Jack up reliability analysis: An overview

Abstract

Jack up is a mobile unit used for oil and gas exploration and production in offshore fields. On demand, the unit is moved and installed in a given location and used for a period up to 12 months before being un-installed and moved to another location. Due to its mobility and re-usability, when the unit is offered for use in a given offshore location, its suitability in terms of safe operation is evaluated in accordance with the guidelines of Site Specific Assessment (SSA) of jack up. When the unit failed safety assessment criteria, the guideline recommended that it is re-assessed by increasing the complexity of the assumptions and methods used. Reliability analysis theories are one of the frameworks recommended for the safety assessment of the units. With recent developments in uncertainty and reliability analysis of structures subject to stochastic excitation, this study aims at providing a review on the past developments in jack up reliability analysis and to identify possible future directions. The results from literature reviewed shows that failure probabilities vary significantly with analysis method used. In addition, from the variants of reliability analysis approach, the method of time dependent reliability for dynamic structures subject to stochastic excitation have not been implemented on jack ups. Consequently, suggestions were made on the areas that need further examination for improvement of the efficiency in safety assessment of the units using reliability theories.

Keywords

jack up / reliability analysis / uncertainty analysis / review of jack up

Introduction

Jack-ups are self-elevating mobile units that operate in relatively shallower water depths. When they are first introduced more than six decades ago, jack up units operates in water depths below 70 m. However, due to their added advantage of mobility and fixity and coupled with increasing need to exploit oil and gas in deeper waters, the demand for the units results in their continuous employment for operations in deeper waters and increasingly more severe environmental conditions. Thus, the design and analysis of these units must always be done in accordance with the recommendations of set aside codes of practice and checks are carried out to ensure that such designs have met various strength, stability and safety conditions set in the design framework [¹]. In addition, due to the fact that jack up units operates in several different offshore locations during their service life, it becomes difficult or near impossible to design the units based on specific environmental conditions. Consequently, when a jack up is offered for operation at a given location, Site Specific Assessment (SSA) is performed in order to match environmental conditions from the new location with the unit’s structural conditions with the view of establishing a safe operating level. The guidelines for the jack up site specific assessments was published in Technical and Research (T&R) bulletin by the Society for Naval Architects and Marine Engineers (SNAME) [²]. To calibrate safety level in both design and SSA of jack up, reliability analysis theories are employed by incorporating reliability criteria in the design and assessment framework [³,⁴]. Reliability analysis of structures is concerned with the estimation of the probability of failure of a structural element or the structure as a whole [⁵]. Although no specific method have been given for the reliability analysis, the report in T&R emphasized that state of the art and consistent methods should be applied in establishing target reliability levels.

Although jack up units have been studied for several decades, no literature was found to have reported the review of the developments in their structural reliability analysis. The study in [⁶] which reviewed the reliability methods for offshore steel structures only focused on fixed structures in general without emphasis on specific types. With recent developments in the analysis and representation of ocean wave loading coupled with developments in the area of reliability analysis methods for dynamical structures subjected to stochastic loads [³,⁷], this study aims to review the reliability of jack up units during operations with a view of identifying areas that require further examination to improve the safety assessment of the units.

Subsequently, this review is divided into Sections as follows; Section 2 discussed the variants of reliability analysis and their solution approach; Section 3 analysed existing literature reports on jack up reliability analysis; Section 4 presents analysis of findings from the literature; Section 5 presents conclusions from findings and recommendations for future studies.

Structural reliability analysis

Reliability theories are concerned with the treatment of uncertainties in engineering problems and with the methods for assessing serviceability as well as safety of structures. They deal with the measurement or calculation of the probability of failure of a structural element, structural member or the structure as a whole [⁸]. It aims at estimating the probability of failure of a system with reference to a specified performance criterion which account for uncertainties arising from the model description [⁵,⁹]. Such analyses are grouped into time dependent and time independent problems.

Time independent reliability analysis

In time independent also known as time invariant reliability problems, the load is assumed to be deterministic and material degradation is non-existent and the failure probability for a system in a given failure mode is given by [⁹]:

(1)P_{\mathrm{f}}=p[G(X)\le \mathrm{0}]={\displaystyle \underset{G(X)\le \mathrm{0}}{\int }f(X)\mathrm{d}x}.

And the failure domain for the system in general is given by:

(2)G(X)={\displaystyle \underset{k}{\cup }}{\displaystyle \underset{i\in C_k}{\cap }\{g_i(x)\le \mathrm{0}\}},

where the function g_i(x) is the i^th limit state function of failure, C_k is the index set for the k^{\mathrm{t}\mathrm{h}} main cut set, G(X) is the failure function (where G(X)>\mathrm{0} is a safe state while G(X)\le \mathrm{0} is a failure state), f(X) is the joint probability density function of load and resistance. Such problems are relatively easy to evaluate as the reliability index can be found from the joint density function [¹⁰] or as shown in Fig. 2, as the minimum distance from origin of the transformed failure function [¹¹,¹²].

Solution approach for time independent problems

For time independent reliability problems, the solution of probability integral is straight forward and involves the determination of joint probability density functions of the load (R) and resistance (S) using appropriate statistical distribution models such as in [¹⁰]. In Fig. 1, the probability density function of the limit state function in Eq. (1) is shown. The statistical parameters of the limit state function are determined from the distribution of load and structural resistance. From the figure, the probability of failure is obtained as the total area of the failure region or using Eq. (3) as:

(3)P_{\mathrm{f}}=\mathrm{\Phi }\left({\displaystyle \frac{{\mu }_{\mathrm{R}}-{\mu }_{\mathrm{S}}}{\sqrt{{\sigma }_{\mathrm{R}}^{\mathrm{2}}+{\sigma }_{\mathrm{S}}^{\mathrm{2}}}}}\right).

Another solution approach for time independent problems is by evaluating the statistical parameters of the uncertain random variables describing the load and resistance of the system. This is followed by transformation of the limit state function and determination of design point by using the standard procedure of FORM, SORM as discussed in [¹³] or by simulation techniques such as Monte Carlo Simulation (MCS)methods as in [¹⁴,¹⁵]. For a set of random variables [X] in the physical x-space, there exist a corresponding set of uncorrelated random variables [Z]in the normal z-space. The limit state in the x-space can be mapped out on the corresponding limit state surface in the normal z-space as shown in Fig. 2, and the first order reliability index \beta can be defined as the minimum distance from the origin to the failure surface on the normal z-space. Thus, the probability of failure is approximately given as;

(4)P_{\mathrm{f}}=\mathrm{\Phi }(-\beta ).

In a typical Monte-Carlo simulation, a large number of samples N are usually simulated from the distributions of the random variables [X] and the limit state function is evaluated for each of the sample set. The number of samples n_f that leads to the violation of the limit state function is counted and the crude estimate of the failure probability is given as

(5)P_{\mathrm{f}}={\displaystyle \frac{n_{\mathrm{f}}}{N}}.

Either of these solution approaches is employed when failure probability is sought for some set of structural and environmental situation at a given time. When the evolution of those failure probabilities over a given time are investigated for a random process, then the time interval can be discretised into smaller time instants and at each instant, the random processes are replaced with their corresponding random variables and the instantaneous probability can be obtained using the time invariant methods.

The Above reliability analysis procedures are usually referred to as “traditional” methods as they are used to determine the failure probabilities of engineering systems when extreme conditions are used as design input. It is assumed in such frameworks that the results of such deterministic analyses represent all scenarios of structural system strength and loadings.

Time dependent reliability analysis

When some of the design random variables in the reliability analysis are subject to change in properties or strength with due course of time, then the effect of time will have to be featured in the reliability analysis. This becomes time dependent reliability problem. The time dependency in this form of reliability analysis may come from material properties which may deteriorate over time due to fatigue [¹⁶,¹⁷], crack growth [¹⁸‒²⁰] or corrosion [²¹,²²] or from environmental loading such as wind generated waves [²³]. Time dependent problems as the result of such uncertainties are considered as random processes. A time dependent reliability analysis is employed to estimate the probability that the system will perform its intended function during a particular period of time. Time dependent probability or cumulative probability of failure is defined [²⁴]:

(6)P_{\mathrm{f}}(\mathrm{0},T)=P\{\exists t\in [\mathrm{0},T]:g(x,y(t),t)\le \mathrm{0}\},

where the time dependent limit state function g(x,y(t),t)=\mathrm{0}depends on the vector X=[X_{\mathrm{1}},X_{\mathrm{2}}\mathrm{........}{X}_n] of n input random variables, the vector Y(t)=[Y_{\mathrm{1}}(t),Y_{\mathrm{2}}(t)\mathrm{.....}{Y}_m(t)] of m input random processes and explicit time t. In such a case, failure occur when g (.)≤0 at any given time; t\in [\mathrm{0},T]. In time dependent problems, failures are estimated at discrete time instants which may or may not be related. In cases where time instant are considered not related, then when a probability of failure is sought at a given time instant for a time dependent limit state, the random processes are replaced with their corresponding random variables and the instantaneous probability can be obtained using the time invariant methods [²⁵,²⁶]. However, in purely time dependent problems, the failure at any instant of time is considered to be related to the time histories of failure events and the relationships between those time instants is incorporated in the reliability solution [²⁷].

Solution approach for time dependent problems

The structural response of engineering systems that are subjected to stochastic loadings such as wind, wave or earthquake loads is also a stochastic process. During the excitation, the structural response continuously makes transitions from safe to unsafe regions of limit state (Fig. 3). The rate of occurrence of these transitions also known as crossing rates are essential in uncertainty assessment of such systems. Whereas the deterministic approach for the uncertainty treatment is still exclusively used in engineering applications, it is well known that it may not lead to an optimum design of such systems. However, at the expense of an increased system model complexity and computational effort, stochastic methods do provide the possibility of an optimum system design through a more rational uncertainty treatment.

Stochastic mechanics theories are those theories that accounts for the spatial variability of material properties and randomness in the source of excitation. These theories have seen a lot of developments within the last three decades as one of the most powerful tools in the computational aspect of stochastic mechanics. Stochastic Finite Elements Method (SFEM) has been employed as an extension of the traditional finite elements methods in the solution of systems in which finite elements with random characteristics are involved [²⁸]. There are two fundamental stages involved in the rational treatment of uncertainty using the theory of SFEM. The first stage is the quantification of the uncertainty using mathematical theories of stochastic functions. This is followed by the propagation of the uncertainty through the structure and the subsequent assessment of the stochastic system response [²⁹].

Uncertainty modelling

Ocean wave phenomena falls within the category of multi-scale problems [³⁰,³¹] due to different scales in its features from its generation by winds, propagation across various basins to their interactions with costal systems when fully developed. Modelling of such multi-scale problems is achieved using partial differential equations with multi-scale features in the coefficients. A fully developed sea is a stochastic process which can be modelled using its short and long term statistical properties and is used in the analysis and design of offshore structures [²]. The interaction between such waves and offshore structures can be studied in the framework of SFEM.

In the framework of SFEM, the first stage of uncertainty treatment of systems is the representation of the stochastic fields which is the mathematical description of the uncertain input of the structural system. Such uncertain input may be mechanical, geometrical or loading behaviour of the system. Example of mechanical and geometric uncertain input parameters includes Poisson ratio, Young’s modulus, yield stress as well as physical systems cross sections. In the case of composites such as CNT/Polymer [³²], these parameters include molecular interactions, agglomerations, CNT diameter at Nano scale etcetera [³¹–³³]. In practical engineering applications, these quantities usually exhibit non-Gaussian behaviour [²⁸]. However, due to simplicity and lack of readily available data, a general Gaussian assumption is often made.

Propagation of uncertainty through the structure occurs in different scales of length. For jack up structures, ocean wave loads acts in a relatively short period of time ranging from several seconds to few hours. On the other hand, material and geometrical uncertainties take periods up to several years to make significant impact [³⁶]. However, jack up operates in a given offshore locations for a period ranging from few months to one year. This is relatively short in comparison with the period of propagation of material and geometric uncertainties. Consequently, reliability of jack up during operation focuses mainly on the uncertainties inherent in load. Subsequently therefore, uncertainty analysis for Gaussian field is discussed.

Although several approaches exist, two most widely accepted methods for simulating a Gaussian random field are the Karhunen-Loeve expansion (KLE) and spectral representation approach. The basic mathematical approach in the two methods is briefly presented below.

One most widely used approach to expand a stochastic field u(x,t)is the spectral representation method in which the series summation of trigonometric functions with amplitudes and random phase angles are used. The amplitudes are determined from the power spectrum of the given stochastic field. The mathematical representation of the field is given as:

(7)u(x,t)={\displaystyle \sum _{i=\mathrm{1}}^N\{{\alpha }_i\cos [k_{i}x-{\omega }_{i}t]+{\beta }_i\sin [k_{i}x-{\omega }_{i}t]\}},

where the variables {\alpha }_nand {\beta }_nare independent Gaussian distributed random variables with zero mean and standard deviation \deltagiven as:

(8)\delta =\sqrt{S({\omega }_i)\mathrm{d}\omega },\hspace{0.17em}\mathrm{a}\mathrm{n}\mathrm{d}\hspace{0.17em}\mathrm{d}\omega ={\displaystyle \frac{{\Delta }_u}{N}}.

In the above expression, {\Delta }_u is the cut-off frequency defining the region of concentration of the power spectral density function. Such representation is always assumed to be Gaussian when N\to \inftydue to central limit theorem.

Karhunen-Loeve expansion (KLE) is a special case of stochastic field representation in which orthogonal functions are selected and used as the eigenfunctions in the Fredholm integral of the second kind. The general expression of a stochastic field u(x,t) using KLE is given as:

(9)u(x)=\bar{u}(x)+{\displaystyle \sum _{n=\mathrm{1}}^N\sqrt{{\lambda }_n}{\xi }_n{\phi }_n(x)}.

The function \bar{u}(x)is the mean value of the field and is mostly assumed to be equals to zero. The eigenvalues {\lambda }_n and the eigenfunctions {\phi }_n are obtained from the solution of the integral equation cast with the auto covariance function C_{\mathrm{f}\mathrm{f}}(x_{\mathrm{1}},x_{\mathrm{2}})of the field given as:

(10){\displaystyle {\int }_D{C}_{\mathrm{f}\mathrm{f}}(x_{\mathrm{1}},x_{\mathrm{2}}){\phi }_n(x_{\mathrm{1}})\mathrm{d}{x}_{\mathrm{1}}={\lambda }_n{\phi }_n(x_{\mathrm{2}})}.

KLE is mostly suitable for representation of a strongly correlated stochastic field in which fewer terms corresponding to the bigger eigenvalues are sufficient to capture the essential random fluctuation of the field. The number of such terms is indicated by the rate of decay of the eigenvalues. Figure 4 shows the rate of decay of eigenvalues for a stochastic representation of an ocean wave surface elevation process. The time domain wave profile is shown in Fig. 5 in which the two methods of KLE and spectral representation are used in the simulation of the field. More details of these results can be found in [³⁷].

Probability of failure estimate

In the analysis and design of systems subject to stochastic excitation, the desired quantity is the probability that the stochastic response R(t) will remain within the prescribed threshold throughout the duration of the excitation. In addition, the distribution of the first time of limit state violation or fist passage across the specified threshold or barrier is of interest. The first passage or first excursion probability, which is the probability that at any time t, the response quantity R(t) will violate the set criteria in the limit state equation and is defined as [³⁸]:

(11)P_{\mathrm{f}}(t)=\mathrm{1}-(\mathrm{1}-P_{\mathrm{0}}){\mathrm{e}}^{-\alpha t}.

The probability P_{\mathrm{f}}(t) is the probability of failure or the probability of crossing at time t during the excitation period. P_{\mathrm{0}} is the instantaneous probability of failure at timet=\mathrm{0} and \alpha is the limiting decay rate of the first crossing probability. The approximation to such failure probability for different settings of the threshold criteria is accomplished by considering independent crossing events according to Poisson process [¹⁵,³⁹] or by accounting for the statistical dependence between crossings [⁴⁰,⁴¹].

One of the famous methods that are simple and more accurate for treatment of uncertainty in stochastic problems is the use of Monte Carlo simulation (MCS) [⁴²]. The method involved random simulation of the stochastic input and solving the system N_MC times to obtain large population of structural response (r). Response variability can then be studied using simple statistical relations as [³³]:

(12)E(r_i)={\displaystyle \frac{\mathrm{1}}{N_{\mathrm{M}\mathrm{C}}}}{\displaystyle \sum _{I=\mathrm{1}}^{N_{\mathrm{M}\mathrm{C}}}{r}_i(j)}

(13){\sigma }^{\mathrm{2}}(r_i)={\displaystyle \frac{\mathrm{1}}{N_{\mathrm{M}\mathrm{C}}}}\left[{\displaystyle \sum _{i=\mathrm{1}}^{N_{\mathrm{M}\mathrm{C}}}(r_i^{\mathrm{2}}-N_{\mathrm{M}\mathrm{C}}{E}^{\mathrm{2}}(r_i)}\right]

The accuracy of MCS method depends on the population size N_MC and consequently, for large systems, it becomes nearly infeasible. To reduce the computational cost in the solution of large dimension problems, two main variants of the MCS method were developed. For example, methods such as the importance sampling [⁴³], subset simulation [⁴⁴] and line sampling [⁴⁵] were proposed whereby instead of solving the system a large number of times, the simulation of the random input is controlled in such a way that only representative random variables are used without significant loss in accuracy. On the other hand, meta-modelling techniques such as the use of Polynomial response surface [⁴⁶], gradient-enhanced Kriging [⁴⁷] and Artificial Neural Networks (ANN) [⁴⁸] are used to develop a model that describe the relationship between the structural input and the resulting output in such a way that when the stochastic input is simulated, the desired population of the response can be obtained without recourse to the structure thereby significantly reducing the computational cost. Such modelling techniques are used to investigate the sensitivity of the input variables to the structural output using schemes such as in [⁴⁹].

Reliability of jack up platforms

Despite the requirements of reliability levels in the safety assessment of jack up units, not much development in their reliability studies have been reported in literature when compared with works reported for other type of offshore structures. From the literature search conducted, studies on jack up are generally accomplished by considering the dynamic effect in the structural response. The dynamic response is obtained either by using the dynamic amplification factor [⁵⁰,⁵¹] or by full time domain analysis using analytical [⁵²] or numerical schemes [⁵³,⁵⁴].

Reliability analysis of a jack up unit is generally accomplished by making a general assumption of the wave surface elevation process (whether regular or random). This is followed by analysis and representation of the uncertainty in the wave by either deterministic approach of extreme values or by the spectral approach in representation of the surface elevation and the selection of reliability solution method. For example, the study in [⁵⁵] considered the random wave surface elevation process and developed a jack up reliability analysis framework by considering uncertainty inherent in the random load. The surface elevation was represented using Fourier coefficients and trigonometric functions in the modified new wave approach initially presented in [⁵⁶]. By selecting crest heights in the short term sea state of few hours, the probability distribution of the extreme wave heights are modelled using Raleigh distribution and used to simulate several waves. These waves represent extreme wave events and the maximum response from each wave was evaluated in dynamic analysis of a jack up unit. The probability distribution function of the extreme response within the sea state was estimated. From the several numbers of structural analyses using the numerous waves simulated, the distribution function for the limit state is also evaluated and the failure probability was estimated from the number of failures and the total number of analyses performed.

This framework was used to investigate the effect of direction of propagation of ocean waves [⁵⁷] and the irregularity and non-linearity of the waves [⁵⁸] on the failure probability of jack up. Various approaches for reliability of jack up adopted as reported in the literature are categorised as pointed earlier and are discussed under the following sub sections.

Wave process assumption

Earlier studies on the reliability of jack up used the simplified assumption of regular wave in simulating the sea surface elevation to determine the wave load [⁵⁰]. This assumption is also used in approximating the wave load in more recent studies when the reliability is sought from uncertainty in foundation models [⁵⁹,⁵³]. However, most of the studies used the assumption that the wave is irregular and random and the sea state is represented by linear superposition of trigonometric functions using the spectral representation method. In the random wave assumption, several single waves are randomly simulated and used depending on the reliability solution approach. For long term analysis, statistics of the extremes of these short waves and their corresponding structural response are evaluated and used to determine the long term load or response distribution as used in the studies of [⁵²,⁵⁴,⁶⁰‒⁶⁴]. The simulated random waves were also used to determine the statistical properties of the wave parameters for use in the reliability analysis as reported in [⁶⁵] or in selecting the extreme wave episode for use in evaluating an extreme response as adopted in [⁶⁶].

Source of uncertainty considered

Apparently due to the shorter duration required for jack up operation as compared to its design life, most of the studies on jack up reliability found in the literature focused on the operational uncertainty from wave loading and foundation models. The studies that considered structural material uncertainty evaluated the lifetime probability of jack up failure from fatigue due to material degradation [⁵²,⁶⁷] fracture and crack propagation [⁵⁴,⁶²,⁶⁸]. Many of the studies in literature [⁵⁰,⁵²,⁶⁰,⁶³,⁶⁹] have considered uncertainty from wave load and evaluated probabilities by varying those parameters that describe ocean wave process. Whereas the study in [⁶¹,⁶⁴] considered the uncertainties from both wave loads and foundation models, the study in [⁷⁰] incorporated horizontal seismic loads in the uncertainties from wave loads and foundation models and the framework was used to estimate jack up failure probabilities in the North Sea [⁷¹]. The study of [⁵³,⁵⁹,⁶⁵] developed the reliability analysis for jack up with uncertainty from foundation models by evaluating the reliability indices using different models that describe soil structure and foundation fixity of the structure.

Reliability solution method used

From the literature surveyed, the reliability analysis of jack up units was evaluated by two different approaches.

The first approach aims to evaluate the probability that a predetermined response value of interest is exceeded from the distribution of extreme response data. This was achieved either by repeated simulation of random sea surface wave elevation and modelling of extreme load and response values by fitting of models such as Hermite models [⁵⁰], Weibull’s models [⁵²,⁶³] or by developing analytical or empirical models that describes the relationship between extreme environmental parameters, load and structural response [⁶⁰,⁷²]. In other studies such as [⁶¹,⁷³,⁷⁴], the wave of a given short duration and crest height was selected using statistical methods to represents storm event of few hours duration. This representative wave was simulated randomly by controlling the crest height and then used to determine the response of a simplified jack up model. Peak values of this response due to each random wave was used to determine the statistical parameters of the response probability distribution which was in turn, used to determine the probability of exceeding a given response value.

The second approach as found in the literature is the reliability analysis approach that was aimed at determining the failure probability of the jack up unit in a given failure mode by defining the failure criteria. This is achieved by considering both the deterministic and uncertain variables from load and response of interest and formulating the limit state function of the failure mode. Reliability indices are then obtained by solving the limit state function using standard reliability analysis methods such as FORM, or SORM [⁶⁹,⁷⁴,⁷⁵]. Failure probability of jack up using such approach was evaluated by considering uncertain structural material variables [⁵⁴,⁶²] and uncertainty in foundation models describing the soil-structure interaction [⁵³,⁵⁹,⁶⁴] where the reliability indices were evaluated for various set of the uncertain design variables. In the study of [⁶⁶], the wave parameters that most probably results in an extreme response are selected by using the theory of crossing and FORM analysis. The most probable wave due to these parameters was then used to determine the probability distribution of the extreme response and the subsequent probability of exceedance.

Figure 6 shows three results of reliability analysis of jack up unit using different approaches in which the uncertainty in wave loading is considered. In the study of [⁶¹] shown in blue line and [⁵⁵] shown in yellow line, the statistical distribution of extreme jack up response were developed and the probability of exceeding a specified response value was computed for different type of jack up models. In the study of [⁶¹], a simplified jack up model with beam elements representing the legs and hull is used. The structural properties of a real jack up unit were used to define the beam elements. A foundation model was selected in the analysis and the wave was simulated using a PM wave spectrum with H_s = 12 m and T_Z = 10.83 m. The study of [⁵⁵] used a full jack up model in which all the structural elements are used. The spud can foundation was also modelled and its soil interaction was represented using a foundation model. The wave was simulated using JONSWAP spectrum with H_s = 17.47 m and T_Z = 17.22 m. Although the two studies used the same reliability approach, the results shows significant differences in the failure probabilities. This may be due to the difference in the wave spectrum model as well as the type of structural model used in the analysis.

In the study of [⁶⁶] shown in red line, critical wave parameters were selected and used to evaluate the probability of exceeding extreme jack up response using FORM. The simplified model with the same structural and material properties as those in [⁶¹] are used. The wave was simulated using JONSWAP spectrum with H_s = 12 m and T_z = 10 m. Whereas the structural model as well as the wave parameters is the same in the two studies, the significant difference in probability of failure can be due to the different methods of reliability analysis applied as well as the selection of the spectrum model.

Analysis of findings from literature

Trend of publication in jack up reliability

Table 1 presents the summary of publications on jack up reliability studies found in the literature and published over the last three decades. It can be seen that there was an increase in the studies in the second decade when compared to the first.

However, within the current decade, the attention given to jack up reliability studies seems to be growing as up to seven (7) articles are found in the literature. This is despite the fact that the decade is just in its second half.

Figure 7 shows the classes of reliability approach and the number of publications of jack up reliability using each approach found in the literature. From the figure, it can be seen that the approach in establishing the reliability of jack up using the statistical distribution method with seven publications have lower number of articles compared to the standard reliability approach using FORM or SORM, Monte Carlo simulation or response surface methodology with 16 publications. However, it can be seen that there is no publication that used the time dependent reliability solution method for stochastic systems.

Limitation of the studies

Time dependent reliability analysis approach in jack up reliability studies is absent in the literature. Presently, the reliability analysis of jack up methods cannot evaluate the evolution of the failure probabilities over time. They cannot accurately estimate the probable time to first failure event with respect to time history of limit state transitions. In time dependent reliability analysis, failure probabilities are evaluated at any given time by considering time histories of the structural response due to random stochastic loading. In addition, although all the literature surveyed considered the dynamic effect in jack up structural response, the reliability analysis methods adopted in the studies cannot produce failure probabilities in which the influence of dynamic response due to stochastic excitation is included in the reliability solutions.

Conclusions recommendation and future works

Conclusions

The developments in jack up reliability analysis for the design and safety assessments are reviewed. This is to provide a quick glance at the methodologies adopted and to identify possible areas that require improvements to enable state-of-the-art reliability methods to be used in design and safety calibration. Whereas reliability analysis are grouped into time dependent and time independent, the review shows that only the time independent approach is employed in establishing jack up failure probabilities. This approach only enables the probability of failure to be evaluated for extreme conditions. However, such approach may not lead to accurate estimation of failure probabilities. In dynamically responding structures such as jack up, extreme load or response may not necessarily results in an increased failure probability, rather, the time histories of the structural excitation have to be considered. Time dependent reliability analysis enables the determination of the evolution of the probabilities over the time duration of the excitation by incorporating the random transitions made by the limit state from safe to unsafe regions of failure.

Recommendation and future works

From the literature surveyed, it have been established that the structural reliability analysis are grouped in to time dependent and time in-dependent. For dynamically responding structures subjected to random excitation, time dependent reliability schemes are required to establish the probability of failure of such systems and to also establish the probable first time to failure. However, these approaches have not been implemented on jack up structures. This is apparently due to the fact that the time dependent reliability schemes involves a rigorous uncertainty and structural dynamic analysis requirements which hitherto made them unfeasible to implement on jack up.

From the findings of this literature survey therefore, the following stages are recommended for developing a framework of time dependent reliability analysis of jack up structures.

1.Since the state of the art time dependent reliability analysis schemes for dynamic systems subject to stochastic loading such as in [⁴¹,⁷⁶] requires structural dynamic analysis to be performed for each term of the load representation, then the Karhunen-Loeve expansion using the eigenfunctions of PSWFs can be used in the wave representation. In this way, it is shown in the study of [⁷⁷] that as fewer as six (6) terms are required to represent a wave, thus allowing the wave load to be given in few expansion terms.

2.By performing the structural dynamic analysis of the jack up using any acceptable numerical scheme, the time domain structural response of interest can be obtained. Consequently, the determination of the spectral moments can be performed using the numerical schemes such as in [⁷⁶]. Hence, the crossings and decay rates can be determined as required。

3.The time dependent reliability solution methods such as in [³⁸] in which crossing rates are shown to be capable in the solution of most engineering problems of similar nature takes into account, the statistical dependence between crossings, hence, this solution method can be used to study the response of the jack up and develop it s failure probabilities.

References

[1] API. Recommended Practice for Planning, Designing and Constructing Fixed Offshore Platforms—Working Stress Design, 2007

[2] SNAME. Guidelines for Site Specific Assessment of Mobile Jack-Up Units, 2008

[3] Qiu Z, Huang R, Wang X, Qi W. Structural reliability analysis and reliability-based design optimization: Recent advances. Science China. Physics, Mechanics & Astronomy, 2013, 56(9): 1611– 1618

[4] Onoufriou T. Reliability based inspection planning of offshore structures. Marine Structures, 1999, 12(7–8): 521–539

[5] Ditlevsen O, Madsen H O. Structural Reliability Methods. Copenhagen: Department of Mechanical Engineering, Technical Unversity of Denmark, 2007

[6] Onoufriou T, Forbes V J. Developments in structural system reliability assessment of fixed steel offshore platforms. Reliability Engineering & System Safety, 2001, 71(2): 189–199

[7] Ching J, Au S K, Beck J L. Reliability estimation for dynamical systems subject to stochastic excitation using subset simulation with splitting. Computer Methods in Applied Mechanics and Engineering, 2005, 194(12–16): 1557–1579

[8] Sudret B, Defaux G, Pendola M. Time-variant finite element reliability analysis—Application to the durability of cooling towers. Structural Safety, 2005, 27(2): 93–112

[9] Andrzej S, Nowak K, Collins R. Reliability of Structures. Boca Rato: CRC Press, 2012

[10] Bai Y, Cao Y, Kim Y, Yang Y, Jiang H. Time-dependent reliability assessment of offshore jacket platforms. Ships and Offshore Structures, 2015

[11] Shin J, Lee I. Reliability analysis and reliability-based design optimization of roadway horizontal curves using a first-order reliability method. Engineering Optimization, 2015, 47(5): 622–641

[12] Gaspar B, Bucher C, Guedes Soares C. Reliability analysis of plate elements under uniaxial compression using an adaptive response surface approach. Ships and Offshore Structures, 2015, 10(2): 145–161

[13] Rackwitz R. Reliability analysis—a review and some perspective. Structural Safety, 2001, 23(4): 365–395

[14] Yang Y, Yao J P, Yang X J, Yang Z B. The MonteCarlo method in application of fatigue life reliability analysis. Applied Mechanics & Materials, 2013,  395-396: 822–825

[15] Hu Z, Du X. Time-dependent reliability analysis with joint upcrossing rates. Structural and Multidisciplinary Optimization, 2013, 48(5): 893–907

[16] Dong W, Moan T, Gao Z. Fatigue reliability analysis of the jacket support structure for offshore wind turbine considering the effect of corrosion and inspection. Reliability Engineering & System Safety, 2012, 106: 11–27

[17] Guo T, Frangopol D M, Chen Y. Fatigue reliability assessment of steel bridge details integrating weigh-in-motion data and probabilistic finite element analysis. Computers & Structures, 2012, 112–113: 245–257

[18] Feng G Q, Garbatov Y, Guedes Soares C. Fatigue reliability of a stiffened panel subjected to correlated crack growth. Structural Safety, 2012, 36–37: 39–46

[19] He W, Liu J, Xie D. Probabilistic life assessment on fatigue crack growth in mixed-mode by coupling of Kriging model and finite element analysis. Engineering Fracture Mechanics, 2015, 139: 56–77

[20] Yang J, Zhang W, Liu Y. Existence and insufficiency of the crack closure for fatigue crack growth analysis. International Journal of Fatigue, 2014, 62: 144–153

[21] Qin H, Zhou W. Reliability analysis of corroding pipelines considering the growth and generation of corrosion defects. Materials and Joining; Risk and Reliability, 2014, 3: V003T12A010

[22] Melchers R E. The effect of corrosion on the structural reliability of steel offshore structures. Corrosion Science, 2005, 47(10): 2391–2410

[23] Schellin T E, Perić M, el Moctar O. Wave-in-deck load analysis for a Jack-up platform. Journal of Offshore Mechanics and Arctic Engineering, 2011, 133(2): 21303

[24] Andrieu-Renaud C, Sudret B, Lemaire M. The PHI2 method: A way to compute time-variant reliability. Reliability Engineering & System Safety, 2004, 84(1): 75–86

[25] Bisaggio H C, Netto T A. Predictive analyses of the integrity of corroded pipelines based on concepts of structural reliability and Bayesian inference. Marine Structures, 2015, 41: 180–199

[26] Lee L S, Estrada H, Baumert M. Time-dependent reliability analysis of FRP rehabilitated pipes. Journal of Composites for Construction, 2010, 14(3): 272–279

[27] Rice S. Mathematical analysis of random noise. Bell System Technical Journal, 1944, 23(3): 282–332

[28] Stefanou G. The stochastic finite element method: Past, present and future. Computer Methods in Applied Mechanics and Engineering, 2009, 198(9–12): 1031–1051

[29] Ghanem R G, Spanos P D. Stochastic Finite Elements: A Spectral Approach. New York: Springer, 1991, 1–222

[30] Panasenko G. Multi-Scale Modelling for Structures and Composites.New York: Springer, 2005

[31] Presho M, Målqvist A, Ginting V. Density estimation of two-phase flow with multiscale and randomly perturbed data. Advances in Water Resources, 2010, 33(9): 1130–1141

[32] Shokrieh M M, Rafiee R. Stochastic multi-scale modeling of CNT/polymer composites. Computational Materials Science, 2010, 50(2): 437–446

[33] Sakata S, Okuda K, Ikeda K. Stochastic analysis of laminated composite plate considering stochastic homogenization problem. Frontiers of Structural and Civil Engineering, 2015, 9(2): 141–153

[34] Sriramula S, Chryssanthopoulos M K. Quantification of uncertainty modelling in stochastic analysis of FRP composites. Composites. Part A, Applied Science and Manufacturing, 2009, 40(11): 1673–1684

[35] Ghasemi H, Rafiee R, Zhuang X, Muthu J, Rabczuk T. Uncertainties propagation in metamodel-based probabilistic optimization of CNT/polymer composite structure using stochastic multi-scale modeling. Computational Materials Science, 2014, 85: 295–305

[36] Qin H, Zhang S, Zhou W. Inverse Gaussian process-based corrosion growth modeling and its application in the reliability analysis for energy pipelines. Frontiers of Structural and Civil Engineering, 2013, 7(3): 276–287

[37] Idris A, Harahap I, Ali M. Efficiency of trigonometric and eigen function methods for simulating ocean wave profile. Indian Journal of Science and Technology, 2017, 10(4)

[38] Vanmarcke E H. On the distribution of the first-passage time for normal stationary random processes. Journal of Applied Mechanics, 1975, 42(1): 215

[39] Hu Z, Du X. Reliability analysis for hydrokinetic turbine blades. Renewable Energy,  2012,  48(6): 251–262

[40] Valdebenito M A, Jensen H A, Labarca A A. Estimation of first excursion probabilities for uncertain stochastic linear systems subject to Gaussian load. Computers & Structures, 2014, 138: 36–48

[41] He J. Numerical calculation for first excursion probabilities of linear systems. Probabilistic Engineering Mechanics, 2009, 24(3): 418–425

[42] Chan W K. Theory and Applications of Monte Carlo Simulations.Rijeka: InTech, 2013

[43] Papaioannou I, Papadimitriou C, Straub D. Sequential importance sampling for structural reliability analysis. Structural Safety, 2016, 62: 66–75

[44] Schneider R, Thöns S, Straub D S. Reliability analysis and updating of deteriorating systems with subset simulation. Structural Safety, 2017, 64: 20–36

[45] Lu Z, Song S, Yue Z, Wang J. Reliability sensitivity method by line sampling. Structural Safety, 2008, 30(6): 517–532

[46] Zhao W, Fan F, Wang W. Non-linear partial least squares response surface method for structural reliability analysis. Reliability Engineering & System Safety, 2017, 161: 69–77

[47] Ulaganathan S, Couckuyt I, Dhaene T, Degroote J, Laermans E. Performance study of gradient-enhanced kriging. Engineering with Computers, 2016, 32(1): 15–34

[48] Chojaczyk A A, Teixeira A P, Neves L C, Cardoso J B, Guedes S C. Review and application of artificial neural networks models in reliability analysis of steel structures. Structural Safety, 2015, 52(3): 78–89

[49] Vu-Bac N, Lahmer T, Zhuang X, Nguyen-Thoi T, Rabczuk T. A software framework for probabilistic sensitivity analysis for computationally expensive models. Advances in Engineering Software, 2016, 100: 19–31

[50] Jensen J J, Mansour A E, Pedersen P T. Reliability of Jack-up platform against overturning. Marine Structures, 1991, 4(3): 203–229

[51] Yasseri S F, Mahani R B. Overturning reliability analysis of Jack-up platforms using spreadsheet. In: Proceedings of the Asme 32nd International Conference on Ocean, Offshore and Arctic Engineering. New York: ASME, 2013

[52] Leira B J, Karunakaran D. Site-dependent reliability of a mobile Jack-up platform. Marine Structures, 1995, 8(2): 151–169

[53] Azadi M R E A. The effect of possible spud-can punch through on the reliability index of neka drilling type Jack-up platform. Journal of Civil Engineering and Science, 2012, 1(3): 80–89

[54] Shabakhty N. System failure probability of offshore Jack-up platforms in the combination of fatigue and fracture. Engineering Failure Analysis, 2011, 18(1): 223–243

[55] Mirzadeh J, Kimiaei M, Cassidy M J. A framework to efficiently calculate the probability of failure of dynamically sensitive structures in a random sea. Ocean Engineering, 2015, 110: 215–226

[56] Tromans P S, Anatruk A R, Hagemeijer P. New model for the kinematics of large ocean waves application as a design wave. In: Proceedings of the First International Offshore and Polar Engineering Conference. New York: ISOPE, 1991

[57] Mirzadeh J, Kimiaei M, Cassidy M J. Performance of an example Jack-up platform under directional random ocean waves. Applied Ocean Research, 2016, 54: 87–100

[58] Mirzadeh J, Kimiaei M, Cassidy M J. Effects of irregular nonlinear ocean waves on the dynamic performance of an example Jack-up structure during an extreme event. Marine Structures, 2016, 49: 148–162

[59] Azadi M R E. Influence of spud-can-soil interaction modeling and parameters on the reliability index of neka drilling Jack-up platform. In: ASME 2008: Safety Engineering, Risk Analysis, and Reliability Methods. New York: ASME, 2013

[60] Van de Graaf J W, Tromans P S, Vanderschuren L, Jukui B H. Failure probability of a Jack-up under environmental loading in the central north sea. Marine Structures, 1996, 9 (1): 3–24

[61] Cassidy M J, Taylor P H, Taylor R E, Houlsby G T. Evaluation of long-term extreme response statistics of Jack-up platforms. Ocean Engineering, 2002, 29(13): 1603–1631

[62] Shabakhty N, Boonstra H, Gelder P V. System reliability of Jack-up structures based on fatigue degradation. Safety and Reliability, 2003, 1437–1780

[63] Karunakaran D, Leira B J, Moan T. Reliability analysis of drag-dominated offshore structures. In: The Third International Offshore and Polar Engineering Conference. Singapore: International Society of Offshore and Polar Engineers, 1993

[64] Frieze P A, Morandi A C, Birkinshaw M, Smith D, Dixon A. Fixed and Jack-up platforms: Basis for reliability assessment. Marine Structures, 1997, 10(2-4): 263–284

[65] Dier A F, Morandi A C, Smith D, Birkinshaw M, Dixon A. A comparison of jacket and jack-up structural reliability. Marine Structures, 2001, 14(4–5): 507–521

[66] Jensen J J, Capul J. Extreme response predictions for jack-up units in second order stochastic waves by FORM. Probabilistic Engineering Mechanics, 2006, 21(4): 330–337

[67] Shabakhty N. Comparing fatigue reliability of jack-up platforms based on selected nonlinear stress models. In: Proceedings of the 27th International Conference on Offshore Mechanics and Arctic Engineering. New York: ASME, 2008

[68] Shabakhty N. Fracture reliability of Jack-up platforms under extreme environmental loads. In: Proceedings of the ASME 30th International Conference on Ocean, Offshore and Arctic Engineering. New York: ASME, 2008

[69] Manuel L. A Study of the N Onlinearities, Dynamics, and Reliability of a Drag-Dominated Marine Structure. Palo Alto: Stanford University, 1992

[70] Azadi M R E. Reliability study of neka Jack-up platform under SH-seismic waves, sea-waves and current with considering spud-can-soil interaction. Asme International Conference on Ocean, 2009: 505–514

[71] June J. OMAE2010-20003. 2016

[72] Morandi A C, Smith I A A, Virk G S. Reliability of Jack-ups under extreme storm conditions. Marine Structures, 2001, 14(4–5): 523–536

[73] Cassidy M J, Taylor R E, Houlsby G T. Analysis of Jack-up units using a constrained new wave methodology. Applied Ocean Research, 2001, 23(4): 221–234

[74] Cassidy M J. Non-Linear Analysis of Jack-Up Structures Subjected to Random Waves. Oxford: University of Oxford, 1999

[75] Daghigh M, Hengst S, Vrouwenvelder A, Boonstra H. Reliability Analysis of Dynamic Structures. Scientia Iranica, 2003, 10(1): 1–12

[76] Valdebenito M A, Jensen H A, Labarca A A, He J, Au S K, Beck J L. Estimation of first excursion probabilities for uncertain stochastic linear systems subject to Gaussian load. Probabilistic Engineering Mechanics, 2014, 24(3): 418–425

[77] Sclavounos P D. Karhunen-Loeve representation of stochastic ocean waves. Proceedings Mathematical Physical & Engineering Sciences. 2012,  468(2145): 2574–2594

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag GmbH Germany