Long-term correlation structure of wave loads using simulation

Marine Structures 24 (2011) 97–116

Contents lists available at ScienceDirect

Marine Structures journal homepage: www.elsevier.com/locate/ marstruc

Long-term correlation structure of wave loads using simulation Martin Petricic*, Alaa E. Mansour 1 Mechanical Engineering Department, University of California, Etchevery Building, UCB Campus, Berkeley, CA 94720, United States

a r t i c l e i n f o

a b s t r a c t

Article history: Received 2 January 2011 Received in revised form 28 January 2011 Accepted 30 January 2011

This paper proposes a new method for combining the lifetime wave-induced sectional forces and moments that are acting on the ship structure. The method is based on load simulation and can be used to determine the exceedance probabilities of any linear and nonlinear long-term load combination. It can also be used to determine the long-term correlation structure between these loads in the form of the long-term correlation coefficients. They are essential part of the load combination procedures in design and strength evaluations as well as in the fatigue and reliability analysis of ship structures. The simulation method treats the non-stationary wave elevations during the ship’s entire life (long-term) as a sequence of different stationary Gaussian stochastic processes. It uses the rejection sampling technique for the sea state generation, depending on the ship’s current position and the season. Ship’s operational profile is then determined conditional on the current sea state and the ship’s position along its route. The sampling technique significantly reduces the number of sea state–operational profile combinations required for achieving the convergence of the longterm statistical properties of the loads. This technique can even be used in combination with the existing long-term methods in order to reduce the number of required weightings of the short-term CDFs. The simulation method does, however, rely on the assumption that the ship is a linear system, but no assumptions are needed regarding the short-term CDF of the load peaks. The load time series are simulated from the load spectra in each sea state, taking into account the effects of loading condition, heading, speed, seasonality, voluntary as well as involuntary speed reduction

Keywords: Long-term load combination Sampling Correlation coefficients Simulation Wave-induced loads

* Corresponding author. 485 Ohlone Way, Apt. 301, Albany, CA 94706, United States. Tel.: þ1 510 705 3065. E-mail addresses: [email protected] (M. Petricic), [email protected] (A.E. Mansour). 1 Tel.: þ1 510 643 4996. 0951-8339/$ – see front matter
2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.marstruc.2011.01.003

98

M. Petricic, A.E. Mansour / Marine Structures 24 (2011) 97–116

in severe sea states and the short-crested nature of the ocean waves. During the simulation procedure, special care has been given to maintaining the correct phase relation between all the loads. Therefore, time series of various load combinations, including the nonlinear ones, can be obtained and their correlation structure examined. The simulation time can be significantly reduced (to the order of minutes rather than hours and days) by introducing the seasonal variations of the ocean waves into a single voyage simulation. The estimate of the long-term correlation coefficient, obtained by simulating only a single voyage with the correct representation of seasonality, approaches the true correlation coefficient in probability. This method can be applied to any ship and any route, or multiple routes as long as the percentage of the ship’s total lifetime spent in each of them is known. A study has been conducted to investigate the effects of ship type, route and the longitudinal position of the loads on the values of the correlation coefficients between six different sectional loads; vertical, horizontal and twisting moments, as well as shear, horizontal and axial forces. Three ocean-going ship types have been considered; bulk carrier, containership and tanker, all navigating on one of the three busy ship routes; North America–Europe, Asia– North America and Asia–Europe. Finally, the correlation coefficient estimates have been calculated for five different positions along the ship’s length to investigate the longitudinal variation of the correlation coefficient.
2011 Elsevier Ltd. All rights reserved.

1. Introduction During the ship’s lifetime, which is typically a period of twenty-five years, its structure will be subjected to various non-stationary stochastic loads. There are many ways to categorize these loads, but a very natural one can be obtained by looking at a long time record of, e.g., normal stresses in a longitudinal structural member of the ship’s hull structure. A simple Fourier transformation of such a signal would reveal that its variance is mainly distributed around three frequency bands. The first one is very close to zero frequency, the second one is close to the wave encounter frequency (around 0.5 rad/s), while the third one is in the range of high frequencies (around 4 rad/s). Hence, the stationarity and load frequency become the basis for categorization as follows:  Quasi-stationary loads (stillwater and thermal loads);  Low-frequency non-stationary loads (wave-induced loads);  High-frequency non-stationary loads (springing, slamming/whipping, machinery induced vibrational loads). Low-frequency wave-induced loads are associated with the rigid body motion of the vessel in waves. High-frequency wave-induced loads arise due to slamming and hydroelastic behavior of the vessel known as springing. Ship designer needs to find a realistic estimate of the total combined load effect to which the structure will be subjected in order to properly design it. The estimate that is too conservative will subsequently result in an over-designed structure that has a smaller payload. On the other hand, if the estimate of the total combined load effect is lower than its true value, the resulting ship structure might be unfit for service, expensive to maintain or unsafe to operate. If all the loads acting on the ship structure were deterministic and stationary (or quasi-stationary), then simple addition of the loads would be sufficient to obtain the total combined load effect at a certain point on the structure. That value could then be used in the design of that particular structural member. However, due to the stochastic and non-stationary nature of the ocean waves during the ship’s lifetime, both the low- and high-frequency loads will also be stochastic and non-stationary. In order to properly

M. Petricic, A.E. Mansour / Marine Structures 24 (2011) 97–116

99

combine them, methods of classical statistics and time series analysis have to be used. The main focus of this paper is to present a new method for combining the sectional low-frequency wave-induced loads, which are the most significant group of stochastic loads. However, the method can be applied to any other stochastic load as long as its transfer function is known. A number of methods to combine the extreme load values can be found in the literature ([1–11]). Most of them depend on the knowledge of linear relationship between two individual loads, which is represented by the correlation coefficient r. It is important to distinguish between the short- and long-term load combinations. The former corresponds to the time interval of up to 3 h during which the sea state, represented by the significant wave height, HS, zero crossing period, T0, and the prevailing wave direction, b, does not change. Therefore, the wave elevations can be represented by a stationary Gaussian process. On the other hand, long-term load combinations refer to the entire lifetime of the ship. Here, the stationarity of input (wave elevations) can no longer be assumed since the ship encounters many different sea states at many different speeds, V, and relative headings, a. Short-term load combinations in the extreme design sea state, for a certain loading condition, are important for the structural strength evaluations, especially the limit state calculations. Long-term load combinations are, however, more important in the structural reliability analysis and fatigue calculations where all loads encountered by the ship during her lifetime need to be taken into account. The following Sections 1.1 and 1.2 provide a short description of the existing short-term and long-term methods respectively. 1.1. Short-term load combinations Mansour [2] studied the short-term combinations of two extreme loads of the form

fc ¼ f1 þ Kf2

f1 > f2

(1)

where f1 and f2 are the characteristic design values of individual loads and fc is the characteristic value of the combined load. Characteristic value can be defined as the expected extreme load value in N peaks or the most probable extreme load value in N peaks or the extreme load value with a certain probability of exceedance in N peaks. K is the load combination factor that depends, among other things, on the correlation coefficient between individual load components r12. In the case of stationary Gaussian sea,the expression for the K-factor is as follows:

K ¼

1=2 i mr h  mc 1 þ r2 þ 2r12 r 1 r

where r ¼ s2 =s1 ; mr ¼

(2)

ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ln½ð1321 Þ1=2 N1  ln½ð132c Þ1=2 Nc  , r12 is the short-term correlation ; mc ¼ 2 1=2 2 1=2 ln½ð132 Þ

N2 

ln½ð131 Þ

N1 

coefficient between two individual loads, r is the ratio between the standard deviations of individual loads, 31, 32 and 3c are the bandwidths of the response spectra of the individual loads and of the combined load respectively, and N1, N2 and Nc are the number of peaks of the individual loads and the combined load respectively. If the individual and combined load processes are narrow-banded and have similar central frequencies, then mr and mc can be taken as unity. This is usually the case when combining stresses resulting from the vertical and horizontal bending moments. The short-term correlation coefficient can be calculated using the following expression for the long-crested waves:

912 ¼

1

s1 s2

Z 0

N

  < H1 ðuÞH2 ðuÞ Sx ðuÞdu

(3)

where H1(u) and H2(u) are the frequency response functions (transfer functions) of the two loads and “*” denotes the complex conjugate of H2. Sx(u) is the input sea spectrum. Similar expressions can be derived for the short-crested waves as well as for the combinations of three extreme loads. It can be shown, [2], that the combination factor K is always greater than or equal to the correlation coefficient between the two loads, with equality being established only for K ¼ r12 ¼ 1. Since the correlation coefficient represents the overall linear relationship between the two loads and K

100

M. Petricic, A.E. Mansour / Marine Structures 24 (2011) 97–116

represents the linear relationship between the extreme values of these loads, it is evident that the load values are more correlated at the extreme value level than at the overall level. This is a very important consideration in the structural strength analysis. It is also shown in [2] that the three widely used load combination methods; the Peak Coincidence method (PC), the Square Root of the Sum of Squares method (SRSS) [3] and the Turkstra’s Rule (TR) [4] are just special versions of the K-factor method. The PC method assumes perfect correlation between the loads, in which case K ¼ r12 ¼ 1 and fc ¼ f1 þ f2. The SRSS q method, ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi on the other hand, assumes no correlation between the loads (r ¼ 0). In this case fc ¼ f12 þ f22 . Turkstra [4] proposed that the maximum extreme load can be found as the maximum of the values obtained by combining the maximum value of each individual load with the expected values of the corresponding ”point-in-time” values of the other loads. Mathematically, this is expressed as:

fc ¼ maxff1 þ E½x2 jf1 ; f2 þ E½x1 jf2 g

(4)

where E½x2 jf1  and E½x1 jf2  denote the conditional expectations of “point-in-time” load values x2 and x1 given the occurrence of the extreme load values f1 and f2 respectively. It can be shown that, in the case when mr ¼ mc ¼ 1, Eq. (4) simplifies to fc ¼ f1 þ r12 f2 . If we compare this expression with Eq. (1), we can see that TR uses K ¼ r12. This is also intuitively obvious since the linear relationship between any two load values, including the case when one of them reaches its maximum value, is given by the correlation coefficient. Obviously, the PC method overestimates the value of the combined extreme load, while the SRSS and the TR underestimate it. It is also worth mentioning that in the case that mr ¼ mc ¼ 1, Eqn. (1) can be re-casted into a more familiar form used by many classification societies:

fc ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f12 þ 2r12 f1 f2 þ f22

(5)

All the load combination methods mentioned so far, including the K-factor method, apply only to the short-term load combinations in a single sea state when the assumption of stationarity holds.

1.2. Long-term load combinations Today, numerous methods for predicting the combined long-term loads exist. Probabilistic longterm analysis can be found in Baarholm and Moan [5] or Jensen et al. [6]. It consists of finding the longterm cumulative distribution function (CDF) of the combined response peaks (e.g., longitudinal stress peaks in the shear strake) by statistically combining many short-term CDFs:

FRLT ðrÞ ¼ ¼

R R R R HS T0 V a

R R R R HS T0 V a

FRST ðrjhs ; t0 ; n; aÞ$fHS T0 ;V;a ðhs ; t0 ; n; aÞwdhS dt0 dvda

FRST ðrjhs ; t0 ; n; aÞ$fHS T0 ðhs ; t0 Þ$fV;ajHS ;T0 ðv; ajhS ; t0 ÞwdhS dt0 dvda

(6)

where FRLT ðrÞ is the long-term CDF of the combined response peaks; FRST ðrjhs ; t0 ; n; aÞ is the shortterm conditional CDF of the combined response peaks given a sea state, speed and relative heading; fHS T0 ;V;a ðhs ; t0 ; v; aÞ is the joint probability density function (JPDF) of HS, T0, V and a; fHS T0 ðhs ; t0 Þis the JPDF of HS and T0 only; fV;ajHS ;T0 ðv; ajhS ; t0 Þ is the conditional PDF of V and a given a sea state and w is the weighting factor which expresses the relative rate of response peaks within each sea state. It can be quite difficult to obtain the analytical expressions for fHS T0 ;V;a ðhs ; t0 ; v; aÞ, and especially for FRST ðrjhS t0 ; v; aÞ as a function of the random variables involved. The situation becomes even more difficult if we introduce new variables such as the area of the ocean, season and the loading condition. Therefore, the integral in Eq. (6) has to be discretized into individual sea state–operational profile combinations whose number can become quite large depending on the number of variables and their resolution. Additionally, to facilitate the calculations in Eq. (6), the independence of relative heading and the sea state is usually assumed. This is, however, not true for the ships operating on a single route where the wind blows predominantly in one direction. Short-term conditional CDF of the response peaks given a sea state, speed and relative heading, FRST ðrjhS t0 ; v; aÞ, needs to be known for the

M. Petricic, A.E. Mansour / Marine Structures 24 (2011) 97–116

101

combined response. If the response is Gaussian and narrow-banded, then its CDF exactly follows the Rayleigh distribution [1]: r 2

FRST ðrjhS t0 ; v; aÞ ¼ 1  e2m0

(7)

where m0 ¼ s2c is the zero moment of the combined response spectrum and is equal to the variance of the combined response. It depends on the values of HS, T0, a and V through the combined frequency response function. Today, the combined frequency response function can be calculated using the stateof-the-art seakeeping software, but only for linear response combinations. Moreover, in reality, the combined response is rarely narrow-banded, even if the wave elevations are narrow-banded. This is especially true for the case of ship springing where a second peak at higher frequencies in the vertical bending moment spectrum exists. In that case, the short-term CDF of the combined response should be represented with the Rice distribution [1]: "pffiffiffiffiffiffiffiffiffiffiffiffiffi #

  pffiffiffiffiffiffiffiffiffiffiffiffiffi r2 r FRST ðrjhS ; t0 ; v; aÞ ¼ F pffiffiffiffiffiffiffi  1  32 $e2m0 $F 3 m0

1  32 $r pffiffiffiffiffiffiffi 3 m0

(8)

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffi Ru RN where 3 ¼ 1  m22 =ðm0 m4 Þ; FðuÞ ¼ N ð1= 2pÞexpðz2 =2Þdz; mn ¼ 0 un SðuÞdu. 3 is the bandwidth parameter, F(u) is the standard Gaussian CDF and mn is the nth moment of the combined response spectrum. For a perfect narrow-banded process with 3 ¼ 0, Rice distribution reduces to Rayleigh probability law. Since 3 depends on the zero, second and fourth moment of the combined response spectrum, using Rice distribution instead of Rayleigh entails more computation. Also, there are numerical difficulties in determining the fourth moment of the load spectrum since it does not converge. Eq. (6) can only be used to find the long-term CDF of the loads and their linear combinations. A similar, but separate weighting of the short-term correlation coefficients has to be performed in order to obtain the long-term correlation coefficients between various loads. One such method has been described in [7]. Authors of this paper caution against using Eq. (5) to indirectly calculate the long-term correlation coefficient when the extreme values of the individual and the combined load are known. This is because Eqn. (5) is only valid in the short-term case when the loads are stationary random processes. Other long-term methods can also be found in the literature ([5],[7–11]). A short summary of these methods is given in [12]. Most of them introduce additional assumptions that decrease the overall accuracy in an attempt to reduce the number of necessary calculations in discretized version of Eq. (6). Next section will present the new method for long-term load combination that is based on simultaneously simulating multiple realizations of the individual load time series that are adequate (in the statistical sense) representations of the long-term load random processes. Long-term time series have been composed of multiple short-term load simulations. The method includes the effects of different sea areas, seasons, ship’s velocity/heading profiles, ship’s type, route, loading condition, short-crested nature of the ocean waves and it does not pose any restrictions on the spectral density bandwidth of either the input or the output. A computer program has been developed that efficiently simulates multiple voyage time series of the individual loads taking into account their correlation structure. Therefore, long-term CDFs of both the individual load peaks and the combined load peaks can be found, as well as the long-term correlation coefficients between them. Nonlinear load combinations can also be investigated by combining the point-in-time values of the individual loads. The rejection sampling technique has been used in order to significantly reduce the number of short-term load simulations in various sea state–operational profile combinations. These points will be further explained in Section 2.2. The simulation method does, however, use linear transfer functions for the loads, which means that nonlinearities in individual loads, like the nonlinearities in the vertical bending moment, cannot be simulated. The methods of incorporating the nonlinear transfer functions into the simulation algorithm and analyzing their effect are the objectives of an ongoing investigation. 2. Simulation procedure 2.1. Statistically sufficient realizations of the load random processes Wave-induced loads acting on a ship’s hull are generated by a common stochastic process: the ocean waves. This commonality of input is an important source of their correlation. Wave elevation

102

M. Petricic, A.E. Mansour / Marine Structures 24 (2011) 97–116

during the ship’s lifetime is a non-stationary stochastic process and that creates difficulties in statistical analysis because many of the methods and theorems of classical statistics and time series analysis are only valid if the process is weakly stationary (has constant first and second moment over a period of time). In order to apply spectral analysis in the frequency domain, long-term wave elevations can be decomposed into a sequence of time periods during which the stationarity assumption is satisfied. Observations of the wave elevations at a certain area of the ocean have shown that this random process is Gaussian and weakly stationary over a period of up to 3 h. Therefore, multiple responses to the wave elevations can be found using the linear filter analysis in the frequency domain for each short-term period. All known sources of randomness have to be modeled in order to simulate the stochastic loads (outputs). For a ship navigating between two different ports with a constant loading condition pattern, the shortest time series of each load that is generated by the computer model (simulator), would have to be one-year long to correctly account for all the randomness of the input process. Only then can the effects of seasonality in the wave elevations be accounted for. Therefore, every one-year record represents a different realization of the underlying stochastic process and they all have the same statistical properties. Still, generating one-year time series for each load is computationally very expensive. Since the only additional randomness that is not included in a single voyage simulation (both legs) comes from the seasonal changes in the wave elevation, these effects can be artificially incorporated into a single voyage. This point will be explained later. Thus, time series of different loads obtained by simulating only one voyage, with seasonality effects included, can be used to obtain the unbiased estimate of the correlation coefficients between the individual loads according to the following expression:

Pn R ¼ "

i ¼ 1 ðxi

Pn

j¼1



xj  x

 xÞðyi  yÞ

2 Pn

k ¼ 1 ðyk

 yÞ

#1=2

(9)

2

where R is the unbiased sample estimate of the true correlation coefficient r, n is the number of simulated load values, xi and yi are the individual load values of two different loads whose correlation is being estimated and x and y are their mean values. If R is calculated from the load time series that are statistically adequate representations of the corresponding long-term random processes, as explained above, then it can be proven [12], using Slutsky’s lemma [13], that R (for any two loads) approaches its true value, r, in probability. This means that sufficiently accurate estimates of the correlation coefficients between the individual loads can be obtained from only a single voyage simulation. This is because the number of simulated load values is very large even for a single voyage simulation (on the order of 106). Multiple simulations are needed only if the variability of the estimate R is needed. 2.2. Simulation Three different ship types have been analyzed in this work: containership, product oil tanker and bulk carrier all navigating on one of the three busy shipping routes: the North Atlantic (NA) route between Boston and Southampton, the North Pacific (NP) route between Long Beach and Singapore and the Europe-Asia (EA) route between Barcelona and Singapore. The containership has been fully loaded on both legs of the voyage, while the tanker and the bulk carrier have been fully loaded on the eastern leg, and in ballast on the western leg of the voyage. This is in accordance with the usual ship loading patterns, i.e., oil and iron ore have a one-way transportation pattern. For simplicity, changes in the loading condition due to the consumption of fuel and provisions have not been taken into account. Such effects can easily be included by using transfer functions corresponding to the actual mass distribution of the ship at different stages of the voyage. Multiple routes can also be accounted for, provided that the percent of the ship’s total lifetime spent on each route is known. For example, if the ship spends two thirds of her lifetime on route A and one third on route B, then the shortest load time series, that adequately represents all known randomness, must be composed of two complete voyages on route A and one voyage on route B. Any simulated time series has to contain load values from both routes in proportion to their lifetime probabilities.

M. Petricic, A.E. Mansour / Marine Structures 24 (2011) 97–116

103

In order to simulate the wave loads, an adequate statistical model of the ocean waves is needed. For that purpose, the area of the ocean in which the ship is navigating has been first divided into different zones according to the Global Wave Statistics (GWS) Worldwide and European databases [14]. Then, the ship’s route has been accurately plotted using the charting software Netpas Distance [15]. For example, on the Boston - Southampton route, the ship is traveling through areas 8, 9, 15 and 16 from the Worldwide database and areas 23, 24 and 25 from the European database. The letter database has been used for parts of the ocean close to the European continental shelf because it has a better resolution and accuracy than the Worldwide database. Distances in each ocean area have been accurately calculated using Netpas Distance. The ship routes on a world map with outlined sea areas are shown in Fig. 1. Wave data for each area is given in the form of scatter diagrams representing the normalized number of observations of each combination of HS and T0 at discrete intervals. For each season (winter, spring, summer and fall), eight scatter diagrams are given corresponding to eight different prevailing wind/wave directions (N, NE, E, SE, S, SW, W, NW). A total of 768 scatter diagrams have been used for all areas on the three routes. Next, a joint probability density function (JPDF) of HS and T0 has been fitted to every scatter diagram. The JPDF used in this work was the one proposed by Ochi [16]. He found that for certain ocean area (A), season (S) and wave direction (b), the conditional PDF of the significant wave height, fHS jA;S;b ðhS ja; s; bÞ, can be accurately represented by the generalized gamma distribution over its entire domain. The conditional distribution of T0 given HS, A, S and b can be represented by the log-normal distribution. Therefore, the conditional JPDF of HS and T0, given A, S and b, is given by:

fHS ;T0 jA;S;b ðhS ; t0 ja; s; bÞ ¼ fHS jA;S;b ðhS ja; s; bÞ$fT0 jHS ;A;S;b ðt0 jhS ; a; s; bÞ ( )   c cm cm1 1 ½lnðt0 Þ  mðhS Þ2 l hS exp  ðlhS Þc $pffiffiffiffiffiffi exp  ¼ GðmÞ 2psðhS Þt0 2½sðhS Þ2

(10)

where c, m and l are the parameters of the generalized gamma distribution and have to be estimated from the scatter diagram data. Parameters of the conditional log-normal distribution depend on

Fig. 1. Different shipping routes considered in this work.

104

M. Petricic, A.E. Mansour / Marine Structures 24 (2011) 97–116

HS: mðHS Þ ¼ a1 þ a2 HSa3 ; sðHS Þ ¼ b1 þ b2 expðb3 HS Þ, where the parameters a1, a2, a3, b1, b2 and b3 also have to be estimated from the scatter data. The JPDF in Eq. (10) has been fitted to the data in each of the 768 scatter diagrams by using the maximum likelihood estimation (MLE) to find its nine parameters. The MLE is based on the assumption that the form of the JPDF is known (Eq. (10)) apart from the uncertainty about its parameters. The first step in the MLE is to form the likelihood function:

LðqÞ ¼

Y

fHS ;T0 jA;S;b ðhS ; t0 ja; s; b; qÞ;

q˛Q

(11)

Over all combinations on HS and T 0 from the scatter table:

where q ¼ [c, m, l, a1, a2, a3, b1, b2, b3] is the vector of unknown parameters and Q is the parameter space (e.g. c, m, l and s(HS) all have to be greater than zero). The main idea of the MLE is that L(q) will be relatively larger for values of q near the ones that generated the data. Therefore, parameters are found by maximizing L(q). This was performed in Matlab [17] using the nonlinear constrained optimization routine fmincon. Since the data in the scatter diagrams is right and left censored, special attention has been paid to account for this in the MLE. After having obtained these nine parameters at eight different wind/wave directions, b, for each area and season, a piecewise cubic Hermite polynomial has been fitted to each parameter in order to get the explicit dependence of the conditional JPDF in Eqn. (10) on b. Finally, the conditional JPDF of HS, T0 and b, given area and season, can be found as:

fHS ;T0 ;bjA;S ðhS ; t0 ; bja; sÞ ¼ fHS ;T0 jA;S;b ðhS ; t0 ja; s; bÞ$fbjA;S ðbja; sÞ

(12)

The conditional PDF of b, given area and season, fbjA;S ðbja; sÞ, has been estimated from the wind direction data in the Global Wave Statistics. For NA route, westerly directions have been found to be more frequent then the others. For the NP and EA routes, big seasonal wind direction variations have been observed. The foregoing analysis needs to be performed only once. When the explicit dependence of the nine parameters on b is found for each area and season, the model is then saved on a computer for later use. Given area and season, a triplet (HS, T0, b) can be sampled from the conditional JPDF in Eq. (12) using the rejection sampling technique (for details see, e.g., Davison [13]). The idea is to first independently sample HS, T0 and b from uniform distributions. Then, each generated sample is accepted with a probability that is proportional to the value of the conditional JPDF in Eqn. (12) for that sample. It is important to emphasize that the triplets sampled using rejection sampling are independent. In reality, the time series of HS, T0 and b have a non-zero autocorrelation function, at least for small lags. This means that the consecutive sea states are not independent. However, for the purpose of determining the estimates of the correlation coefficients or the long-term distributions of various loads, the correct (realistic) sequence of sea states is irrelevant. To correctly incorporate the seasonality effects in one voyage, seasons have been sampled randomly between winter, spring, summer and fall with the probabilities corresponding to the percent of time that the ship spends navigating in each season. In this work equal probabilities have been assumed. This way, each simulated voyage record will have the correct proportions of data from each season. Ship’s heading was determined using the Netpas Distance software for each stage of her voyage. Changes in the ship’s heading due to rough weather have not been considered in this work. This is a topic of an ongoing investigation. However, involuntary and voluntary speed reductions that depend on the significant wave height and the relative heading between the ship and waves have been considered. Table 1 shows the speed profiles used in this work for different ship types (speed is given in knots). Other speed profiles can easily be specified. Weather routing has been partially considered in this work by limiting the highest possible sampled significant wave height to 20 m. This is similar to the procedure used by Jensen et al. [6] where the scatter diagram has been modified depending on the value of the significant wave height. HS, T0 and b have been sampled from Eqn. (12) based on the current position of the ship (area) and the randomly determined season. The relative heading (a) that the ship makes with the prevailing

M. Petricic, A.E. Mansour / Marine Structures 24 (2011) 97–116

105

Table 1 Velocity profiles for all three ships used in this work. Relative wave direction

Wave height [m] <2

2–4

4–6

6–8

8–10

>10

(a) Containership Head Bow Beam Quartering Following

25.60 25.60 25.60 25.60 25.60

25.60 25.60 23.00 25.60 25.60

23.00 23.00 20.00 23.00 23.00

15.00 15.00 15.00 15.00 20.00

10.00 10.00 10.00 10.00 15.00

10.00 10.00 10.00 10.00 10.00

(b) Tanker and bulk carrier Head Bow Beam Quartering Following

15.00 15.00 15.00 15.00 15.00

15.00 15.00 13.75 15.00 15.00

13.75 13.75 12.50 13.75 13.75

11.25 11.25 11.25 11.25 12.50

10.00 10.00 10.00 10.00 11.25

10.00 10.00 10.00 10.00 10.00

wind/wave direction (b) has been calculated as a ¼ g – b, where g is the ship’s heading in that area and voyage leg (see Fig. 2). Consequently, the ship’s speed has been automatically determined from Table 1. In this work, the stationarity assumption has been taken to hold for a period of up to 2 h. Based on that time period, low-frequency, wave-induced loads have been simulated (according to the procedure that will be described below) and the distance traveled by the ship has been calculated. This procedure has then been repeated until the ship has reaches a new area, and from then on, until all the areas have been traversed and booth legs of the voyage have been completed. Therefore, load time series for entire voyage are composed of many 2 h long stationary time series simulated in each sea state.

Fig. 2. Nomenclature of the relevant angles for short-crested waves.

106

M. Petricic, A.E. Mansour / Marine Structures 24 (2011) 97–116

All six sectional low-frequency wave-induced loads have been simulated in this work; longitudinal force (LF), shear force (SF), horizontal force (HF), torsional moment (T), vertical bending moment (VBM) and horizontal bending moment (HBM), all at amidship (except for the case when the longitudinal position of the loads has been intentionally varied). Other loads can be simulated using the same procedure provided that their transfer functions are known. Wave elevation stochastic process has been represented in the frequency domain by the ISSC twoparameter spectrum. Cosine squared spreading function has been used to account for the short-crested nature of the ocean waves. Load transfer functions, both the amplitude and the phase part, have been calculated using SEAWAY [18] software. It is a frequency domain ship motion program based on the modified strip theory that includes the effects of forward speed. The transfer functions have been calculated for each loading condition, each relative heading angle (in ten deg. increments) and for every speed specified in Table 1. The output spectra for each individual load is calculated according to the well-know expression for linear filters:

Sy ðue Þ ¼

Z p=2

p=2

Sy ðue ; mÞdm ¼

¼ Sx ð ue Þ

Z p=2 2 p=2

p

Z p=2

p=2

jHy ðue ; a  mÞj2 Sx ðue ; mÞdm

cos2 ðmÞjHy ðue ; a  mÞj2 dm

(13)

where index x denotes the input (ocean wave elevation), y denotes the output (any of the sectional forces and moments), S(ue) is the one-sided encounter spectral density, m is the angle between the wave component under consideration and the prevailing wind/wave direction and a is the relative angle between the ship velocity vector and the prevailing wind direction (see Fig. 2). ue is the encounter frequency of the waves that is measured on board a moving ship and is equal to ue ¼ u þ ðu2 =gÞVcosðaÞ where u is the absolute wave frequency, g is the gravitational acceleration and V is the speed of the ship. The integral in Eq. (13) can be calculated numerically. According to the spectral representation theorem (for details see, e.g., Shumway and Stoffer [19]) any stationary time series of a random process can be taught of, approximately, as a random superposition of independent sines and cosines oscillating at various frequencies and having amplitudes that are in proportion to their underlying variances. They are, in turn, determined by the spectral density of the random process. Therefore, wave elevation time record taken on board of a ship can be expressed as a Fourier integral and approximated with a finite sum of independent periodic components as follows:

xðtÞ ¼

Z ue;MAX ue;MIN

Xðue Þcosðue t þ fÞdue z

N X



Xi cos ue;i t þ fi

(14)

i¼1

where the summation is conducted over N equally spaced encounter frequencies between ue,MIN and ue,MAX of the wave spectrum, X is the amplitude of the component oscillating at frequency ue and f is the random phase angle having a uniform distribution on the interval [p, p]. This ensures the independence of individual components. Also, wave elevation x(t) expressed as a sum in Eq. (14) is approximately a Gaussian stochastic process according to the central limit theorem. The amplitude of a certain periodic component in the time series is, in the case of gravity waves, equal to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xi ¼ 2Sx ðue;i ÞDue;i where Due ¼ ðue;MAX  ue;MIN Þ=N For any linear filter, represented by its transfer function, H, the periodic component of the output will have the same frequency as its corresponding input component. However, its phase will change by the phase part of the transfer function and its amplitude will change by a factor of jHj. Therefore, the simulated output time series can be obtained as follows:

LFðtÞ ¼

ffi N qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X



2SLF ue;i Due;i cos ue;i t þ fi þ fLF;i

(15)

i¼1

SFðtÞ ¼

ffi N qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X



2SSF ue;i Due;i cos ue;i t þ fi þ fSF;i i¼1

(16)

M. Petricic, A.E. Mansour / Marine Structures 24 (2011) 97–116

HFðtÞ ¼

ffi N qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X



2SHF ue;i Due;i cos ue;i t þ fi þ fHF;i

107

(17)

i¼1

TðtÞ ¼

N qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X



2ST ue;i Due;i cos ue;i t þ fi þ fT;i i¼1

VBMðtÞ ¼

ffi N qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X



2SVBM ue;i Due;i cos ue;i t þ fi þ fVBM;i

(18) (19)

i¼1

HBMðtÞ ¼

ffi N qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X



2SHBM ue;i Due;i cos ue;i t þ fi þ fHBM;i

(20)

i¼1

where SLF, SSF, SHF, ST, SVBM and SHBM are the one-sided spectral densities of the sectional loads, calculated using Eqn. (13), fi is the random phase angle (identical for all six loads) and fLF,i, fSF,i, fHF,i, fT,i, fVBM,i and fHBM,i are the input–output phase differences for each load. For example, the torsion input-output phase difference is given as fT;i ¼ tan1 fJ½HT ðue;i Þ=<½HT ðue;i Þg. This way, the correct relationship between the loads is maintained. Since the encounter spectra can have singularities for certain combinations of the ship’s speed and relative heading, the product SLoad(ue,i)Δue,i can be replaced by SLoad(ui)Δui for sufficiently small Δui. This is possible because the total variance of all periodic load components inside an infinitesimal encounter frequency band is equal to the total variance of all the load periodic components inside the corresponding infinitesimal absolute frequency band, i.e. SLoad(u)du ¼ SLoad(ue)due. Having obtained the six load time series for the whole voyage (with seasonality correctly represented), the estimates of the correlation coefficients can be calculated using Eq. (9), as well as the longterm empirical probability distributions of all six loads and their combinations. An important consideration when simulating load values is how closely must two consecutive load values be spaced in time in order to accurately represent the time series. According to the Shannon’s Theorem (see, e.g., [20]), a continuous time series is completely described if the values are generated with the frequency that is at least twice as large as the maximum frequency, ue,MAX, of a periodic component that is present in the series. 2ue,MAX is called the Nyquist frequency. It has been found in this work that for all load encounter spectra, ue,MAX < p rad/s. Therefore, generating load values at the frequency of 2p rad/s or 1 Hz, has been found sufficient. Another consideration is the number of encounter frequency intervals N. This number must be sufficiently large to avoid any periodicities in the simulated time series and to ensure its approximate normality according to the central limit theorem. Normal q–q plot can be used to verify if there is sufficient reason to discard the normality hypothesis of the output time series. N 100 has been found to satisfy both criteria (non-periodicity and apparent normality). Sea state and operational profile generation using rejection sampling technique that was described in this section can also be used in combination with the discretized version of Eq. (6) to find the longterm CDF of a linear load combination. In s similar manner, it can also be used to find the long-term correlation coefficients between various loads. Instead of weighting the short-term CDFs or correlation coefficients over every possible sea state–operational profile combination, the weighting is performed over a limited number of such combinations until the statistical convergence is achieved. Rejection sampling ensures that these short-term states appear in the long-term record with relative frequencies that correspond to their true occurrence probabilities. If Eq. (6) is used, then going in the time domain and actually simulating each load from its spectral density is not necessary. However, authors believe that the simulation method offers two advantages over the existing long-term methods. First, the load simulation eliminates the need for assuming the model for the distribution of the short-term CDF of the load peaks based on the bandwidth of their spectrum as discussed in Section 1.2. Even though the use of Rayleigh CDF produces more conservative results when the load spectral density is not narrowbanded, the authors believe that it is better to be more accurate than conservative. Second major reason for load simulation is the possibility of combining the loads in a nonlinear manner by combining their point-in-time values. This is relevant in the calculations of the CDFs of e.g. Von Mises stresses, or in the calculations of the CDFs of an entire limit state function in the reliability analysis.

108

M. Petricic, A.E. Mansour / Marine Structures 24 (2011) 97–116

Table 2 Main particulars of all three ships used in this work.

Length B. P. [m] Breadth [m] Draught (full load) [m] Draught/trim (ballast) [m] Block coefficient Deadweight (full load) [t] Deadweight (ballast) [t] Deck section modulus [m3] Side section modulus [m3]

Containership

Tanker

Bulk carrier

283.30 32.20 11.26 N/A 0.70 68240 N/A 50.00 85.00

282.89 49.00 15.00 8.16/2.21 0.84 172007 89044 N/A N/A

283.00 45.00 16.00 7.81/2.91 0.81 168743 77991 N/A N/A

The simulation procedure described in this section has been implemented into efficient computer program using Matlab [17]. The total simulation time of all six loads and for one voyage is a few seconds on an ordinary personal computer. The complete flowchart of the simulation procedure is given in the Appendix. Simulation results are described in the next chapter. 3. Results Main particulars of the three cargo ships that were analyzed in this work have been chosen in such a way to facilitate the comparison. In other words, all three ships have very similar lengths between perpendiculars. Special care has been taken to model the realistic (existing) ship forms with realistic loading conditions using the SEAWAY ship motion software. The main particulars of all three ships are shown in Table 2. In order to investigate the ship route effects on the long-term correlation coefficients, the containership has been analyzed navigating on all three routes. Subsequently, in order to analyze the ship type effects, all three ships have been analyzed navigating on the EA route. Of course, much more ship type-route combinations have to be analyzed to fully determine how these two factors affect the long-term load combinations. This is an objective of the current investigation. Fig. 3a shows 1000 sea states (combinations of T0 and HS) sampled from the Area 27 in winter. The sampled data has been superimposed on the contour plot of the fitted JPDF of T0 and HS for Area 27 (all directions). The validity of the rejection sampling procedure can be observed, since the largest number of sampled sea states comes from the range with highest probabilities. The third sampled variable, the

Fig. 3. Sampled sea states for Area 27 in winter. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).

M. Petricic, A.E. Mansour / Marine Structures 24 (2011) 97–116

109

wind/wave direction b, is shown in Fig. 3b. Its empirical PDF is plotted together with the empirical PDF of b-s from the GWS. It can be seen that the sampled b-s agree well with the GWS wind/wave data for that area and season. This is a further validation of the rejection sampling. Similar validation has been performed for all other areas and seasons. 3.1. Basic results First, the basic findings from the application of the simulation method will be explained using the example of the containership navigating on the NA route. More results for this ship and route have been published in [12]. An example of the simulated time series of the vertical bending moment for one whole voyage is shown in Fig. 4a. Time series is approximately 130 long with load values spaced 1 second apart. Different sea states are clearly visible. Fig. 4b shows a scatter plot of the VBM and the HBM. Some degree of liner relationship exists, as will be shown later in this paper. Due to the periodic components with very low encounter frequencies that are present in the loads, the periodicity of the time series is not an issue. The effects of length of each of the load time series on the variability of the correlation coefficient estimate are shown in Table 3. For simplicity, only the sectional moments have been considered in this part of the study, although similar conclusions can be drawn for sectional forces as well. In each case a total of 50 simulations have been performed but each time generating load time series of different lengths (1 voyage, 5 voyages.). As it was stated in Section 2.1, and proven in [12], the estimate of the correlation coefficient approaches the true value in probability as the length of the time series increases. This can be seen from the fact that the variability of the estimate, represented by the coefficient of variation (d - ratio between the standard deviation and the mean value), decreases as the time series length increases. At the same time, the mean values of the correlation coefficient estimates (mR) quickly converge to distinct values for each load combination. Therefore, even fifty consecutive simulations of one voyage are sufficient to accurately estimate the true (population) correlation coefficient using the mean value of its estimates. Torsion and VBM, as well as VBM and HBM are found to be significantly positively correlated. On the other hand, torsion and HBM are practically uncorrelated. Since one voyage consists of approximately 130 individual short-term time series, a total of 6500 sea state–operational profile combinations were sufficient to achieve the convergence. Considering that sea state was described using three random variables (HS, T0 and b), that the operational profile was described using four additional variables (season, area, speed and loading condition) and considering their resolution, a total number of possible sea state–operational profile combination in this work is on the order of 108. This is the total number of weightings needed if Eq. (6) is used. However, rejection sampling reduces this number to 6500 which significantly speeds up the calculations.

Fig. 4. Some basic outputs from the simulation (containership, NA).

110

M. Petricic, A.E. Mansour / Marine Structures 24 (2011) 97–116

Table 3 The effects of the length of the simulated time series. No. of sim.

Length

mR(T-VBM)

dR(T-VBM)

mR(T-HBM)

dR(T-HBM)

mR(VBM-HBM)

dR(VBM-HBM)

Sim. time [min]

50 50 50 50 50

1 voyage 5 voyages 10 voyages 25 voyages 50 voyages

0.20 0.20 0.20 0.20 0.20

0.279 0.133 0.106 0.063 0.044

0.04 0.05 0.04 0.04 0.05

1.954 0.870 0.782 0.394 0.250

0.26 0.26 0.26 0.26 0.26

0.292 0.117 0.088 0.067 0.034

9.1 33.2 66.7 167.2 371.7

Fig. 5a shows a comparison of the simulation results with the long-term probability of exceedance of individual VBM peaks in sagging condition, obtained by the simplified numerical long-term analysis (both linear and nonlinear) as described in Jensen et al. [6]. Jensen’s nonlinear procedure includes the effect of slamming. However, both methods described in [6] use simplified expressions for the transfer functions, the combined scatter diagram for North Atlantic, and they rely on the assumption that the output spectra are narrow-banded. As expected, nonlinear analysis gives the largest probabilities of exceedance, while the simulation results lie between the linear and nonlinear results. It was shown in [6] that, in the case of no weather routing and EA route, the nonlinear analysis overestimated the exceedance probability compared to the full-scale measurements. This means that even though simulation procedure relies on the linear transfer functions and cannot take into account the nonlinearities in the VBM or slamming, it produces reliable results. An advantage of the simulation method is that any load combination can be obtained by directly combining the point-in-time values of the corresponding loads. For example, combined longitudinal stress in the shear strake, due to both the VBM and the HBM, can be obtained by simply adding the two time series (divided by the corresponding section moduli). This way, long-term exceedance probabilities can be obtained for the combined stress peaks. The total longitudinal stress will not be symmetric with

Fig. 5. Examples of the analyses that can be performed with the simulation method. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).

M. Petricic, A.E. Mansour / Marine Structures 24 (2011) 97–116

111

respect to the ship’s centerline because of the positive correlation between VBM and the HBM. This can be observed in Fig. 5b. The long-term empirical CDF’s of the combined stress peaks for port side (PS) and starboard (SB) are given along with the Weibull probability distribution fitted to each of them. For this particular ship and route one can expect more fatigue damage in the shear strake on the port side. Literature review has revealed that there is very little information on the long-term correlation coefficients between different loads. The ones identified in this study pertain only to the vertical and horizontal bending moments. Hovem [8] proposed that rVBM-HBM ¼ 0.1 based on calibration with longterm characteristic load values and is valid for the long-crested waves only. ISSC [21] recommends rVBM-HBM ¼ 0.32. This value is valid for Series 60 hull with length 283 m and for NA region. Simulation method yields values of 0.26 for short-crested waves and 0.31 for long-crested waves (both values are for the analyzed containership in NA). ISSC value matches closely with the simulated values, especially for the long-crested waves. On the other hand, the value proposed by Hovem is somewhat lower. Real comparison is not possible, however, because the value in [8] is given without reference to ship type or the area of navigation. 3.2. Ship route effect To investigate the ship route effect on the values of the estimate of the long-term correlation coefficient, R, containership has been analyzed traveling on all three routes (NA, NP, EA). Apart from the correlation of three sectional moments that has already been mentioned for containership on the NA route, Fig. 6a shows all sixteen mutual correlations (the correlation matrix is symmetric). It is evident that the LF and the VBM are significantly positively correlated (RLFVBM ¼ 0.72). This is as expected, considering that both loads reach their maximum at the same relative wave angles; in beam and following seas. Fig. 6b and c also show full correlation matrices, but for the containership on other two routes. Fig. 6d shows a graphical comparison of the more significant values of R. It is evident that the ship route has a moderate effect on the values of R. However, the sign of the correlation coefficient estimates does not change with the ship’s route. Another important observation is that the loads acting on the containership navigating on EA route appear to be, in general, less correlated than the loads encountered on the other two routes. This is probably because of the big seasonal variations of the wind/ wave direction and the wave climate of the Indian Ocean. Since the ship usually changes several routes during its lifetime, it can be expected that the values of the correlation coefficient estimates will lie

Fig. 6. Full correlation matrices for the containership on all three routes (based on 100 voyages) – the ship route effect.

112

M. Petricic, A.E. Mansour / Marine Structures 24 (2011) 97–116

Fig. 7. Comparison of the sagging VBM peaks (during one voyage)for the containership navigating on three routes. Green line represents the EA route, blue line NP route and the red line represents the NA route. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).

somewhere between these. More investigation of route combinations is needed, though. It is also worth mentioning that the HF is always negatively correlated with all other loads, except with the HBM. It is interesting to compare the probability of exceedance of individual sagging VBM peaks for all three routes. Fig. 7 shows this comparison. The North Atlantic is widely regarded as the harshest part of the oceans. It is always taken as the design ocean area for the ships that are classified for unrestricted navigation. Thus, it is not surprising to sea that the VBM has the highest probability of exceedance when the ship is on the NA route, and the lowest probability of exceedance when the ship is on the EA route. The differences between the three routes are, for higher values of the VBM, more than one order of magnitude large. 3.3. Ship type effect All three ship types have been compared on the EA route in order to investigate the effect of the ship type on the values of the correlation coefficient estimates. The results for the containership navigating on the EA will be repeated for clarity purposes and the results for the oil tanker and the bulk carrier will be added. Fig. 8 is very similar to Fig. 6, but shows the effect of the ship type on the values of R. It can be seen that this effect is more significant than the ship route effect. Although some differences exist, the correlation matrix for the bulk carrier is more similar to the one for the oil tanker, than to the one for the containership. This can be explained by the fact that both the oil tanker and the bulk carrier have similar hull forms, displacements and speeds. On the other hand, containership has a very fine form with lower block coefficient. It is designed to carry cargo that is partly stowed on its deck. Therefore, it has a much higher center of gravity than the other two ships. Additionally, it has significantly higher cruising speed, which also affects the load transfer functions. 3.4. Longitudinal position effect Fig. 9 shows the correlation coefficient estimates between different loads at various positions along the length of the containership navigating on the EA route. This figure shows only four of the more interesting load combinations. It can be seen that the correlation between loads significantly changes with their position on the ship. It is interesting to observe that the correlation between the vertical

M. Petricic, A.E. Mansour / Marine Structures 24 (2011) 97–116

113

Fig. 8. Full correlation matrices for the all three ships on the EA route (based on 100 voyages) – the ship type effect.

Fig. 9. Effect of the longitudinal position of the ship (based on 100 voyages).

bending moment and the shear force becomes very high at one quarter and three quarters of the ship’s length. This is significant because shear force is largest at approximately these locations. It can also be observed that other load correlations become significantly positively correlated at three quarters of the ship’s length. Finally, very high correlation between some loads near the aft of the ship might not be so significant because global loads are close to zero near the ship’s perpendiculars. More ship types and ship routes have to be analyzed to fully understand the effect of the longitudinal position of the loads. 4. Conclusion The simulation method presented in this paper can be used to estimate the long-term correlation coefficient between different loads or their combinations in an accurate and efficient manner. Long-term exceedance probabilities, as well as other statistical properties of the individual load peaks and their combinations can also be found. The agreement with the probabilistic (numerical) methods for obtaining the long-term CDFs of response peaks has been found satisfactory. The simulation method has been used

114

M. Petricic, A.E. Mansour / Marine Structures 24 (2011) 97–116

to find the combined longitudinal stresses in the shear strake of a containership navigating on the NA route. The stresses in the shear strake on PS and SB side are not symmetric due to the positive correlation between the VBM and the HBM. If the ship is navigating only on that route, more fatigue damage can be expected on the PS due to the higher stress cycles. Some of the advantages of the simulation procedure are:  Sampling of the short-term sea state–operational profile combinations greatly reduces the simulation length necessary to achieve the convergence. It also reduces the number of weightings when existing long-term methods are used;  During the simulation, the correct phase relation between the loads is always preserved which enables the estimate of the correlation coefficient to be obtained directly from the generated samples;  Since the actual time records of each load are generated, linear and nonlinear load combinations can be investigated by directly combining their point-in-time values. Also, long-term statistics such as the exceedance probabilities of the load responses and their combinations can be found;  The developed method is very flexible in terms of the randomness that can be modeled. In the present study different sea areas, seasons, routes, ship types, speed/heading combinations, loading conditions, longitudinal positions of loads and the short-crested nature of the ocean waves have been taken into account;  No assumption is needed regarding the bandwidth of the input or the output spectra;  Different routes and loading conditions can easily be included in the simulation according to the procedure described in the Section 2.2;  It is efficient and fast needing only a few seconds to generate the entire voyage time series for all loads;  Correlation coefficient estimate can be obtained based on only a number of single voyage simulations that include the effects of seasonality. It has also been found in this work that the values of the long-term correlation coefficient estimates are moderately sensitive to the ship’s route and very sensitive to the ship type and the position along the ship’s length. The sign of the correlation is the same for all ship routes, but starts to differ when different ship types are analyzed. The highest level of linear relationship occurs between the LF and the VBM. The most important load combination in ship design, the one between the VBM and the HBM at amidship, is affected by both the ship type and the ship route. These two loads have positive correlation in all cases. It is highest for the NA route and for the bulk carrier. It is lowest for the EA route and the containership. The comparison of the long-term exceedance probabilities for the VBM peaks confirms the fact that the NA is harsher than the other ocean areas. Not only are the VBM and the HBM larger in magnitude in this area of the ocean, they are also more positively correlated, resulting in larger combined stress cycles. This needs to be taken into account when designing a new ship. 4.1. Future work Future work will focus on:  Investigation of extreme loads using simulation;  Investigation of the correlation structure between global loads and secondary loads due to water pressure;  Determination of the long-term distributions of various nonlinear load combinations;  Investigation of the effects of springing and slamming using simulation (work in progress);  Investigate the effects of weather routing;  Simulating nonlinearities associated with individual loads;  Expanding the parametric study. Acknowledgment The first author would like to acknowledge the generous financial support of the Fulbright Foundation which sponsored his studies at the University of California at Berkeley. Without the support this work would not have been possible.

M. Petricic, A.E. Mansour / Marine Structures 24 (2011) 97–116

Appendix – Simulation Flowchart

115

116

M. Petricic, A.E. Mansour / Marine Structures 24 (2011) 97–116

References [1] Mansour AE. An Introduction to Structural Reliability Theory, Technical Report No. SSC-351, Ship Structure Committee, Washington, DC, January 1989. See also URL http://www.shipstructure.org. [2] Mansour AE. Extreme loads and load combinations. J Ship Res 1995;39(1):53–61. [3] Mattu RK. Methodology for Combining Dynamic Responses, NUREG-0484, rev.1, office NRR, U.S. NRC, 1980. [4] Turkstra CJ. Theory of Structural Safety, SM study No. 2. Ontario, Canada: Solid Mechanics Division, University of Waterloo; 1970. [5] Baarholm GS, Moan T. Efficient estimation of extreme long-term stresses by considering a combination of longitudinal bending stresses. J Mar Sci Tech 2002;6(3):122–34. [6] Jensen JJ, Pedersen PT, Shi B, Wang S, Petricic M, Mansour AE. Wave Induced Extreme Hull Girder Loads on Containerships, in: Proceedings of SNAME Annual Meeting, Houston, October 2008, pp. 128–52. [7] Shin YS, Kim B, Fyfe AJ. Stress combination for fatigue analysis of ship structures. J Offshore Mechanics Arctic Eng 2005; 127:175–81. [8] Hovem L. Loads and Load Combinations for Fatigue Calculations - Background for the Wave Load Section for the DNVC Classification Note: Fatigue Assessment of Ships, DNVC Report No. 93–0314, Høvik, Norway, 1993. [9] Funaki T, Hibi S, Shimizu H. A note on the correlation coefficient between wave-induced stress. J Soc Nav Archit Jpn 1996; 180:575–89 (in Japanese). [10] Kawabe H, Shibazaki K. Combination of the fatigue damage produced by several wave-induced loads based on the correlation coefficient method, in: Wu YS, et al. (Eds.), Proceedings of the PRADS 2001, Shanghai, China, September 2001, pp. 1089–95. [11] DNV, Fatigue Assessment of Ship Structures, Classification Notes No. 30.7, Høvik, Norway, October 2008. [12] Petricic M, Mansour AE. Estimation of the Long-term Correlation Coefficients by Simulation, in: Proceedings of the ASME 2010 29th International Conference on Ocean, Offshore and Arctic Engineering, OMAE2010, Shanghai, China, June 2010. [13] Davison AC. Statistical models. Cambridge: Cambridge University Press; 2008. p. 77–9. [14] BMT, Global wave statistics online, http://www.globalwavestatisticsonline.com; December 2010. [15] Netpas. Netpas distance software, http://www.netpas.net; December 2009. [16] Ochi MK. Ocean waves: the stochastic approach. Cambridge: Cambridge University Press; 1998. p. 132. [17] MATLAB version R2010a. Natick, Massachusetts: The MathWorks Inc.; 2010. [18] Journée JMJ, Adegeest LJM. Theoretical Manual of Strip Theory Program SEAWAY for Windows, Report 1370, Delft University of Technology, Delft, The Netherlands, September 2003. [19] Shumway RH, Stoffer DS. Time series analysis and its applications with R examples. New York: Springer; 2006. p. 534–54. [20] Madsen H. Time series analysis. Boca Raton: Chapman Hall/CRC; 2008. p. 80. [21] ISSC, Report of Committees 3 and 10, in: Proceedings of The Fifth International Ship Structure Congress, Hamburg, Germany, 1973.