structural analysis of impact. Similar
limits were suggested for the requalification of in-service
components [5].
The strain limits in the literature fall into two different
classes. The first type addresses the instability that occurs when
the ultimate tensile load is reached. This type of criterion limits
the membrane (and sometimes also the membrane and bending)
strain across the load-bearing cross section (wall thickness).
Such a criterion is needed if large deformation effects are
inadequately considered in the analysis, e.g. if the analysis
employs small (infinitesimal) strain theory, or if beam or shell
elements are used that cannot account for local changes in cross
section.
The other type of strain based failure criterion addresses
the local strain when the ductility of the material is exhausted.
For ductile materials, this occurs at a strain level well beyond
the uniform strain at the ultimate load. The strain limit
addresses ductile tearing that occurs when voids open up inside
the material at high strains under the influence of tensile
hydrostatic stress. When the effects of material hardening and
change in geometry are correctly represented in the analysis,
plastic instabilities are also reflected correctly, and strain limits
of the first type are not necessary. Such an analysis does not
address whether the material is capable of reaching the local
strains that would be predicted at the point of instability.
An analysis methodology (using FEA) considers nonlinear
geometry effects adequately if it satisfies the following test:
• Develop a model of a simple tension bar specimen, which
could be a single rectangular element. Use the element type
as in the intended analysis, along with all the same solution
options.
• Specify a bilinear material model with yield strength σy and
plastic hardening modulus (slope, tangent modulus) KH.
Apply a purely axial displacement at the end of the model
and track the tensile load. The ultimate load should be
reached at a strain, εinst, tensile, of
theory active? Does the element support the use of true stress
and strain? Is the FEA stress-strain curve correct?
Ultimate load analysis will predict the failure load of a
structure assuming that the material has enough ductility to
accommodate the plastic strains that occur anywhere in the
structure up to the point where the ultimate load has been
reached. Generally, for the materials and manufacturing
methods that were traditionally permitted for the design of
pressure vessels, this is the case. For this reason. the plastic
instability analysis following Section III, Appendix F, does not
require the evaluation of plastic strains. The trend to more high
strength materials may be responsible for a heightened
sensitivity towards failure by excessive plastic strain. The main
area of concern would be either any notches or severe
discontinuities, or any other areas of local plastic strain
concentration where the stress trixiality is elevated.
Positive (tensile) stress triaxiality causes materials to fail at
a lower plastic strain, since it increases the propensity for void
formation in the material, the first stage of ductile rupture.
When formulating strain-based criteria, it is therefore important
to consider the level of hydrostatic stress (the sum of the three
normal stresses) relative to the shear stress that induces plastic
flow. Elevated tensile hydrostatic stresses, besides reducing the
ductility, also facilitate crack propagation. On the other hand,
compressive hydrostatic stresses result in increased ductility
and less tendency to crack initiation. Thus, if the hydrostatic
stresses are compressive, strain limits need not be imposed.
The strain-based local failure criterion given in Section
VIII Div. 2, Paragraph 5.3.3, is a criterion of the local type. It
deals with the exhaustion of ductility. The limit for the
allowable peak strain, εph, is given by
(4)
KH
• Run the analysis. At the ultimate load, the load-elongation
(or, equivalently, engineering stress-strain) plot will exhibit a
maximum. Verify this maximum occurs at the strain from (4).
If the analysis runs correctly, a stress-strain curve as in Figure 2
results. In the example analyzed in the figure, σy / KH = 0.5 was
chosen. The analysis predicted the tensile instability at a
displacement of 0.646 in with a gauge length of 1 in. The true
strain at instability from analysis is then εinst, tensile =
ln(1+0.646/1) = 0.498. The predicted strain from eq. (4) is 0.5,
which is in excellent agreement. If this behavior is not
obtained, the model does not correctly account for all of the
effects needed to predict the ultimate load and the model should
be reviewed for the following: Is “finite” deformation / strain
⎛ ⎡ 100 ⎤
⎞
e ⎤
⎡
⎜⎜ ln ⎢
⎥, 2 ln ⎢1 + 100 ⎥, m2 ⎟⎟ for ferritic
RA
100
−
⎦
⎣
⎦
⎝ ⎣
⎠
steels
⎛ ⎡ 100 ⎤
⎞
e ⎤
⎡
⎜⎜ ln ⎢
, 3 ln ⎢1 +
, m2 ⎟⎟ for austenitic
⎥
⎥
⎣ 100 ⎦
⎝ ⎣100 − RA ⎦
⎠
stainless steel
RA Minimum percent reduction in area from material
specification
e
Minimum percent tensile elongation from material
specification
The above lists only two important materials; more are
contained in the Code [1], which also contains a method to
incorporate the effect of cold work. The allowable peak strain
decreases as the triaxiality ratio of the local stress increases.
The allowable plastic strain (correctly) approaches zero as a
purely triaxial stress state (Tr = 0) is approached.
ε inst, tensile = 1 −
σy
−
α sl ⎛ Tr
⎜
ε ph = e 1+ m2 ⎝ 3
−
1⎞
⎟
3⎠
εf
(5)
where
αsl
m2
Tr
2.2 for ferritic steels, 0.6 for austenitic stainless steel
0.6(1 - R) for ferritic steel, 0.75(1 – R) for austenitic
stainless steel
σ +σ2 +σ3
triaxiality ratio = 1
σ eq
εf
3
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The strain ε f is the uniaxial stress at failure. From a
tension test, the best representation of this strain is obtained
from the reduction in area measured after rupture, which should
be used as the preferred parameter to calculate ε f . The
reduction in area is representative of the local elongation of the
material once necking (strain localization) has begun.
Assuming plastic incompressibility, the remaining area at
tensile rupture is inversely proportional to the local elongation.
The use of reduction in area from a tension test is still
conservative. Equation (6) makes it clear that the stress state at
rupture is presumed to be uniaxial in the neck, when it is well
known that the formation of a neck gives rise to a triaxial stress
state. Therefore, if a simulation of the tension test were
undertaken, the analysis would predict a local triaxiality higher
than 1, and the criterion would then (conservatively) indicate a
failure strain below ε f .
The elongation, e, either at the ultimate tensile load, or at
rupture, is a much inferior measure of the rupture strain, since
it cannot reflect the local strain in the neck. Therefore, Section
VIII Div. 2 uses a factor greater than one on the true strain
calculated from the elongation e. This can be used if the
reduction in area is not specified.
For the case where neither the reduction in area nor the
tensile elongation are specified, Section VIII Div. 2 requires the
use of the estimated power-law stress-strain curve exponent, m2
without factor, which is expected to be quite conservative,
since the strain limit is now the strain at the tensile instability.
The strain limit applies to the effective (von Mises) true
plastic strain, εp,eq. The strain limit for local failure is plotted in
Figure 3 as a function of the triaxiality ratio. The limit
decreases exponentially with increasing triaxiality. The Figure
shows the similarity of the Section VIII Div. 2 strain limit to a
similar limit on local strain that was previously given by
Cooper [5].
It is noted that, in order to evaluate the local strain
criterion, the stress-strain curve needs to extend beyond the
tensile instability strain. If the curves are obtained from a
typical tension test, this information would not be available,
because the conversion of measured elongation to true strain
becomes invalid beyond the instability. Beyond this point, the
strains in the tensile specimen start to localize, and the average
strain over the gauge length does not represent the local strain.
This could be an issue if measured stress-strain curves are used.
Section VIII Div. 2, paragraph 5.2.4.4 contains, therefore the
provision “… A true stress - strain curve model that includes
temperature dependent hardening behavior is provided in
Appendix 3.D. When using this material model, the hardening
behavior shall be included up to the true ultimate stress and
perfect plasticity behavior (i.e. the slope of the stress - strain
curves is zero) beyond this limit.” It is recommended that this
same technique be followed when measured true stress - true
strain curves are used.
The factored load at which the peak strain that is limited
must be obtained is lower than the factored load for which the
absence of plastic collapse must be demonstrated (1.7 versus
2.4) per Table 5.5 of [1]. Also, the required load is only
pressure, static head and deadweight acting simultaneously.
The presumed reason is the local character of the criterion
FE ANALYSIS OF BURST DISK TESTS
Tests and Previous Analysis
Cooper et al. reported on a large number of tests of burst
disk geometry specimens [4] under pressure loading up to the
point of burst. The disks consist of a thick annular rim and a
flat uniform-thickness central plate. The outer rim is clamped.
The relevant thin part of the disk starts out as a flat plate. As
the disk is pressurized, the central plate bulges upward, until
the final failure occurs by rupture, when the central region has
a more or less spherical shape. The location of rupture was
observed to be sometimes at the center of the disk and
sometimes beside the thick rim (Figure 1).
An advantage of the burst disk tests is that they were
performed with different materials, some with stainless steel as
an example of an extremely ductile material, and some with
higher strength carbon steel. Jones and Holliday [6] performed
an analysis of some of these tests on type 304 austenitic steel
and A533 low alloy steel. The same materials are selected for
the present study.
These two materials are of interest because the austenitic
material is very ductile, and was observed to fail by cracking at
the center of the disk after extensive plastic deformation into a
nearly hemispherical shape. The low alloy steel has lower
ductility. The failure of the disks made from this material
occurred near the rim where the local bending deformation is
large. This indicates that exhaustion of ductility was the reason
for the failure, which would correspond to the local strain
criterion controlling the failure.
Material Properties
To obtain a reference solution, a solution was obtained first
with a material model that was fitted to the data reported in [4].
The stress-strain curve was created based on the reported yield
strength, ultimate tensile strength and strain hardening
exponent, n. The curve was then represented in the form of the
Voce [7] relationship that is the basis for the ANSYS [8]
mutlilinear-isotropic hardening model. It is noted that the
difference between isotropic and kinematic hardening is rarely
important for monotonic loading when little unloading occurs.
Some specific geometries (such as some formed heads) may
exhibit reverse plasticity as loading increases, and require the
use of kinematic hardening
The computationally more efficient isotropic hardening is
used in the present analysis. The Voce stress-strain relationship
has the following functional form:
(
σ eq = σ y + R0ε p + R∞ 1 − e
−b ε p
)
(6)
The parameters used for the present runs were
304L: σy = 33.9 ksi, R0 = 124.2 ksi , R∞ = 49.5 ksi, b = 3.76
A533: σy = 74.9 ksi, R0 = 109.5 ksi , R∞ = 19.7 ksi, b = 30.0
A second set of runs was performed with the Section VIII
Div. 2 generic stress-strain curves and Code minimum tensile
4
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properties. For these runs, the following set of parameters was
used:
304L: σy = 25.0 ksi, σu = 70.0 ksi, m2= 0.482
A533: σy = 70.0 ksi, σu = 100.0 ksi, m2= 0.133
The true stress - true-strain curves are compared in Figure
4. Note that the Section VIII Div. 2 curve was truncated at the
true ultimate stress in the finite element model. Figure 7 shows
the complete multi-linear stress - strain curve for 304L that was
used in the analysis. The truncation at the ultimate stress as
well as the elimination of the slight “bump” in the Section VIII
Div. 2 curve are evident.
FE Model
Although an axisymmetric FE model could have been
used, a 3D FE model was developed as being more
representative of the models that might be used in practical
cases when more complicated geometries are being
investigated. The model was meshed using ANSYS SOLID186
20-noded hexahedral elements, taking care to apply mesh
refinement in the areas where high strains are expected, i.e. at
the center of the disk and at the transition from thin plate to
thick rim (Figure 5).
The top and bottom surface of the rim were constrained in
all three degrees of freedom. Pressure was applied to the
bottom of the central thin disk-shaped region as well as to the
interior surface of the rim and to the transition fillet (Figure 6).
The nonlinear geometry (NLGEOM) option was chosen to
include large deformation and large strain effects. The disk was
pressurized until convergence was lost after several bisections
of the load step to find the instability pressure.
All intermediate load steps were stored to enable postprocessing to check the local strain limit. The ANSYS element
table functions were used to evaluate the ratio of the local
equivalent plastic strain over the strain limit, eq. (5). The strain
limit is a function of location because it depends on the
triaxiality ratio, Tr. The local strain criterion was evaluated at
all elements. Using element table plotting, a quick overview of
critical regions can be obtained. The criterion is satisfied if the
ratio of equivalent plastic strain over strain limit is everywhere
les than or equal to one.
The evaluation of the local strain limit was reported to be
fairly sensitive to mesh refinement. In regions of high strain
gradient, most notably around the fillet between the center and
rim of the disk, mesh refinement to about 10 elements through
the thickness was therefore provided.
When the local strain limit was found to be more
controlling than the instability criterion, the strain limit was
evaluated at successively lower pressure solutions until the
pressure level was found where the strain limit was just
satisfied. .
RESULTS
“Best Estimate” Instability Analysis
The “best estimate” run, where the measured material
properties were used in the analysis, proceeded to instability for
both materials. The results are summarized in Table 1. For the
304 material, the predicted ultimate pressure matched the
experiment. At the ultimate pressure, the thin region of the disk
assumed an almost hemispherical shape, also in agreement with
the test. The result suggests that the experimentally observed
cracking at the center of the disk occurs after the ultimate load
has been reached during the phase of unstable deformation. The
true plastic strain at the ultimate load is about 80%. Rupture
occurs at an even higher local strain.
On the other hand, the FE prediction of the ultimate
pressure for the A533 material was significantly higher than the
experimental burst pressure. This is consistent with the
observed local failure near the rim, the zone of maximum
strain, which suggests that ductility exhaustion was responsible
for the burst. FE predicts the maximum local strain at the
ultimate load to occur near the rim, and being well above 100%
at that point. This strain must be above the rupture strain of this
material under the near plane strain constraint conditions at the
rim
The present analysis predicts slightly higher ultimate
pressures than those obtained in [6], and the values predicted
for the two materials are more similar. This is due to the use of
non-linear hardening in the stress-strain curve as opposed to
linear hardening in [6]. Also, the curves used in [6] were
perfectly-plastic beyond the ultimate strength.
Section VIII Div. 2 Plastic Analysis
The runs that used the Section VIII Div. 2 stress-strain
curve all resulted in a conservative prediction of the
experimental burst pressure, Table 1. The burst pressure
prediction for 304L, Figure 8, was about 20% conservative,
whereas the predicted burst pressure for A533, Figure 9, was
within less than 1% of the experimental one.
The conservatism against the “best estimate” FE analysis
was about 20%, coming from the combined effect of a lower
stress-strain curve, which is consistent with the Code minimum
tensile properties, and from a lower tensile instability strain.
Section VIII Div. 2 Strain Limit against Local Failure
The plot of the plastic strain relative to the plastic strain
limit, Figure 10, shows that the 304L material has not reached
the limit even at the ultimate pressure since the plotted ratio
does not exceed unity. Therefore, this material is not limited by
ductility exhaustion, in agreement with the experimental
evidence.
The A533 material, on the other hand, reached the strain
limit at a pressure of 2030 psi, Figure 11. This is less than half
of the predicted ultimate pressure. Although the required
structural factor for satisfying the strain limit is 1.7, whereas
the structural factor on the ultimate pressure is 2.4, the local
failure criterion (strain limit) is still the more restrictive one for
A533. The most limiting location is near the rim, so failure
would be predicted here. This is indeed the location where the
failure occurred in the tests.
Section VIII Div. 2 Allowable Pressure
The maximum allowable pressure is obtained as the lower
of the ultimate pressure divided by 2.4 or the pressure where
5
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the local plastic strain reaches the strain limit (or the ratio of
the two strains is one)divided by 1.7. For the 304L material,
this leads to an allowable pressure of 2200 psi, controlled by
the ultimate (collapse) load, Table 2. For A533, the allowable
pressure is 1190 psi, which is controlled by the plastic strain
limit (local failure criterion). These pressures have a reasonable
margin of 3-4 against the experimental failure pressures.
However, according to the plastic limit analysis performed by
Jones and Holliday [6], the allowable pressures based on limit
analysis are 136 psi for 304L and 199 psi for A533. These
values are lower than those given by plastic analysis by a factor
of 6 for A533 and by a factor of 15 for 304L. This result is
discussed further in the next section.
Table 1: Comparison of Burst Pressures from FE and
from Experiments
Burst Pressures [psi]
Material
FE Best
Estimate
FE Sect.
VIII Div. 2
FE Jones /
Holliday [6]
Test [4]
304L
6800
5270
6140
6800
A533
7070
5220
5680
5300
Table 2: Section VIII Div. 2 Allowable Pressure, Margin
to Test and Allowable Pressure from Limit Analysis
Allowable Design Pressure
Material
Sect. VIII
Div. 2 EP
[psi]
Limiting
Criterion
Test /
Sect. VIII
Div. 2
Limit
Analysis
[psi] [6]
304L
2200
Collapse
3.09
140
A533
1190
Local
Failure
4.45
200
DISCUSSION
Burst Pressure Prediction from Section VIII Div. 2
Plastic Analysis
The results in Table 2 show that the Section VIII Div. 2
elastic-plastic analysis methods perform very well overall. The
term “elastic-plastic analysis” refers here to a combination of
the ultimate load analysis in Article 5.2.4 and the local failure
criterion (strain limit) in Article 5.3.3.
For the present example, the generic stress-strain
relationship from Article 3D.3 represented the best estimate
stress-strain curve very well, based on Figure 4. To determine
the generic stress-strain curve, the Code minimum yield and
ultimate strength were used in the present study. It is expected
that designers would use this method (Code minimum
properties) most of the time, since the actual material stressstrain curve would not be known at the design stage.
Any issues on the numerical side aside, the stress - strain
curve is largely responsible for the quality of the ultimate load
prediction. The provisions in Section VIII Div. 2 result in
additional conservatism by mandating that the stress-strain
curve must be perfectly plastic beyond the true ultimate
strength. This conservatism is appropriate at this point where
there is little information about the continuation of the true
stress - true strain curve after localization (necking). This
provision is conservative if the equivalent strain at tensile
instability is exceeded locally. In the burst disk example, this
occurs particularly with the bending strain near the rim. If the
material hardening that actually occurs is underestimated, then
the support through bending is underestimated.
The burst disk simulation confirms that the ultimate load
analysis by itself is sometimes not sufficient to predict failure.
The strain limit in Article 5.3.3 is discussed in detail below.
Strain Limit
The results from the FE simulation of the burst test show
that the strain limit is useful and needed for some cases
involving high strength material with limited deformability and
high local deformation. The strain limit in Section VIII Div. 2
performed very well, being controlling for A533, but not for
304 austenitic steel. With regard to the prediction of the failure
load, the strain limit is adequately conservative for A533 when
the specified tensile reduction in area or elongation is used for
the uniaxial failure strain. On the other hand, if the parameter
m2 would be used, the strain limit would become very
conservative. The use of a factor on m2 for the purpose of
establishing the uniaxial failure strain, similar to what is
applied for the tensile elongation strain, could be considered in
future Code editions.
Based on the present A533 example, the strain limit
appears to imply a larger margin to failure than the ultimate
load analysis. This is explained by the local character of the
criterion. Failure does not actually occur when the first little
volume of material exceeds the limit, but when a significant
fraction of wall thickness is affected. This appears to be the
reason that the structural factors in Table 5.5 of [1] for Design
conditions are lower for the evaluation of the local criterion
than for the ultimate load analysis. It also suggests that seeking
to resolve the true peak strain with FE in situations with a steep
strain gradient is unnecessary, since it does not provide a result
that is meaningful for the desired failure criterion.
Use of the Section VIII Div.2 Method for Design
The design of pressure vessels has been based on two
principles, to achieve margin against burst or plastic failure and
to avoid large-scale (gross) plasticity. Although many designers
feel that the first goal, margin against burst or plastic failure,
should be the primary one, traditional Code methods have
focused on avoiding large-scale plasticity. Avoiding gross
plasticity is an important principle because it allows the vessel
to function in the shape in which it was designed.
At the extreme end of a gross-plasticity based approach is
the “double elastic slope” method of plastic analysis in Section
III, NB-3228.3 [3]. This method is based on the yield stress and
limits the plastic deformation of the structure at “collapse” to
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be less than or equal to the elastic deformation. The ultimate
strength of the structure never enters the evaluation.
The Allowable Stress S or Design Stress Intensity Sm are
defined to provide minimum margins against both the yield
strength and the ultimate strength. Therefore, the primary stress
limits (elastic analysis) and limit-load analysis provide
protection against gross plasticity, but also provide a nominal
margin against burst. The term “nominal” is used deliberately;
there is no quantifiable margin against burst or failure because
these methods are based on small strain analysis, and therefore
provide no reliable estimate of the true failure that occurs after
significant plastic deformation. In this sense the opinion that
these methods “imply” a margin of 3.5 or 3 against burst or
failure is strictly speaking incorrect, though often
approximately true in practice.
The Section VIII Div. 2 plastic analysis methods from
Article 5.2.4 and 5.3.3 represent the other extreme, as they are
based only on the margin to burst and not on an evaluation of
gross plasticity.
The present burst disk problem illustrates the possible
consequences. The burst disk is essentially the same as a flat
head of a pressure vessel. Consider the disparity of the Section
VIII Div. 2 allowable pressure and the pressure from limit
analysis (Table 2). The reason for this disparity lies in the
difference in the failure modes that are considered. The limit
analysis result considers failure in bending and arrives at a low
allowable pressure. However, if some moderate plasticity is
allowed to occur, the pressure load will be increasingly
equilibrated by membrane stresses, which is a more effective
mechanism. This is “geometric strengthening”, and is similar to
the way a clothesline with no bending stiffness at all can carry
lateral loads as an axial load after sagging.
When designing a flat head by limit analysis, it will
function as a flat head in operation. When designing it with the
Article 5.2.4 / 5.3.3 method, it may function as a dished head
after the first pressurization, when the pressure given by limit
analysis is exceeded significantly. This is not an issue of safety
since the margin against failure is preserved, but may not
preserve the intent of the design. It is most certainly something
the designer should be aware of and prevent if required. This
leads to the conclusion that the Section VIII Div. 2 method in
Article 5.2.4 / 5.3.3 does not give guidance on how to avoid
gross plasticity. If designers intend to avoid gross plasticity,
they need to devise their own methods to do so is addition to
using 5.2.4 and 5.3.3, or use one of the other methods of Article
5.2 to design certain components. A clearer statement of the
limitations of the method in the Code, similar to what was done
for limit analysis, would be helpful.
For component designs that involve significant geometric
softening, typically those that involve compressive stresses, the
elastic-plastic method may provide additional margin to plastic
collapse over traditional designs, e.g. with limit analysis or
even with elastic methods. For these conditions, the elasticplastic method would be preferred. Article 5.2.3.2 of [1] on
limit load analysis explicitly directs users to Article 5.2.4 for
these cases.
It is noted that the plastic analysis methods allow higher
primary stress levels in components than the more traditional
ASME Code methods. This can affect the degree to which a
design based on non-cyclic analysis will also pass cyclic
criteria. It is to be expected that components designed with
these elastic-plastic methods will more often be limited by
ratchet and fatigue analysis if in significant cyclic service. They
may also be less flaw-tolerant than traditional designs, and thus
it is more important that potential in-service degradation
mechanisms are considered by means like corrosion allowance,
material selection or cladding, for example. It may also be
necessary to prescribe more rigorous inspection regimes for
critical vessels.
Serviceability Criteria
Users should be aware of the heightened importance of
serviceability criteria when the elastic-plastic analysis method
per Article 5.2 is used for the design of vessels. Unlike other
design methods in the Code, these methods design purely
against ultimate failure and not against “excessive plastic
deformation”. The advantage of this is that the users can decide
on what “excessive” means for their particular application. For
a flange, “excessive” could mean a much smaller deformation
or strain than for a head with no attachments. On the other
hand, it essentially requires users to specify deformation limits
for many design details. For general cylindrical and spherical
shells away from discontinuities, the load factors will generally
prevent excessive plasticity. However, for other details, in
particular heads of other than hemispherical shape, significant
plastic deformations are likely if only the elastic-plastic rules
are used. Such details, and likely many others, will require
user-defined serviceability criteria.
Therefore, the Design Specification may need to contain
more serviceability criteria than are traditionally imposed. In
fact, it will be good practice to specify serviceability criteria for
all interfacing dimensions, both for direct interfaces such as
nozzles, flanges, etc. as well as for the general dimensions of
the vessel. Also, manufacturers may wish to use their own
default limits for plastic deformations in elements like heads.
Presumably, if a manufacturer designs a flat, dished or
torispherical head, there is a reason for the selected shape,
which would likely make it undesirable for that shape to be
obliterated upon pressurization.
Interaction with Ratcheting Provisions
The elastic-plastic design methods allow in many cases a
significantly higher level of primary stress than e.g. Section III
or the “old” Section VIII Div. 2 (pre-2007). For example, just
for a shell under pressure, the difference could be as much as
15% for the difference between the Tresca and von Mises yield
criterion plus 10%-20% for the load factor of 2.4 against burst
versus about 2.7 to 3 from the traditional wall thickness
formulas that use Sm as the limit on general primary membrane
stress.
In the absence of significant thermal load, this should not
cause any problem. However, as the level of primary stress
rises, the ratchet limit on cyclic secondary, particularly thermal,
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stresses decreases significantly. Again, this would not be a
problem since an analysis against ratcheting is also performed.
However, users may not expect that in some cases the ratchet
analysis may become more restrictive than the analysis against
plastic collapse, and may well set the required wall thickness. It
is expected that this would be the case more often than in the
previous ASME Code designs if the reduced wall thickness
from the elastic-plastic analysis against collapse is fully taken
advantage of. This also points to the need for special care in the
ratcheting assessment.
CONCLUSIONS
1. The Section VIII Div. 2 elastic-plastic analysis methods of
Article 5.2.4 and 5.3.3 are meant to provide margin against
plastic failure (collapse or burst). Failure occurs either
because the ultimate (highest sustainable) load is reached
or because the ductility is exhausted and rupture occurs.
2. The burst disk example showed that both the ultimate load
evaluation and the strain limit against rupture are needed
for the range of materials and geometries addressed by the
Code. The Section VIII Div. 2 methods performed well in
predicting the failure load and failure mode with moderate
conservatism.
3. These evaluation methods do not address the amount of
plasticity that occurs under Design loadings. For
geometrically strengthening components like the
pressurized burst disk, pressurization to the operating
pressure could lead to substantial permanent deformations
from the as-designed geometry. Other examples would be
vessel heads that are not spherical. For these types of
component, other design methods may be preferable.
4. In the absence of Code guidance to avoid permanent
deformations in elastic-plastic design, designers may need
to impose their own limits. Serviceability limits should and
Rim
5.
need to be used frequently to prescribe acceptable
deformations.
The use of the elastic-plastic method can result in more
rational designs. However, designers and owners or
preparers of the design specification should be aware that
additional care in design, operation and maintenance of
these vessels might be needed.
REFERENCES
[1] ASME Boiler and Pressure Vessel Code, 2007, Section
VIII, Division 2, Alternative Rules.
[2] Kalnins, A. and Updike, D.P., 1997, Ultimate Load
Analysis for Design of Pressure Vessels, ASME PVP
Conference Vol. 353, pp. 289-293.
[3] ASME Boiler and Pressure Vessel Code, 2007, Section III,
Division 1, Subsection NB and Appendices.
[4] Cooper, W.E., Kottcamp, E.H., and Spiering, G.A., 1971,
Experimental Effort on Bursting of Constrained Disks as
Related to the Effective Utilization of Yield Strength,
ASME Paper 71-PVP-49.
[5] Cooper, W.E., 1981, Rationale for a Standard on the
Requalification of Nuclear Class 1 Pressure Boundary
Components, EPRI Report No. NP1921, Electric Power
Research Institute.
[6] Jones, D.P., and Holliday, J.E., 2000, Elastic-Plastic
Analysis of the PVRC Burst Disk Tests with Comparison
to the ASME Code Primary Stress Limits, Journal of
Pressure Vessel Technology, v 122, n 2, p 146-151
[7] Voce, E., 1955, A Practical Strain-Hardening Function,
Metallurgia, v 51, pp. 219-226.
[8] ANSYS, 2005, ANSYS 10.0 Online Documentation,
ANSYS Inc., Canonsburg, PA
.
Failure Regions
Rim
1/8 in
1 in
Central Thin Region
∅ 6 in
∅ 10 in
Figure 1: Burst Disk Geometry (sketch)
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0.6
Predicted Strain at Ductile Tearing,
ε ph
Carbon (Ferritic) Steel
n = 0.2
m 2 = 0.3
0.5
αs = 2.2
0.4
EPRI
Sc VIII Div. 2
0.3
0.2
0.1
0
1
1.5
2
2.5
3
3.5
4
4.5
5
Triaxiality Ratio Tr
Figure 2: Load-Displacement Curve from FE
Simulation of Tension Test
Figure 3: Comparison of Section VIII Division 2 and
EPRI [5] Limits on Ductile Tearing Strain
(a)
(b)
Figure 4: True Stress - True Strain Curves, “Best Estimate” FE vs. Section VIII Div. 2 Model,
(a) 304 Austenitic Steel, (b) A533 Low Alloy Steel
Figure 5: Meshed FE Rupture Disk Model
Figure 6: FE Boundary Conditions and Pressure
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80
160
304L - Section VIII Div. 2 Material Model
140
304L - Section VIII Div. 2 Material Model
70
FE Multi-Linear Representation
True Stress [ksi]
True Stress [ksi]
FE Multi-Linear Representation
60
120
100
80
60
40
50
40
30
20
Entire Curve
20
Low Strain Region
10
0
0
0
0.2
0.4
0.6
0.8
0
True Strain
0.025
0.05
0.075
0.1
0.125
0.15
True Strain
Figure 7: FE Stress-Strain Curve for Section VIII Div. 2 Model for 304L, Multi-Linear Representation
Figure 8: Deformed Shape and Plastic Strain of 304L
Burst Disk at Ultimate Pressure – Section VIII Div. 2
Material Model
Figure 9: Deformed Shape and Plastic Strain of A533
Burst Disk at Ultimate Pressure – Section VIII Div. 2
Material Model
Figure 10: Plastic Strain / Strain Limit for 304L Burst
Disk at Ultimate Pressure
Figure 11: Plastic Strain / Strain Limit for A533 Burst
Disk, Pressure at Strain Limit
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