Fatigue Analyses of Offshore Structures

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 .

resistedcyclesstressofNumberappliedcyclesstressofNumberDamageFatigue =

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

i i

i

NresistedNappliedDamageTotal

1

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

SCFSS alno ×= min

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:

22

41

21 DACDVCq MD ρπρ +=

ρ – water density ~ 1,025 CD – Drag Coefficient ~ 0.7 CM – 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.

DamageTotalLifeFatigue 1

=

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:

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅

−−−

− ⋅⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅⋅⋅=

2

5.0exp4

52

45exp)( p

p

www

pwwwgwS

σ

ηη γα

parameterpeakness

wwif

wwifparameterwidthspectral

g

wHtconssPhilipdgeneralize

gravityofonacceleratig

Twfrequencypeakspectralangularw

TperiodwavetsignificanorperiodpeakTperiodwaveT

Twfrequencywaveangularw

Where

p

p

ps

ppp

ZP

w

w

−

>=

<=−

⋅−⋅

=→−

−

=→−

−−

=→−

γ

σ

γαα

π

π

09.0

07.0

))ln(287.01(165tan`

2

2:

2

42

Where:

w – angular wave frequency → WT

w π2= ;

TW – wave period; TP – peak period or significant wave period TZ

wP – angular spectral peak frequency → P

P Tw π2

= ;

g – acceleration of gravity;

α – generalized Philip’s constant → ( )( )γα ln287.01165

2

42

⋅−⎟⎟⎠

⎞⎜⎜⎝

⎛ ⋅⎟⎠⎞

⎜⎝⎛=

gwH PS

σ – spectral width parameter = 0.07 if w < wP = 0.09 if w > wP γ – peakness parameter The Pierson-Moskovitz spectrum appears for 1=γ

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:

σRMS = ∫∞

0

5.02 ))()(( dffSfH ηη

Every σRMS has an associated average period Tz:

Tz = σRMS / ∫∞

0

5.022 ))()(( dffSfHf ηη

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 = dsssN

sN

RMS ⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−∫

∞ 2

02 22exp

)( σσ

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 .481 -91. 0.774 0.7 0.639-106. 0.44-92.8 0.248 -2.2 RAO DP 16. 0.383 91.1 0.395-93.5 0.66 2.5 0.936-117. 0.508-94.5 0.268 -2.4 RAO DP 14. 0.244 91.1 0.204-109. 0.497 6.8 1.77179.6 0.547-97.1 0.258 -2. RAO DP 12. 0.079 83.4 0.064-86.9 0.262 15.9 0.27 10.1 0.498-103. 0.193 -1.5 RAO DP 10. 0.069-64.3 0.051106.8 0.080 69.2 0.234-70.2 .097-166. 0.018 17.9 RAO DP 8.5 0.036 1.9 0.022-177. 0.033148.7 0.026-111. 0.102 8.6 0.036179.9 RAO DP 6.0 0.012 -8.4 0.011-156. 0.006-175. 0.01 -6.3 0.008 -6.1 0.012 167. RAO DP 4.0 0.003167.1 0.004 21. -141. 149.1 -160. 0.003 19.2 RAO DP 3.0 6.9 -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 7.81 10.0 135.0

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 V2 / f + 4 X / ( 2 + X2 )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 Va 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:

σRMS = ∫∞

0

5.02 ))()(( dffSfH

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.

M=200kg

φ 1m

Fy = 350 EI = 134 kN m2 (φ 21/2 std)

3m

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, Pdynamic, 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.

6/52

6/522

2

6/52

2

26222222

22

2

2

3

22

)20252(432

4540

18001800)452(

454400015,0)2(

4

104)1(1

)2()1(1

)785.0005.0(89.14

1..

005.0401000125,0

110016

89.142713433

785.04

fS

ffV

fX

ff

fXX

fVkS

rrrrHn

fSvHndevstaY

dVdVdVVmCdPdynamic

kPamkgfVPstatic

mkN

LEIK

mDA

+=

===

+⋅

⋅⋅

=+

⋅⋅

=

⋅⋅+−=

⋅⋅+−=

Δ⋅⋅⋅⋅=

=××=⋅⋅⋅=

===

=×

==

==

−

∑

ε

γ

π

0

50

100

150

200

250

0 0.5 1 1.5 2 2.5

f (Hz)

Sv

Considering a numerical integration with N = 128 intervals and varying f between 0 and 2 yields:

015625.0128

2==Δf

For |Hn|2 this produces:

26

2226222

37.1104

37.11

1104)1(

1

37,1289.14

21

21

⎟⎠⎞

⎜⎝⎛⋅⋅+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎥⎦⎤

⎢⎣⎡−

=⋅⋅+−

=

===

−

−ffrr

Hn

mkf

ππ

0

50000

100000

150000

200000

250000

300000

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

f (Hz)

Hn2

0

100

200

300

400

500

600

700

800

900

1000

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

f (Hz)

Hn2

The final integration is given by the equation below:

( )

( )∑

∑

= −

=

−

−

+⋅⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎦⎤

⎢⎣⎡⋅⋅+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎥⎦⎤

⎢⎣⎡−

⋅⎥⎥⎦

⎤

⎢⎢⎣

⎡

+⋅⋅⋅

⎥⎦⎤

⎢⎣⎡⋅⋅+⎟

⎟⎠

⎞⎜⎜⎝

⎛⎥⎦⎤

⎢⎣⎡−

03125.0;96875.3

0 6/522

622

015625.0;984375.1

06/52

7

26

22

2025237.1

10237.1

1

000104,0

015625.020252

43210154

37.1102

37.11

1

f

f

fff

fff

These calculations can be easily performed with an EXCEL spreadsheet:

mdevstaY 0117.00305.089.14

1. ==

The static deflection is:

cmest 27.589.14785.01

=⋅

=δ

The Rayleigh distribution is given by the equation below:

[ ] ∫∞

−=>

λσ

σ

σλσ dAeAAP A 22 2/

2

for which the final total deflection is given by 5.27 + 3 x 1.17 = 8.79cm The corresponding dynamic magnification factor is:

67.127.579.8

==DMF

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:

0 exp)( >⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛−= SSSFS

γ

δ

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.

( ) γδ /1R

R

LnNS

=

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:

⎟⎟⎠

⎞⎜⎜⎝

⎛+Γ⋅

⋅+⎟⎟

⎠

⎞⎜⎜⎝

⎛+Γ⋅

⋅= zr

CNzm

AND o

Tm

T ,1,1γ

δγ

δ 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;

⎟⎟⎠

⎞⎜⎜⎝

⎛+Γ zm ,1

γ and ⎟⎟⎠

⎞⎜⎜⎝

⎛+Γ zr

o ,1γ – are incomplete gamma functions.

Incomplete gamma functions are defined as:

( ) ( ) ( )zaadtetza ota ,,

0

1 Γ−Γ=⋅=Γ ∫∞

−−

( ) ∫ −− ⋅=Γz

tao dtetza

0

1,

Where: γ

δ ⎟⎟⎠

⎞⎜⎜⎝

⎛= QS

z, 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.

The safety and the life are not the same as those for which the tables

were established, so first this must be corrected. This correction is

linear and related to the number of cycles:

Factor = 20 x 1 / (25 x 2 ) = 0.40.

This value could also have been obtained form table 5.8 given on the

previous page.

This value should be used in table 5.5, two pages above, to obtain

the stress reduction factor for the weld type and “in air environment”

established. Based on linear interpolation between 0.8 and 0.9 that

table yields a value of (0.733 + 0.741)/2 = 0.737.

Linear interpolation is also used in table 5.3 to obtain (208 + 174.6)/2

= 191.3MPa.

The final value is then obtained by multiplying this value by 0.737,

which yields 191.3 x 0.737 = 141MPa.

Attention is drawn to the fact that this value is only limited for

thicknesses up to 25mm. A further reduction based on the equation

given below is applicable for plate thicknesses exceeding that value.

Corrected Stress = Original Value x (25mm / greater thickness)0.25

For a 30mm plate, for instance the corrected value would be:

Corrected Stress = 141 x (25/30)0.25 = 134.7MPa