Uncertainty Modelling and Fatigue Reliability Calculation of Offshore Structures with Deteriorated Members

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 stiffnessand deterioration-uncertainties and the updated system load-vector is stated in terms of deteriorationand 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 deteriorationand 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 1 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], 0 0 0 tot nb Hs,Tz D T E[ dD ] f ( h,t ) dh d T λ ∞ ∞ = ∫ ∫ t (1) where T is the period of lifetime, T0 is the mean zero-crossings period of the stress process, 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, nb E[ dD ] Hs,Tz f ( h,t ) is the joint probability-density-function of significant wave height and zero-crossings period of waves ( s H and ). The mean damage, , 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, z T nb E[ dD ] 0 0 1 tot Hs,Tz m D T E[ dD ] f ( h,t ) dh d T ∞ ∞ = ∫ ∫ t (2) where is the period of stress maxima and 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. m T E[ dD ] 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 Copyright © 2004 by ASME 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, ( 1 1 o [T ] [ I ] [ r ] [ k ] − − = + ) (3) in which [ r and 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, ] o [ k ] T o o T o [ k ] [T ] [ k ] [ k ][T ] { p } [T ] { p } = = = (4) where 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, , are represented by a single uncertainty parameter o { p }