Spectral analysis and stochastic fatigue reliability calculation of offshore steel structures

Dr. H. Karadeniz TECHNISCHE UNIVERSntH Laboratorium voor Scheepshydromøchanlca Archief

Meketweg 2 262B CD Deift

ToL 015-786873- Faz 018-781836 April 1983

THDeIft

DeIft University of Technology

Spectral analysis and

stochastic fatigue

reli-ability calculation of

offshore steel structures

Department of Civil Engineering

Applied Structural Mechanics Division

Report on Óffshore Structural Analysis

SPECTRAL ANALYSIS AND STOCHASTIC FATIGUE RELIABILITY CALCULATIO1

OF

OFFSHORE STEEL STRUCTURES

by Dr. H. KaTradeniz

SUMMARY

I - Introduction

2 - Spectral Participation Factors (SP?) 3

-

Generalized (Modal) Wave Forces 4

-

Calculation of Modal Wave Forces

5 - Calculation of Transfer Functions between Modal Wave Forces and the Water elevation

6 - Spectral Statements of Modal Wave Forces and Calculation of Spectral Moments of Structural Response Outputs

7 - Formulation of Probabilistic Fatigue Damages

8 - Uncertainties in the Cumulative Probabilistic Fatigue Damage

9 - Probabilistic Fatigue Reliability Calculation

10 - Some Calculation Aspects related to Numerical Differentiation of the Failure Function

II - Calculation Scheme for the Reliability Analysis 12 - Analysis of an Example Structure

APPENDIX APPENDIX APPENDIX

-CONTENTS

REFERENCES ₈₉

I :Calculation of Influence Regions for Concentrated Load 70

Parameters and, for Modal Forces of Members

II :Statementsof the LinearizedDrag Force Coefficient 80

III :Orthogonal Transformation of Correlated Random ₈₆ Variables 2 5 8 14 22.. 26 33 39

46

52 55 61

In this rèport,a refined spectral calcuiaion of fied jacket type offshore structures, to be constructed in deep water environments,

is first presented.. For this càlculation, the modal analysis and the concept of the spectral participation factors (SPF) are used. In order to apply.the modal analysis and to calculate (SPF) , the

generalized (modal) wave forces are determined by means of the so-called load influeflce regions as functions Of the frequency.

In the text, the concept of the. (SPF) 'is briefly outlined. Then,

the Ñod'al wave forces of structural members for deep water condition are formulated, and the calculation procedure by using the load

influence regions is presented. in proceeding sections, calculations of trañsfer functions of wave forces and the response sectràl moments are explained.

For the stochastic fatigue 'reliability analysis, the formulation of the stochastic cumulative fatigue damage is presented in a later

stage of the text; Uncertainties existing in this damage are explained, then a brief sununay of the fatigue reliability calculation is givèn

ith an efnphasis on pràctial applications, In the lást,, stage of the

report, thé aiculatíon algorithm and the coñcerning computer programs are eplai-ned. Finally, in order tö demonstrate the application of the methods and, the calculatiOn techniques üsed in this report, anéxample

I - INTRODUCTION

Structures to exploit offshore enetgy resources are mostly designed to survive within àn expectéd period of tme. The design analysis of such structures is a difficult task due to he fact that they are built in oceans wher high uncêrt-äiniês exist nd environmental cnditions

re. random Therefore, a reliable design calculation for these strúctures

should be carried Out by ustn.g the stochastic analysis. The time varying environmental conditions produce dynamic stresses in structutal members arid joints that primarily cause a fatigue failure in the long term model of sea states. However, in thè shdrt term model of sea states a

structural failure can also occur due -o ari exceptional case of environ-mental loading that exceeds the ultimae-límit capacity of structures.

In general, offshore structures -are subjected to different environmental

[4] .

loads which may be induçed by,

Winds, - Waves,

Currents, - Earthquakes.

Among these loads, waves-and earthquakes are the most important sources of structüral excitations. Wind loads may be important during the towing of parts Of the structural ensemble, and in the operational case, these forces are considered unimportant in the analysis. Current loads are also considered ünimportant in the sructurl dynamic analysis as compa±ed with wave and eartquake loadings. The earthquake loadings must be takén

into account in the analysis when structures are built in tectonic offshore fields. In this report,only the wave loadings are taken into

account, and the formulation of the analysis is based on this loading case.

The stochastic analysis of offshore structures requires the spectral function of stress variations that can be calculated in terms of the sea spectrum by means of some transfer

functions

between the random sea-surface elevation and the stress at a structural point considered. The sea spectrum is given analytically, e.g. Pierson-Mokowitz and Jönswap spectrum, as functions of some parameters that are btained from some Observed data.

Transfer functions mentioned above may be considered in different steps. The most considerable step may be the one between the random sea-surface elevation and the wave forces actiñg on structural members. Thewave forces are calculated by using the Morison' equation as the drag and the inertia terns. The drag term is a non-linear function of the water velocity ànd the inertia term is a linear function of the water

acceler-ation . In order to apply the spectral analysis, the non-linear drag force

term must be linearized. A linearization procedure is presented in a previous report {11, and therefore, it is not explained in this report

once more. The linear wave theory (Airy wave theory) is essentially used to find out transfer functions between the water elèvation and the water velocity vector. The detail of this ïs explained in ref. [1]. The wave forces obtained frOm the Morison's equation are distributed along

lengths of structural members. The function of the distribution in space is obtained from the linear wave theory as exponential in the vertical direction and harmonic in the horizontal direction, for an arbitrary orientated member in the sea. However, for the load ensemble-process of

the finite element application, the equivalent joint wave forces of structural members need to be calculated. Then, by using the ensemble load vector and the structural transfer function matrix, the response displacement vector of the structure can easily be calculated in general. When the modal superposition technique is applied, which is the case used in this report, the calculation of the response displacements becomes easier. In this case., however, the generalized (modal) forces are used in the response analysis. An exact calculation of these môdal forces without using ensemble load vector and without calculating the equivalent joint loads of members is presented in this report. If the modal analysis is applied to the structural response calculation, the

spectrum of stress variations can easily be obtained by using the concept of spectral participation factors. This concept is presented

in ref. [1] in detail and briefly outlined alsô in this report. Having. obtained the stress spectrum, the statistical measures of stress variat-ions (variance and zero crossing period) can easily .bè calculated from some integral statements of the spectrum. A procedure to carry out inte-grations involved in these calculations is presented in a pìevious

-.4-As it is mentioned before, the fatigue collapseis the dominant failure Of offshore structures in the long-tetm period. The cumulative damage maybe calculated by using the PaÏmgrén-Miner's rule as the sum of

ratios between numbers of stress cycles at stress ranges applied and the corresponding numbers of stress cycles to failure. This cumulative damage can analytically be stated as fùnctions of the so-called fatigue

parameters, that are obtained from the experimental data., and the

statistical measures of stress variations, that are calculated by means of the spectral analysis. In the final statement of the cumulatie

fatigue damage, there are some uncertainties that may arise from different sources. Among these, uncertainties in the modelling of random sea waves, uncertainties in the observed data, uncértainties in the structural modelling and uncertainties in the eetimental data may bementioned in general. These uticertainties can be taken into account when the fatigue reliability analysis is carried out, y which an expected life time for

the. structural. survival may be estimatEd.

In this tepôrt, the subjects described above _{are studied with the emphasis} on the application to fixed offshore jacket type structures _Theoretical

formulations are presented in detail, and the solution algorithms for practical purposes are explained. Finally, an example for the purpose of a demonstration is also worked out.

2 - SPECTRAL PARTICIPATION FACTORS (SPF)

In the previous report[h], the calculation of spectral participation factors has been discussed in detail, assuming that wavé forces are

linearly distributed along the lengths of structural rnember. Under this assumption a fine structural mesh for the finite element division

is required. In the probabilistic analysis of.structures using afine mesh, or increasing structural degrees of freedom, gives considerable

rise to thé calculation cost. Therefore, the assumption of using linear wave force distribution along thé lengths of members is not suitable. In this report, añ eff:icient calculation of spectral

participation factors, which will be denoted bySPF henceforth, with the exact istr-ibution of wavé. forces is presented. For this purpose, however,

a general formulation änd the meaning of SPF are first explained.

The overall dynamic equation of a structural system is given131 b

[K].D} + [C] {D} + 2.1)

where [K], [C] and [M] are respectively overall stiffness, damping and mass matrices of thé structural system. D} and {P} are respectively the overall displacement and load vectors. A dot denotes a time derivative. By using the mode superposition technique, (2.1) can be separated into a number of differential equations. One of these equations, for the eigen mode k, can be written as,

Zk + 2k 0k Zk + W.Zk = m (2.2)

in which,

damping ratio, Wk : eigenfrequency,

Zk .: generalized,, or modal, displacement stated in the time domain,

generalized mass, and

generalized, or modal, force of the mode k.

Structural displacements can now be obtained by superposing the contri-butions of eigenmodes as written by,

q

{D) = (2.3)

where denotes the eigenmöd vêctôi, änd q denotes the numbér öf eigenmodes included. The generalized ass is simp1y stated by

=

{qI T _[M]

q}

_(2.4

and the genetalied force is,

k (25)

By Using the Fourier transformatibn of zk(t) and fk(.t), (2. 2) can be

expressed in the frequency domain, as written by, in general,

zk(w) = (w) (2 6)

wherè zk(w) and fk(w) are respêctivel57 FoUrier transform pair of

zk(t).and fk(t), and' hk(w) is the freuency response function the eigerunode k. this fùnctioh is also knö..iti as the strütural transfer

functiOn which is stated by,

k(t) h (w)

k

w2)

_{+ i.(2k Wk} w)

Here, the purpose is to calculate an Output spectrum, e.g. spectrum of displacements. It is now conidered that zk(t), k = 1,2,...,q are multiple inputs and {D(r)} is the output vector of a lineat system,

see fig. 2.1.

Modal suprposition

Fig. 2.1 - MODAL SU RI'OSITÍON PROCESS

It follows from (2.3) that the spectra of thé suãturàl displacements cañ be written as, in the mat-rix fonñ

{D(t)}

k=Ir=1

r

q

q

= E Sk(W) .{Fk} {F.

lç=L r=1

where {}'' is the transpositiono the eigenvéctor

f&};

The term

sk(w). is the so-called spectral participation

factor

which states the cross-spectrum of thegeneralize4 displacements zk(t) and z(t). The

spectrum of any output which is linearly related to the displacement

vector [D(t)} may be obtained in the same fäshion given by- (2.8). or example,

the spectral matrix of internal fOrces can be written a,

in which fF} and fF} are the internal force vectors calculated from the eigepmode vectors and

{Ç

respectively.

By using (2.6), the sttethnt of SPF can easily be written as,

Skr()) = h(w),

Sf fo

(2.10)

where () denotes the.complex conjugate and s (w) denotes the

cross-kr

spectrum of the generaled forces

Iti order to calculate s (w) , the. generalized (modal) forcés need to be

calculated first. The calculation of these forces is discussed in the following sectiOns in detail.

.3 - GENERALIZED (MODAL) WAVE FORCES

The statement of the generalized force for the eigenmode k has been

given y (2.5). Thè meaning of this statement may be explained by writing the extetnàl energy (work) of the structural system. The total work done by external forcés may be stated as, iÍ vectorial notätioi,

w = D} {p) _(3.')

Having introduced the expression of D}, given by (2.3), into (3.1) it js written that,

k= I

(3.2)

As it might be recognized that the term

{k}T.P} is the géneralzed

(rnodal) force for the mode k, thus (3.2) mar cbnveniently be writtén as,

q

k=I k k

(3.3)

Now, it can be cönciudéd from (3.3),withthehelpof(2.5), that

k may be considered as the wok (energy) done by external forces under the shape configuration of the eigenmode k. Thu, it is considered here as the over.iil mOdal work of external forces. This modal work (energy) can be obtained by superposing the individüal thodal wòrk Of structural members. Thus,

=

(3.4)

where is the modal work corresponding to the structural methbèr j, and N is the number of structural members loaded. The calculation of f . and f

f

kj k

is the main concern here, but first, a brief suiay of wave forces is presented for the sake. of understan4ing of catcuiatiòn-détails as well as for the completeness of the text, see also ref. [.1].

The wave force per unit lengt.h of a sructural member s given by Norison's equation which is stated by,

= CD .1w!. + CM. (3. 5.)

where denotes the absolute value,.: is the vector of water Velocity which is normal to the structural member. CD and are defined by, respèctivély,

CD . p. Cd . D

nD2

CM -

_TT

in whiòh p : density of water,

Cd : drag force cöeficient,

inertia fOrce coefficient,

-m -

-D : diameter of the sttuct-ural membet.

Having linearized the drag force term, (3.5) may conveniently be written

as,

{p} = .{w} + CM .{}- (3.7)

where A is a coefficient obtained from the linearization of the drag

force term. The vector of wave forces given by (3.7) can be stated. i-fl the

frequency domain as trittefl by, see réf. [I],

-{p(a)} = {iÇ(w)} . _rì(ui) (3.8)

where n(w) denotes the water e'levatioti in the frequency domain, and H(w) } is the vector of transfer functions between the water eLevation Ti and the

wave force vector {p}. This vector isgiven by, ee ref. [Ii,

.R. _{[T] .[H]. {e} (3.9)}

in which i

/T

, x is the horizontal coordinate méasùrêd in thé directiOn

of wave propagation andm

is the wàve

number. R and [T] ae respectivély,

R = CD . A + i ü) CM

!C

CÇ

-CC

-ccy

i-Cy

CC

_cyc

i-4

[T] =

where c, c

and are the cosines of the direction of the structural

member The matrix [H] in (3 9) denotes a diagonal matrix defined by,

(3.6a)

(3.6b)

3. lOa)

-

Io

--a

where,H(w) and H(w) are given by, see ref. j

w H (w) = -. . cosh (mZ + md) x sinli (md) (3. lOd) w

-.1

H (w) = . - . sinh (mZ + md) z sinh (md)

in which d denotes the wàter depth, Z denotes the vertical coordinate measured from the still Vater level. Finally, the vector {.O} in (3.9) is

given by, cos

fo] =

siñ (3. ¡0e)

where is the angle of án idividual wave direction measured from the

glQbal X axis, see also fig.. (3.1).. Th wave number th is dependent On the frequency by the relation writen as,

m.g.tanh (md) (3. lOf)

In thé case. of deep water condition, as it is assumed in this report, the statement (3.9) may be simplified as written by

H(w)}

= w.e _{.R .[T] . {O},} :

(311)

and the wave number rn can simply be Obtained frofn (3.IOf) äs-,

w2 m =

g (3. 12)

For an arbitrary member in. water, the coordinates x and Z can be expressed as

functions of the axial coordinate of the membe±-. These fÙntions eire written

by,

H(w)

O o

EH] o

H(w)

(3. lOc)

main wave

ditectiOfl

CH

(w)}

= wE .R.e

m(c_iL).s

.{TJ.0

pfl I

in which R is given by (3IOa) and E1 is defined by,

.Zi

-Fig. 3.1 AN ARBITRARY STRUCTURAL MEMBER IN WATER AND THE COORDINAT-E AXIS

Having introduced (3.13a) iflto. (3.11),

it

may bewritten that,

(3 14)

(3.15)

ZZ

+CZ

S

where x1 and Z1 are thé coordinates of the member end ( 1), see fig. (3.1),

= cx

cos

c1

+ cy .

Sifl

(3. 13b)

From (3.14)

it

can be seen that the function R.e Z x

determines the distribution of wave forces along the leiigth of a structural member.. This function will bé denoted by y(s).

Thus,

y(s) = R.e

Here, R is also a function of s, becaúse the coefficient A of the linearized drag force, see (3.lOa), is an exponential. function Of the vertical coordinate Z, for further detail see ref. [1].

Having introduced (3.16) into (3.14), the vector Of transfer fúnctions

will be,

{H(w)}

= w. y(s).E1. [TLfO} (3.17)

Since y(s) is an exponential functioñ of s, it is exponentially interpolated between the

mmbèr

ends as written by;

R1. _{= CD.Al +} LW

(3. 19)

R2 _{CD.A2 ÍWCM}

The parameter a

written in

(3.18) is complex, frequency dependent function of the individual wave direction and he direction of the structural member. This parameter is defined by,

-

12 -

-(3.16)

u. = m.(c i.L )

Z x

The wave force distribution obtained from the Mòrisòn's equation in the freqùency domain is flow fully determiied by theañs of (318), (3.1) and

(3.8).

(3.20)

y(s)

e'-

{(eeY

e (easl) (3. 18)

in which Z is the length .of the rnembe, R1 and R2 te the values of R at the member ends (i) and (2). respectively. These values are, from

The generalized force, or the so-called modal energy, of a structural thember is now calculated by.inegrating the distributed rodal energy over the length of the member. Thus, for the eigenmode shap k, thé generalized force or the so-called modal energy of the member j can be

Fig. 3.2 DEFORMATION PATTERN OF THE MEMBER j UNDER THE CONFIGURATION OF THE EIGENMODE k

expressed as,

J{p}.

ds

(3.21)

where is the length of the member {u} is the deformation vector due to the eigenmode shàpe k at a point between the member ends, e.g. at the. point Q in the figure, in the direction of applied forces, see fig. (3.2). In order to carry out thè integration given by (3.21), the vector luk) must be stated in terms of the aual coordinate s This

subject and the ca1clation Of genealized forces will be discussed in the following seátion.

deformation pattern due to the

4 - CALCULATIOÑ OF ÑODAL WAVE FORCES

As it is stated in the previous sectiOn the vector _{uk} denotes a deformation} vector due to the eigenmode shape k at a point between the member ends.From

the solution of the eigenvalue problem ôñly the deformations _{of the member} ends aré obtained, and they form the so-called eigenmode vector. The usual way to determine internal displacement _{or deformations, is to use polynomial} interpolation functions between the meniber ends. These functions are also known as the shape functions. Thus, thé displacement vector U}k in the local coordinate system of a member may be stated as,

{u } = [N].{d

ok

where [NJ denotes the shape functiôn márix defined by,

[N1 O O -N3 N2 O O

_N41

[o Ñ1 N3 O O N2 N4

o]

[N.] =

and d }kiS the nodal displacement,or igenrnode, vector f the ,member in

the local coordinate system. Since wave forces are normal to structural members, the axiâl and torsional deformations of the member are not included

in the above stàtements. Because, thes deformations do not produce any external wrk (energy) Under wave loadings, the components of the

d o6 14

-o5

X Fig. 4.1 - COMPONENTS OF THE NODAL

ANPINTERNAL DEFORMATION VECTORS OF A

STRUCTURAL MEMER IN HE LOdAL COORDINATE SYSTEM

(4.1)

(4.2)

vectors

fu }

and {d } are..shown in fig. (4I). Thé shape functions

ok

pk

[71

written in (4.2) are obtáine4 to bE,

[t] =

s-N4=

--o

Having introduced (4.1) into (4.4) the dformaion vector in thé.globàl êoordiñate system becomes,

.2

s2

s2 +

2,2

s3

In order to calculate the external work (energy) , thé deformation änd the force vectors must be consistent with each other Since wave forces are stated in the global coordinate system, the deformation vector given by(4.1) will be ttànsforrnedto the same coordinate system. This

transforrnat-io1ima) be written as,

[tl.fu}k

.

(4.4)

where [t] is the transformation matrix defined by,

Cy

_cc

v9

_c

However, it is worth noting that the eigenmode vectot

fd

still iñ the local coordinate system of. the member.

(4.3)

(4.5)

in (4.6) remains 2. 2,2 fuk = [t]. [N]. {d}k

(4.6)

= [TIT. lUk}1 = [T]T. {Uk}2

[}T}

{+}

[T]T.{ßk}2 16

-The vector of dist-ributéd wave forces 1h the frequency domain has been given by (3.8), and the transfet function vector b (3.17). Having introduced (3.17) into (38) the force vector becomes.

{p(w)} = y(s).w.E1 [TI .{O .r)(w) (4. 7)

The genêralied (modal) force o the nithber j cah now be calculated by

means of (4.6), (4.7) and (3.21). The explicit statément of this modal force in the frequency dothain is writtèn as,

f(w) =

E1.(d.

(JY(). [NIT.dS) {}T

[T].fO} . n(w) (4.8)

Having simplified (4.8), the generalized (modal) force can be expressed conveniently as written by,

fkj(w)

(q1.1T q2}T

3;f3}T:

q4. T).Ø}fl(w) (4.9)

in which the scalar functions q. (i = 1,2,3,4) may be considered as to be the coñcentrated load parameters at the mémb&r ends. The general statement of thesé parameters is,

= w.E1.

_JY().N.ds

(4.10)

o

The vectors (tS.} (j = 1,2,3,4) in (4.9) are the transfOrmed eigenmode

vectors o the member j. These vector are in the same direction of the watér velocity Vector which is normal to the thember, and defihed by,

where the matrix [T] is the same as defined by (3.lOb), and the matrix

[Te] is obtained às , -.

(4.11)

The vector.s {uk}12 and in. (4.11) are respectively displacements

and rotatiòns due to the eigenmode shape k at the ends of the member j in the globa.i coordinate system. The cmponents of these vectors are shown in fig. 4..

Fig. 4.2 - COMPONENTS OF THE EIGENMODE VECTOR OF A MBER IN THE GLOBAL COORDINATE SYSTEM

From (4.9) and (4.11) it can be seen that the concentrated load parameters q3 and q4 represent the influencé of cônsistent bending mOments on the modal force of the member, because the vectors {} and 6+} inlude only the rotational components of the eigenmode vector. Having introduced the

notátion,

g1()

}T (4.13)

where i =, 1,2,3, and 4, the generalized (modal) force given by (4.9). may

q1 = L

q2 = L .{P1(w) q3 =

2

IP(W)

-ihere Q denotes the length of the member.

FrOm (4.18) it may be concluded -that the functions F.(w) and G.(w) (i=-1,2,3,4) are considered to be nondimensional influeúce regions of the force parameters

and P2(w). Câlculàtion of the ñondimeñsionäl influence regions will be presented in detail in Appendix I. However, in order- to ùnderstand-their meanings a simple example is given below.

18

-=(q1.g1()

q2ig2() + q3.g3(q) + q4. (e)). w) (4.14)

-Now attention is given tO the vecto of transfér funct'ionsfH (w)} given

by (3.11). The frequency dependent term of this vector wilJ be denoted by

P(w). Thus,

P(w) = w.. e

me-ix)

. R (4.15)

which may be considered as a distributed wave force parameter.- The values of P(w) at the member ends (I) and(2) are respectively denOted by P1(w) and P2(w), or explicitly they re writtt as,

= W.E - (4.16a)

= W.E2

. R2 (4. 16b)

where E1 is the same as defined b-y (.15),. R1 and R2 are also the- sa:me as

given by (3.19), and E2 is defined

br,

E2 = e m(Z2-i.x2) (4.17)

in -which Z2 and x2 are respectively he välues of coordinates Z and c

at the hiember end (2 )

-The concentrated loa4 parameters defined by (4.10) can be written in terms of the distributed wave forcé paratheters at the member ends- as stated by,

F1(w) + P2(w) .

G(w)}

F2(w) + G2(w)} F3(w) + F4(w) + P2(w) . G4(w)} (4.18)

Assuming that

_{Q1, Q2, Q3}

and _Q4 are the equivalent nodal forces and

bending moments (consistent loads) of a two dimensional member subjected to alinear force distribution, see fig. (4.3).

p

(1-

_'+ . .P2

mìti;ai;iU;IIIiIiIIIiIL.

z

Fig. 4.3 - CONSISThNT LOADS OF A SL MBER SUBJECTED TO A LfNEAR

FORCE DISTRIBUION

By using the shape functions given by (4.3) the consistent loads are calculated arid the results are written below.

Ql =

+ Q2 = .

P.2, +

(4. 1.9) Q3 = . P1. Z + . P2. z2 Q = (1. 9 + .!5 P2.z2)

As it can be seen, the first. tems on the right. hand side in (4.9) are

the contributions of P1,, añd the coefficients of P1 are considered as to

be the influéncé regions (lengths and areas) of P. The propottions o

these. influence regions to the whple reg-ions ( and 92) ar,e the so-called

nondimensional influence regions of P1. The same rule applies also to P2. The influence regions concerning this example are demonstrated in table

20

-Table 4.1 - INFLUENCE REGÍONS OF LINEARLY DISTRIBUTED FORCES ON A SIMPLE MEMBE-R

g1() +

F2(w).2())

= .(G1(w).1 +

It is worth noting that the first terms of the functions i1(w) and

i(w) (terms with 9V), as it can be seèn from (4.21), aré the contributions of the equivalent nodal forces, and ti-le second terms (terms with 2) are

the contributions of the, equivalent nodal benditig moments. Hiving

introduced the

älues

of the distributed wave force parameter given by (4.16) intò (4.20) fk(w) can be stated in an alternative form as written

by,

fk.(w)

_(1.1

- R2.W2)fl(w) (4.22)

where the functions W1 and W2 are defined as,

(P3(w).3()

(4. 2 I a)

(G3.3)

+ (4.2 lb) Consistent load

Influéncé iegiôn Nóndiménsional inflüence

-,. regIon, PI P2 Pl P2 Ql

79,

9, : 7 -. 3 3 -3 .7 -. r' 1. 9,2 1 9,2 I 20 1 30 20 30 r'

L9,2

. 1 9,2 30 20 30

Now turning back to the main concern that the generalized (modal) force of an individual member given by (4.14) canfurtherbe simplified b.y introducing the àoncentrated load parameters given by (4.18) into (4.14). Having carried out this simplificatioñ fk.(w) can be stated as,

fkj(w)

=(PiW).Î

+

P2(w)(w)).fl(w)

(4.20)

where the functions 1(w) andt4.2(w) aré obtained to be.,

+ 9,2

W1 = wE1,1(w)

(4.23a)

W2 = wE2(w)

(4.23b)

in which E1 and E2 are the same as defined by (3.15) and (4.17) respect-ively. The functions R1 and R2 in (4.22) are also defined by (3.19).

Having obtained the indivç1ual generalized (modal) forces, or the so-called individual modal energies, the total modal force (modal energy) corres-ponding to the eigenmode k may be calculàted by using (3.4). This modal

force can be stated simply a,

= Hf (4.24)

where H (w) is called the transfer function between f and the water

fkn

k.

elevation fl. In general, it is a complex function which may be written in

terms o the real and be imáginary components as,

H (w) = Re H - (w) i. Im H (w) (4.25)

where the first term on the rihthänd side dertotés the real part, the second one detQtes the imaginary part, and i = ici. In order to calculae the modal force stated in the frequency domain, the transfer functon

H (w) needs to be calcUlated fitst.. Its calculation is presented in fkn,

5 - CALCULATION OF TRANSFER FUNCTIONS BETWEEN MODAL WAVE FORCES AND THE WATER ELEVATION

As it is mentioned in the j,revious section, the modal force fk(w) in the

frequency doman i calculated by means of Hf (w) and n(w). Since '1(w)

is known as the input of the random process, the calculation of the transfer function H (w) is thernain concern of thi section. The.

transfer function fot the whole structural system,cor.responding to the eigenmodek, can be obtained from the transfer functions f br tructural

thembérs simply: as in terms of thé reâl and the imaginary parts. Thus, it follows from (3.4) and (4.25) that,

N Re H (w) = E Re H (w) J 3 N Im Hf-(W) = E Im Hf (W)

jI

j

in which the real and the imaginary parts of the transfer fùnction for the member j are simply written from (4.22) as,

Re Hf (w)=ReR1ReW1 + ReR2ReW2 - Im R11m W1 -

R21m W2

(5.2a)

Im .1(w)=ReI1Im W1 + Im R1ReW1 + ÌeR.IÌn W2 + Tm R2ReW2 (5.2h)

Having introduced .the real and thé imainary parts of the fúnctions R1 and

R2., see (3.19),. into (5.2), the transfer fùñction becomes,. as real and

imàginary parts, 22 -ReRffl(w)=CD!(Al.ReWJ + A2..ReW2)_w.C.(Ïm WI + n w2) Im Hf À21m W2)+w..(ReW1 + ReW.2 ) J.

In (5.3), CD and. CM re defined by (3.6), A1 and A2 are the values of the coefficient A obtained fröm the linearization of the drag force term at thè member ends (see (3.7)), thé real ärid the imaginary parts of the functiOns W1 and W2 are obtainèd by usiñg (3.15), (4.17) and (4.23) a

to be,

(5 ..3a)

ReW1 =

w.èmZ1.(cosx1.Re

1(w) + sin(mx1),Im

i1(w)

Im W1= w.e1.(cos(mx1),Im

1(w) -

sin(mx1)Re

i1(w)

ReW2 =

w.emZ2.(cos(mxR

Im W2=

W,emZ2.(cos(Im

(a) - sin(mx2)Re )2(W)

A = f(O

)G

(H ) o u S X 2(w) + sin(mx2)1 (5.4a) (5. 4b) (5.4c) (5. 4d)

where m is th wave number defined by (3.12), Z1 and Z2 are the vertical coordinates of the member ends, x1 and x2 are the horizontal coordinates of the member ends in the wave direct-ion. The real and the

imaginary parts of the functions 1(w) and 2(w) can be obtained from (4.21). The calculation of these functions is presentéd in Appendix I. Here, attention is given to the calculation of the coefficients

A1 and A2 written in (5.3).

The coefficient A stated in (3.7) is obtained from the linearization process of the nonlinear drag force term. The detail of this process is explained in a previous report[1I. The general statement of the coefficient A is given as, see ref.[1],

(5.5)

where f(O ) is a function of the main wave direct-ion, and G (H ) denotes

o - - -- u s

the standard deviation of the water velocity corresponding X t-othe

unidirectional wave in the horizontal directioñ. ThIs deviatiòn for .deep - water condition is calculated from the statement, as a function of the

sign-ificant wave height,

in which S(w) is the unidirectional sea spectrum. The values of A at the

member ends (A1 and A2), t-hus -at Z = Z.1 and Z = Z, can be calculated by using (5.6) and (5.5) as functions of the significant wave height H. -However, in practice, these values may be approximated to some- knôwt functions of H. Thus, it can be stated- by using vectorial notations that,

Gu(Hs)

%1JW22mZS(W)dW

(5.6)

Re Hf (w)

_{= CD{bI)i{F(HS)} -CIm kj}

Im Hfw) = CD{b2}.{F(HS)} +wC Re

in which the vectors b

L

and b }. are defined as,

2j

{b11.

=

gew1,a1} + ReW2a2

f

_2j

b

ImW{a } + ImW{a

I 1 2 2 24 -1 = {a1}T.{F(H)} (5.7 a) {a2}T.{F(H)} _(5.7b)

where the vector {a1} and fa21j are to be calculated by using some approxi-mation points of H, and {F(H)) denots a vector of the assumed (known)

functions of H. The calculation of {) and and the determination

of {F(H)} are presented in détail in Appendix Il. Now, by introdu:ing (5.7) into (5.3), it càn be statéd thät,

and the scalar is defined-as the real and the imaginary parts written

by,

Re Re W1 + Re W2 (5. 9c)

Ith

kJ Im WI + Im W2 (5. 9d)

Here, j denotes a structural member and k denotes an eigenmode

In the determination Of the drag and the inertia force coefficients, CD and CM, there are sOrne ùncertainties that should be taken intO account when a reliability analysis is carried out. In

otdér

to represent these uncertainties, the

drag

and the inertia force coefficients may be stated

in tetns of their thean values, which tnay vary from methber to thethbe, and the

so-cal1ed uncertainty aàmetets that aré assuthéd to be independent of members. Thus, the concerning coefficients are expressed as,

CD = aD.PCD CM =

4CM

(5.8a) (5.8b) (5.9a) (5.9b)

where

D and are the uncertainty parameters (stochastic parameters), and

CD änd are thé mean values of CD and CM respectively. It

is. obvious from (5.10) that the mean values of aD and are' both equal to

unity. By introducing (5.10) into (5.8), the transferfunction for the member j becomes, as thé real and the imaginary parts,

Re Hf

_{= D1JCDbI}}

fF(H)}

- MÜ)

CMm

(5.1la)

3 3

Ith Hf (w)

_{aD.CD{b2}!fF(S}

₊ (5.Ilb)

.1

J

The transfer function for the whole structural system, for the eigenmodek, can now be calculated by us'iñg (5.11) into (5.1). The real and the imaginary parts of this transfer function áré stated as,

N = z -i {b2}. j=1 CD N Re =

.E Wc

Re j=1 N Im k =

WICN Im

j=1 Re H (w) = a CB }T {F'(H )}

_{- M.Im}

k11 D 1 s k Im H

(w) = a B

}T {F(H )} +

Re Q

D 2 s k

where the vectors _BI}k and {B2k are defined, by, from (5.1),

The real and the imaginary parts of

k in (5.12) are defined by, also from (5.1),

(5. 13a)

(5. 13b)

(5. 14a)

(5. 14b)

The transfer function of the modal wave force for an eigenmode, Hf

-is now completely determined. Then, the spectra of modal wave forces can easïl.y be calculated from the linear relation (4.24) between fk(w) and

26

-6 - SPECTRAL STATEMENTS OF MODAL WAVE FORCES AND CALCULATION OF SPECTRAL MOMENTS OF STRUCTURAL RESPONSE OU PUTS

In the previous section, the transfer function of the modal wave force for an eigenmode is determined. Since the purpose is to calculate the spectral participation factors, they càn be Obtained in terms of spectra of modal wäve forces änd structural transfer functions, see (2.10).

Strûctural transfer fuñctions are prev.ously determined f toxi the eigerivaiue

solutiOn of the structuie, and for the eigenmode k, a structural transfer function is given bî (2.7). Therefore, this .setiön is devoted to the spéctra of mOdal wave forces and, to explicit statements of the spectral patticipatiotì factois. By Using the spctral relationJ8 between an input and an outpUt of à linear system, the cross-spectrum of the modal wave forces fk(w) and f(w) corresponding to the eigenmodes k and r may

simply be written as., from (4.24),

Sfkfr(W)

=

H(W)

S(w)

(6.1)

wheré denOtes the complex conjugate and S(u) is the spectral function

of

the watet elevation (sea spectrum). The real and the imaginary parts of the transfer function corresponding to the eigenmode k are stated by (5.12). Now, by introdùcing (5.12), for eigenmodes k and r, into (6.1) the real and the imaginary parts of the cross-spectrum of the mo4al wave forces ara obtained as to be,

Re Sf f (w)

(.F(H)}T[BX].cF(H)

D

{F

)}{GX)

+

c.Qx).S(w)

(6.2a) I

Sf

f (w)

=(.{F(HS)}T[BY1.fF(HS)}

+ +

r

. (6.2b)

where the matrix notations are,

[B}

={B,l}k.{BI)T

+

2k2r

(6.3a)

[Bi] =Bl}k.{B2}T

-r

(6.3b)

{G}

Ré

+ Re Qk{B2r - Im r'ik

Em

G} = Re

_r'ik

+ Im

r2k

- Re Qk{BI

and the scalar notations are,

Re Qk.Re Q + Im Qk.Im r Q. - Im Qk.Re Q Sf f (w) = Re Sf f (w) + iIm S (w)

kr

kr

kr

Re Sk (w) Re s (w) - Yk(w). Im Sf f (w).

kr

kr

Hàving obtained the real and the imaginary parts of the cross-spectrum, in general, it is stated that,

(6. 4a)

(6.4b)

(6.5a)

(6.Sb).

(6.6)

By changing k and r from I to q (number of eigenmodes included), all cross-and auto-spectra of the modal wave forces are obtainéd. It can be shown through (6.2) and (6.5) that the cross-spectrum of two modal forces is Hermitian, that is to say Sf

f

(w) = s (w) in which dehötes the çomplex conjugate. Thus, the auto-spera of moa forces are all ral. This 'cañ also be verified from the statements through (6.2) and (6.5). Therefore, it is necessàry to calculate only the half pärt of the spectral matrix of, the modal forces, namely for the variation of k I to q àñd r k to q.

ft is worth noting that,as a result òf the harmonicvátiätiôn of thi

ôross-spectra of joint wave loads with respect to the frequency(seé fig.6.I),

the cross-spectrum, of the generalized forcés,. Sf

f (w),becdmes afiuctuatitig function of w. This feature must be taken into acunt when a réliability analysis is carried out.

A general statement of the spectral participation factors is piesented by

(2.10)

in

terms of the spectra of the modal forces and the structural trans-fer functions. Thus, b introducing (6.6) into (2.10) the real part o, the spectral participation factors becomes,

(6.7)

whére Xk(w) and Yk(w) are the real and the imaginary. pärt

of (h(w).h(w)).

For completeness, X.K(w) and Yk(w) are explicitly written below.'

X (w) (w

w2).(w2

w2)

_{+ 4k}

_r

Wk Wr C)2

mkr ((2_2)+

₄

_ww).((w

w2)2 4 w w2. Y (w) = kr

in,'

(cw

-

28

-w(w2-w2)-Ç

w.(ww2)

+ 4 w

)o((ww2)2+

4 w2

)

Fig. 6.1 - CROSS-PECTRUN, OF WAVE FORCES AT DÌFFERENT STRUCTURAL JOtNTS (REAL PART)

(6. RLI)

in which ni.

and mr are the generalized masses, Wk and w

are the

eigefi-frequencies, and

_k

and

are the damping ratios corresponding to the

eigeninodes k and r respectively.

Only the iéaI parts of the spectral. part.cipation factors aré needed.

Because, in the structural response analysis

the

real eigenmodes

aré considered. therefore, the imaginary parts of the spe.çtra. participation

factors are not requiréd. The real parts will be denote4 by Skr(W)

hence-forth. As it can be seen from

(67)skr(W)

=

rk'

because Xk(w) is

synmietric and Yk(w) is anti-yrmnetric, see (6.8), with respect to k and r

Now, by introducing (6.2) into (6.7) the spectral participation fâctots

become,

. -Sk(ü))

(c.{F(H

in which

[FDI {GDM}=

krX

=

kr

Q)Ç

-SD(u)) {F(H)}..tFDIfF(H)}Sflfl(w) SDM(w) CF(H SM(w) = QMSnn(w)

]{F(H)

+ c.o.{F(H

)}T.{GDM} ₊

_QMSfln()

(6.9)

In this way, the, spectral participation factors are

alculated

and, as

lt iS explained in Section

2,

the speçtra of any output can easily be

calculated by means of these fáctors for example, see ('2.8) and (2.9).

The calculation procedure of a stress spectrum is.outlined in fig.

6.2.

(6. 12a)

(6. 12b)

(6. 12c)

or

kr

can simply be written as,

kt

= ODSD(w) + ctD.aMSDM(w) 2..S,(w) . (6.11)

wheré the spectral functons

SD(w), SDM(w) and

SM(w) are respectively

defined by,

[B.»

(6. lOa)

r

.{c}

(6

lOb)

-WATER ELEVATION

WAVE LOADS

(MORISON1s EQ.)

STRUCTURE

MODAL. DISP. (GEN..DisP,) SPECTRUFI OF. GEN.DISP. (SPECTRAL PART.FAC.

)

Fig. 6.2 CALCULATION PROCEDURE OF STRESSSPECTRUM BY USING SPECTRAL PARTICIPATION FACTORS

SPECTRUM OF STRESSES

EIGENVALUE SOLUTION

(NAT.FREQ. AND MODE SHP,.

However, in the stochastic analysis of structures, statistical measures of response outputs are required. These statistical measures can be

calculated in terms of the spectral moments of structural response outputs

(spectral thoment of ±erö, second and fourth order) so that they rnust be

calculated first. The calculation of these mments can simply be carried

out f the moments of the spectral participation factors are known, as similar to the calculation of spectra of respone outputs by the use of spectral participation factors. The problem, now,. turns out to calculate the moments of the spectral participation factors. In general, they are calculated from, for unidirectional sea spectrum,

(C)k

=

I

W5kr)(W

o

or for multidirectional sea spectrum,

pkr

2

(6. 13a)

(6. 1 3b)

in which O denotes the variable of the directional distribution of the sea spéctrum, that is the angle between an individual and the main wave directions as defined by O see fig. 3.1. In (6.13), p denotes the order of moments, thus p = 0,2,4. The directidnal distribution of the sea spectrum is discussed elswheré - [111 and it is not the subject of this report. In general, the multidïrectiönal sea spectrum is stated

as,

S(wO) = D(8)

S(w)

(6.14)

where D(0) denotes the directional distribution. In this report, the following statement for D(0) is adopted, see also ref [11].

D(8) =

cos0

(6.15)

where -. O . Outside of this region D(0) is assumed to be zerO.

By introducing (6.11) into (6.13), the moments of the spectral

partici-pation factors can be stated as,

32

-where

Dkr'

ÌMkr

and (MM)k are respectively the moments

of

tite

.spectral fûnctions SD(w), SDM(w) and SM(w), see (6.12). These moments

ate calculated in. the same fashion as (6.13). llnving calcui:tLed tite

moments of the spectral participation factors, the spectral moments of response otputs may be s'tated as,

[m]

= k=1

r!IFkT

(6. 17)

in whih the vectors {Fk} and F} ar the output vectors obtained from

the eigenmode shapes k and r respectively. An integration procedure to

calculate response spectral moments is presented in ref. 1121 in detail.

Therefore, in this report, n special attention is given to carry out tite integration given by (6.13).

Due to continuously variation of the wave fordes Òf f shore structures confront mostly the danger of the fatigue collapse within a service life time. However, there may also be somé ecceptiotial loading cases that a structural failure occurs due to the violation of thé ultimate limit capacity f structures. In this report, such exceptiçnal cases are not discussed. Only the case of fatigue phenomenon is studied, and the

stochastic fatigue reliability analyis is presented. In the ànalysis, thére are some Uncertainties that cannot be avoided. These uncertainties are taken intO account by means of sdme stochastic variables. A detailed survey of them will be presented latei. First, a formulation of the

stochastic fatigue damage is presentad for abetter understanding of the probabilistic fatigue reliability anlysis.

RANI)OM SEA ELEVATION

RANDOM WAVE FORCES

I

RANDOM STRESS VARIATION e RANDOM FATIGUE DAMAGE

7 - FORMULATION OF PROBABILISTIC FATIGUE DAMAGES

Due to cyclic stress variations in the structural members and joints,

fatigue damages occur when the number of stress cycles reaches a certain value at a certain stress range. The nature of the fatigue damages are explained elsewhere, see e.g. ref. [13], atd it is not the

subject of this report. The commonly uséd criterion for the fatigue damages is given as the Palmgren-Miner's rule to be the sum of the

fractions of stress cycles. The damage is analytically stated as,

n(S.) =

i=I

N(S.)

(7.1)

where n(S.) : number of stress cycles at the stress range

N(S.) number of stress cycles to failure at the stress range

q applied stress alternations

According to the Palmgren-Minér's rule the damages at different stress ranges are accumulated, and when thé cumulative damage reaches unitya jôint, or a member, is said tO be failed. For offshore structures the

stress range S. is a random variable, see fig. 7.1, so that the mean valìé of the fatigue damage D wl1 be uséd in the reliability analysis.

34

-FOr à sea state, the fatigue damage due to one stress cycle at a stress range S. maybe written as, (see also ref. [14]), from (7.1),

d

N(S.) (7.2)

which is a fúnction of the random variable S.. The mean value of this

115]

individual damage is stated as1

E[dD.

fN(Sj)

f5.(s) ds (7.3)

-o

where f5 (s) is the probability density function of the stress range S.

for one 1 sea state. After a number of stress cycles within one sea state, the expected fatigue damage will be,

TH f5..(s)

=

N(S.) ds (7.4)

o i

where TH duration of the sea state considered,

T :, zero crossing period of sress cycles in the sea state

The expected damage

.

during a sea sate is obtained as a function of the sigúificant wave height H. Because, the calculations of T and E[dD.}

given by (7.3) require spectràl momets of stress variations that they can only be cálcul-ated as functions of H. In the reality, the significant wavE height is â random variable, and the probability density function of

this variable is assumed to be known às (H). Thus, the mean value of with respect to the significant waé 1ìeght is stat-ed as, from (7.4)4

E[Th]

I

-2.ß

dsf

(H)dH

i i T i H

j0 o j0 N(S.) s

(7.5)

Aòcording to the PalmgrenNiner's rule stated by (7.1) the expected damages for all sEa states within a service life time will be accumulated, -and -the cumulative, or the total, damage will be showïi by Dtt. Thus,

n

s

D = E _E[D..]

tot . i

where n denote thé number of sea states applied.

It is now assumed that all sea state haVe the same duration as TH. within the service life time T. Then, the totàl fatigue damage becomes,

D = n ( 7.7)

tot s

-and the number of sea states applied will be,

n5

=

T

(7.8)

By introducing (7.8) and (7.5) intô (77) Dtot gill bé obtained as,

jj

. ds)

o -o

(7.6)

(7.9)

in which T is the service life time considered. Here, it is assumed that the stress range S. in ä sea state is double of the stress amplitude Si at a time. It is also assumed that., for the computational convenience., the stress variation is a narrow band Gaussian process. However, this assumption is only true if the eigenfrequencies Of the structure are not far from the predominant frequency of the sea waVes, which. means that the peaks of the structural transfer funétioñs and the peak of the sea spectrum are close Lo each other. In the case of high eigénfrequencies,

that- -is to -say, iñ the case of semi-static behaviour, the narrow band

stress process is obtained for large valUes of the significant wave height. Otherwise the assumption -of the narrow band stress process is conràdictory to the reality. Nevertheless, this assumption is used in this report si-ncé broad band stress process causes some analytical difficulties,at leást for the time being, and requires an experimental fatigue model. Under these

-assumptions can be showrJ4' that the probability dnsjt- f-unctión of the

36

-0.0 0.5

This. functioh is given by,

s -fA (s) e s S,. ₂ 'L s I.0 1.5 2.0 f(s) / RAYLEIGH DISTRIBUTION 2.5 3.0

Fig. 7.2PROBABILITY DISTRIBUTION OF STRESS

_{.ALITUDES IN ONE SEA STATE}

s'a

_s

(7. 10)

in which .G denoté thé atiáñce of :t1i stress vatiations. The number

of

tres

ycles to fáiiute, N(S.), cánhly be detetmiried, at present, ftom

xperimerital data with sorne uncertainties, see ref. [16]. The theories of the fracture mechanics are, so far, no applicable to the complex joints of offshore steel structures to find a more fundamental expression for N(S.) analytically. Therefore., experimentally determined SN curves are commonly used for N(S.). These curves are generally stated" as, see

fig. 7.3. . ,

CfSh

(7.11)

whére Cf and k are the fatigue parameters to be detérmined from experimental data, and denotes the hot-spot stréss rangé at a s uctùral joint. It

änd /

N(.) = C (2SCF'

i f

fc7

¶ 1r41 m 2 o = SCFS. (7. 12). i

where SCF means the stress concentration factor calculated by the fihite element method, and S. is the nominal stress range used in the statements above.

log Sb log N = log Cf - klog Sh

(7. 14)

(7. 15b)

log Cf log N

Fig. 7.3 - AN S-N CURVE IN LOGARITHMIC SCALE

In terms of the stress amplitudes, (7.11) may be stated as, since S. = 2..,

(7. 13.)

Now, by introducing (7.10) and (7.13) into the general stätement given by (7.9), the total mean fatigue damage becomes,

cok

tot =

(2vSCF)k

_ro

+).f

;-f o s

In this statement, and T0 remain in the integral sign, because these

measures are functions of the sianificant wave height, aria they are calculated by means of the spectral analysis explained previously. The variance,

corresponds to the zero moment of the stress spectrum, and T is calculated from thesecond and zero moments of the stress spectrum, see ref. [4]. Thus,

02 _{= m (7.15a)}

- 38

in which ni, and ni are the zero and the second spectral moments of the

o 2

stres variation at a structural joint. Calculation of these moments is simply carried out, as similar to (6.17), by using the moments of the spectral participation factors given by (6.16).'The integral part of the mean fatigue damage given by (7.14) eau only be carried out numerically in

such a wa that fo a hurnber of signficant wave heights a and T are

first calculated, and then the numerical integration is applied. För the aÍiatiön of the fatigué parameter k and the parameters of the probability density function of the. significant wáve height,

H (H), the previously

calculated a and T remain unchanged hen the relibility analysis is

carried out. In the ábove statement of the fatigue damage, see (7. 14),

thére are some uncertainties that must be taken intò accOunt in the reliability analysis These uncertainties are briefly discussed in the following section.

8 - UNCERTAINTIES IN THE CUMULATIVE PROBABILISTIC FATIGUE DAMAGE

The probabilistic fatigue damage is formulated as given by (7.14) for a narrow band stress process. However, in this formulation there are some parameters that can not be determined exactly in. practice.. Only their average values can be mentioned when the deterministic analysis is concerned. In the probabilistic analysis,these parameters are con-sidered as random variables with known or assumed probability distribution functions. The uncertain parameters in the mean fatigue damage may be

classified as:

- fatigue parameters,

- parameters of the probability density function of H. (parameters of

H (H)),

- parameters affecting a and T through the spectral analysis. S Since there is no theoretical relation between stress ranges and number of stress cycles to failure in offshore applications, for the time being, experimentally determined relations (S-N curves) are necessarily used in practice. The parameters of S-N curves, namely Cf and k, are Obtained from the linear regression analysis. of some fatigue tests data for some simple structural joints.under simple loading cases in the air cándition mostly. These S-N curves (after some modification) are used for the analysis of complex structural joints in. the sea environment where conditions are different than air conditions, and also the loading

cases are more complex than simple cases. Besides these environmental and loading conditions, the fatiguetests data for a particular joint are randomly spread that in the determinatiOn of S-N curves üncertain-ties cannot be avoided. In practice, there are only a few number of

experimental data for tubular joints ùnder the same conditiOns, at present, that the S-N curves based on these data include naturally higher uncer-tainties. By increasing the number of experiments and. refining the experimental conditions, uncertainties may be sothewhat reduced, but cannot completely be avoided. Therefore, both fatigue parameters are considered here as stochastic variables in the reiibility analysis. FOr further details about this subject ref. [16] should be consulted.

Since the fatigue phenomenon is an occurence in the long term duration of structures in the sea, a long term distribution for the significant wave height is used in the fatigue reliability analysis. This

distribut-ion is Ñostly reprèsented by a three parameters Weibull distributdistribut-ion, see ref. [2], as written by,

0.00 Ç) s 40 -WEIBULL DISTRIBUTION A = 0.60 tn. B = 1.67 Tn; d 1.21

pig. 8.. 1 - PROBABILITY DENSITY FUNCTION ÓF THE SIGNIIÇANT WAVE HEIGHT F0R LONG-TERM- DISTRZBUTION

(H-A\ =

_-e

\B)

where FI >A, and A, B and C are the parameters of the distribütion determined froth the data of one year observation; 'the probability density funétion

f H isöbtaitied horn (8.1) as stated by, see also fig. 81,

s

fH(H) ₌

î

()CH

(8.2)

In this stäternent A, B and C are obtained from some obsetve4 data that un-certainties in these parameters are always exist, and these uncertainties aré taken hito account in the reliability analysis by considering that the

There are also some other uncertainties in the mean fatigue damage that arise with the statistical measures, O and T, throughout the spèctral analysis. These uncertainties may be Considered due to - Foundation parameters (spring constants)

- Structural parameters (member thicknesses, damping ratios, masses) - Loading parameters (drag and inertia force coefficients, sea spectrum)

The tnertainties in the mean fatigue damage are,shówn together in Fig. 8..2.

FATIGUE PARAMETERS ( Cf , k )

1

FOUNDATION PARAMETERS -SPRING CONSTANTS UNCERTAINTIES STATISTICAL MEASURES OF STRESS VARIATION ( o , T0 ) STRUCTURAL PARAMETERS -MEMBER THICKNESSES -I)AMPIÑG RATIOS

-MASS OF THE DECK -ADDED MASSES

Fig. 8.2 - UNCERTAINTIES IN THE MEAN FATIGUE DAMAGE

Fixed jacket type structures are built on piles that in thé analysis structure-pile interactions should be taken into accOUnt for an exáct câlculation. The interaction problem causes,however, sothe extra compüt-atiônai difficulties, especially, in the stochastic analsis. TherefOre, in practice, the effect of piles are adequately represented by a massless spring system with some uncertainties that may arise in the calculation

PARAMETERS OF f (H)

Ns

( WEIBULL DIST.- A;B,C)

e

LOADING PARAMETERS -DRAG FORGE COEF. -INERTIA FORCE COEF.

[K] =a[K

g .g gp

where [KgpI denotes the mean value of [Kgl 42

-procedure of the spring constants. These uncertaintiés may be considered iñ the andlysis if the spring constants are taken to be stochastic variables. Further, it is àssuméd that thé pring constants in different directions are fully coreìated. Thus, an independent variable is used ih thé analysis to tpresent uncertainties of the foundation effécts. This variàbié is denoted by g With a mean value equal to I. Then, the stochastic diagonal stiffness matrix f r the foundation effects is written as,

In the class of structural pära7metets methber thicknesses, damping

ratios and masses may be mentioned. Uncertainties in the memker thick-nesses are dueto geometrical imperfe&ions that may occur during the fabricätion process. These uncertainties may be best represented in the analysis by conidering thät the, diffêreñt thicknesses are independent stochastic variables. the drawback of this consideration may be that the computation time is increased considerably. Therefore, it is necessarily assumed here, for the time being, that all member thick-nesses are fully correlated.

Thus, a single stochastic parameter with a méan value equal o I is

assUmed to represent uñcertaiñt-ies of the member thicknesses. The thickness of a member is, then, stated as,

t. = i t

whére denotes à common stochastié 'ar-iable for all member thicknesses

and i. dénotes the mean value of th& thickness ofmembe,r i. t].

The damping mechanism of structures is not yet exactly known to cal-cuÌate the damping ratiOs analyticairy. Therefore, experimentally

determined damping ratios are used iñ the structural response analysis. The éxperimentally determined damping ratios include uncertainties

that cannot be avoided, especially in the sea environment. These. un-certainties are taken into account by.using stochastic' damping ratiós in the analysis. For the computational convenience, it is assumed that damping ratios for different éigénmodes aré fully correlated, and their uncertainties are represented by 'in the reliability

(8.3)

analysis. The mean value of this stochastic variable is also equal to 1. Thus, the damping ratio corresponding to the eigenmode k is written as,

=

(8.5)

where j is the mean value of

A possible variation in the mass of the deck is taken into account in the reliability analysis by considering that the deck mass is a stochastic variable in which uncertainties are represented by ad. Besides the deck-and structural masses offshore structures are also subject to hydrodynamic masses produced by the, interaction between surrounding water and struct-ures (added masses). in the calculation of the added masses uncertainties are inevitable. These uncertainties are also represented by a stochastic variable in the analysis. This variable is denoted by

Uncertainties in the wave loadings and in the spectral modelling of the random sea waves are consdêred in the class of loading parameters. Wave loadings are calculated from the Morison's equatiOn às explained

in previous sections in terms of tiìe drag and inertia forces. The drag force coefficient CD varies with the Reynold's number (Re =v.D/\) and also with the Keulegan-Carpenter number (NKC

inertia force coefficient CM varies with the Keulegan-carpenter number (for detail, see ref. [4]). The design values of these coefficients may show some dicrepancies for different structural mernbers and also

include uncertainties. These uncertainties are considered in the analysis by stochastic parameters OED and OEM of the drag and inertia force coeffi-cients respectively The definition of these parameters is given by

(5.10), and more detail can also be found in section 5.

Random sea waves are represented by some empirical spectral functions of the water elevation known as the sea spectrum. In a sea spectrum there are parameters that are determined from the analysis of wave records in which uncertainties exist. These uncertainties can be con-sidered in the analysis if the parameters of the sea spectrum are taken to be stochastic variables. The most commonly used sea spectrum is the PIERSON-MOSKOWITZ, and thé JONSWAP spectrum, for detail, see ref. [4]. In this report, the Pierson-Moskowitz sea spectrum is adopted. This spectrum may be stated [4] as, see also fig. 8.3,

= y .T/D)while the

4a.

g2

a.g2

. e Hz 4 S

₍₎

nil w5

where

a

denotes the parameter of the spectrum and H is the significant wave height, g is the acceleration of the gravity.

-

44

-= 0.0081

H

=4.30rn

s

g = 10.0 rn/sec2

Fig. 8.3 - PIERSON-MOSKOWITZ SEA SPECTRUM

The Paran1eteris considered here as a stochastic variable to represent uncertainties in the sea spectrum. The mean value of this variable for the North Sea is taken as 0.0081.

The stochastic variables considered in the probabilistic fatigue relia-bility analysis are presented in table 8.1.

(8.6)

Table 8.1 - STOCHASTIC VARIABLES OF THE RELIABILITY ANALYSIS

Stochastic var i able

Description of stochastic variables

a g Foundation Member thicknesses o. a Added masses

Mass of the deck

Damping ratios

Sea spectrum (PiersonMoskowitz)

Drag force coefficients

Inertia force coefficients

A

B C

Parameters of Weibull distribution for H

s

Cf

k

Fatigue parameters N =(cf.s_k)

I

2.a2

f (z) - . e Z

- 46

-9 - PROBABILISTIC FATIGUE RELIABILITY CALCULATION

Methods to calculate structural reliability aré so ar establihed, and explained elswhete h181 In order to understand the terminology, a brief su!1ary of the reliability calculation is also presented here. In general, strüctures are -sùbject to sóme applied iMading states, and when the appliéd loads exceed the lOad bearing capacity of the structure (strength), the structure is said to be failed. From this statement a structurai failure function can be defined as,

Z = R-S (9.1)

wheré R denotes the structural loading capacity and S denotes applied loads, assuming that both paraméters are random variables with probability density functions fa(r) and respectively.

In the case of independent R and S, thé probability of the structural safety (commonly used the reliability) can be stated as,

'k

= i

-'r

{Rs}

(9.2a) or = I

f(f:R(r)d)

(9.2b)

This statement, of the structural reliability corrésponds to the level III

calcu1àton that it isused only for checking the validity of othersimplified

reliability methods. Another form of (9.3) may be written iñ terms of the failure function Z,assuming that its probability density function is known as f(z). Thus,

'k

= i - 1'r

{zO}

(9.3a) or I _-

fzz).d

(9.3b)

Now, it is assumed here that Z is normally distributed from which f(z)

is written by,

(z_uz)2

By using the transformation,, z

+

, the reliability can be

stated as,

ck= i -c(-)

where is the so-called reliability index defined by,

,,,u11,i,,Ill1I,/If1I/Il1lIIIIiI!1IIiiiI1IIIIIIII/i

z<o

failure region

Fig. 9.1 - PROBABILITY DENSITY CURVE OF THE FAIURÈ FUNCTION

The réliability calculation turns out, now, to calculate the reliability

index , hence to calculate and az.

In general, these statistical measures can be calculated from [15j,

.uz

= z.f(z).dz .00 =

J(z_i.tz)2

fzz

Normal density cùrve

The drawbáek of (9.7) is such that the failure function Z is mOstly a. multi-variate nonlinear function with respect to the stochastic vriables

(design variables), and therefore,these integrations are practically

(9.5)

(9.6)

änd(.) represents the standard normal distribution function. The

definition of ß (the reliability index) is also demonstrated in fig. 9.1.

(9.7a)

= n Z . .OE i 1=1 i t(*)

[n

E [G!

)) ] j=1 j - 48

impossible for design purposés. In ordér to overcome the integration difficulties a linearized failúré funcUon is used for an aproximate calculation of and . A brief suiary of the calculation is pre-sented here. The failure function Z in the spáce of deèign variables maybe stated in general as,

Z = g(X, X2,..., (9.8)

where X.(i = 1,2,..., n) are the design Variables which are assued to be independent and normally distribútéd.

y using the first-Order Taylor series expánsion,at some points of X., the function z becomes,

*

Z g( ) + E (X.-X,. ).g(X )

where (X* = X,...,X*) deflotes linearization poiiit in the variable

space and g!(X*) denotes derivatives óf g() at the linarization point. For more information see ref [19] By introducing (9 9) into (9 7),

thé statistical mèàsures of Z will be obtained as1 n * = g + E

_{X.Xi ).g(X )}

i=1 i =

[X1

og.*)]l

where lix and

denotes the mean and the standard deviation of an individual design Variable. The staúdárd deviation òaîi also be stated as,

where ct. is the sensitivity coefficient defined by,

aj

a ogt(x*)

The square of

.

denôtés the cOntribution of an individual desigi variablé to the vatiance of the failure function. Thus,

(9.9)

(9. lOa)

(9. lOb)

(9 11)

n * * E ).g(X ) 13 i=1 L

t.

. * E

a.a

.g.!(X ) i=1 n --= I (9.13) i *

If the linearization point is chosen at the boundary surface, g(X

=0

will be obtained. In this case, the reliability index 13 becomes,

from which the relation,

-)gT(x*)

O

will be obtained. The only conditi6n tO satisfy (9.15) is,

- 13'a. = O

from which the linearIzation point (design point) on the failure surface is obtained to be,

X

=

-This method to calculate the reliability index 13 is known as the level II

advanced first-order second-moment method, and requires än iterative solution. The algorithm of the iteration is summarized às follows:

Start from the mean values (X?

=

Calculate and o frOm (9.10)

Calculate a., a. =o .g(X*)/a.

i

i xi i z

Calculate 13, 13

Calculate new design point from (9.17)

Repeat from (2) until a required convergence, is obtained

Check if the design point is on the failure surfacé, g(X*) = O

In the report, this method LS used to calculate the reliability index It is worth noting that, when the failure function fluctuates with respect to the design variables that affect the natural frequencies, this method does not give a convergence. In this case, however, a combination of

(9.14)

(9.15)

(9.16)

Z = 1 - D.

- tot

50

-level III and levél II may be suggested, for detail ref. [201 should be consulted. For linear or, slightly non-linear failure fùnction, the advanced first-order second-moment mehod gives exact results. In the

cäsé Of thé nofl-northal distributiOn o design variables., methods to cal-ulate the reliability index are explained elswhere, see for example ref. [18], and they are not considee'd in this repott

In order to apply the method outlined above to the fatigue reliability, first, a failure function needs to b _{defined. The mean total' fatigue} damage has been gieú by (7.14). Wheii this damage reâches Unity,

it is assumed that failure occurs. Thüs, a failure fúnction may intuiti-vely be defined as!

By definition, if Z < O, the failure region, if Z = Û, the failure

surfacé, and if Z > _{O the safe region is obtained. Thefailure function}

defined by (9.18) is highly nnlineàr with respect to the stochastic variables considered. To reduce the nònlinearity for the application

of the level II methods, an alternative definition of the ãilure function

is suggested. to use the natural logárithm of the fatigue damage as to be

Z = 2,n D

to t

(9 18)

(9.19)

which satisfiés the above failure condition. This definition is used in

this report Having introduced (7 14) into (9 19) the failure function

becomes,

Z=n Cffl

[fef()dH]

(9.20)

The désign variablés of this failure function are presented in section 8, and listed in table 8.1.

En the reLiability calculation Rn

Cf.iS

used instead of the fatigue

parameter Çf as a stochastic variable-, with a normal distribution.

This variable (Ln Cf) is correlated wIth tle other fatigue parameter k

In the analysis, these two variables are first transformed to paitof un-correlated (independent) variables b using orthogonal transformation. The detail of such trañsformation is presented in Appendix III. Then, the