June 2009/ John Wægter
Note 5.1
Stress range histories and Rain Flow counting
Introduction......................................................................................................................................2
Stress range histories........................................................................................................................3
General .........................................................................................................................................3
Characterization of irregular fatigue loading ...............................................................................4
Stress range histogram .................................................................................................................5
Long-term exceedance diagram ...................................................................................................5
Stochastic processes.....................................................................................................................7
Rain flow counting...........................................................................................................................9
Cycle counting .............................................................................................................................9
Basic principle in Rain Flow Counting......................................................................................10
References......................................................................................................................................12
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Stress range histories and Rain Flow counting
Introduction
In the previous sections we have considered most of the elements of a fatigue design. The starting
point has been a given structure and a known loading from which we are able to construct a relevant
stress history as basis for the fatigue design.
When we use the nominal stress approach, a particular type of detail is assigned to a particular
fatigue class with a given S-N curve. The design stress is determined as the nominal stress adjacent
to the weld under consideration and can often be determined using beam theory. Alternatively, we
may determine the nominal stress based on the results from a coarse mesh FE analysis. In some
cases the effects of macro-geometric features (e.g. a hole in a plate) must be considered so that the
relevant stress for fatigue design is
σ local = SCFmacro ⋅ σ nom
(1)
where SCFmacro is the stress concentration from macro-geometric effects.
When we use the hot spot stress approach, we determine a local stress at the hot spot that considers
the influence of the structural discontinuities due to the geometry of the joint, but excludes the
stress raising effects (notch) from the weld itself. Hot spot stresses are in principle structural
stresses (geometric stresses) based on the theory of shells determined on the surface at the hot spot
of the component which is to be assessed. When the stress concentration is captured in simple
tubular joints using parametric stress concentration formulas we apply the relationship given in
Equation (2) taking σ nom as the beam stresses in the brace considered.
σ hotspot = SCF par ⋅ σ nom
(2)
where SCF par is the stress concentration from parametric stress concentration formulas.
Alternatively, we may determine the hot spot stress using FE analysis, based on shell or solid
elements. The nominal stresses in Equation (2) are then the stresses without considering the stress
raising effect due to the joint geometry, e.g. the rightmost stresses in Figure 1.
Since the SCF in Equation (2) in case of membrane stress σ m differs from the SCF in case of
bending stress σ b , the total stress is in principle given by a combination of stresses as indicated in
Equation (3).
σ hotspot = SCFm ⋅ σ m + SCFb ⋅ σ b
(3)
2
Figure 1
Determination of hot spot stress
For some structural details with pronounced plate bending, load shedding may justify a reduction in
the contribution from the bending stresses to the hot spot stresses in Equation (3).
Typically the hot spot stresses (Figure 1) will be taken as either the extrapolated principal stresses,
or the extrapolated resulting normal to weld stresses found from the FE analysis.
In the above considerations we have assumed that a relevant stress history for the fatigue design
was available. In the following sections we will focus on the details of establishing a representative
stress history and use the Rain Flow Counting Method as a basis for the calculation of the fatigue
damage.
Stress range histories
General
Real life structures prone to fatigue will experience stresses that vary with time, often in a very
complicated manner. These stress histories are generally the results of an irregular load history and
the dynamic response of the structure considered.
In some cases the dynamic response is negligible and the response becomes quasi-static and the
stress history closely follows the external loading. In other cases the dynamics influence the stress
history greatly.
In both cases the basis for the fatigue calculation is the long-term stress distribution in the detail
considered.
The methods considered apply for high cycle fatigue design, which for convenience may be defined
as a fatigue design for more than 104 cycles. The distinction between high cycle fatigue and low
cycle fatigue is reflected in most design standards as high cycle fatigue design standards normally
give S-N curves starting with N = 104 cycles.
For low cycle fatigue design the stress range concept is not immediately valid, since typically the
low cycle fatigue strength is governed by (large inelastic) strains.
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Characterization of irregular fatigue loading
For a typical offshore structure the fatigue load history spans a period of 20 years corresponding to
about 108 wave load cycles (assuming an average wave load period of 6 sec.).
Figure 2 shows the terminology used for irregular loading histories, and the most important
concepts are explained below.
For practical design the time history must be reduced to a manageable format still retaining the
characteristics of the loading, and the stress history must be broken down into individual stress
ranges with an associated number of stress cycles. Different approaches for this purpose will be
discussed in the following.
Figure 2
Definition of terms related to irregular load histories
The most important concepts are
Reversal
the derivative of the load-time history changes sign.
Peak
the derivative of the load-time history changes from positive to negative
sign.
Valley (or trough)
the derivative of the load-time history changes from negative to positive
sign.
Range
is the algebraic difference between successive valley and peak loads
(positive range) or between successive peak and valley loads (negative
range). Sometimes the range may have different definitions depending
on the counting method used.
Mean crossing
or zero crossing, is the number of times that the load-time history crosses
the mean load level during a given duration of the history. Normally
only crossings with positive slopes are counted.
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Stress range histogram
The stress range histogram also denoted the stress range spectrum, is a representation of stress
ranges and the associated number of cycles. It may directly be used as the basis for the fatigue
damage calculation, see Figure 3.
In practice the spectrum is approximated by a manageable number of stress range blocks
characterized by ( Δσ i , ni ) . For 108 cycles or more, 20 stress levels will normally be sufficient,
while a lower number of stress range blocks may be sufficient for a smaller number of cycles. All
cycles in a stress range block should be associated with the mean of the stress ranges in the block.
Figure 3
Simplification of stress range spectrum
The damage induced by each stress range block is determined as its contribution to the Miner’s sum
and is found from
⎛n⎞
ΔDi = ⎜ ⎟
⎝ N ⎠i
(4)
based on the appropriate S-N curve and Δσ i .
Long-term exceedance diagram
In the following the distribution of stress ranges from wave load will be derived based on a
simplified long-term exceedance diagram for wave heights.
It has been found that the distribution of the individual wave height is close to linear if H (wave
height) is plotted as a function of logN (N is the number of waves exceeding H), see Figure 4.
5
Wave height
H100
log N
log N100
Figure 4
Simplified long-term exceedance diagram for wave heights
In Figure 4, H100 is a wave height with a return period of 100 years, which means that H100 on
average is only exceeded once every 100 years. N100 is the total number of waves in 100 years.
The relationship between H and N can be found to be
H=
− H100
⋅ log N + H100
log N100
⎛
log N ⎞
H = H100 ⋅ ⎜ 1 −
⎟
⎝ log N100 ⎠
(5)
Alternatively the following relationships apply
H − H100 =
− H100
⋅ log N
log N100
⎛
H ⎞
log N = log N100 ⋅ ⎜ 1 −
⎟
H100 ⎠
⎝
N = N100
⎛
H ⎞
⎜ 1−
⎟
⎝ H100 ⎠
(6)
In Equation (6) above N is thus the number of waves exceeding a given wave height H during 100
years.
If we now assume that the wave force is proportional to the wave height (true for inertia load
dominated wave load), and the corresponding wave induced stress is proportional to the wave force,
then the distribution in Figure 4 could be turned into a long term exceedance diagram for wave load
induced stress range simply by scaling the values of the Y-axis to produce Δσ instead of H. These
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considerations may be generalized to consider a general long-term exceedance diagram for stress
range Δσ , as shown in Figure 5.
Δσ ref is a stress range only exceeded once during the reference period, and N ref is the total number
of stress ranges during the reference period. In analogy with Equations (5) and (6) the following
relations apply
⎛
log N
Δσ = Δσ ref ⋅ ⎜ 1 −
⎜ log N ref
⎝
N = N ref
⎛
Δσ
⎜ 1−
⎜ Δσ ref
⎝
⎞
⎟⎟
⎠
(7)
⎞
⎟
⎟
⎠
(8)
In Equation (8) N is the number of stress ranges exceeding a given stress range Δσ during the
reference period.
Stress range
Δσ ref
40
20
log N
0
N=
Figure 5
1
1
N 40
2
1
10 10
N 20
log N ref
2
Long-term exceedance diagram for stress range
From Figure 5 it can be seen, as an example, that N 20 stress range values exceed a stress range of
20, while only N 40 stress range values exceed a stress range of 40. We can therefore conclude that
there are ( N 20 − N 40 ) stress range values between a stress range of 20 and 40. Continuing along
these lines, we can turn the information given in Figure 5 into an equivalent stress range histogram
with blocks similar to Figure 3.
Stochastic processes
In practical design work the stress history is often found to be irregular and in some cases also
random. The stress range history can therefore be considered a stochastic process, where we
describe the process by its statistical properties, i.e. mean value and standard deviation.
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A process is said to be stationary if the statistical properties do not vary with time. Many processes
may be considered stationary provided the time period considered is short enough. This is e.g. true
for the sea surface elevation, which normally is considered stationary within time intervals of three
to six hours. In the following we only consider stationary processes, and the value of the process at
time t is denoted x(t).
The probability density function p(x) is given by
p( x ) ⋅ dx = prob( x ≤ x (t ) ≤ x + dx )
(9)
The cumulative distribution function P(x) is given by
x
P ( x) = prob ( x(t ) ≤ x) =
∫ p( x) ⋅ dx
(10)
−∞
The expected value is given by
∞
E ( x ) = ∫ x ⋅ p ( x ) ⋅ dx
(11)
−∞
The expected value of the process is equal to the mean value of the process. Often the coordinate
system is chosen so that the mean value is equal to zero. An example would be measuring the sea
surface variation relative to the mean sea water level.
The energy spectrum of the process S (ω ) can be found directly from sampled values of the process
using Fast Fourier Transform (FFT). When performing FFT on a time signal we transform the
process given by x(t) in the time domain into its equivalent representation in the frequency domain
X (ω ) , from which we can derive the energy spectrum S (ω ) .
It is normal to distinguish between narrow band processes and broad band processes, and to
characterize a process with its energy spectrum S (ω ) as shown in Figure 6.
S (ω )
S (ω )
ω
Narrow band process
Figure 6
ω
Broad band proces
Stochastic processes
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A stationary stochastic process may be considered composed of infinitely many harmonic
components, each of different frequency. Let us as an example consider a wave spectrum derived
from the surface elevation x (t ) .
The energy of a harmonic wave is proportional to the square of its amplitude, and the energy
spectrum (wave spectrum) shows how this energy is distributed on the various frequency
components as given below. It should be noted that the energy within a frequency band Δω is equal
to that of a sine wave with amplitude ai as indicated on Figure 7.
x (t )
t
S (ω )
1
2
ai2 = S (ωi )Δω
ai
Ti =
Δω
ω
2π
ωi
ωi
Figure 7
A stochastic process x(t ) and its energy spectrum S (ω )
Rain flow counting
Cycle counting
With narrow band time series as shown in Figure 8a, individual stress cycles can easily be
identified and counted, e.g. by counting one stress cycle for each zero crossing (with positive slope)
and take the stress range as the difference between the peak and the valley values.
In broad band time series, where large cycles are interspersed with small cycles with varying mean
level, Figure 8b, the question of what is meant by a cycle and the corresponding stress range
becomes less evident. It is therefore necessary to use a cycle counting method that in an
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unambiguous way breaks the stress history down into individual cycles which can be summed up
into a stress range distribution.
σ (t )
σ (t )
t
t
a) Narrow band process
Figure 8
b) Broad band proces
Time series from stochastic processes
Several methods have been proposed for general cycle counting of broad band processes, generally
leading to different results in terms of accumulated fatigue damage. Cycle counting procedures that
give the most correct physical representation of the fatigue process are therefore often preferred.
For narrow band time series the choice of counting method seems less critical, and most generally
accepted methods lead to quite similar results.
For high cycle fatigue the most frequently used cycle counting methods are the Reservoir Counting
Method and the Rain Flow Counting Method. As the Rain Flow Counting Method is the most
frequently used cycle counting method in the industry, the following will focus on the details of this
method.
Basic principle in Rain Flow Counting
The method is designed to count reversals in accordance with the material’s stress/strain
relationship including hysteresis loops. Thus, complex loading situations with local stress reversals
can be accounted for in a consistent way. The principle may be illustrated by the strain history given
in Figure 9 and the corresponding stress-strain path.
ε
σ
4
4
2
2
2′
3
1
Figure 9
3
t
1
2′
2-3-2´ closed hysteresis loop
ε
Strain history and corresponding stress
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The strain history in Figure 9 consists of a large stress range 1-4, with a small closed cycle 2-3-2´
embedded. A basic principle in rain flow counting is to count a cycle each time a hysteresis loop is
closed. Thus, the total count in Figure 9 is a half cycle with stress range 1-4 plus a full cycle with
stress range 2-3.
To illustrate the principle further a more complicated stress history is considered in Figure 10. This
example is an extension of the strain history given in Figure 9.
Figure 10
Rain flow counting based on a more complicated strain history
The rain flow method has originally obtained its name from an analogy of rain falling down a
pagoda roof. The time series considered X (t ) is converted to a point process of peaks and troughs
as shown in Figure 11, and the process is considered a sequence of roofs with rain falling on them.
Rules for rain flow counting:
1. A rain flow is started at each peak and trough.
2. When a rain flow path started at a trough comes to a
tip of the roof, the flow stops if the opposite trough is
more negative than the one the flow started from.
3. For a flow started at a peak it is stopped by a peak
which is more positive than the one the flow started
from.
4. If the rain flowing down a roof intercepts flow from
an earlier path, the present path is stopped.
Figure 11
Rain flow counting of process X (t )
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The rain flow count applied to the process in Figure 11 results in
•
•
half cycles of trough generated stress ranges
half cycles of peak generated stress ranges
In both cases the stress ranges are found as the projected distances on the stress axis.
Figure 12 shows the details of the rain flow count of the process from Figure 11.
Half cycles of trough generated stress ranges
1-8, 3-3a, 5-5a, 7-7a, 9-10, 11-12 and 13-14
Half cycles of peak generated stress ranges
2-3, 4-5, 6-7 10-12b, 12-12a and 8-(9)-13
Total count
Full cycles: 2-3-3a, 4-5-5a, 6-7-7a, 9-10-12b(9) and
11-12-12a(11)
Half cycles: 1-8, 8-13 and 13-14
Figure 12
Rain flow count details
It should be noted that for X (t ) sufficiently long, it can be shown, Ref. /3/, that any trough
originated half cycle will be followed by another peak originated half cycle for the same range. This
is also the case for short stress histories if the stress history starts and ends at the same stress value.
Although the rain flow counting method widely is considered superior to other counting methods
for fatigue calculation, a basic criticism of the method as used above has been that the fatigue
damage procedure cannot account for the sequence of the stress ranges. This criticism is justified
especially since fatigue tests have revealed that the sequence of the stress ranges in some cases is of
importance. However, the effect of sequence has been found to even out in fatigue calculations
where many time histories are considered, and currently no better procedures are available.
References
Ref. /1/
Wægter, J.: Fatigue design based on S-N data. Ramboll Oil & Gas, Denmark. 2009.
Ref. /2/
Almar-Næss et. al.: “Fatigue Handbook – Offshore Steel Structures”. 1985. Tapir,
Norway.
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Ref. /3/
Wirsching, P. H., Mohsen Shehata, A.: “Fatigue Under Wide Band Random Stresses
Using the Rain-Flow Method”. Journal of Engineering Materials and Technology.
July 1977.
Ref. /4/
ESDEP WG 12 Fatigue. Lecture 12.1: Basic Introduction to Fatigue.
http://www.kuleuven.ac.be/bwk/materials/Teaching/master/toc.htm.
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