REMAINING
LIFE ANALYSIS FOR A PRESSURE VESSEL SUBJECTED TO CYCLIC LOADS BASED ON FRACTURE
MECHANICS
G.
Duvenhage: University of Pretoria, Pretoria, 0002 and Engineering
(Mechanical) Division, Sasol Synthetic Fuels, P0 Box 600, Secunda 2302, South
Africa
J.
Wannenburg: University of Pretoria, Pretoria, 0002, and
Laboratory for Advanced Engineering (Pty) Ltd, P0 Box 30536 Pretoria 0132,
South Africa
First
published in: Int.J.Fatigue Vol. 17, No. 7, pp477 – 483, 1995.
Keywords:
Remaining life analysis, Pressure vessel, Fracture mechanics
Fracture
mechanics has evolved into an engineering tool able to solve problems related to
the safety of structures containing defects. In practice, however, a wide
choice of parallel methods as well as unsolved discrepancies in the theory
requires engineering judgment and pragmatism. In this paper a practical case
study is presented, concerning a maintenance strategy assessment performed on a
welded pressure vessel. The pressure vessel is analyzed in terms of
leak-before-break arguments, proof test logic, fatigue initiation life, defect
initiation and unstable growth critical sizes, fatigue crack propagation that
leads to fracture or leakage, as well as probabilistic methods based on defect
size distributions and NDE detection probabilities. The paper deals with some
background to the theories employed, the different steps of the assessment methodology,
as well as with the interpretation of the result to design a practical and safe
maintenance strategy.
INTRODUCTION
The continuous use of process
equipment well beyond its intended lifespan and with the minimum downtime is of
growing importance, owing to economic constraints and environmental concerns.
It is the responsibility of fracture mechanics analysts to assess the risk
involved with the operating of equipment and to propose effective fracture
control measures.
In
past years fracture mechanics and fatigue analysis techniques have evolved into
efficient engineering tools, which can be used to assess fitness for purpose of
structures. Various different techniques exist and new ones are proposed
continually and need continued improvement to ensure it stays valid well into
the future. In this paper these concepts are applied to a case study,
demonstrating the uses and associated problems of fracture mechanics.
Table
1 summarizes the various fitness-for-purpose concepts and the related
arguments that may have relevance to the typical pressure vessel case study.
These concepts have all been evaluated in terms of the present case study. The
logical methodology followed during the assessment is depicted in Figure 1. This
methodology is also maintained for the ordering of this paper.
Concept |
Argument |
Analysis Required |
Leak-before-break |
Demonstrate that critical surface defects will not become critical through-thickness
defects when fracturing leading to leaking will occur before catastrophic
failure. |
Fracture analysis for through-thickness defects. |
Proof test logic |
Demonstrate that the largest defect that can exist after a proof test will
not propagate to critical size at work pressure before next proof test. |
Fracture analysis at proof test and working pressures, as well as
propagation analysis. |
Fatigue life |
Demonstrate that calculated fatigue life of structure is far in excess
of required vessel life. |
Fatigue (stress-life) analysis. |
Critical defect sizes |
Demonstrate that calculated critical defect sizes are larger than what
can safely be detected with NDE techniques. To be used with scheduled
inspections. |
Fracture analysis. (initiation and stable growth) |
Damage tolerance |
Demonstrate the sufficient propagation life (from NDE-detectable size
to critical size) exists such that a safe number of inspections can be
scheduled during propagation life. |
Fracture analysis and propagation analysis. |
Probabilistic methods |
Demonstrate that the probability of failure associated with any of the
above concepts is acceptable. |
Statistical analysis of defect sizes and detection probabilities. |
Table 1:
Fitness-for-purpose concepts and related arguments
DESCRIPTION OF CASE STUDY
The pressure vessel involved in
this case study is utilized in a pressure swing process. The purpose of the process
is to purify a contaminated hydrogen-rich feed stream for re-usage in a
reactor. Contaminants are removed by means of an adsorbent placed in the
vessel. Adsorption of contaminant occurs at high pressure and the adsorbent is
then cleaned at a low pressure. The pressure cycle is between 110 kPa and 2290
kPa and is repeated every 15 minutes. The operating temperature is 32^{0}C.
The
vessel was manufactured from a low-carbon pressure vessel steel and was fully
post-weld heat treated.
The
system has been operating for 12 years, and after recent failures of similar
hydrogen-containing vessels it was decided to increase the inspection
frequency. At the same time a project was initiated to perform a detailed
integrity assessment based on state-of-the-art fracture mechanics principles,
to arrive at a safe and optimal maintenance strategy.
ANALYSIS INPUTS
Stresses
Two
types of stress need to be determined. Primary stresses, which are caused by
actual loading conditions such as internal pressure and weight, are calculated
by means of finite element analysis. Figure 2 shows a model of the
vessel used in the finite element analysis. Secondary stresses refer to the
residual stresses caused by the welding. Direct determination and measurement
of these stresses are nearly impossible, and therefore literature was used to
assess the levels of these stresses parallel and transverse to the weld.
Stress-strain properties
The
material's stress-strain properties are used to generate a material-specific
failure assessment diagram, whereas the yield and ultimate stresses are needed
in the assessment of limit loads and residual stresses:
s_{
y} = 360 MPa
s_{
u} = 504 MPa
The
stress-strain properties are represented by a Ramberg-Osgood relation of the
following form:
_{} (1)
_{} (2)
Toughness
properties
Assessment documents require toughness to be expressed in terms of the
plane strain value K_{ Ic} . Owing to the thickness limitations
imposed it can be difficult to do a valid test, as the specimens become too large.
This problem can be alleviated in two ways. One is to do the test on
substandard specimens and to adapt the result for thicker specimens. Another
method is to convert J data into K_{Ic}. J data are less
stringent on the thickness limitations and smaller specimens could therefore be
used. This method is recognized to give a good representation of the toughness
data.
Specimens
were prepared so as to measure accurately the toughness properties of the weld
material and the heat affected zone. The J - Da
curve was measured by a compliance technique utilizing a clip gauge to
measure crack mouth opening displacement. Figure 3 shows the load vs
crack mouth opening displacement of the test. Crack extension is calculated
from the slopes of the partial unloading lines. The value of these slopes is
known as the compliance value. Data manipulation was done as outlined in ref.
6. Owing to the small specimen size, a linear representation of the J - Da
curve was obtained, as shown in Figure 4. From the J - Da
line the initiation toughness was obtained. These elastic-plastic toughness
data were then converted to linear elastic data. The initiation toughness as
well as the increase in toughness due to ductile tearing is shown in the
following:
K_{IC}
= 233 MPa m^{½} (3)
J_{C}(da)
= 238.1 + 681.7(da-0.02) (4)
_{} (5)
Crack
propagation rates
The
measurement of crack growth rates, especially where environmental conditions could
influence the results, is heavily influenced by loading rate and test time. It
is a known fact that a hydrogen environment would increase the crack growth rate
substantially and should therefore be anticipated.
A modified Forman equation
was chosen to represent the crack growth data. This allowed for the
incorporation of a stress ratio in the analysis. Experimental data from
literature were used in the calculation of constants.
The crack propagation rate
is represented by:
_{} (6)
where
da/dN is measured in mm cycle ^{-1 }and DK
in MPa m ^{½}.
NDE
results
Thorough NDE examination as part of statutory
inspections has been carried out on the vessel over a number of years. Some 140
defects have been detected, with sizes of up to 120 mm length and 3 mm in
depth. Unfortunately, only the maximum lengths and depths of defects found
during each inspection were recorded.
ANALYSES
Fatigue life analysis
A
fatigue life analysis based on the stress life approach was followed to obtain an
estimate of the expected life of the vessel. The method used is outlined in
ref. 4 and is considered as an alternative to BS 5500 appendix C.
The
fatigue curves, or S-N curves, are two deviations below the mean,
representing a 97.7% probability of survival. Comparisons of these S-N curves
and fatigue test data obtained from cyclic pressure tests on welded vessels
indicate that they are conservative for pressure vessels, but not excessively
so.
The
minimum life obtained for any joint was 32.2 years. This did not include
effects such as overloading, stress relief, environment and welding processes,
and it could therefore be judged as a conservative estimate of the vessel life.
Critical flaw size analyses
Low-carbon
pressure vessel steel as used in the manufacturing of this vessel behaves in a
ductile manner. This would require an elastic-plastic analysis, which accounts
for the plasticity associated with ductile fracture. A linear-elastic analysis
in this case would be unconservative. Any fracture analysis is, however, just
as good as the quality of the input data, and the non-availability of geometry
factors for an elastic-plastic analysis restricts its use.
This problem is overcome by the now well-known two criteria approach, in
which failure is predicted as a function of brittle fracture and plastic
collapse. This concept is graphically presented in a failure assessment
diagram, as shown in Figure 5.
The
two criteria are represented on the failure assessment diagram as a normalized
crack driving force K_{R} = K/K^{C} on the vertical
axis, with a brittle failure limit at K_{R} = 1; and a normalized
stress S_{R} = s/s_{y
}on the horizontal axis, with a plastic collapse limit at S_{R}
= 1. Assessment points falling inside the diagram are considered safe.
In
regions of high stress, near the yield limit, plasticity prevails, and the
horizontal line (i.e. linear elastic fracture mechanics) is not valid. Models have
been developed in which the fracture line in regions of high stresses has been
modified to account for plasticity. The limit for yield failure has also been
increased to allow for work-hardening in high-stress areas. With the use of the
failure assessment diagram it is therefore possible and valid to conduct an
elastic-plastic analysis without the elastic-plastic geometry factors and
without being overly conservative.
Three
recognized methods exist for the calculation of critical defect sizes: the CEGB
R6 method, British Standard Institution PD-6493 document, and the EPRI
elastic-plastic analysis methodology.
The
R6 and PD-6493 methods use elastic geometry factors in the analysis. Plastic
effects are incorporated by employing a failure assessment diagram. The EPRI
method is a full elastic-plastic analysis using elastic-plastic geometry
factors and material properties. The application of this method is limited
owing to the unavailability of geometry factors.
Through-thickness
flaws: An analysis was performed using CRACKWISE^{TM}
to determine the critical through-thickness defect sizes. The analysis
performed corresponds to a PD-6493 level 2 analysis. The minimum
through-thickness flaw was calculated to be 410 mm. This length is
approximately 12 times the average thickness of the vessel.
Surface
flaws: An analysis was performed to
calculate the critical surface flaw dimensions. The results are represented as
a graph of critical crack depth vs length. Flaws on the inside and the outside
of the vessel were considered.
The
critical defect analysis was conducted based on the CEGB R6 methodology. A
category 3 analysis employing a material-specific failure assessment curve
(level 2 in R6) was used. The limit crack sizes for initiation and stable crack
growth were calculated. A graphical representation of the analysis process is
shown in Figure 6.
Figure
7 shows the combined results of tolerable flaw sizes for all the different
positions and flaw orientations considered on the vessel, based on initiation
toughness at working pressure.
Axial
flaws in the circumferential weld were found to be the smallest allowable cracks.
As it was considered impractical to impose a maintenance strategy that
differentiates between different flaw locations or orientations, it was decided
to use these calculated values as the minimum allowable for the rest of the
vessel. Figure 8 shows the graph for critical depth vs length for initiation
and stable growth analysis performed on axial cracks in the circumferential
welds.
Crack propagation analysis
Various
scenarios exist in which cracks can grow to produce cracks that could cause
failure. The calculated critical flaw sizes showed that large cracks could
exist without causing fracture failure. It was therefore decided to base the
crack propagation analysis on cracks that could propagate to produce a leak.
Leakage of hydrogen could cause a fire hazard and would therefore also
constitute a failure. It would therefore be appropriate to determine the time
that a known flaw will take to start leaking. A conservative approach would
therefore be to analyse the propagation of a small circular flaw until
breakthrough.
Crack
propagation analyses are very dependent on the initial crack sizes. It was
therefore decided to do a reverse calculation, in which the end size is known
and the initial size is calculated as a function of the period in service,
assuming that the defect would be semicircular throughout its life. The
semicircular assumption would be conservative in the practical situation where
the surface breaking length is measured and the depth is assumed to be half of
this length. Figure 9 shows the crack length as a function of time to
produce a leak.
These
crack growth calculations should be conservative, as they do not account for
the crack retardation that occurs after an overload, as in the case of a
pressure test, and the worst case stress situations were used.
Probabilistic analysis
For
the purpose of a probabilistic damage tolerance argument to be discussed later,
it was necessary to obtain a mathematical description of the statistical
distribution of defect sizes that can be expected in the vessel. It has been
found in previous studies^{6 }that a log-normal distribution may be
used for this purpose:
_{} (7)
with
y = ln x being normally distributed (mean = m_{
y} , variance = s_{y}^{2}),
x = defect size (either depth or length).
By definition:
_{} (8)
The mean and variance of the natural logarithms of both the lengths and
depths of defects found during inspections should be calculated. As these data were
not available for the present case, a distribution function was used that was
obtained from inspections done on similar welds on similar thickness material.
The log-normal probability density function for defect lengths assumed to be
valid for this case is shown in Figure 10.
The
cumulative probability function is shown in Figure 11. From this graph
it is possible to read off the total fraction of defects that will be smaller
than a certain length. (e.g. 92% defects will be smaller than 40 mm in length)
Results
for various analyses have been presented. These results were subsequently
applied to different possible fitness for purpose arguments.
Fatigue life
The
results of the fatigue life analysis indicated that the vessel may potentially
fail due to a fatigue mechanism before the intended end of the design life.
This precluded the possibility of demonstrating fitness for purpose for the
total vessel life by ensuring a defect-free vessel through a once-only
inspection. Other fitness-for-purpose arguments are therefore required.
Proof test logic
Proof
test logic implies the calculation of the maximum time between inspections,
based on the time it would take for the largest crack that could exist after a
proof test to propagate to a critical crack size at working pressure, causing
failure of the vessel.
A
leak at proof test would be considered as failure of the vessel; therefore the
calculation is performed for surface flaws only.
The
crack propagation analysis is done using geometry relations that address growth
in the depth and length directions separately. A piecewise linear integration is
used to assess the crack propagation.
The
time to failure for surface flaws ranged from 17 days (1647 cycles) for shallow
cracks to 12 days (1080 cycles) for deep cracks. Figure 12 shows the
growth of two cracks from critical at proof test to critical at working
pressure.
The
results of this argument showed that a pressure test will not be valid for
demonstrating fitness for purpose for the prolonged use of the vessel.
Leak-before-break
The
results of the critical defect size analyses indicated that the vessel would
leak before catastrophic failure (the critical size of a through-thickness
defect was 410 mm, whereas a surface flaw 410 mm long would have to be 17 mm
deep before breakthrough would occur) but, because the leaking of hydrogen from
the vessel could not be permitted, leak-before-break could not be considered as
a valid fitness-for-purpose concept.
Critical flaw sizes
The
critical flaw size results indicated that it is safe to operate the vessel with
defects of easily detectable sizes. This makes it viable to employ damage
tolerance principles to demonstrate and maintain fitness for purpose for the
vessel.
Damage tolerance
It
was concluded that the failure mechanism to be guarded against would be the
propagation of a small surface flaw until leakage occurs. Inspections should
therefore be scheduled such that flaws that are large enough to propagate to
leakage before the next inspection would be detected and repaired during each
inspection Figure 9 provides the information necessary for this
decision.
A
deterministic argument in this respect would be based on a decision as to what
size crack would be large enough not to be missed during inspection. It could
be argued that a crack with a length of 25 mm would always be detected, and
from Figure 9, inspections every 20 months would be prescribed.
With
a probabilistic argument, it would be possible to calculate confidence levels
for such a decision. From Figure 11 it may be seen that 80% of all defects
will be smaller than 25 mm. This fact, together with an estimation of detection
probabilities, would make it possible to calculate the probability that a crack
that will grow to leakage before the next inspection will remain in the vessel
after an inspection.
Owing
to the lack of sufficient information, a detailed probabilistic analysis could
not be performed in detail for the present case. However, from the literature
and the estimated distribution of defect lengths assumed above, the following
calculation may be performed.
The
probability that, if a defect exists, it will grow to leakage before the next
inspection and remain in the structure after inspection and repair, is
calculated as
P_{fd}
= P(a_{c} < a) (1 - P_{d}) (9)
with P_{d} = probability that
a defect will be detected by NDE.
The
probability that the vessel contains a critical defect after inspection and
repair is given by:
P_{f}
= 1 - (1 - P_{fd})^{N} (10)
with
N = number of defects in the vessel before inspection.
P(a_{c}
< a) can be obtained from Figures 9 and
11.
For
a length of 25 mm (corresponding to the time to the next inspection of 20
months from Figure 9) the probability for a larger crack to exist will be 20%
(from Figure 11).
A
typical value for P_{d} cited in the
literature is 90%. Some 4-6 defects have been detected on average per
inspection, which gives an indication of the value to be used for N.
Using
these values, a probability of failure (leakage) could be estimated as 11%.
Decreasing the time to inspection to 5 months (corresponding to a crack length
of 35 mm from Figure 9) would change this to 6%.
These
results seem to be unacceptable. However, there are two important factors that
make these calculations overly conservative. First, the assumption that a
detected flaw would be semicircular implies very conservative growth rates. To
obtain more realistic results would, however, require knowledge of the
two-dimensional size distributions of defects to be expected in the vessel,
which is currently not available.
Second,
it is a known fact that detection probabilities are a function of defect size
(a larger defect would be more easily detected than a smaller defect), which
has not been taken into account, again owing to the lack of information on
which to base more realistic estimates.
CONCLUSIONS
This
paper presented a practical case study to demonstrate the use of various
fracture mechanics analysis techniques, arguments and methods, to develop a
maintenance strategy for a welded pressure vessel.
Material
toughness properties were determined using the unloading compliance technique,
yielding J - Da curves.
These tests are difficult to execute, and confusion reigns in the literature
concerning the interpretation of the results, but it was demonstrated that the
R6 interpretation of these results yields useful engineering values for
toughness.
Critical
defect size analyses were performed according to the R6 method and the PD-6493
method. The analyses demonstrated that realistic results could only be obtained
by performing an elastic-plastic analysis, incorporating stable crack growth.
Fatigue crack propagation analyses were performed, which demonstrated the
importance of engineering pragmatism concerning the choice of initial crack
size.
A
leak-before-break situation was shown to exist for the vessel, but fitness for
purpose could not be proven using
this result, as leaking may cause an explosion. Proof test logic was followed,
which also could not demonstrate fitness for purpose.
It
was found that the vessel may be maintained using damage tolerance principles.
The limiting scenario that would govern the inspection schedules was found to
be the propagation of surface cracks through-thickness to cause leakage.
Inspections scheduled for every 20 months would require the detection and
repair of all defects longer than 25 mm.
A
probabilistic assessment of the risk of leakage, should the above schedules be
maintained, yielded very conservative results. It would be possible to obtain
more realistic results, should more complete inspection results be recorded in
future.
REFERENCES
1) ‘Guidance on methods for assessing the
acceptability for flaws in fusion welded structures', PD 6493, British
Standards Institution, 1991
2) Kumar, V., German, M.D. and Shih, C.F. 'An
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Report NP 1931,1981, Palo Alto, California.
3)
Mime, I., Ainsworth, R.A et al. Int. J. Pressure Vessels Piping 1988, 32, 105
4) Assessment of vessels subject to fatigue:
alternative approach to method in appendix C, Enquiry Case 5500/79: May 1986 to
BS 5500:1988, British Standards Institution.
5) Duvenhage,
G.H. Fitness-for-purpose assessment of a pressure vessel subjected to cyclic
loads based on fracture mechanics methodologies, Masters Degree Thesis,
University of Pretoria, 1994.
6) Wannenburg, J.A. Fracture mechanics methodology
for the integrity assessment of welded structures, Masters Degree Thesis,
University of Pretoria, 1992.
7) Broek, D. 'The Practical Use of Fracture
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