Uncertainty Modelling and Fatigue Reliability Calculation ... - CiteSeerX

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Proceedings of OMAE04 23rd International Conference on Offshore Mechanics and Arctic Engineering June 20-25, 2004, Vancouver, British Columbia, Canada

OMAE2004-51403

UNCERTAINTY MODELLING AND FATIGUE RELIABILITY CALCULATION OF OFFSHORE STRUCTURES WITH DETERIORATED MEMBERS H. Karadeniz Department of Civil Engineering Delft University of Technology DELFT / The NETHERLANDS

ABSTRACT This paper presents formulations and procedure of an efficient calculation of stress spectra and fatigue damage of offshore structures with deteriorated members in the uncertainty space. Calculation modeling of member deteriorations is represented by equivalent spring systems, which can be determined on basis of damage detection and stiffness degradation, with a deterioration uncertainty parameter. Redistributions of the member and system stiffness matrices and the load vectors are expressed in incremental (decremental) forms. The updated system stiffness-matrix is sated in terms of stiffness- and deterioration-uncertainties and the updated system load-vector is stated in terms of deterioration- and loading-uncertainties. Using the Neumann expansion solution technique, the inversion of the updated system stiffness matrix is expressed in terms of uncertainty parameters so that the reliability iteration can be performed without requiring repetitive inversion of the stiffness matrix. The deterioration- and uncertainty-update of the stiffness matrix requires resolution of the eigenvalue problem. This problem is reformulated in terms of uncertainty variables and an efficient solution algorithm is presented. An extra uncertainty parameter is used in structural transfer functions to represent damping uncertainties. Having expressed wave forces as functions of uncertainty variables, formulations of transfer functions of displacements and member internal forces are presented in the uncertainty space, which enable the reliability calculation to be efficient and fast. Apart from uncertainties of structural and loading origins, uncertainties arising from environmental origin, which appear in the spectral-analysis, are summarized. These are related to the modeling of random waves and wave-current interactions as well as to the long-term probability-distribution model of the significant wave height. Uncertainties in SCF, damage model (S-N line), nonnarrowness of the stress process, long-term probability

distribution of sea states and in the damage at which failure occurs (reference damage) are considered in fatigue-related uncertainties. An example is presented to demonstrate the application of the approximate analysis procedure to the mean value response analysis of deteriorated structures. INTRODUCTION Deterioration of structural components in the long term is a time dependent process under which the limit state functions of the reliability calculation may not be known explicitly at different times, since they might be time variant as well [1]. A simple reliability analysis of deteriorating structural system is based on the reliability of its structural components, which constitute a failure sequence for the structure. In the reliability analysis, a model of time dependent deterioration function is defined, which is based on simulation results of case studies of structural deteriorations due to corrosion [1], fatigue damage, plastic deformations, etc. All these deteriorations mechanism reduces member kinematic connectivities and cross-sectional properties, and thus it leads to a member-resistancedeterioration. In practice, time dependent exponential functions may be used to model corrosion-deterioration in marine environment [2]. In general, similar deterioration functions can be defined for other deterioration types on statistical basis of detected existing deteriorations. Since the degradation mechanisms are uncertain, experimental data are lacking, and thus a time dependent degradation function should be treated as stochastic. However, it has been reported that the variability in time dependent degradation function is of minor importance when compared to mean value degradation, and thus it is practically assumed as deterministic [7]. Deterioration affects dynamic properties of structures such as natural frequencies and modal damping due to energy dissipation through the defects [3]. Repeated cyclic loading such as due to wave and earthquakes develops hysteresis in the inelastic response range

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Copyright © 2004 by ASME

of structures that causes deterioration in structures [4], [5] and [6]. As it is reported in [4], under imposed constant-amplitude inelastic displacement cycles, stable hysteresis loops with constant energy dissipation at each cycle produce stiffness degradation while hysteresis loops with reduced cyclic energy dissipation produce both stiffness and strength deterioration, from which energy-based damage models can be defined [4]. The stiffness degradation occurs due to geometric effects and closely related to ductility. It can be accurately modeled by the pivot rule [5]. The strength degradation occurs due to weakening or partly loss of yield capacity and it can be modeled by reducing the capacity of the yield moment [5]. One other source of degradation is the local buckling, which causes degradation in cross-sectional stiffness properties and strength [6]. In offshore structural engineering, fatigue degradation becomes an important issue in the long term, since it causes partial or complete failure of structures due to a continuous damage accumulation in a random fashion. Because of random loading and environmental conditions as well as uncertainty in fatigue damage-mechanism in ocean environment, design analysis of structures can be carried out only at the mean value level, or more precisely, a reliability assessment based on fatigue degradation may be performed [8]. Even in the case of mean value analysis, it is of great interest to know the deviation of damages from the mean value level if a condition, e.g. an uncertainty parameter of the analysis, is slightly changed. In this paper, a physical damage-based deterioration of members due to fatigue and other phenomena (corrosion [9], large localdeformations [10], yielding and ovalization of member crosssections [11], etc…) is considered in the spectral probabilisticfatigue-damage calculation. In general, fatigue damage contains a great deal of uncertainties that should be considered in the design to assess a safety factor, which depends on the choice of uncertainties, their statistical descriptions and degrees of importance. The mean cumulative fatigue damage in an offshore structural member can be analytically calculated from, see e.g. [12,13], ∞ ∞

Dtot = T

λ

∫ ∫ T0

E[ dD ]nb f Hs,Tz ( h,t ) dh dt

(1)

0 0

where T is the period of lifetime, T0 is the mean zero-crossings period of the stress process, E[ dD ]nb is the mean damage due to one cycle of a narrow banded stress process, and λ is a damage correction-factor due to non-narrow banded nature of the stress process in a short-term sea state, f Hs ,Tz ( h,t ) is the joint probability-density-function of significant wave height and zero-crossings period of waves ( H s and Tz ). The mean damage, E[ dD ]nb , can be generally obtained in terms of statistical characteristics of the hot-spot stress process and parameters of the fatigue model used. The damage correctionfactor λ is defined empirically. An alternative fatigue damage formulation for non-narrow banded stress process was presented by Karadeniz [14] as given by,

∞ ∞

Dtot = T

1

∫ ∫ Tm

E[ dD ] f Hs ,Tz ( h,t ) dh dt

(2)

0 0

where Tm is the period of stress maxima and E[ dD ] is the mean damage due to one cycle of a non-narrow banded stress process in a short term sea state, which are both functions of stress statistical characteristics. Stress statistical characteristics are calculated by using spectral analysis while fatigue damagemodel is determined experimentally [15]. The fatigue damage calculation presented in this paper is based on Eq.(2). Modeling of corresponding uncertainties and identification of their importance are significant issues in the reliability analysis of offshore structures. A reduced uncertainty modeling [16] can also be used for a simple reliability calculation. In this model, most of uncertainties in stress spectral functions are represented by a single uncertainty parameter. In general, uncertainties in response characteristics are associated with modeling of structures and random wave environment. However, there may be some other uncertainties occurring during the response process and in the long term. Such uncertainties are closely related to deterioration of structural members, which can be incorporated in the analysis by successive modification of member connectivities. Deteriorations of structures, or structural members, may influence response results considerably, and therefore, the inherent uncertainties should be taken into account when a reliability analysis is carried out. A sophisticated design should include progressive damages in structural members or components during the lifetime or assumed service-period of structural functionality. In the long term, fatigue damages are the most important occurrences in structural components and constitute one of main design criteria. Their cumulative feature in time causes some deterioration in structural members that can affect the functionality performance of structures. In the traditional design procedure, the damage-based deterioration is not considered normally as a progressive fatigue-failure-process since the structural lifetime is usually determined on the basis of failure of the weakest member. The traditional analysis produces conservative results since, after failure of one or more members, the structural system can still function as long as a full collapse mechanism is formed. In the following sections, analysis modeling of deteriorated-members and the corresponding solution-algorithm of spectral responses of offshore structures are presented with representative uncertainty parameters. MODELING OF DETERIORATED-MEMBERS Members possessing any kind of changes in the original forms can be considered as deteriorated members. In reality, the deterioration is a continuous cumulative process. But, for the simplicity, it is treated here as a series of successive deteriorating stages. Its effect can be taken into account in the analysis by representing deteriorated members as being flexibly connected at joints [17]. The damage-based crack-deterioration problem is non-linear such that, under compression, the

2


member can perform full functionality, and under tension, it loses load carrying capacity somewhat as depending on the degree of deterioration (crack size), which is the case of wave loading (dynamic loading). In this study, it is fundamentally assumed that, under both tension and compression, member joint connectivity performs the same behavior, which allows for the deterioration to be represented by equivalent spring systems at member ends (flexible connections). The analysis model of a flexibly connected member is as shown in Fig.1. In this case, the stiffness and mass matrices, as well as the consistent load vector of a member can be formulated in incremental forms [18,19]. The increments mentioned here do not indicate a non-linearity. They indicate deviations from the original values due to joint flexibilities that may occur during the response, which is assumed to be linear elastic at a specific time station. In practice, the spring system of joint flexibilities can be determined analytically from damages of members on the basis of previous analysis or experimentally from statistics of available damage measurements. At each analysis step, a new set of spring system can be determined and the analysis is repeated with the new spring setup. For this purpose, a member transformation matrix is determined [17] as given by,

[rj]

y

j

[k] : stiffness matrix of (i-j)

z Fig.1 - A flexibly connected member (spring-beam element).

(

[T ] = [ I ] + [ r ] −1 [ ko ]

)

−1

(3)

in which [ r ] and [ ko ] are the spring and original stiffness matrix of the member. By using transformation, the stiffness matrix and the load vector of the deteriorated member can be stated as, [ k ] = [T ] [ ko ] = [ ko ][T ] T

{ p } = [T ] T { po }

)

−1

(5a)

in which [ µr ] and [ µko ] are the values of [ r ] and [ ko ] at the mean value uncertainty parameters. The uncertainty parameter X T is defined as being the ratio of uncertainty parameters of [ ko ] and [ r ] as written by, XT = ( X K / X r )

(5b)

where X K and X r are uncertainty parameters of the stiffness and spring matrices, [ ko ] and [ r ] . The linearized form of Eq.(5a) can be obtained from the Taylor series as written by, [T ] ≅ [ µT ] ( [ I ] − YT ( [ I ] − [ µT ] )

)

(6a)

in which [ µT ] is the transformation matrix at the mean value of X T , and YT is a shifted uncertainty parameter of the deterioration with a mean value of ( µYT = 0 ), which is defined by, YT = ( X T − 1)

(6b)

(

[ko] : stiffness matrix of (io-jo)

io

(

[T ] = [ I ] + X T [ µr ] −1 [ µko ]

[ µT ] = [ I ] + [ µr ] −1 [ µko ]

x i

Eq.(3) can be rewritten as,

From Eq.(5a), [ µT ] is written as,

jo

[ri]

mean value is equal to 1, i.e. ( µ X T = 1 ). With this definition,

(4)

where { po } is the original load vector. It is worth noting here that joint masses are used in this study, and therefore, member mass-redistribution is unrelated. The spring matrix [ r ] in Eq.(3) is determined from member deteriorations, and thus, it contains deterioration uncertainties. It is further assumed that all uncertainties in the transformation matrix, [T ] , are represented by a single uncertainty parameter X T of which the

)

−1

(7)

The matrix [ µT ] is bounded between [ 0 ] and [ I ] matrices. Thus, [ 0 ] ≤ [ µT ] ≤ [ I ] . When [ µT ] approaches [ 0 ] the member loses totally its stiffness capacity and when [ µT ] approaches [ I ] the member does not posses any deterioration and it functions with full stiffness capacity. Having introduced Eq.(6) into Eq.(4), the member stiffness matrix and load vector can be obtained in terms of uncertainty variables. The member stiffness matrix is stated in the decremental form as, [ k ] = X K ( [ µ k ] − YT [ ∆µ k ] )

(8a)

where [ µk ] and [ ∆µ k ] are defined as, [ µk ] = [ µT ] T [ µko ]

(

)

(8b)

[ ∆µ k ] = [ I ] − [ µT ] T [ µk ]

in which X K is a parameter, which represents uncertainties in the original stiffness matrix. Similarly, the load vector is stated in the decremental form as, { p } = { pµ } − YT { ∆ pµ }

(9)

in which { pµ } and { ∆ pµ } are defined as, { pµ } = [ µT ] T { po }

(

)

{ ∆ pµ } = [ I ] − [ µT ] T { pµ }

3

(10)


where { po } is the load vector of the original member (before deterioration), which contains loading uncertainties, and consequently, { pµ } and { ∆ pµ } also include loading uncertainties. The uncertainty parameters, X K and YT , are dependent. From Eqs.(5b) and (6b), their relation can be expressed as, YT = ( X K / X r ) − 1

(11)

where X r is the uncertainty parameter of the spring matrix [ r ] , which is dependent on deterioration. CALCULATION ALGORITHM, TRANSFER FUNCTIONS OF DISPLACEMENTS AND MEMBER FORCES The system stiffness matrix and load vector are obtained from the assemblage of member stiffness matrices and load vectors and they can be stated in the same form of members as given by Eq.(8a) and Eq.(10). Thus, they are written as, [ K ] = X K ( [ µ K ] − YT [ ∆µ K ] ) { P } = { Pµ } − YT { ∆ Pµ }

(12)

in which [ µ K ] , [ ∆µ K ] , { Pµ } and { ∆ Pµ } are obtained from the assemblage of [ µ k ] , [ ∆µ k ] , { pµ } and { ∆ pµ } for members respectively. In the direct spectral analysis method, the stiffness matrix needs to be inverted for each deterioration stage of the structure, which leads to enormous amount of computation time. The Neumann expansion solution technique [20] has been adopted to avoid the repeated inversion of the system-stiffness-matrix during an iterative analysis of the reliability assessment and deteriorating structures. For this purpose, the stiffness matrix is rewritten in the form of matrix product as, [ K ] = X K [ µ K ] ( [ I ] − YT [ ∆ K ] ) [ ∆ K ] = [ µ K ] −1 [ ∆µ K ]

(13)

From the Neumann expansion solution technique, the inversion of [ K ] can be expressed as, N tot ⎛ n⎞ −1 (14) ⎜ [ I ] + ∑ (YT [ ∆ K ] ) ⎟ [ µ K ] ⎜ ⎟ n =1 ⎝ ⎠ where Ntot is the total number of terms included in the series expansion. The error introduced by this solution technique depends on the degree of (YT [ ∆ K ] ) . If this term is small

[ K ] −1 =

1 XK

enough, which is usually the case in practice, then the Neumann expansion solution technique converges and can be applied successfully. The related error can be obtained as, error = (YT [ ∆ K ] )

Ntot +1

proximity of its mean value the error still remains small. This approximate inversion is valid only for the reliability iteration. However, for iterations of successive deterioration processes, a similar inversion process for [ µ K ] −1 can be applied as explained in [21]. In the following sub-sections, calculation algorithms of some items required in the spectral analysis are presented.

(15)

Apart from the degree of [ ∆ K ] , which is [ 0 ] ≤ [ ∆ K ] ≤ [ I ] , if the deviation of YT is in the close

a) Eigenvalue solution Calculation of natural frequencies and mode shapes of the structure are calculated in this section by using the approximate inversion of the stiffness matrix explained above. The structural eigenvalue problem is given as, [ K ]{ φ } = ω 2 [ M ]{ φ }

(16a)

where ω is the natural frequency and { φ } is the eigenvector. Alternatively, the eigenvalue problem can be written as, [ K ] −1 [ M ]{ φ } = λ { φ }

(16b)

where λ is the eigenvalue defined as ( λ = 1 / ω 2 ) . In these equations, [ M ] denotes the mass matrix. It is assumed in this study that all masses of the structure are transferred to joints to constitute [ M ] . Uncertainties in the mass matrix are represented by a single parameter X M with the mean value of 1, i.e. µ ( X M ) = 1 . Thus, it is stated as, [ M ] = X M [ µM ]

(17)

in which [ µ M ] is the mean value of [ M ] . Having introduced Eqs.(14) and (17) into Eq.(16b) the eigenvalue problem can be stated as, Ntot XM ⎛ n⎞ −1 ⎜ [ I ] + ∑ (YT [ ∆ K ] ) ⎟ [ µ K ] [ µ M ]{ φ } = λ{ φ } (18) ⎜ ⎟ XK ⎝ n =1 ⎠ Using power iteration this equation can easily be solved as explained below. 1- Start with { φ } = { φo } where { φo } is the eigenvector corresponding to the mean-value-uncertainty-parameters. 2- Calculate { X o } = [ µ K ] −1 [ µ M ]{ φ } , { Y1 } = [ ∆ K ]{ X o } and { Yi +1 } = [ ∆ K ]{Yi } where ( i = 0,1,2,...,( N tot − 1 ) ). Ntot ⎛ ⎞ n ⎜ { X o } + ∑ YT {Yn } ⎟ ⎜ ⎟ n =1 ⎝ ⎠ 4- Normalize { Z o } so that λ = max .({ Z o } ) .

3- Calculate { Z o } =

XM XK

5- Set { φ } = { Z o }norm and go to step 2 for further iteration until a required convergence is achieved. Higher eigenmodes can be calculated similarly by using deflation of the mass matrix. The calculation algorithm is the same as presented above, except { Z o } in item 3, which is modified to be,

4


Ntot ⎛ ⎞ n ⎜ { X o } + ∑ YT {Yn } − {W } ⎟ ⎜ ⎟ n =1 ⎝ ⎠ where the vector {W } is defined as,

{ Zo } =

{W } =

XM XK

µmj

m −1

∑

j =1

( µkj − YT ∆µkj )

{φ j }

(19a)

(19b)

in which m denotes the eigenmodes to be calculated, µ kj ,

∆µkj and µmj are defined respectively as,

µkj = { φ j } [ µ K ]{ φ j }

(23a)

µkm = X M µmm in which µkm , ∆µ km and µmm are calculated from,

∆µkm = { φm }T [ ∆µ K ]{ φm }

∆µkj = { φ j } [ ∆µ K ]{ φ j } T

(19c)

µmj = { φ j }T [ µ M ]{ φm } where { φm } is the eigenmode vector to be

(YT [ ∆ K ] ) ,

calculated. In

only one power iteration

suffices usually. In this case, the generalized stiffness for an eigenmode is calculated from, Ntot ⎞ X2 ⎛ (20a) k = M ⎜ ko + ∑ YTn kn ⎟ ⎜ ⎟ XK ⎝ n =1 ⎠ in which ko and kn are defined as, ko = { X o }T [ µ K ]{ X o }

(20b)

kn = { Yn }T [ µ K ]{ X o }

The generalized mass is calculated from, Ntot N tot Ntot ⎞ X3 ⎛ m = M2 ⎜ mo + 2∑ YTn mn + ∑∑ YTn + A mnA ⎟ ⎜ ⎟ XK ⎝ n =1 n =1 A =1 ⎠ where mo , mn and mnA are defined as,

(21a)

mn = {Yn }T [ µ M ]{ X o }

(21b)

mnA = {Yn } [ µ M ]{YA } T

In these equations, for an eigenmode, { X o } and { Yn } are as defined in step 2 of the calculation algorithm presented above. b) Structural transfer function In response calculation of structures, dynamic contribution is usually included in the analysis by using the modal analysis technique. The damping and frequency dependent contribution of an eigenmode to the response is defined as the structural transfer function, which can be stated [16] for the eigenmode m as, ⎛ ⎞ ωm2 1 ⎜ (22) αm ( ω ) = − 1⎟ km ⎜⎜ ωm2 − ω 2 + 2iζ mωmω ⎟⎟ ⎝ ⎠ in which km , ωm and ζ m are respectively the generalized stiffness, natural frequency and damping ratio for the mode m.

)

(23b)

µmm = { φm }T [ µ K ]{ φm } where { φm } is the eigenvector for the mode m calculated from aforementioned algorithm. The generalized stiffness and natural frequency, km and ωm , are functions of the uncertainty parameters, X K , X M and YT . It is assumed here that uncertainties in damping ratios are represented by a parameter, X ζ . The structural transfer function α m ( ω ) can be stated in terms of uncertainty parameters as, 1 hm ( ω ) αm ( ω ) = XK

(24a)

where hm ( ω ) is, hm ( ω ) =

mo = { X o }T [ µ M ]{ X o }

(

km = X K ( µkm − YT ∆µ km )

µkm = { φm }T [ µ K ]{ φm }

T

practice, for small

In this equation, the natural frequency is calculated from ( ωm2 = km / mm ) , in which km and mm (generalized mass) may be calculated from Eq.(20a) and (20b), or more generally from,

⎛ ⎞ 1 ωm2 ⎜ − 1⎟ ( µkm − YT ∆µkm ) ⎜⎜ ωm2 − ω 2 + 2iX ζ µζ mωmω ⎟⎟ ⎝ ⎠

(

)

(24b)

which is function of uncertainty parameters X K , X M , YT and Xζ . c) Transfer functions of wave forces In the spectral analysis, wave loads are calculated from the linearized Morison's equation, which is stated in terms of drag and inertia force components. In the uncertainty space, the consistent force vector of an original member (not deteriorated) can be written [16] as, { po } = { h pη o } η ( ω ) (25) { h pη o } = ( X cd X D ) { hcdo } + X cm X D2 { hcmo }

(

)

where η ( ω ) is the water elevation, X cd , X cm and X D are the parameters of uncertainties in the drag and inertia forces, and in the member diameters due to marine growths, { hcdo } and { hcmo } are the vectors of transfer functions of the drag and inertia force terms. For deteriorated members, transfer functions of consistent forces can be calculated from Eqs.(9) and (10) as written by, { h pη } = { h p } − YT { ∆h p }

(26a)

In this equation, { h p } and { ∆h p } are calculated from,

5


(

)

{ h p } = ( X cd X D ) { hcd } + X cm X D2 { hcm }

(

)

{ ∆h p } = ( X cd X D ) { ∆hcd } + X cm X D2 { ∆hcm }

(26b)

in which { hcd } , { hcm } , { ∆hcd } and { ∆hcm } are calculated from, { hcm } = [ µT ] T { hcmo }

( ) } = ( [ I ] − [ µ ] ){ µ

{ ∆hcd } = [ I ] − [ µT ] T { µcdo } T

T

{ H Dη ( ω )} = [ K ] −1{ H Pη ( ω )} +

N mod

∑ α r (ω )h frη ( ω ){ φr }

(31)

r =1

where α r ( ω ) , { H Pη ( ω )} and h frη ( ω ) are as given by

{ hcd } = [ µT ] T { hcdo }

{ ∆hcm

d) Transfer functions of displacements In general, transfer functions of system displacements can be written in vector notation [16] as,

cmo

(27)

}

For the whole system, the vectors { hcd } , { hcm } , { ∆hcd } and { ∆hcm } of members are assembled to form { H CD } , { H CM } , { ∆ H CD } and { ∆ H CM } of the system load vector. These vectors are deterministic. The vector of transfer functions of system loads can be sated as, { H Pη } = X CD ({ H CD } − YT { ∆ H CD }) + X CM ({ H CM } − YT { ∆ H CM })

(28a)

where X CD and X CM are uncertainty parameters defined as, X CD = X cd X D

Eqs.(24a), (28a) and (29) respectively for the eigenmode r , and N mod is the total number of eigenmodes included. The first term of Eq.(31) is the quasi-static contribution and the second one is the dynamic contribution. Thus, { H Dη ( ω )} = { H Dη ( ω )}sta + { H Dη ( ω )}dyn

(32)

where the static contribution is calculated by using Eqs.(14) and (28a) as written by, 1 (33) { H Dη }sta = X CD { H DDη }sta + X CM { H DMη }sta XK

(

)

in which { H DDη }sta and { H DMη }sta are respectively drag and inertia force contributions, which are functions of the uncertainty parameter YT only. They are calculated from, Ntot

(28b)

X CM = X cm X D2

Eqs. (28a) is the required vector of transfer functions of wave forces of the whole structural system in the uncertainty space. In the response analysis, generalized forces are also required to calculate dynamic response contribution using the modal analysis technique. They can be calculated by using Eq.(28a) and the eigenvectors of deteriorated structure, which are determined as explained above. Thus, for the eigenmode m , transfer function of the generalized force f m ( ω ) can be written as, h fmη ( ω ) = { φm }T { H Pη } (29)

{ H DDη }sta = { H Dη }o + ∑ YTn { H Dη }n n =1 Ntot

{ H DMη }sta = { H Mη }o + ∑

n =1

(34) YTn { H Mη }n

in which { H Dη }o , { H Dη }i , { H Mη }o and { H Mη }i

are

calculated from the expressions, { H Dη }o = [ µ K ] −1 ({ H CD } − YT { ∆ H CD } ) { H Dη }i = [ ∆ K ]{ H Dη }i −1

(35)

{ H Mη }o = [ µ K ] −1 ({ H CM } − YT { ∆ H CM } ) { H Mη }i = [ ∆ K ]{ H Mη }i −1

which is a function of uncertainty parameters X K , X M , YT , X CD and X CM . It is also stated explicitly in terms of uncertainty terms X CD and X CM as,

where ( i = 0,1,2,...,Ntot ) . By using Eqs.(24a) and (30) the dynamic contribution can be written as,

h fmη ( ω ) = X CD h fCD ( ω ) + X CM h fCM ( ω )

{ H Dη }dyn =

(30a)

in which h fCD ( ω ) and h fCM ( ω ) are functions of X K , X M and YT only, which are calculated from, h fCD ( ω ) = { φm }

T

h fCM ( ω ) = { φm }

T

({ H CD } − YT { ∆ H CD }) ({ H CM } − YT { ∆ H CM })

(30b)

Since { H CD } , { H CM } , { ∆ H CD } and { ∆ H CM } are deterministic vectors, during the reliability iteration transfer functions of wave forces will be calculated easily from Eqs.(28a) and (30a).

1 XK

N mod

∑ hr ( X CD ( h fCD )r + X CM ( h fCM )r ){ φr }

(36)

r =1

where hr , ( h fCD )r and ( h fCM )r are respectively calculated from Eqs.(24b) and (30b) for the eigenmode r . Having introduced Eqs.(33) and (36) into Eq.(32) transfer functions of system displacements can be written in terms of drag and inertia contributions as, { H Dη } =

(

1 X CD { H DDη } + X CM { H DMη } XK

)

(37a)

where { H DDη } and { H DMη } are calculated from,

6


{ H DDη } = { H DDη }sta +

N mod

{ H DMη } = { H DMη }sta +

∑ hr ( h fCD )r { φr } r =1 N mod

(37b)

∑ hr ( h fCM )r { φr } r =1

which are functions of uncertainty parameters of stiffness, mass and damping, i.e. X K , X M , YT and X ζ , only. Transfer functions of member displacements are extracted from the system displacements, which are given by Eqs.(37a) and (37b). For a member, they can be stated similar to Eq.(37a) as, { hdη } =

(

1 X CD { hdDη } + X CM { hdMη } XK

where { hdDη } and { hdMη }

)

(38)

are the drag and inertia

components of member displacements, which are extracted from Eq.(37b). e) Transfer functions of member forces Transfer functions of member internal forces are calculated from transfer functions of member displacements and fixed-end member forces [16] as stated by, { hFη } = [ k ]{ hdη } + { h pη }

(39)

where [ k ] is the member stiffness matrix given by Eq.(8a), { hdη } is the vector of transfer functions of member displacements given by Eq.(38) and { h pη } is the vector of transfer functions of member fixed-end forces given by Eq.(26a). The transfer functions of member internal forces can also be stated in terms of contributions of drag and inertia terms as written in vectorial form by, { hFη } = X CD { hFDη } + X CM { hFMη }

(40a)

where the drag and inertia components, { hFDη } and { hFMη } , are calculated from, { hFDη } = ( [ µk ] − YT [ ∆µk ] ) { hdDη } + ({ hcd } − YT { ∆hcd }) { hFMη } = ( [ µ k ] − YT [ ∆µ k ] ) { hdMη } + ({ hcm } − YT { ∆hcm })

(40b)

in which [ µk ] and [ ∆µ k ] are calculated from Eq.(8b), { hdDη } and { hdMη } are as defined in Eq.(38), and { hcd } , { hcm } , { ∆hcd } and { ∆hcm } are calculated from Eq.(27). Transfer functions, and consequently spectra, of stresses at member-ends can be easily calculated from member internal forces in terms of uncertainties from structural and loading origins.

UNCERTAINTIES IN THE FATIGUE DAMAGE Uncertainties in the cumulative damage given by Eq.(1), or Eq.(2), can be generally classified into two categories as a)

uncertainties related to spectral analysis and b) uncertainties related to fatigue phenomenon. Apart from uncertainties of structural and loading origins, there are also uncertainties from environmental origin in the spectral-analysis category [16]. These are related to the modeling of random waves and wavecurrent interactions, which can be imbedded in parameters of wave spectral functions and the current velocity as well as in the long term probability-distribution model of the significant wave height. A detailed explanation of these uncertainties can be found in [16]. Uncertainties in the mean damage of one stress cycle, E[ dD ] , see Eq.(2), which are not related to the spectral analysis, are considered in the fatigue related category. They are encountered mainly in stress concentration factors ( SCF ), fatigue model, damage correction or non-narrowness factor of the stress process and in the reference damage (damage rate at which failure occurs) [16]. In practice, stress concentration factors ( SCF ) are determined from some empirical expressions [22], and therefore, they contain inevitable uncertainties. Practically used fatigue models (S-N models) are usually based on statistical analysis of experimental data, which display a wide range of scatter leading to deviations from idealized mean lines. Uncertainties in the parameters of S-N models, C j and k j , are represented in the damage by two correlated variables. The non-narrowness factor of the stress process is empirically determined on the basis of numerical experiments [14] using rain-flow cycle counting algorithm so that parameters of this factor are uncertain and they are represented by independent variables in the damage. The criterion of the fatigue failure is defined as the condition that failure occurs when the cumulative fatigue damage Dtot , which is given by Eq.(1), or Eq.(2), reaches a value D f . This limit value of the damage is called as the "reference damage". It displays a large variation in practice [23], and therefore, it should be treated as an independent uncertainty variable. Detailed studies of the fatigue-related- uncertainties and a fast reliability-calculation have been presented in [16] assuming that all uncertainties related to spectral analysis have been represented by a stress-spectral-shape uncertaintyparameter except uncertainties in the drag and inertia waveforce-components, which have been represented by their own uncertainty parameters. The reason of this simple uncertainty representation of the stress spectral shape has been to avoid repetitive execution of the spectral analysis, which is the most time consuming part of the whole calculation process. It seems from a numerical investigation in [16] that the uncertainty of the stress-spectral-shape contributes about 20% to the reliability index. In the current study, essential transfer functions are formulated for an efficient spectral analysis technique by splitting up the uncertainty of the stress-spectralshape into more specific uncertainty origins as the stiffness and mass matrices and also structural deteriorations. This enables the reliability analysis with more detailed uncertainties to be carried out with a minimum effort of computation time. In the

7


following example, effects of deteriorations on mean value responses are presented by applying the approximate analysis procedure explained in this study. EXAMPLE In this section, the approximate analysis procedure presented in this paper is demonstrated for mean value responses by an example jacket type structure. The purpose of this example is to demonstrate the approximate analysis procedure only at the response level, since the calculations of the fatigue and reliability have not been completed at the submission of this paper. Therefore, a simple deterioration model (plastic hinges) is assumed for this reason. The topology and 3D-calculation model of the structure are as shown in Mtop=4800 ton.

+10.0 m. SWL

26.0 m.

20 19

+0.0 m.

15

36.0 m.

a) Topological data

11

7 3

9 10

8 4 6 2

5 1

b) 3D-calculation model

Fig. 2 – An example jacket type structure

Fig.2. It is assumed that the structure is fixed at the pile tips (joints 1,2,3,4 in Fig. 2). Member dimensions, mass and material properties are as presented in Table 1. A unidirectional wave in a horizontal-diagonal-direction is used. The drag and inertia force coefficients of the Morison’s equation are respectively cd = 1.3 and cm = 2.0 . The water depth, significant wave height and wave period are respectively h = 50 m. , H s = 2.5 m. and Tz = 6.5sec. In the analysis, P.M. Table 1 - Member dimensions, mass and material data of the example structure. Members Diameter Thickness Legs of the Deck 1.50 m. 16 mm. Legs of the Jacket 1.50 m. 16 mm. Horizontal and vertical bracings 1.50 m. 14 mm. Plain bracings (horiz. diagonals) 1.20 m. 12 mm. Material Steel Mass of the Deck 4800 ton Density of water 1024 kg/m3 Added mass coefficient 0.90

b) Incremental mode shape due to deterioration.

Fig. 3 – First and incremental natural mode shapes of the example structure.

sea spectrum is used. Two natural modes are used with the same damping ratios of ζ = 0.01 . For the deterioration, it is assumed that plastic hinges are developed at both ends of the jacket legs, i.e. at joints 5-16 of the legs shown in Fig.2b. Natural mode shapes and frequencies of the original (without plastic hinges) and deteriorated (with plastic hinges) structures are calculated. The first normalized natural mode shape of the original structure and its increment due to deterioration

({ ∆φ } = { φ }det er − { φ }origin )

13 14

12

- 25.0 m.

- 50.0 m. - 55.0 m.

17 18

16

a) First natural mode shape of the original structure.

are

shown

in

Fig.3.

The

normalized maximum value of the first natural mode shape {φ}1 (Fig. 3a) equals 1.0 and that of the incremental mode shape {∆φ}1 (Fig.3b) equals 0.03. The first and second mode shapes are the same in orthogonal directions with the same natural frequencies. Natural frequencies obtained from complete and approximate analyses are summarized in Table 2. Approximate frequencies of the deteriorated structure are calculated by using single power iteration explained above in the sub-section of Eigenvalue solution. As it can be seen from Table 2 – Natural frequencies of the example in (rad/sec.) Original structure (complete analysis) Deteriorated structure (complete analysis) Deterior. structure, 1 term Neumann Expansion Deterior. structure, 2 terms Neumann Expansion Deterior. structure, 3 terms Neumann Expansion

2.83470 2.75910 2.77807 2.76335 2.75909

Table 2, 3 terms Neumann Expansion with one power iteration produces practically the same result as that obtained from the complete analysis. Transfer functions and spectral values of the displacements at the top are calculated for the given sea state for the purpose of demonstration. Transfer functions and spectra of displacements of the deteriorated structure are illustrated in Fig.4 and Fig.5 respectively, and the spectra of displacements of the original structure are illustrated in Fig.6. Spectral moments and mean periods of displacements are presented in Table.3 where m0, m2 and m4 are respectively zero, second and fourth spectral moments, T0 and Tm are mean periods of zero crossings and maxima in seconds. From the examination of Table 3, it is seen that Neumann Expansion solution produces reasonable results if higher terms are included.

8


CONCLUSIONS This paper presents uncertainty modeling and formulations of loading- and response-transfer functions of deteriorated offshore structures. Updated system stiffness-matrix and loadvector due to deteriorations are expressed in terms of representative uncertainty variables in the incremental (decremental) forms. This allows the stiffness matrix to be inverted approximately using the Neumann expansion solution technique in terms of power functions of the uncertainty parameter of deteriorations. Based on this approximation, the eigenvalue problem is reformulated and an efficient solution algorithm is presented to calculate updated natural-frequencies and mode-shapes by using the power iteration technique. Transfer functions of response spectra are formulated explicitly in terms of uncertainty variables of the structure and wave loading so that stress statistical characteristics can be easily calculated during the iteration of the fatigue reliability assessment. From a numerical investigation, it reveals that a few terms of the Neumann expansion are sufficient to produce

2.11E-3 1.89E-3 1.68E-3 1.46E-3

1.03E-3 8.09E-4 5.92E-4 3.75E-4 1.58E-4 -5.91E-5

0.40

1.30

2.19

3.09

3.98

0.663

Fig.6 –

X Y Z

0.515

6.67

7.57

8.47

Spectra of displacements at the top of the original structure

0.2000E-03 0.3077E-04 0.5169E-06

0.1566E-02 0.2068E-03 0.2575E-05

m4

0.1256E-01 0.1580E-02 0.2006E-04

T0

Tm

2.245 2.423 2.815

2.219 2.273 2.251

Deteriorated structure (complete analysis)

X - Displacement Y - Displacement Z - Displacement

0.368

5.78

Table 3 – Spectral moments and mean periods of displacements.

0.589

0.442

4.88

Frequency (rad/sec)

Displ. m0 m2 Original structure (complete analysis)

0.736

X Y Z

0.294

0.2046E-03 0.5705E-04 0.4450E-06

0.1515E-02 0.3881E-03 0.2041E-05

0.1148E-01 0.2824E-02 0.1714E-04

2.309 2.409 2.934

2.282 2.329 2.168

Deteriorated structure, 1 term Neumann Expansion solution X Y Z

0.221 0.147 0.074 0.000

X – Displacement Y – Displacement Z – Displacement

1.24E-3

0.1921E-03 0.4254E-04 0.4294E-06

0.1441E-02 0.2839E-03 0.1938E-05

0.1107E-01 0.2076E-02 0.1655E-04

2.295 2.432 2.958

2.267 2.324 2.150

Deteriorated structure, 2 terms Neumann Expansion solution 0.67

1.54

2.40

3.27

4.14

5.00

5.87

6.74

7.61

X Y Z

8.47

Frequency (rad/sec)

Fig. 4 - Modulus of transfer functions of displacements at the top of the deteriorated structure

0.1964E-03 0.5205E-04 0.4364E-06

0.1458E-02 0.3520E-03 0.1978E-05

0.1108E-01 0.2562E-02 0.1672E-04

2.307 2.416 2.951

2.279 2.329 2.161

Deteriorated structure, 3 terms Neumann Expansion solution X Y Z

0.1979E-03 0.5537E-04 0.4408E-06

0.1464E-02 0.3758E-03 0.2009E-05

0.1109E-01 0.2731E-02 0.1691E-04

2.310 2.412 2.943

2.283 2.331 2.166

2.22E-3

acceptable results, and also a single power iteration for the eigenvalue solution results in appropriate natural frequencies and mode shapes. The method presented in this paper is very suitable for iterative calculation procedures such as used in the reliability assessment.

1.99E-3 1.77E-3 1.54E-3 X – Displacement Y – Displacement Z – Displacement

1.31E-3 1.08E-3 8.56E-4 6.29E-4 4.01E-4 1.74E-4 0.66

1.53

2.40

3.26

4.13

5.00

5.86

6.73

7.60

Frequency (rad/sec)

Fig. 5 - Spectra of displacements at the top of the deteriorated structure.

8.47

REFERENCES [1] Li, C.Q. 1995, “Computation of the failure probability of deteriorating structural systems”, Computers & Structures, Vol.56, No.6, pp.1073-1079. [2] Qin, S. and Cui, W., 2002, “A new corrosion model for the deterioration of steel structures in marine environments”, Proceedings of the 1st. ASRANet International Colloquium, 8-10 July, Scotland, U.K. [3] Razak, H.a. and Choi, F.C., 2001, “The effect of corrosion on the natural frequency and modal damping of reinforced

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[15] de Back, J. and Vaessen, G.H.G., 1981, Fatigue and corrosion fatigue behaviour of offshore steel structures, ECSC Convention 7210-KB/6/602, Final Report, Dept. of Civil Engineering, Delft University of Technology, Delft. [16] Karadeniz, H., 2003, “A fast calculation procedure for fatigue reliability estimates of offshore structures”, Proceedings of the 22nd. International Conference on Offshore Mechanics and Arctic Engineering (OMAE2003), June 8-13, Cancun, Mexico. [17] Karadeniz, H., 1994, “An algorithm for member releases and partly connected members in offshore structural analysis”, Proceedings of the 13th. International Conference on Offshore Mechanics and Arctic Engineering (OMAE-1994), Vol.1, pp.471-476, Houston, Texas, USA. [18] Karadeniz, H., 1999, “Probabilistic modelling of flexiblemember connections in offshore structural analysis”, Proceedings of the 18th. International Conference on Offshore Mechanics and Arctic Engineering (OMAE1999), Paper No.: OMAE99/S&R-6024, July 11-16, St. John’s, Newfoundland, Canada. [19] Karadeniz, H., 2001, “Uncertainties in spectral fatigue damages of offshore structures”, Proceedings of the 20th. International Conference on Offshore Mechanics and Arctic Engineering (OMAE-2001), Paper No S&R OMAE01-2118, June 3-8, Rio de Janeiro, Brazil. [20] Ghanem, R.G. and Spanos, P.D., 1991, Stochastic finite elements – A spectral approach. Springer Verlag, Berlin. [21] Karadeniz, H., 2003, “Calculation of response spectra of structures with deteriorating members”, Proceedings of 13th International Offshore and Polar Engineering Conference, Vol. IV, pp.480-486, Honolulu, Hawaii, USA. [22] API RP 2A, 1989, Recommended Practice for Planning, Designing and Constructing Fixed Offshore Platforms, 18th. edition, American Petroleum Institute, Washington DC. [23] Schutz, W., 1981, "Procedures for the prediction of fatigue life of tubular joints", Proceedings of International Conference on Steel in Marine Structures, Paris, France, 58 October.

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