Calculation of mean outcrossing rates of non-Gaussian processes with stochastic input parameters — Reliability of containers stowed on ships in severe sea

Probabilistic Engineering Mechanics 25 (2010) 206–217

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Calculation of mean outcrossing rates of non-Gaussian processes with stochastic input parameters — Reliability of containers stowed on ships in severe sea Ulrik Dam Nielsen ∗ Department of Mechanical Engineering, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark

article

info

Article history: Received 9 February 2009 Received in revised form 14 October 2009 Accepted 2 November 2009 Available online 10 November 2009 Keywords: Non-Gaussian processes Mean outcrossing rate First-order reliability method (FORM) Monte Carlo simulation Decision support systems for ships Stowed containers Racking failure

abstract Mean outcrossing rates can be used as a basis for decision support for ships in severe sea. The article describes a procedure for calculating the mean outcrossing rate of non-Gaussian processes with stochastic input parameters. The procedure is based on the first-order reliability method (FORM) and stochastic parameters are incorporated by carrying out a number of FORM calculations corresponding to combinations of specific values of the stochastic parameters. Subsequently, the individual FORM calculation is weighted according to the joint probability with which the specific combination of parameter values is expected to occur, and the final result, the mean outcrossing rate, is obtained by summation. The derived procedure is illustrated by an example considering the forces in containers stowed on ships and, in particular, results are presented for the so-called racking failure in the containers. The results of the procedure are compared with brute force simulations obtained by Monte Carlo simulation (MCS) and good agreement is observed. Importantly, the procedure requires significantly less CPU time compared to MCS to produce mean outcrossing rates. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction Deck stowed containers on ships are often exposed to damages due to large forces in the containers and in the container securing arrangements, when the ship operates in rough weather, e.g., [1,2]. The forces are governed by the wave- and wind-induced accelerations, and most classification societies have explicit formulas to derive the design forces. With regards to operational conditions of a ship, it is of interest to evaluate the reliability of a stack of containers against damage or loss. On the assumption of a Gaussian process with linearity between ship responses and wave excitations, Mansour et al. [3] derived closed-form expressions for a fast evaluation of the probability of failure of a stack of containers stowed on a ship. The risk of failure of a given event can also be expressed in terms of the mean outcrossing rate, defined as the average rate at which the underlying process exceeds a given threshold. In general, ship responses are non-Gaussian processes. This means that closed-form expressions cannot be derived for the evaluation of the risk of, say, breaking of the container end-wall — the so-called racking failure — in connection with the operation of container ships. Another complication under real, operational conditions is that many governing parameters (significant wave height, wave

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direction, material properties, etc.) are not known exactly. Instead, the parameters may be specified with uncertainty, so that the specific parameter is described in terms of a random variable with a given probability density function (PDF). As mentioned by, e.g., Gaidai and Naess [4] it can be a difficult, or a very time consuming, task to calculate mean outcrossing rates of stochastic processes. The straightforward approach to evaluate mean outcrossing rates, considering non-Gaussian processes, implies brute force time series simulations using, e.g., Monte Carlo simulation (MCS), which is often also used in combination with extreme predictions [5–9]. Recently, Nielsen et al. [10] developed a procedure based on parallel system analysis for the evaluation of mean outcrossing rates in which the governing processes may be non-Gaussian and highly non-linear with respect to the waves and, furthermore, the processes may depend on stochastic input parameters. The parallel system analysis is based on the firstor second-order reliability method (i.e. FORM or SORM), which requires the formulation of a limit state to describe the governing problem. The main advantage in using parallel system analysis when compared with brute force simulations is less computational time in the evaluation of the limit state(s). Moreover, the driving conditions in terms of the most probable stochastic wave process will be a direct outcome of the parallel system analysis. The present interest in mean outcrossing rates derives from risk-based decision support systems for safe operation and navigation of ships, e.g., [10–14]. Decision support systems are intended

U.D. Nielsen / Probabilistic Engineering Mechanics 25 (2010) 206–217

to give advice to the ship master on, say, course and speed changes in severe sea to avoid critical response levels, which for example could be related to the forces in containers and container securing arrangements. Decision support systems as such will not be dealt with in any further details in the article, but it is important to keep the systems in mind as the overall context, since actual decision support may be based on the evaluations of mean outcrossing rates under given conditions. For decision support systems to be useful in practice, there is a need for fast evaluations of the mean outcrossing rate. The parallel system analysis, [10,15], is a step towards this goal. However, as it is reported, the parallel system analysis has not proved as efficient as initially expected and, moreover, there are situations in which the FORM associated optimisation problem of the parallel system analysis is too difficult to solve, so that no solution can be obtained (with the applied software). In the present article, FORM is also used as the basis for evaluating mean outcrossing rates of non-Gaussian processes, taking into account uncertainties in the governing (input) parameters. The derived procedure — the FORM approach — is, however, based on a more intuitive reasoning than was the parallel system analysis, which also makes the present approach more robust in producing solutions to the associated limit state problem. Basically, the procedure consists of a number of FORM calculations that are carried out for combinations of parameter values, so that a parameter study of the stochastic input variables forms the basis of the calculation of the mean outcrossing rate. The approach can, e.g., be applied to evaluate the reliability of lashed containers against different modes of failure (racking and corner post compression and tension, respectively), which are all characterised by non-Gaussian processes. In the article, the approach will be illustrated by an example, where the results are directly compared with corresponding brute force time series simulations. The literature on level crossings of Gaussian and non-Gaussian (vector) processes is large, e.g., [16–22] to mention but a few. Schall et al. [23] specifically studied the aspects in relation to ergodicity assumptions in reliability calculations of offshore structures, where focus is also directed towards outcrossing rates obtained by FORM/SORM techniques and it is pointed out that rigorous formulations can require substantial numerical effort. Der Kiureghian [24] pioneered in the application of FORM to deduce outcrossing rates of non-Gaussian processes and the method has since been studied in a variety of contexts. In particular, the method has recently been applied to study wave loads on ships and offshore structures and wind turbines, both for evaluation of operational conditions and for design evaluation with a focus on extreme predictions, e.g., [25,26,14,27–29]. The application of FORM offers the insight into the actual driving processes; here, the stochastic wave elevation. Although this insight can be very useful, the present article will not touch on this and, instead, reference is made to the literature, of which the above can be mentioned among others. The article is organised into 6 sections. Sections 2–3 outline the equations of ship motions and, based on the motions, the forces in containers and container securing arrangements are determined. Reliability assessment can be carried out with the means of mean outcrossing rates and Section 4 derives the procedure that can be used to calculate mean outcrossing rates for non-Gaussian processes, taking into account uncertainty in any input parameter. The procedure is illustrated by a numerical example in Section 5 and, finally, Section 6 summarises and concludes. 2. Equations of ship motions In six degrees of freedom, the ship motions are surge, sway, heave, roll, pitch, and yaw, relative to the centre of gravity (COG) of the ship. In this article, only the heave, w , roll, ϕ , and pitch, θ , motions will be considered.

207

A simplified model for the roll motion of a ship is applied; see [30]. The model accounts for heave, but leaves out direct coupling terms to pitch, yaw, surge and sway. The damping term is modeled by a combination of a linear, a quadratic and a cubic variation in the roll velocity. The governing equation is given by

ϕ¨ + 2β1 ωϕ ϕ˙ + β2 ϕ| ˙ ϕ| ˙ +

β3 ϕ˙ 3 (g − w) ¨ GZ (ϕ) Mϕ + = . ωϕ rx2 Ixx

(1)

The heave motion and the wave-induced roll moment Mϕ are taken to be linear functions of the wave elevation, and the linear closed-form expressions given by Mansour et al. [3] are applied. The mass moment of inertia about the longitudinal axis is denoted by Ixx . The cross term mass moment of inertia Ixz is assumed to be small, and pitch is thus only included through the static balancing of the vessel in waves in the calculation of the GZ -curve. Similarly, the sway, yaw, and surge motions are ignored, since the vertical motions have the largest influence on the instantaneous GZ -curve. The wave elevation H (X , t ) is in the present study assumed to be normally distributed which means that, for long-crested waves, the variation with space X in the direction of vessel propagation and time t is given by the sum H (X , t ) =

N X [Vi ci (X , t ) + Wi di (X , t )]

(2)

i =1

where the variables (Vi , Wi ) are uncorrelated, standard normal distributed, while the deterministic coefficients are determined as ci (X , t ) = σi cos(ωi t − ki X ) di (X , t ) = −σi sin(ωi t − ki X )

(3)

σ = S (ωi )1ω. 2 i

The wave frequency ωi and the wave number ki are related through the (deep water) dispersion relation. S (ωi ) is the wave spectrum, which is taken as a Pierson–Moskowitz (PM) spectrum, controlled by S (ω) = Aω−5 exp −Bω−4 A=

Hs2 4π



2π Tz




4 and B =

1

π



2π

4

(4)

Tz

in which Hs is the significant wave height and Tz is the zeroupcrossing period. The instantaneous GZ -curve in irregular waves can be estimated from the numerical results for a regular wave with a wave length equal to the length L of the vessel and a wave height equal to 0.05L. This procedure is described in [30], where the instantaneous wave height h(t ) along the length of the vessel and the position of the crest xc are determined by an equivalent wave procedure somewhat similar to the one used by Kröger [31]. It is clear that the formulation in Eq. (1) is very simplistic, but the model is well suited to illustrate phenomena such as parametric rolling, resonance excitation and forced rolling. Hence, it is possible to identify which mode is the most probable for a given combination of sea state, speed and heading. Broaching and dynamic rolling (where a strong coupling to surge exists) cannot be modelled by Eq. (1). However, the model has recently been enhanced to include surge motion, see [32]. In summary, the motions of the ship are included as heave, roll and pitch. The heave and the pitch motions are described by linear theory and reasonable approximations for the RAOs may be obtained from the closed-form expressions found in [3]. Roll is taken as a non-linear motion, so that a time-domain solution of Eq. (1) is established. 3. Determination of container forces Mansour et al. [3] outline explicit formulas for the determination of the forces on containers stored onboard a ship. The content

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The vertical COG for each container is taken to be 1/3 of its height above the bottom. The containers are assumed identical with height h, breadth b and racking flexibility KC . A lashing system consisting of two identical rods placed between the bottom of the lowest container and the bottom of the third lowest container is assumed. Each rod makes an angle ψ with the y-axis and the flexibility of the rod is denoted KL . Only wave- and gravity- induced loads are considered here, and long-crested waves are assumed, although the analysis could readily be extended to short-crested waves at the expense of increased calculation time. Wind forces will not be considered, but could be introduced in the analysis by modelling the fluctuation in wind speed in a similar manner as Eqs. (3) and (4). In relation to extreme predictions of the load on floating offshore wind turbines, Jensen [29] shows how to include wind loads in an analysis making use of FORM.

mn h

3

m2 h

m1

(x, y, z)

3.1. Racking force 0

The racking force on the lowest container in a stack of containers can be written

b

R(t ) = Rϕ ϕ(t ) + Ra a0 (t ) (6) where a0 is the transverse wave-induced acceleration at z = z0 . That is, by double differentiation with respect to time t,

Fig. 1. Container stack secured by a lashing system [3].

of this section is almost identical to a similar section in Mansour et al. [3]. For general information about containers and container securing, The Standard and Lloyd’s Register of Shipping has published a so-called ‘Master’s Guide to Container Securing’ [33]. The external forces on a stack of n containers stored on deck consist of inertia forces from the wave-induced accelerations, the gravity forces and the wind-induced forces. The resulting internal forces in a single container depend on the external forces and the stiffness of the containers including a possible lashing arrangement. Guidelines have been issued by the major classification societies for the calculation of these internal forces. In most cases, the results are given in explicit, analytical form due to the rather simple structural arrangement. For instance, Lloyd’s Register of Shipping [34] presents explicit formulas for the tension in the lashing rods as well as the racking and compressive/tension forces in the containers for a stack of secured containers. Similarly worked examples can be found, e.g., in DNV Classification Notes 32.2 [35] and ABS Rules [36]. The guidelines use linear wave-induced inertia forces but nonlinear transverse gravity forces. However, as sin(ϕ) ≈ ϕ

(5)

for angles of roll of interest (the error at 30◦ is 5% on the conservative side), in the present procedure the linearized form of the transverse gravity load is applied and likewise the normal component W cos(ϕ) is replaced by the weight W , independent of the motion. It is noted that the last approximation is non-conservative as regards the corner post tensile force. The present procedure for calculating the forces in the containers and the lashing system deviates from the guidelines from the classification societies in the way the maximum wave-induced motions and accelerations are determined. In this article, the motions are determined according to Section 2. In the following the racking load R and the compression/tensile corner post load P are determined for a stack of n containers, placed with the centre of the bottom of the lowest container at location (x0 , y0 , z0 ) measured from the origin of a coordinate system located at the COG of the ship. This right-hand coordinate system has the x-axis pointing forward in the longitudinal direction and the z-axis positive upwards. The mass of each container is denoted mi ; i = 1, . . . , n, starting from the lowest container. The general arrangement of a stack of containers is illustrated in Fig. 1.

a0 (t ) = −z0 ϕ( ¨ t ).

(7)

The coefficients Rϕ and Ra only depend on the structural characteristics of the container stack system: n 1 X Rϕ = m1 g + g mi − Kg 6 2 i=2

1

m1

Ra =

12

+

m1

m2 3

+

n 1X

2 i=3

! mi

n 1X

m2

(8)

!

mi z i − K z1 + z2 + mi z i 2 i =2 12 3 2 i =3 where g is the acceleration of gravity, 6

z1 +

n 1X

m1

zi = z0 +

h 3

(9)

+ (i − 1) · h

(10)

and K =

2KC 2KC + KL

.

(11)

3.2. Corner post tension The corner post tension force PT (>0) in the lowest container can be written n

1 X PT (t ) = − g mi + Pv · av (x0 , y0 )(t ) + Pϕ · ϕ(t ) + Pa · a0 (t ) 4 i =1 (12) where the acceleration in the z-direction is av (x0 , y0 , t ) = − w( ¨ t ) + y0 ϕ( ¨ t ) − x0 θ¨




(13)

and the coefficients are given as Pv =

n 1X

4 i =1

Pϕ = g

Pa =

mi

n h X

2b i=1 n h X

(14)

 mi

i−



2



3 2

− Kg

2h b

"

m1 12

+

m2 3

+

n 1X

2 i=3

# mi

(15)



mi zi i − 2b i=1 3 " # n 2h m1 m2 1X −K z1 + z2 + mi zi . b 12 3 2 i =3

(16)

U.D. Nielsen / Probabilistic Engineering Mechanics 25 (2010) 206–217

random variables of X, e.g., vessel speed, relative wave heading, significant wave height, strength of container securing equipment. A straightforward calculation procedure is based on brute force time series simulations applying, e.g., MCS. In this way, similar realisations to that in Fig. 2 can be used to count the number of outcrossings. There are, however, other procedures which can be used to calculate the mean outcrossing rate, where the one, which will be studied in this article, is based on the FORM. The deduction of crossing rates of (vector) processes has been previously studied with a focus on FORM, e.g., [19,24]. However, it has not been considered how to take into account uncertainties in the governing (input) parameters, with an application to decision support systems for ships.

Fig. 2. Time variation of general (non-Gaussian) process.

3.3. Corner post compression For the corner post compression force PC (<0), the corresponding results are Pv = −

n 1X

4 i =1

Pϕ = g

mi

n h X

2b i=2 " m1

×

12

h X 2b i=2 " m1

×

12

i−

mi

+

n

Pa =

(17)



m2 3

+

 mi z i z1 +

i− m2 3

2



 − Kg

3

n 1X

2 i=3 2

z2 +

b

− tan(ψ)

4.2. Probabilistic approach using FORM The use of FORM requires the set-up of a limit state function g (Z (t , X), ζ ) and, in general, a limit state can be defined by

(



g (Z (t , X), ζ ) =

# mi



3

2h

 −K n 1X

2 i =3

(18) 2h

− tan(ψ) b #

mi z i .

209

< 0; = 0; > 0;

unsafe domain failure surface safe domain.

(20)

With attention drawn to Fig. 2, the limit state function is given by g ≡ ζ − Z (t , X).



(19)

The results for the racking force and the corner post tension and compression can easily be extended to two lashing systems yielding formulas in agreement with Lloyd’s Register of Shipping [34]. 4. Reliability assessment in terms of mean outcrossing rates In Section 3, it was seen that the forces on the container(s) depend on the motions of the ship as well as on the position and the mass of the individual containers. In a real (and irregular) sea, the ship motions are stochastic and the container forces need, therefore, to be determined by a probabilistic approach. Moreover, the actual strengths of the containers and of the securing arrangements themselves will be associated with uncertainty. In the following, a probabilistic approach is derived for the evaluation of the reliability of lashed containers stowed on a ship. To associate number(s) to the reliability assessment, the evaluation focuses on mean outcrossing rates relative to a given threshold. In general, the measure of an outcrossing rate can be interpreted as an equivalent measure to ‘probability of exceedance per time unit’. 4.1. Outcrossing rates Fig. 2 illustrates the behaviour in time t of a general response Z (t , X) that depends, in some way, on the parameters of the vector X. It is assumed that the response depends non-linearly on the wave excitation. In the figure, a threshold value, say ζ , has been indicated by a dashed horizontal line. Thus, the number of level crossings relative to the threshold value can be counted, which means that the outcrossing rate for the specific period of time can be obtained. Obviously, it would be possible to obtain an estimate of the mean outcrossing rate ν¯ of the process Z (t , X) if a vast ensemble of realisations similar to that in Fig. 2 were considered. This article focuses on the calculation of the mean outcrossing rate ν¯ = EX [ν + (X)], taking into account uncertainties in the

(21)

In the specific situation, considered herein, Z (t , X) may represent, say, the racking force, Eq. (6), whereas ζ would represent the breaking strength related to racking failure. Moreover, it should be realised that the process Z (t ) depends on the time-invariant set of W = (Vi , Wi ), which characterises the wave field, governed by Eq. (2). Thus, Z(t ) ≡ Z (t , X, W)

(22)

where it will be assumed that all the (random) variables in X can be modelled as stochastic parameters, each with a known PDF. One way to calculate the mean outcrossing rate of Z(t , X, W) has been presented in [10]. Therein, the problem is solved by a parallel system analysis, in which successive use of FORM in a nested analysis of so-called ‘inner’ and ‘outer’ loops leads to the solution. The approach suggested herein is somewhat similar to the parallel system analysis, although it can be argued that the present solution is based on a more direct and intuitive approach. Basically, the approach is constituted by successive use of FORM to a conditional event followed by an integration to make the overall solution unconditional, so that the expected value of the mean outcrossing rate is obtained. 4.2.1. Conditional event In accordance with Der Kiureghian [24] and, e.g., [25], the realisation of the process that exceeds the given threshold at time t = T0 is sought for a given outcome of X = x. Hence, the event of failure is formulated as a conditional time-invariant reliability problem g (Z (t = T0 , W|X = x) , ζ ) < 0

(23)

to which the solution is given by the design point W = (Vi , Wi∗ ), where the design point is the shortest distance from the origin to the limit state surface, g = 0, in the 2N-dimensional hyperspace of W. In this way, the first-order reliability index β is defined by ∗

v u N uX
β=t Vi∗2 + Wi∗2 .

∗

(24)

i=1

The solution, leading to the design point (Vi∗ , Wi∗ ), of this problem can be approximated by use of FORM, e.g., [37,38]. The FORM

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U.D. Nielsen / Probabilistic Engineering Mechanics 25 (2010) 206–217

analysis can be carried out by standard reliability software codes, e.g., [39], where Z (t = T0 , W|X = x) is calculated for a number of combinations of (Vi , Wi ) until the design point is reached. Then, based on the design point, a so-called critical wave episode, Jensen and Capul [25], can be determined by Eqs. (2) and (3) with (Vi , Wi ) = (Vi∗ , Wi∗ ). This means that the critical wave episode is defined as the most probable wave episode leading to exceedance of the limit state. It is important to emphasise that the processes considered are assumed to be stationary and ergodic. The only requirement to the time T0 is, therefore, that the period [0; T0 ] is sufficiently long, so that the initial conditions at t = 0 do not influence on Z (T0 ). Based on the design point, it can be shown that the outcrossing rate is given by, e.g., [25]

v u N uX
1 1 ν(ζ |X = x) = exp − β 2 t αi∗2 + α¯ i∗2 ωi2 2π 2 i=1

In the example that follows below this will, among other things, be studied. Another point which is worth to study further is whether all kinds of variables — ergodic as well as non-ergodic — can be included in the averaging performed by the integration expressed in Eq. (26). Theoretically, Eq. (26) is correct for ergodic variables but not (necessarily) for non-ergodic variables such as the breaking strength of container equipment. This aspect has been discussed in some detail by Schall et al. [23]. In their study, computational schemes for structural reliability with respect to ergodic sea load processes, including sequences of sea states, and simple nonergodic variables are investigated, and it is foreseen that some of the findings reported by Schall et al. [23] may be useful with regard to further work. However, this work is an entire study by itself and beyond the scope of the present article.



{αi∗ , α¯ i∗ } =

(25)

{Vi∗ , Wi∗ } . β

Thus, the outcrossing rate may be obtained by a single FORM analysis. It is noteworthy that if a SORM analysis is performed, only the one reliability index being the argument to the exponential function should be replaced by the SORM reliability index. The reason is that the α -vector is the direction (i.e. the gradient vector) towards the design point which is the same in FORM and SORM analyses. 4.2.2. Expected value The outcrossing rate in Eq. (25) is calculated conditional on a given (random) outcome of the vector X, which describes the uncertainties in sea state parameters, heading, speed, etc. By definition, the mean outcrossing rate is calculated by evaluating the multi-dimensional integral

ν¯ =

Z

ν(ζ |X = x)fX (x)dx

(26)

all x

where fX (x) is the joint density function of the random vector X. In general, the integral cannot be solved analytically, but must be evaluated by numerical means. In [10], the integral is therefore reformulated, so that it facilitates a nested analysis in which the cumulative distribution function of ν(ζ ) is found. The direct approach is, however, to evaluate the integral in Eq. (26) by brute force simulation. An approximation to this approach is to calculate the conditional outcrossing rate for (appropriate) discretised ranges of X. The mean outcrossing rate ν¯ may then be approximated by calculating the weighted mean of the conditional outcrossing rates, where the weighting is controlled by the probability density of the discretised components of X. Schematically, this can be written

ν¯ =

X

m=

X

ν(ζ |X = xi )

i

fX (xi ).

fX (xi ) m

(27)

i

It is noted that ν(ζ |X = x) is given by the conditional event, cf. Eq. (25). In the specific case, ζ represents the breaking strength and will itself be a stochastic variable. Therefore, ζ needs to be included in the X-vector, so that integration is done (also) over all possible ζ . In practice, Eq. (27) may be evaluated by a parameter study, in which all parameters in X are included as part of the parameter study. Consequently, the questions arise how fine should the hyperspace of parameters X be discretised, and can some of the uncertain parameters, included in X, be neglected in the parameter study without compromising (significantly) on the accuracy?

4.2.3. A word on the computational efficiency The computational efficiency of FORM relies on the fact that the number of required time-domain evaluations of Eq. (6) is limited; to the extent that the minimisation of the distance from the origin to the failure surface in the hyperspace of variables can be carried out. That is, the computational efficiency is independent on the probability level. In contrast, the required number of simulations needed in brute force time series simulation will be directly influenced by the probability level with which the given event is expected to occur. 5. Numerical example—Racking failure 5.1. General information The derived calculation procedure will now be illustrated by a numerical example considering a specific vessel and ship response, and comparisons will be made with brute force time series simulations in terms of MCS. Although different failure modes of the containers have been considered in Sections 3 and 4, the example deals with racking failure only, since the actual (non-linear) response is of less importance for the purpose of illustration. The application of the probabilistic approach using FORM, Eqs. (25) and (27), and the application of MCS, to deduce outcrossing rates can be conveniently carried out by combining the time-domain simulation routine of Eq. (6) with some kind of probabilistic software tool. In the present case, the probabilistic analysis is made by use of the software Proban [39,40] which is a general purpose probabilistic analysis program developed specifically to solve problems within reliability and probability contexts. Basically, FORM yields the solution to a minimisation of a geometric distance in a hyperspace of variables; here, the set of uncorrelated, normal distributed variables (Vi , Wi ) that characterise the wave elevation. In this study, the FORM minimisation is in all cases carried out by the Rackwitz–Fiessler method, which is a built-in function in Proban. If MCS is used to estimate the mean outcrossing rate, the evaluation is based on an ensemble of k simulations of length [0; T0 ] in which the number of outcrossings is counted. However, the counting should not start before stationary conditions are attained, so that the influence of initial conditions is avoided. In the actual calculations, this means that only the last 1Tsimul seconds in each realisation is used to count the number of outcrossings; implying that the initial period 1Tinitial (T0 = 1Tinitial + 1Tsimul ) is neglected. The choice of an appropriate T0 is explored further in the example, but in any case 1Tsimul = 100 s. In the following, T0 will denote the target time, although this notation only has its strict meaning in case of FORM, cf. Eq. (25). In case of MCS, it is, as mentioned above, the last 1Tsimul seconds prior to T0 that are used as the basis for the evaluations.

U.D. Nielsen / Probabilistic Engineering Mechanics 25 (2010) 206–217

211

Table 1 Main dimensions of the considered ship and location of centre of bottom container, and other data. Length, Lpp Breadth, Bmld Draught, Tmean Block coefficient, Cb Metacentric height in still water, GM Location of container, (x0 , y0 , z0 ) Container spring constant, KC Lashing spring constant, KL Mean racking strength, µrack

284.7 m 32.2 m 10.5 m 0.61 0.89 m (0.35, 0.05, 0.04)L 0.275 m/MN 0.329 m/MN 300 kN

5.2. Vessel and container data The example concentrates on a container vessel with main particulars given in Table 1. In the analysis, the calculations apply for a stack of six containers located at (x0 , y0 , z0 ) = (0.35, 0.05, 0.04)L, where L is the length of the vessel. The containers are standard 40’ ISO containers that measure h = b = 2.438 m; relevant data are given in Table 1. At door ends, the container spring constant is KC = 0.275 m/MN according to Lloyd’s Register of Shipping [34]. The container mass is 30 tonnes for each of the three lowest containers and 3 tonnes for each of the three containers on top. In the example, one lashing system is applied, so that it connects the deck (in close vicinity of the corner of the lowest container) to the corner of the bottom of the third lowest container; see Fig. 1. The lashing ropes are made of 19 mm steel ropes with spring constant KL = 0.329 m/MN. In a previous study on cargo securing onboard ships by Andersson et al. [41], it was reported that the probability density for breaking strength of lashing equipment is skewed negatively. For illustrative purposes, a similar probability density will be assumed herein for the racking strength of the containers. The distribution, as reported by Andersson et al. [41], can be modelled by a cubic Hermite transformation [42], as shown in [3], and, in the analysis, the breaking strength is therefore modelled as Srack = µrack 1 + c0 Z1 + c3 (Z12 − 1) + c4 (Z13 − 3Z1 )






Fig. 3. Racking strength (upper plot) as function of normal distributed parameter Z1 , and associated PDF (lower plot) using cubic Hermite transformation. Table 2 Modelling of operational parameters. Parameter Hs Tz

χ U

Significant wave height Zero-upcrossing period Relative wave heading Speed of vessel

Unit

Mean value

Distribution

CoV

(m)

10.0

Log-normal

0.20

(s)

11.0

Log-normal

0.25

( ◦)

120

Normal

0.10

(m/s)

6.0

Log-normal

0.05

(28)

where µrack is the mean racking strength, and Z1 is a standard normal distributed parameter. The coefficients c0 , c3 and c4 are given by

√ c4 =

6κ4 − 14 − 2 36

= 0.075

κ3 6(1 + 6c4 ) q c0 = CoV 1 + 2c32 + 6c42 = 0.0267 c3 =

(29)

with an assumed coefficient of variation (CoV ) equal to 0.025, a skewness κ3 = −2.0 and kurtosis κ4 = 6.0. The mean racking strength, listed in Table 1, is chosen somewhat arbitrarily, although it resembles that reported by Mansour et al. [3] and Murdoch and Tozer [33]. The distribution and the associated PDF of the racking strength is illustrated in Fig. 3. 5.3. Operational parameters and modelling The example considers one set of operational parameters, and uncertainties will be associated to the significant wave height Hs , the zero-upcrossing period Tz , the relative wave heading χ and the vessel speed U. Mean values of the four parameters are given in Table 2 (χ = 180◦ is head sea) and it should be noted that the parameters correspond to a rather severe sea state. The table specifies also the kind of probability distribution which is associated to the individual parameter, including the respective

Fig. 4. Pierson–Moskowitz wave spectrum with Hs = 10.0 m and Tz = 11.0 s. In the example, the cut-off frequencies are ωlow = 0.25 rad/s and ωhigh = 1.25 rad/s, respectively.

CoV . The distributions and the corresponding parameter values in Table 2 do not reflect any data as for justification and it is important to realise that the uncertainty modelling of the variables does not associate to the aleatory uncertainty (the long-term statistics). Instead, the modelling relates to the epistemic uncertainty which associates to observation in a short-term sense. Moreover, it is noteworthy that the parameters are assumed to be uncorrelated. The example deals with irregular, long-crested waves and the stochastic sea is described by a PM wave spectrum. All calculations are carried out for a set of cut-off frequencies corresponding to ωlow = 0.25 rad/s and ωhigh = 1.25 rad/s and with the wave spectrum discretised into 25 components. It should be noted that, within the cut-off frequencies [ωlow ; ωhigh ], the wave frequencies are discretised non-equidistantly, so that the area under the wave spectrum between two adjacent wave frequencies is identical for the whole set of adjacent wave frequencies. This discretisation is chosen to avoid repetition in the time series simulation. Fig. 4 shows the applied PM spectrum, produced with Hs = 10.0 m and

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Fig. 5. Outcrossing rate as function of target time; Case 0, all input parameters (Table 2) are deterministic.

Tz = 11.0 s, and it is seen that the interval of discretised wave frequencies covers the energy distribution well. 5.4. Transient versus stationary conditions—No uncertainty In the analysis, a time-domain solution of Eq. (6) is considered, which means that the evaluation of outcrossing rates cannot be carried out before stationary conditions have been obtained. The issues of transients and ergodicity are given particular attention in [43,44], where it is shown, in relation to parametric rolling of a ship, that considerable time is needed before the transient stage has been passed. The container forces depend directly on the ship motions, which means that the starting time of stationary conditions will be the same independent on the failure mode (racking, corner post compression or tension) in question. Moreover, it is assumed that the time of stationarity is not influenced by uncertainty in the (stochastic) input parameters, so that the length of the transient stage can be evaluated on the basis of a set of deterministic input parameters. Fig. 5 shows the mean outcrossing rate as function of the target time T0 . The outcrossing rate has been calculated by use of FORM applying the mean values of the operational parameters, given in Table 2. In this case, the mean outcrossing rate is calculated directly by Eq. (25). From Fig. 5, it can be seen that initially the mean value of the outcrossing rate behaves erratically and that it does not stabilise before approximately T0 = 400 s, from which point stationary conditions can be assumed with ν¯ FORM = 2.5 × 10−4 s−1 . In Fig. 5, the results of MCS have also been included and, similarly, the result is erratic until about T0 = 400 s. As stationary conditions are attained, the mean outcrossing rate converge to the value ν¯ MCS = 2.4 × 10−4 s−1 . From these results it can, thus, be concluded that it takes about 400 s to pass the transient stage and, furthermore, good agreement between the FORM solution and MCS is observed. In the following, all calculations apply for a target time T0 = 400 s. It is noteworthy that the order of magnitude (10−4 s−1 ) of the mean outcrossing rate roughly corresponds to racking failure once every 3 h. Obviously, this rate is not acceptable under real conditions and, therefore, means should be taken to reduce the rate/risk of failure. However, this discussion is out of the scope of this article. A remark should be given on the reliability of the results obtained by MCS. The reliability of the results may be estimated in terms of the expected CoV . For a zero-one variable z, the CoV obtained from k experiments can be approximated by

√ CoV =

k · σz

k · µz

≈ √

1

k · 1Tsimul · ν

(30)

where the variance of z is given by Var (z ) = σz2 = E [z 2 ] − (E [z ])2 = E [z ] − (E [z ])2 and with the expected value E [z ] = µz = 1Tsimul · ν (µz  1). In the above situation, k = 10,000 ensembles

Fig. 6. Outcrossing rates as function of Hs (upper plot) and PDF of Hs (lower plot).

were simulated (for each target time) and with 1Tsimul = 100 s, CoV = 0.06 at T0 = 400 s, indicating a statistically reliable result. As it was mentioned just above, the outcrossing rate takes, in case of real, practical conditions, an unacceptable high value, so that means should be taken (course/speed changes) to reduce the rate. However, the operational parameters are, intentionally, chosen in this way to facilitate reliable results of MCS, even for a relatively few number of simulations, so that the calculation time is not an issue. In the given situation above, the calculation time at T0 = 400 s using MCS is about 25 min on a standard PC (Dell Latitude D630, 2.40 GHz, 3.50 GB of RAM), which means that about 10 million seconds can be simulated in one hour. The corresponding FORM solution at T0 = 400 s takes about 90 s, which means that it is a factor of 17 times faster. As an example, it is interesting to note that if the significant wave height is reduced to Hs = 8.0 m, the mean outcrossing rate is ν¯ = 1.4 × 10−6 s−1 (T0 = 400 s) obtained both by FORM and MCS. However, in this situation 250,000 simulations were needed for a CoV = 0.17 for the MCS, resulting in a CPU time of 10 h, whereas the FORM solution is established in 2 min! 5.5. Inclusion of uncertainty The influence of uncertainty in the individual parameters and the influence of uncertainty in combinations of parameters can be seen from Table 3. In the table, case 0 represents calculations without uncertainty in any parameter; that is, the results from the previous section. In cases A–E, one parameter is introduced at a time as a stochastic variable, where the modelling follows from Table 2. Cases F–I consider combinations of uncertainty in the input parameters. In all cases, the target time is chosen as T0 = 400 s, which is assumed to be the time from when stationary conditions apply. Before the values of the mean outcrossing rates in Table 3 are discussed, the actual calculation procedure of the probabilistic approach using FORM will be illustrated by considering one of the cases with only one stochastic input parameter. Case B considers the significant wave height as the stochastic parameter and, based on a parameter study on Hs , it is possible to determine the outcrossing rate as function of the significant wave height. In Fig. 6, the outcrossing rate (upper plot) is plotted for a (coarse) range of Hs and in the same figure the corresponding PDF is shown (lower plot) for the same interval of Hs . In the upper plot, for the FORM result,

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213

Table 3 Combinations of stochastic input parameters and the corresponding mean outcrossing rate obtained by the FORM approach and MCS. Case

Srack (kN)

Hs (m)

Tz (s)

χ (◦)

U (m/s)

ν¯ (s−1 ) FORM

MCS

0

300

10.0

11.0

120

6.0

2.5 × 10−4

2.4 × 10−4

A B C D E

Stochastic 300 300 300 300

10.0 Stochastic 10.0 10.0 10.0

11.0 11.0 Stochastic 11.0 11.0

120 120 120 Stochastic 120

6.0 6.0 6.0 6.0 Stochastic

2.8 × 10−4 8.3 × 10−4 2.7 × 10−4 42 × 10−4 2.6 × 10−4

2.3 × 10−4 9.8 × 10−4 3.4 × 10−4 46 × 10−4 2.5 × 10−4

F G H I

300 Stochastic Stochastic Stochastic

Stochastic Stochastic 10.0 Stochastic

11.0 11.0 Stochastic Stochastic

Stochastic Stochastic 120 Stochastic

6.0 6.0 Stochastic Stochastic

55 × 10−4 51 × 10−4 3.2 × 10−4 61 × 10−4

54 × 10−4 52 × 10−4 3.7 × 10−4 64 × 10−4

shown by Jensen [28], and also found by Söding and Tonguc [46] using different arguments. Moreover, on the assumption that the critical wave episode does not change shape but only magnitude for variations in response levels, Eq. (25) reduces to, e.g. [28]



1

ν(ζ ) = C exp − β 2

Fig. 7. Distribution of the outcrossing rate, the full line, obtained from only one FORM calculation (Hs = 12.0 m). A range of FORM results has been included to compare with.

an outcrossing has been calculated at each of the points marked by ‘•’. As it can be observed, it is possible to draw an interpolating polynomial through the points and, in the plot, a cubic spline has been fitted. This is somewhat similar to the idea of Gaidai and Naess [4], where it is noted that, in relation to calculations of mean (out)crossing rate of stochastic processes represented as secondorder stochastic Volterra series, the distribution of the crossing rate can be determined from calculations ‘‘in some discrete points’’ after which spline interpolation can be applied. Returning to Fig. 6, the mean outcrossing rate may be determined by combining the two plots. Hence, each calculated outcrossing rate is weighted according to the probability of the corresponding Hs and hereafter the mean outcrossing rate is obtained by summation; as schematically given by Eq. (27). In this way, the mean outcrossing rate is estimated to be ν¯ FORM = 8.3 × 10−4 s−1 . It is noteworthy that the cubic spline has been extended at both interval ends, so that the tails of the PDF are included in the calculation of the mean outcrossing rate. This is an(other) advantage of the FORM approach when compared with MCS, since, in brute force simulations, the main difficulty is to capture events of very low probability; explored also by e.g. [45,9]. It should be notified that to produce a (proper) distribution of the outcrossing rate it is important that the applied interval of, in this case, Hs covers well the PDF. However, the coarser the interval (with a sufficient number of points for spline interpolation), the faster the production of the distribution of the outcrossing rate. This means that the CPU time, required for the calculation of the mean outcrossing rate, can be significantly reduced by applying spline interpolation to a limited number of FORM calculations (without any major loss of accuracy on the assumption that the FORM calculations are correct/reliable; more on that later). In the specific situation in case B, where the significant wave height Hs is taken to be a stochastic parameter, it is possible to reduce the calculation time to an absolute minimum, since the distribution of the outcrossing rate can be approximated by only one FORM calculation. This has to do with the fact that the wave spectrum studied here, cf. Fig. 4, is of the type S (ω) = Hs2 S¯ (ω), which means that the critical wave episode, defined by the design point (Vi∗ , Wi∗ ), becomes independent of the significant wave height as

2

 (31)

where C is a constant. The invariance of the critical wave episode to the significant wave height means that a change in Hs by a factor r will just change the design point (Vi∗ , Wi∗ ) and hence also the reliability index β by a factor 1/r. As reported by Jensen [28], the overall implication is that calculations only need to be done for one value of the significant wave height. This is illustrated by Fig. 7, where the ‘‘interpolating’’ curve has been constructed from only one FORM calculation; namely, the one which applies to Hs = 12.0 m. In the specific case, the mean outcrossing rate can be found to be ν¯ FORM = 8.2 × 10−4 s−1 . Thus, the calculation time can be reduced to the absolute minimum, and still accurate results are achieved, when the significant wave height is included as a stochastic parameter. It is, however, strictly important to keep in mind that this way of calculating the mean outcrossing rate is only feasible when variations in the significant wave height are considered. It does not hold for any of the other four parameters, since the critical wave episode, i.e. the design point, will depend directly on each of these parameters. Therefore, the four other cases, A, C, D, and E, which consider one stochastic parameter at a time, must be evaluated by constructing spline interpolations to the FORM calculations, as was initially reported for the significant wave height. The results of cases A, C, D, and E are included in Table 3 and the distributions of outcrossing rates and the corresponding PDFs are shown in Fig. 8. In general, the numbers in Table 3 reveal reasonable agreement between the FORM approach and MCS, where the largest absolute, relative deviation of the five cases, is 22% in case A, while the average (absolute) error is 14%. With regard to the results of MCS, the mean outcrossing rate is evaluated on the basis of 10,000 realisations in the individual cases. This means that the estimated CoV takes values between 0.07 (case A) and 0.01 (case D), indicating reliable results. Based on the plots in Fig. 8, it can be seen that the FORM calculations and the associated cubic spline interpolations, in all cases, cover well the range of the respective parameter values of the PDF, so that the tails of the PDF are included in the calculations. For the specific cases, including case B, it is worth to note that some of the distributions of outcrossings can be interpolated on the basis of only a few FORM calculations, as it is the situation when the breaking strength Srack , the significant wave height Hs and the vessel speed U (cases A, B and E), respectively, are stochastic parameters. On the other hand, it requires more calculation points to make accurate interpolations in case of outcrossing distributions that consider uncertainty in, respectively, the zeroupcrossing period Tz and the relative wave heading χ (cases C and D).

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Fig. 8. Distributions of outcrossing rates and corresponding joint PDFs, respectively, of cases A (upper left set), C (upper right set), D (lower left set) and E (lower right set). Note the difference in scales.

It has been seen that in all cases, A–E, considering uncertainty in only one parameter at a time, the mean outcrossing rate differs from the case without uncertainty, i.e. case 0. It is, however, clear that in some cases, as in A, C and E, it has little influence to include the respective parameter as a stochastic variable. The hypothesis is, therefore, that the expected mean outcrossing rate can be estimated reasonable well by including only, say, the significant wave height and the relative wave heading as stochastic parameters, when the probabilistic approach using FORM is applied, although all of the parameters, in principle, should be included as uncertain. In terms of the FORM approach, it should be clear that it is (now) of particular importance to make the discretisation of ‘FORM calculation points’ as coarse as possible, since the computational time increases exponentially with the number of uncertain parameters. For this purpose the preceding evaluations, concerning uncertainty in only one parameter, are useful, since the plots of Figs. 6 and 8 indicate how to make an appropriate discretisation of FORM calculation points. Naturally, it is understood here that the multi-dimensional surface of the distribution of the outcrossing rate in a specific case may be interpolated, so that the joint (multi-dimensional) PDF surface is fully covered. Cases F–I are concerned with uncertainty in two or more of the input parameters. It

should be noted that although it has been seen previously that calculations considering variations in the significant wave height can be reduced to one calculation only, this feasibility is not employed in the following. Case F considers uncertainty in the significant wave height Hs and in the relative wave heading χ , and Fig. 9 shows contour maps of the distribution of the outcrossing rates and the joint PDF, respectively, as function of the two parameters. The distribution of the outcrossing rate has been constructed on the basis of 6 Hs values and 10 χ -values, which means that a total of 60 FORM calculations have been carried out to draw the interpolated surface (shown here as a contour map). When performing the FORM calculations it should be remembered that the ‘‘overall’’ limit state function, hypothetically constructed by the variables {(Vi , Wi ), Hs , χ}, is assumed to be smooth (and in terms of efficiency, it should preferably be differentiable, e.g., [47]). Thus, the preceding FORM solution, i.e. the design point, should be passed as the starting guess to the next FORM calculation in the parameter study to minimise the computational time. The calculation of the mean outcrossing rate follows from Eq. (27), which basically means to combine the contour maps in Fig. 9, and in the present case the mean outcrossing rate is estimated to be ν¯ FORM = 55 × 10−4 s−1 . The value is

U.D. Nielsen / Probabilistic Engineering Mechanics 25 (2010) 206–217

Fig. 9. Contour maps of the distribution of outcrossing rates (upper) and the joint PDF (lower) of case F.

215

Fig. 10. Cuts in the volumes of the distribution of outcrossing rates (upper) and the joint PDF (lower) of case G.

−2%. Previously, for cases with only one uncertain parameter, it included in Table 3, which also presents the corresponding result by MCS as obtained from 10,000 realisations. As noted from the values in the table, the relative deviation between FORM and MCS amounts to 2%, indicating a good agreement. From the maps in Fig. 9, it is interesting to note that the maxima of the joint PDF and of the distribution of the outcrossing rate are distinctly separated. This means that the actual combination of mean values of Hs and χ is not the most critical; a situation that would occur if the maximum of the joint PDF would be located in the lower right corner, so that the maxima would coincide. Case G considers uncertainty in three of the five parameters, and is in this way an extension of case F. Case G includes the breaking strength Srack , in addition to Hs and χ , as stochastic parameter and, due to the dependency of three parameters, the joint PDF and the distribution of the outcrossing rate form volumes in space. Fig. 10 illustrates the variation with the three parameters of the distribution of the outcrossing rates and the joint PDF by showing cuts in the volumes at specific parameter values; the volumes have been produced from cubic spline interpolations. Similar to Fig. 9, it is seen that the maxima of the two volumes do not coincide, implying that the situation (for the chosen set of mean values) is not the most critical. The mean outcrossing rate has been calculated both by the FORM approach and by MCS and, as noted from Table 3, there is a good agreement between the results with a relative deviation of

was noted that the uncertainty in breaking strength was of little relevance to the mean outcrossing rate of the given process, since the results of cases 0 and A deviated little. Similarly, the numbers in Table 3 show that the mean outcrossing rates of cases F and G are almost identical. Of the five cases A–E, the least variation from case 0 was observed for cases A, C and E, which represent cases with the breaking strength, the zero-upcrossing period and the speed of the vessel, respectively, taken as a stochastic parameter. The combination of all three parameters, Srack , Tz and U, as stochastic variables in the analysis corresponds to case H. From Table 3, it is seen that the results of case H are ν¯ FORM = 3.2 × 10−4 s−1 and ν¯ MCS = 3.7 × 10−4 s−1 by FORM and MCS, respectively, with a relative deviation of −14%. As expected, the results do not vary much from case 0 and, as the final case, it will therefore be interesting to see how uncertainty in all parameters influences the result. In case I, all parameters are included as stochastic variables and the mean outcrossing rate can be estimated to ν¯ FORM = 61 × 10−4 s−1 and ν¯ MCS = 64×10−4 s−1 by FORM and MCS, respectively. The agreement between the two results is, as expected, good (−5% relative deviation) but, more importantly, it is seen that the deviation between the mean outcrossing rate of case I varies little from case F. This fact speaks in favour of the hypothesis made earlier. Thus, the mean outcrossing rate may be estimated

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Fig. 11. Mean outcrossing rates as found by FORM and MCS for the different cases.

reasonably well by considering only the most important uncertain parameters in the analysis, which also may be what is intuitively expected. In this way, the computational speed can be increased significantly and, in the specific situation, the relative deviation between case F and case I is about −15%, which is acceptable in the actual context; the difference corresponds to situations with a mean of 20 failures (case F) versus 23 failures (case I) in one hour of operation. On the other hand, the computational time of the FORM approach of case F is only about 1% of that of case I for the specific discretisation of operational parameters (corresponding to 60 versus 4200 FORM calculations). In this connection, it should be remembered that the stochastic parameters are taken to be uncorrelated, which means that all combinations of the parameters must be considered in the parameter study that is carried out. However, in reality, it is physically incorrect to assume all parameters to be uncorrelated since, for example, the significant wave height and the zero-crossing period will usually be highly correlated. Hence, it would be allowable to neglect certain combinations of parameters, which would increase the computational efficiency of the FORM approach. Fig. 11 gives a graphical presentation of the agreements between results of the FORM approach and of the MCS in different cases. As it has already been presented, the agreement between FORM and MCS is good. As a peculiarity, it is rather interesting to note that if the areas of the columns of cases A–E are summed they (approximately) make up the area of the column of case I; that is, the summation of the mean outcrossing rates, Table 3, of cases A–E yields the mean outcrossing rate of case I (to the decimal in case of MCS). The apparent explanation has to do with the stochastic parameters being uncorrelated. This explanation is, however, not viable for cases G and H. 5.6. A discussion on simplifications It should be realised that the presented methodology and, in particular, the studied example build on a number of simplifications. In the context of decision support for navigational safety of ships, it is, therefore, of interest to discuss the usefulness of the obtained results toward practical assessment in regards to reliability. It is the author’s belief that most of the introduced simplifications will, indeed, influence the assessment quantitatively. When it comes to practice, with a focus on real-time, onboard decision support systems, the mathematical and the hydrodynamical models need, therefore, to be based on state-of-the-art techniques, as also reported by Bitner-Gregersen and Skjong [11]. Moreover, it will be

of utmost importance with a proper description of the introduced uncertainty modelling, so that all random variables are introduced accordingly from, say, full-scale measurement campaigns. In this way, the assessment strives to be as quantitatively accurate as possible. Qualitatively, on the other hand, the reliability assessment by the proposed method is believed to be useful in practice. Thus, it is the author’s opinion that the introduced simplifications will be of little significance for the qualitative assessment. However, the assumption about stationary processes does not strictly hold in practice. In future, it will, therefore, be of great relevance to extend the procedure to include non-stationary processes. Furthermore, future work should seek to scrutinize on the aspect of averaging of extreme value distributions as pointed out previously in regards to Eq. (26). Finally, it should be said that a complete discussion about the usefulness of the obtained results toward making practical and useful assessment of the desired reliability is beyond the scope of this article. The intention of the article is to present ideas on means and procedures that are expected to be applicable as underlying models of decision support systems of ships, since the ‘‘basin’’ of models is still not fully elaborated. 6. Summary and concluding remarks The calculation of the mean outcrossing rate of non-Gaussian processes with stochastic input parameters has been discussed in some detail. It has been shown that the mean outcrossing rate can be obtained by a sequence of FORM calculations, so that the uncertainty in input parameters is integrated out by performing a parameter study on the stochastic parameters. The individual results of the parameter study are weighted according to the probability with which the specific combination of input parameters occurs and, subsequently, the mean outcrossing rate is achieved by summation. The derived procedure — the FORM approach — produces results that compare well with brute force simulations in terms of MCS. The FORM approach requires significantly less CPU time than brute force simulations to calculate the mean outcrossing rate; in particular, for events of low probability. The computational efficiency relies on the fact that only a limited number of FORM calculations are needed to cover the (joint) PDF of the stochastic parameter(s). That is, the full distribution of the outcrossing rate on the whole interval of the stochastic parameter(s) may be obtained by spline interpolation using relatively few points (i.e. FORM calculations), as also suggested by Gaidai and Naess [4] although in a different context discussing MCS and saddle point integration. In the article, it was shown that it might not necessarily be of utmost importance to include all uncertain input parameters as stochastic variables, since reasonable results can be achieved by including only the most important parameters in the analysis. This fact may further reduce the computational time of the FORM approach, but it should be underlined that there may be situations, where all stochastic input parameters are equally important and, hence, they should all be included. More studies on this issue (e.g., how to most conveniently decide which parameters are the most important) are planned and so is a study to look further into the peculiarity that the sum of the mean outcrossing rates, taking one parameter as stochastic at a time, is (approximately) equal to the mean outcrossing rate when all uncertain input parameters are included in combination in the analysis. The real justification of the FORM approach is somewhat limited for the specific set of mean values of operational parameters in the studied example, Table 2, since the mean outcrossing rate takes (relatively high) values that can be easily obtained by brute force simulations with acceptable CoVs. The FORM approach should (can) be chosen, and is superior in CPU time, in more realistic

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and practical situations of decision support systems, where the order of magnitude of the mean outcrossing rate should not exceed 10−6 s−1 to 10−7 s−1 , corresponding to failure, respectively, every 12th day to every 116th day (taking such values to be acceptable). Of course, the superiority of the FORM approach becomes even more pronounced in the event of design of ships and offshore structures, where extreme response statistics is desired (e.g., [28, 25,4,9]). The FORM approach has proven to be an efficient tool for calculating mean outcrossing rates. In the studied example, no observations were made of any kind of difficulties with respect to the achieved solution. It will, however, be interesting to apply the approach to other kind of responses as well as to other ranges of operational parameters. Moreover, it might be of relevance to study the convergence of the solution with the number of wave components considered (e.g., [6,44,43]). Acknowledgements The author would like to express his sincere thanks to Professor Jørgen Juncher Jensen for many inspiring discussions and general assistance. Moreover, thanks are due to Professor Peter Friis-Hansen for valuable comments, and, finally, the author thanks an anonymous reviewer who has suggested an additional reference and a discussion about the implications of the introduced simplifications. References [1] France WN, Levadou M, Treakle TW, Paulling JR, Michel RK, Moore C. An investigation of head-sea parametric rolling and its influence on container lashing systems. Marine Technology 2003;40:1–19. [2] Carmel SM. A study of a parametric rolling event on a panamax container vessel. Journal of Transportation Research Board 2006;1963:56–63. [3] Mansour AE, Jensen JJ, Olsen AS. Fast evaluation of the reliablity of container securing arrangements. In: Proceedings of PRADS’04. 2004. p. 577–85. [4] Gaidai O, Naess A. Extreme response statistics for drag dominated offshore structures. Probabilistic Engineering Mechanics 2008;23:180–7. [5] Vinje T. On the statistical distribution of second-order forces and motions. International Shipbuilding Progress 1983;30:58–68. [6] Langley RS. A statistical analysis of low frequency second-order forces and motions. Applied Ocean Research 1987;9:163–70. [7] McWilliam S, Langley RS. Extreme values of first- and second-order waveinduced vessel motions. Applied Ocean Research 1993;15:169–81. [8] Naess A, Karlsen HC. Numerical calculation of the level crossing rate of second order stochastic Volterra systems. Probabilistic Engineering Mechanics 2004; 19:155–60. [9] Naess A, Gaidai O, Teigen PS. Extreme response prediction for nonlinear floating offshore structures by Monte Carlo simulation. Applied Ocean Research 2007;29:221–30. [10] Nielsen UD, Friis-Hansen P, Jensen JJ. A Step towards risk-based decision support for ships—Evaluation of limit states using parallel system analysis. Marine Structures 2009;22:209–24. [11] Bitner-Gregersen EM, Skjong R. Concept for a risk based Navigation Decision Assistant. Marine Structures 2009;22:275–86. [12] Tellkamp J, Günther H, Papanikolaou A, Krüger S, Ehrke K-C, Nielsen JK. ADOPT—Advanced decision support system for ship design, operation and training—An overview. In: Proc. of COMPIT’08. 2008. http://www.anast.ulg. ac.be/COMPIT08/. [13] Nielsen JK, Pedersen NH, Michelsen J, Nielsen UD, Baatrup J, Jensen JJ. et al. SeaSense—Real-time onboard decision support. In: Proceedings of WMTC2006. 2006. [14] Spanos D, Papanikolaou A, Papatzanakis G. Risk-based onboard guidance to the master for avoiding dangerous seaways. In: 6th osaka colloquium on seakeeping and stability of ships. 2008. [15] Nielsen UD. Calculating outcrossing rates used in decision support systems for ships. In: Proc. of intl. mech. eng. congress and exposition. 2008. [16] Ditlevsen O. Gaussian outcrossings from safe convex polyhedrons. Journal of Engineering Mechanics 1983;109:127–48.

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