Fatigue Analysis of Offshore Fixed and Floating Structures
Nelson Szilard Galgoul
January, 2007
Definition The loss of resistance of some materials due to cyclic stresses.
Theory on which it is based None.
Properties commonly accepted
Fatigue is a function of the applied stress range (true);
Fatigue is not a function of the stress level (not really true but commonly accepted)
How it is measured Based on a large number of cyclic tests it has been found that the loss of resistance due to fatigue can be roughly approximated by curves such as the one given below:
Figure 1 – Typical S-N curve
Obviously it is intuitive that these curves vary from steel to steel. The curve given below is a typical curve for concrete steel reinforcement
Figure 2 – Typical curve for concrete steel reinforcement
This next curve is a typical curve for the steel used on offshore jackets (tubular structures).
Figure 3 – Curve given in API-RP2A for tubular joints
It is less intuitive, however, that these curves vary also with the environment (there is less fatigue above water than below), with corrosion and more markedly with thickness (only for thicknesses above 22mm). Basically the information given by these curves is the number of stress cycles that a steel specimen can resist for a given stress range. So if we have a rod with a known cross-sectional area A subjected a tensile force varying between 0 and F the S-N curve gives me the number of cycles N that the rod will resist while subjected to the stress varying between zero and S = F / A.
In order to measure how close the number of cycles of a given stress range brings us to failure a variable called fatigue damage was created .
Fatigue Damage =
Number of stress cycles applied Number of stress cycles resisted
In order to evaluate the fatigue damage caused by stress ranges of different amplitudes Miner-Palmgren have introduced a rule, which says that the damage caused by individual stress cycles may be summed up linearly. One again this is only an approximation but it is valid within engineering accuracy. Nloads
Total Damage =
∑ Nresisted i =1
Napplied i
i
Stress concentration Checking the fatigue damage of a uniform rod subjected to an axial stress range is very simple, but when the structural shape is complicated it is much more difficult to determine the stress variation, because there are stress concentrations, especially when the stress flow changes directions abruptly.
This is shown is figure 5, which contains a typical joint of an offshore platform (the deck – jacket connection, as shown in figure 4), that was modeled in finite elements.
Figure 4 – 3D Model of Fixed Platform whose Deck-Jacket Connection was Investigated
Figure 5 – Joint modeled in finite elements
There are at least 3 common ways to deal with this problem: a) Modeling in finite elements b) Using stress concentration factors
This approach is commonly used for tubular joints, where parametric equations have been developed by several authors based on finite element analyses:
Kuang, Smedley, Woodsworth
DNV
Efthymiou, etc.
These equations vary not only with the geometry of the joint, but also depending on how the loads are applied. This means that the type of joint can only be established after the load distribution within the structure has been determined. This is shown in figure 6.
Figure 6 – Joint classification
In this case the stress range is defined as a nominal stress range multiplied by a stress concentration factor
S = S no min al × SCF
c) S-N Curves with Built-in SCFs The traditional approach for non-tubular cross sections is to include the SCF into the S-N curve. This means that different types of cross-sections with different welding details have different curves. Figure 7 shows differents curves, as given in DnV
Figure 7 – S-N curves for different details Figure 8 shows a sample of how different welding details consider different curves
Figure 8 – Typical welding detail e associated S-N curve
It is important to emphasize that the type of SCF considered in the curves defined above take into account the cross sectional geometry and the type of weld, but any additional cause of stress concentration, such as a hole (see figure 9), or a construction misalignment, must be applied as a multiplier to the stress range as in the previous case.
Figure 9 – Stress concentration
Considering Environmental Loads Up to now we have established how to obtain the fatigue damage, at a given point of the structure, caused by stress cycles of constant amplitude and we have also learned how to obtain the cumulative damage, by adding up linearly the individual damage components, according to Miner’s rule. We are now going to look at the other side of the problem, which is related to finding the stress range values.
In order to do so, however, it is necessary to consider different approaches, which are associated to different types of structures and different types of environmental data.
a) Fixed Platforms In this case the environmental loads are applied directly to the structure. Figure 10 presents a jacket type structure with tubular members loaded directly by wave and current. Normally, in this case, the components of wave and current velocity and wave acceleration (horizontal and vertical) can be determined at any point of the semi-space below the water surface, based on traditional wave theories (Airy, Stokes, Stream Function, etc.). The forces on any member can be determined using the Morison equation.
Figure 10 – Wave forces upon a jacket member Morison equation:
q=
1 2
ρ C D V D + 2
1 4
π ρ C M A D
2
ρ – water density ~ 1,025 C D –
Drag Coefficient ~ 0.7
C M –
Inertia Coefficient ~ 1.7 The Morison equation offers precise results for any kind of
structure, which does not interfere with the wave profile. In practical
terms we can consider this to be true for pipes with diameters up to 3m (this is a very rough limit). In order to calculate the stress range at a given point of the structure, for a specified wave, as it passes through the structure, it is necessary to calculate the response of the structure as this entire wave is stepped through. Normally 18 positions of the wave (offsets of the crest with respect to the origin of the coordinate system) yields good results. This means the structure must be analyzed for 18 load cases, associated to one wave height, one wave period and one wave incidence in order to determine the stress range, which is the difference between the maximum and the minimum of the 18 values. Obviously there are infinite associations of wave height, period and incidence, so some grouping of this data must be performed in order to carry out a cumulative fatigue analysis. There are two traditional ways of doing this, the first of which is very intuitive and we will look at it first, although it is no longer recommended by most codes for offshore structures. Figure 11 contains statistical results of wave data collected at a typical offshore site. Note that this is one of 8 tables with contain wave height x wave period data for one cardinal direction.
Figure 11 – Typical wave data
Considering a minimum of 8 directions, 10 waves per direction and 10 periods per wave and reminding that there must be 18 steps for each wave, this means that the fatigue stresses will require 14400 load cases, in order to determine 800 stress ranges. This is feasible today, but it was prohibitive a few years ago, when computers just couldn’t cope with it, so the first suggestion to simplify it were to
assume that all waves of the same height had the same average period. This cut the data down to 1440 load cases. Typical data of this form is shown in figure 12.
Figure 12 – Typical data for a deterministic analysis
This was still two much, so people either sacrificed precision, by using less waves and steps (5 and 9 for instance cut this number down to 360) or by assuming that waves from opposite directions cause the same stress range (this cuts the cardinal directions down to 4) or by using interpolation functions for the wave height in order to use less waves without losing precision. Fatigue must be checked at all points of the structure where stress concentration may occur. This leads to an enormous amount of data. It is very usual for a fatigue analysis of an offshore structure to require as much as 10GB of storage to perform a fatigue run. In the case of a jacket structure we usually check fatigue at 8 points around the circumference of the joint connection (see figure 13), times 2, because each member has 2 ends and times 2 again because there is stress concentration on both the chord side and the brace side of the connection. If the jacket has 1000 members, this means that 32000 fatigue checks must be performed, dealing with the 1440 stress values that were determined above, leading to 80 stress ranges. This means that 80 x 32000 = 2.56 million damage calculations must be performed for the entire structure (80 for each point).
Figure 13 – Number of points checked around the joint Usually the number of waves per wave height and wave direction is given per year, which means that the inverse of the yearly damage is the life of the structure at each point.
Fatigue Life =
1 Total Damage
Typical results present the point of the structural joint that has the highest stress range and the corresponding fatigue life. The main problems with the deterministic method are related first to the fact that not all waves have the same period and second because assuming all waves are regular does not take into account the stochastic nature of the marine environment. Because of this it has become common practice to perform spectral fatigue analyses instead of deterministic ones.
There are basically two wave spectra that are commonly used in the offshore engineering market: the Pierson Moskovitz, also known in a general form called the ISSC spectrum and the JONSWAP spectrum, which was developed specifically for the North Sea in joint industry study. During many years the ISSC spectrum was said to be valid for the entire world, except for the North Sea, where JONSWAP was used. More recently, however, variations of the JONSWAP spectrum have been found to suit some other parts of the world better than the ISSC curve. For the sake of completeness the equations that govern these spectra are given below:
⎛ 5 ⎛ w ⎞− 4 ⎞ 2 5 ⎟ ⎟ ⋅ γ Sηη ( w) = α ⋅ g ⋅ w− ⋅ exp⎜ − ⎜ ⎜ 4⎜ w ⎟ ⎟ ⎝ ⎝ p ⎠ ⎠
2 ⎛ ⎛ w − w p ⎞ ⎞⎟ ⎜ ⎟ exp ⎜ − 0.5⎜ ⎜ ⎟ ⎟ σ w ⋅ ⎜ p ⎠ ⎟ ⎝ ⎝ ⎠
Where : w − angular wave frequency → w =
2π T w
T w − wave period T P − peak period or significant wave period T Z w p − angular spectral peak frequency → w p =
2π T p
g − acceleration of gravity
α − generalized Philip`s cons tan t → α = σ − spectral width parameter
= 0.07 if w < w p = 0.09 if w > w p γ − peakness parameter
2 4 5 H s ⋅ w p
16
g2
(1 − 0.287 ⋅ ln(γ ))
Where: w – angular wave frequency
w=
2π T W
;
TW – wave period; TP – peak period or significant wave period T Z wP – angular spectral peak frequency
wP =
2π T P
;
g – acceleration of gravity; α – generalized Philip’s constant
H S2 ⋅ wP4 ⎞ ⎛ 5 ⎞⎛ ⎟(1 − 0.287 ⋅ ln(γ )) α = ⎜ ⎟⎜⎜ 2 ⎝ 16 ⎠⎝ g ⎠⎟
σ – spectral width parameter
= 0.07 if w < w P = 0.09 if w > w P γ – peakness parameter
The Pierson-Moskovitz spectrum appears for γ = 1
The wave data is then provided on a statistical basis, where the normal parameters are a significant wave height (average of the 1/3 highest waves) and a statistical period, which is either the peak period (Tp) or the zero up-crossing period (an average value – Tz).
The table given in figure 14 seems similar to that presented in figure 11, but it represents no longer individual waves, but individual sea states with the given statistical periods. The great advantage, in this case, is to try to consider the individual periods of each of these sea states, wherefore waves with the same height also have a range of different periods.
Figure 14 – A wave scatter diagram based on statistical data
Still in this case the number of load cases is very large, so a small approximation has been introduced, based on which it is possible to be much more precise and still consider a smaller number of waves. In general terms something called a transfer function for unit height waves is generated for the whole range of periods that the wave may have. Obviously the variation of wave force with wave height is not linear, but if this function is calculated with the most probable wave height for each period (and then divided by the wave height) the errors incurred will be small and perfectly acceptable within engineering accuracy.
Figure 15 shows a typical transfer function built with 200 wave periods.
Figure 15 – Transfer function for base shear
This figure can be adjusted by a polygonal with only five points as shown below. This means that all the periods and all the wave heights for this one wave incidence can be represented by only 5 load cases.
Figure 16 – Typical transfer function with only 5 points – static only
Unfortunately this curve is not always so smooth, specially when the platform is very slender and the highest natural periods are in the wave period range between 2 and 20 seconds. In this case the curve would show resonance spikes at the given periods and more points would be required. Sometimes the correct representation of a dynamic transfer function may require as many as 30 points. Even in this case, however, the number is not excessive for modern day computational resources.
The fatigue calculations in this case are performed exactly as before. In other words the damage is calculated for each of the individual seastates and summed based on Miner’s rule, but the calculation of the damage for each of the individual seastates is more cumbersome because all the values involved are statistical. For the sake of completeness a summary of the calculation sequence is given below. It is assumed that the wave transfer function H(f) has been determined and the Spectral Density Function (JONSWAP or ISSC) Sηη(f) has already been established.
The calculation performed here are based on established conditions related to the statistical calculations. The first of these conditions is related to the quality of the samples used to establish it. In technical terms it is called a stationary random process, because the average value for a given interval will be the same no matter what the length of time is. For normal fatigue waves this is a very reasonable assumption.
The RMS value σRMS of the stress variation s for a given seastate is given by the following equation: ∞
∫
2
σRMS = ( H ( f ) Sηη ( f ) df )
0.5
0
Every σRMS has an associated average period Tz: ∞
∫
2
2
Tz = σRMS / ( f H ( f ) Sηη ( f ) df ) 0
0.5
Assuming that this given seastate will occur a fraction m of the entire Life of the structure, the corresponding number of cycles will be given by:
N = mLife / Tz
The corresponding damage, assuming a Rayleigh stress distribution is given by:
D=
N
σ
2 RMS
∞
∫ 0
⎡ ⎛ s ⎞ 2 ⎤ exp ⎢− ⎜ ⎟ ⎥ ds N ( s ) 2 2 σ ⎠ ⎥⎦ ⎢⎣ ⎝ s
The total damage will be the sum of the damages of each seastate.
The final result given by the analysis is the life of the structure at all the critical points (joints) where stress concentrations occur.
b) Floating Units There are four main types of floating production platforms: the FPSOs, the Semi-Submersibles, the Spars and the TLPs. Figures 17 through 20 show examples.
Figure 17 – Typical FPSO
Figure 18 – Typical Semi-Submersible
Figure 19 – Typical Spar
Figure 20 – Typical TLP
Obviously everything that was said here for fixed platforms remains valid for floating units, but the problem is how to apply it, because the floating units are not only complex structures, but they may also be changing their positions with respect to the environment. In general all of these types of platforms are moored, but the FPSOs have different types of moorings, which actually change their behavior. Some FPSOs are moored, as shown in figure 21, and considered to be fixed with respect to rotation, but many are pinned to a moored structure called a turret, wherefore they tend to line up with the environmental direction of incidence (see figure 22). In this case most of the waves will come from head seas, but a small percentage will still come from quartering and even from beam seas.
Figure 21 – Mooring Layout for fixed conditions (PETROBRAS 43)
Figure 22 – Turret at the vessel stern
The modeling in this case is done considerably different from what was done for the jacket frame, because the Morison theory is no longer applicable, wherefore a diffraction theory must be considered. A typical 3D diffraction mesh is given in figure 23. 2D diffraction, also known as strip theory, is hardly used any more.
Figure 23 – Typical 3D diffraction mesh
The hydrodynamic analysis here is not the object of this lecture, but just for the sake of completeness, the first step is to calculate the pressure on each of the diffraction elements or panels as the wave passes through the vessel. The integration of these pressures produces the net force on the vessel, which is then used to calculate the vessel motions, treating it as a rigid body with 6 degrees of freedom: three linear displacements (surge front- and backwards, sway sidewards and heave up and down) and three rotations (roll around the longitudinal axis, pitch around the midship transversal axis
and yaw around a midship vertical axis). Water damping plays a very important role in these calculations, because not all motions have restoring forces.
The vessel motions can be divided into two types: the linear wave motions and the second order excursions. The linear wave motions are those produced for each of these 6 degrees of freedom, obtained for waves of unit amplitude, and they are called RAOs (Response Amplitude Operators), which is a traditional terminology for naval architects. Structural engineers would call these curves Transfer Functions, as we did with the wave forces on the fixed platform. Typical RAO curves for quartering seas (the wave incidence on the vessel is 45, 135, 225 or 315 degrees) are given in figure 24.
Figure 24 – Typical RAO curves for quartering seas
The second order motions are long term excursions that the vessel undergoes while it moves along it’s anchor lines. Typical values of excursions are in the range of 10% of the water depth for moored structures subject to storm seastates.
For the specific purpose of this seminar, there are at least 2 different effects to be considered here, that cause fatigue: the first is the vessel deformation and the second the inertial loads induced by the
vessel motions. Both of them require the transformation of the RAOs already determined, as will be seen below.
It was said above that there are “at least 2” effects that cause fatigue, because depending on the structure there can be other sources as well. Some examples will clarify.
Example 1 – FPSO Deck Structure
Let us consider, first of all, the fatigue check of part of the main deck of an FPSO vessel as shown in figure 25.
Figure 25 – FPSO vessel
It is intuitive that the overall behavior of the vessel as a beam will produce varying stresses on it, which are the typical stresses that are
considered in normal ship design. Associated to these are the inertial loads mentioned above, which are normally also considered in ship design.
In normal ship design, the codes usually consider a conservative constant moment, which covers the central part of the vessel and which decays as the section being designed
approaches the two
ends. If, however, one wishes to perform these calculations correctly or better said, more precisely, then the solution is somewhat cumbersome.
The fact that this is an FPSO means that the ballast is gradually changing, not to speak of the abrupt variation when the offloading takes place. It is necessary, therefore, to build a structural model, and calculate the stress variation as the waves pass through it, but this will have to be done for several ballasting conditions. For each of these conditions a transfer function will be determined and a spectral analysis will be performed considering the percentage of the waves that are related to that condition.
The model can be a finite element mesh of the entire vessel, or at least part of it, and it can also be as simple as a single beam extended along the entire vessel length.
It is obvious that this is too much work compared to what is prescribed by the naval architectural rules established by Lloyd’s, ABS, DNV or any other Ship Classification Company.
Because of this, simplified fatigue checks have been developed, which will be discussed ahead. Vessel design can, therefore, be performed based on these precise criteria, but they usually are not.
Unfortunately, however, there are cases in which such simplified criteria are not allowed, because fatigue is a very important design issue.
A second example will show this.
Example 2 – Flare Boom Fatigue
Figures 26 and 27 present two different examples: one where the boom is cantilevering out over the sea and a second with a vertical flare boom.
Figure 22 – PETROBRAS P-50 Platform
Figure 23 – PETROBRAS P-50 Platform
A typical model of the first is presented in figure 24.
Figure 24 – Typical boom structure with multi-flare burners at the top
This is an interesting structure because it is sensitive to fatigue originated by several different sources: -
Fatigue caused by inertial loads generated by the vessel motions;
-
Fatigue caused by the turbulent component of wind;
-
Fatigue caused by vortex shedding;
-
Fatigue caused by the support displacements.
Normally some of these sources are not as important as others. The distance between the supports, for instance, is small compared to the
vessel length and especially in this case, where they are at the bow, so the fatigue related to overall vessel bending is negligible.
Fatigue due to vortex shedding is a local member vibration problem, which is normally prevented by using vortex suppressors.
This leaves us with the first two, which are described below.
Inertial load generation
This first problem is really a matter of determining how the loads will be applied. The vessel motions are determined by the RAOs, so if we are able to relate the RAO motions to accelerations around the structure, we can then generate a transfer function, which relates wave period to inertial loads.
Figure 22 – Transferring motions from the center of rotation to a general point around the structure Assuming in figure 22 that ω is the angular velocity, θ the angular acceleration and a the linear acceleration at the center of rotation, it is easy to show that the x component of inertial force at a general point of the structure, whose distance from the center of rotation is r and whose mass is m, can be given by: Fx = -m ( ax + r θ sinα + ω2 r cosα ) The other component expressions are similar. This means that provided the periods or frequencies are chosen carefully, in order to match all the peaks and valleys of the RAOs, the new inertial force transfer function will be used exactly as the previous one, because the final product will also be member stresses. Stress
variations
will
be
twice
these
values,
because
the
corresponding stresses can be reversed. A typical input file is given below: TOWOPT
MNECLD
MPPPWPOR
65.832 -6.726
12.89XYZ
POSITION RAO DP
20. 0.537 91.6 0.540-90.4 0.849
0.1 0.487-102. 0.374 -92. 0.221 -1.9
RAO DP
18. 0.477 91.4
0.7 0.639-106.
RAO DP
16. 0.383 91.1 0.395-93.5
RAO DP
14. 0.244 91.1 0.204-109. 0.497
RAO DP
12. 0.079 83.4 0.064-86.9 0.262 15.9
RAO DP
10. 0.069-64.3 0.051106.8 0.080 69.2 0.234-70.2
RAO DP
8.5 0.036
RAO DP
6.0 0.012 -8.4 0.011-156. 0.006-175.
RAO DP
4.0 0.003167.1 0.004
RAO DP
3.0
.481 -91. 0.774 0.66
0.44-92.8 0.248 -2.2
2.5 0.936-117. 0.508-94.5 0.268 -2.4 6.8
1.77179.6 0.547-97.1 0.258
0.27 10.1 0.498-103. 0.193 -1.5 .097-166. 0.018 17.9
1.9 0.022-177. 0.033148.7 0.026-111. 0.102
6.9
8.6 0.036179.9
0.01 -6.3 0.008 -6.1 0.012 167.
21.
-141.
149.1
-160. 0.003 19.2
-154.
-66.5
120.
114.9 0.001167.6
WAVE
A001
18 14.60
20.0 135.0
WAVE
A019
18 14.60
18.0 135.0
WAVE
A037
18 14.60
16.0 135.0
WAVE
A055
18 14.60
14.0 135.0
WAVE
A073
18 11.24
12.0 135.0
WAVE
A091
18
10.0 135.0
7.81
-2.
WAVE
A109
18
5.64
8.5 135.0
WAVE
A127
18
2.81
6.0 135.0
WAVE
A145
18
1.25
4.0 135.0
WAVE
A163
18
0.70
3.0 135.0
END
Fatigue caused by wind turbulence
There are several different types of dynamic vibration induced by wind: turbulence, fluttering, galloping, vortex shedding, etc. It is acceptable to say that the wind loads can normally be treated as if they were static, in spite of being variable, because their variations are either small or far from the period of excitation of the structure. In the cases in which their frequencies are near to those of the structure, or the loads begin to change the form of the structure, then dynamic wind loads may become important.
Another important factor when considering the effect of wind is that dynamic response takes time to build up, so it is meaningless to analyze dynamic response with the short period average velocities that are used for inplace analyses. A normal static wind is based on a 3 to 5 second gust, while the dynamic response usually requires at least 10 minutes to build up. This means that the static wind response may be more critical than the dynamic counterpart, because the wind velocities are smaller.
The most common wind spectrum for turbulence is that developed by Harris, whose equation is given below: S(f) = k V 2 / f + 4 X / ( 2 + X 2 )5/6
Where V is the wind velocity (10m above sea level), f is the wind frequency in Hz, k is a roughness coefficient (average about 0.0015) and X = 1800 f / V.
The wind pressure for an average wind speed V is given by the Morison equation: P = 0.5 Cd γ V2
Assuming that V has an average value V a plus a small variation dV due to turbulence. This equation becomes: P = 0.5 Cd γ (Va+/-dV)2 = 0.5 Cd γ Va2 +/- Cd γ Va dV
This equation, where the second order term was neglected, provides both static and dynamic components of wind. The dynamic component would be calculated in a dynamic spectral wind analysis and then added to the static.
A small example is given below just to illustrate the spectral wind calculations.
It was shown above for fixed platforms that the RMS value σRMS of the stress variation s for a given seastate is given by the following equation:
∞
∫
2
σRMS = ( H ( f ) S ( f ) df )
0.5
0
This is obviously applicable to any kind of random variable, so an application showing the it’s use for ultimate limit dynamic wind amplifications is given below.
Example Determine the dynamic magnification of the stress at the bottom of the column support of a roadway traffic sign, whose mass is 200kg, whose plate diameter is 1m and whose CoG is 3m above the embedment point. The wind velocity 40 m/s. The maximum force value shall assume a Rayleigh distribution and a 1.2% probability of being exceeded. Assume also zero damping.
φ 1m M=200kg
Fy = 350 3m
2
EI = 134 kN m 1
(φ 2 / 2 std)
The equations given below calculate A, the plate area, K, the stiffness of the plate column support (wind force required to displace the plate 1m), Pstatic, the static wind pressure, P dynamic, the dynamic wind pressure and finally Ysta.dev, the standard deviation of the dynamic wind displacement. In this equation, Hn is the dynamic amplification factor.
π D 2
A =
4 3 EI
K =
L3
Pstatic =
= 0.785 m 2 3 × 134
=
27
V 2
= 14.89
= 100
kgf
1
∑
2
kN m
= 1kPa
16 m Pdynamic = γ ⋅ Cd ⋅ Vm ⋅ dV = 0,000125 × 1 × 40dV = 0.005dV Y sta.dev. =
14.89 1
2
Hn = S= X = S=
2
2 Hn ⋅ (0.005 ⋅ 0.785) ⋅ Sv ⋅ Δ f
(1 − r 2 ) 2 + (2 ⋅ r ⋅ ε ) 2
k ⋅ V 2 f
⋅
1800 f V
4 X ( 2 + X 2 ) 5 / 6
=
1800 f 40
432 ( 2 + 2025 f 2 ) 5 / 6
=
=
1 (1 − r 2 ) 2 + 4 ⋅ 10 −6 ⋅ r 2
0,0015 ⋅ 40 2
= 45 f
f
⋅
4 ⋅ 45 f (2 + 45 2 f 2 ) 5 / 6
250
200
150
v S 100
50
0 0
0.5
1
1.5
2
2.5
f (Hz)
Considering a numerical integration with varying f between 0 and 2 yields:
Δ f =
2 128
N = 128 intervals and
= 0.015625
For |Hn|2 this produces:
f =
1
k
2π m
=
1
14.89
2π
2 1
2
Hn =
= 1,37
(1 − r ) + 4 ⋅ 10 2 2
−6
⋅ r
2
=
1
⎛ ⎡ f ⎤ ⎜1 − ⎜ ⎢⎣1.37 ⎥⎦ ⎝
2
2 ⎞ f ⎛ ⎞ ⎟ + 4 ⋅ 10− 6 ⋅ ⎜ ⎟ ⎟ ⎝ 1.37 ⎠ ⎠
2
300000
250000
200000
2 n 150000
H
100000
50000
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
1.6
1.8
2
f (Hz) 1000
900
800
700
600 2 n500
H
400
300
200
100
0 0
0.2
0.4
0.6
0.8
1
1.2
1.4
f (Hz)
The final integration is given by the equation below:
1.984375 ; 0.015625
∑ f = 0
⎡ ⎤ 432 ⋅ 154 ⋅ 10 ⋅ ⎢ ⎥ ⋅ 0.015625 2 2 5 / 6 2 2 ⎢⎣ (2 + 2025 f ) ⎥⎦ ⎛ ⎡ f ⎤ ⎞ ⎜1 − ⎟ + 2 ⋅ 10− 6 ⋅ ⎡ f ⎤ ⎢⎣1.37 ⎥⎦ ⎜ ⎢⎣1.37 ⎥⎦ ⎟ ⎝ ⎠ 1
3.96875 ; 0.03125
∑
−7
0,000104 2 2 2⎫ ⎧⎛ ⎪⎜ ⎡ f ⎤ ⎞⎟ 2 5 / 6 − 6 ⎡ f ⎤ ⎪ ( ) 1 2 10 2 2025 − + ⋅ ⋅ ⋅ + f ⎨⎜ ⎢ ⎬ ⎥ ⎢ ⎥ ⎟ 1 . 37 1 . 37 ⎦ ⎠ ⎣ ⎦ ⎪ ⎪⎩⎝ ⎣ ⎭
f = 0
These calculations can be easily performed with an EXCEL spreadsheet:
Y sta.dev =
1
0.0305 = 0.0117m
14.89
The static deflection is:
δ est =
1 ⋅ 0.785 14.89
= 5.27cm
The Rayleigh distribution is given by the equation below: ∞
P[ A > λσ ] =
∫
A
σ
λσ
2
e
− A 2 / 2σ 2
dA
for which the final total deflection is given by 5.27 + 3 x 1.17 = 8.79cm The corresponding dynamic magnification factor is:
DMF =
8.79 5.27
= 1.67
Simplified Fatigue Procedure
The simplified fatigue analysis is also called the allowable stress range method. This method is based on the premise that it is possible to evaluate a long term stress range and compare its maximum value with the allowable stress limit. For this reason the simplified method is classified as an Indirect Method, as it is not necessary to obtain the fatigue life and damage for each point of the structure in order to perform a fatigue design check. In many cases, it is condensed into a pass/fail check on the entire structural model.
Usually, in engineering practice, the simplified method is used for a quick check on fatigue performance, or to assemble a screening check. The screening technique is a quick and conservative check for the fatigue strength of structural details. Actually, if a structural detail meets the screening check, it is considered acceptable and no further checks are required. Nevertheless, all joints that fail screening check are not necessarily under fatigue requirements. It is only an indication that further investigations, using more accurate techniques, must be made on these failing structural details. The screening technique is a good mechanism to determinate the most critical regions, in fatigue terms, on overall structure.
Mathematical Development
The long term stress range distribution may be presented as a two parameter Weibull distribution:
⎡ ⎛ S ⎞γ ⎤ F S ( S ) = exp ⎢− ⎜ ⎟ ⎥ ⎢⎣ ⎝ δ ⎠ ⎥⎦
S >0
Where: Fs(S) – is the probability that the value S will be exceeded S – is the random variable representing the stress range; γ – is the Weibull shape factor; δ – is the Weibull scale factor;
This equation is used in the simplified fatigue analysis, which is based on the cumulative damage rule (Palmgren-Miner),taking into account also the fatigue strength defined by S-N curves. A closed expression for the fatigue damage can be found, based on it.
The Weibull Distribution Parameters are given below:
Scale parameter, or characteristic value:
Let SR – a reference stress range be the major stress range that can occur on a certain number NR of cycles. δ =
S R
( LnN R )1 / γ
Shape Parameter:
The shape parameter can be obtained from a full spectral analysis, used to calibrate the simplified method. After years of experience, however, there is sufficient knowledge of the problem to allow the value to be established for given types of structures. For FPSO modules, for instance, a value of about 0.85 is acceptable.
Fatigue Damage
It can be shown that the closed solution for the fatigue damage considering a two segment S-N curve is as given below: D =
N T ⋅ δ m A
⎛ m ⎞ N T ⋅ δ γ ⎛ r ⎞ ⋅ Γ⎜⎜ + 1, z ⎟⎟ + ⋅ Γo ⎜⎜ + 1, z ⎟⎟ C ⎝ γ ⎠ ⎝ γ ⎠
Where: NT – is the total number of cycles during the design life; A, m – parameters obtained from first segment of the S-N curve; C, r – parameters obtained from second segment of the S-N curve; γ – the Weibull shape factor; δ – the Weibull scale factor;
⎛ m ⎞ ⎛ r ⎞ Γ⎜⎜ + 1, z ⎟⎟ Γo ⎜⎜ + 1, z ⎟⎟ ⎝ γ ⎠ and ⎝ γ ⎠ – are incomplete gamma functions.
Incomplete gamma functions are defined as: ∞
Γ(a, z ) = ∫ t a −1 ⋅ e −t dt = Γ(a ) − Γo (a, z ) 0 z
Γo (a, z ) = ∫ t a −1 ⋅ e −t dt 0
Where: γ
⎛ S Q ⎞ ⎟⎟ z = ⎜⎜ δ ⎝ ⎠ , SQ is the stress range value at which the change of slope of
the S-N curve takes place.
The reference stress range SR is usually set to that related to the storm conditions, so the limiting fatigue life is used as a starting point to determine what the highest allowable SR value would be, so that the given fatigue life is guaranteed.
These calculations are presented, for instance, in the DnV code and copied below:
These curves have all been established assuming a life safety factor of 1.0 and a fatigue life of 20 years. This has led to 100 million cycles. In case the life is different or a different safety factor is desired, correction factors have been provided for each curve, one again both in the air and under water with cathodic protection.
These curves are given below:
An example is presented below: An F3 type detail will be welded on a 25mm plate of an FPSO module. Please determine what the highest allowable stress assuming the Weibull shape parameter to be 0.85. The platform life shall be 25 years and a minimum safety factor of 2 shall be used.
