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Chapter 4 Loads on marine structures 4.1 General For performing reliability analyses of marine structures, certain specific load and strength data are necessary. Prior to estimating the loads acting on ships or marine structures, a statistical representation of the environment is necessary. This includes waves, wind, ice, seismic effects and currents. The last four items are more important for fixed offshore structure than for floating vessels. The environmental information can then be used as input to determine the loads acting on the structure. Typically, an input/ output spectral analysis procedure is used to determine the ‘short-term’ loads, as loads in a specific sea condition (stationary condition). The required transfer function is determined from first- or second-order strip theory using the equations of motion of the vessel, or from a towing tank experiment. In offshore structures, Morison’s equation is usually used to determine the wave load transfer function. Prediction of the loads in stationary sea conditions (spectral analysis) is not sufficient for the reliability analysis. Extreme values and long-term (lifetime) prediction of loads and their statistics are more valuable. For this purpose, order statistics and statistics of extremes play a very important role. Gumbel’s theory of asymptotic distributions is often used in this regard. In the long time prediction, the fatigue loads, i.e., the cyclic repetitive loads which cause cumulative damage to the structure, must also be considered. Methods of combining the loads, such as static and dynamic, including high- and low- frequently loads, should be considered. In nature, many of these loads act simultaneously, therefore their combination must be evaluated for a meaningful reliability analysis. In assessing the reliability of ship structures, two general loading situations may be used: short-term or long-term analysis. At the design stage, if the route of the ship is known and if that route is more or less permanent, then the probability of failure can be predicted using long-term analysis. If, on the other hand, the ship is likely to take a variety of routes during its lifetime, then short-term analysis can be used to obtain the probability of failure under one or more conditions that are considered to be the severest the ship may encounter during its lifetime. The criterion usually used in the short-term analysis is to consider the single most severe sea condition (a sea condition with a specified return period, or more appropriately, a sea condition with a specified encounter probability) and subject the vessel to this condition for a specified period of time. These short- and long-term analyses will naturally produce different final results for the safety margins. Therefore, care must be taken when comparing safety margins of different ships, i.e., the method and criterion used in predicting the loads acting on the ship will have a considerable impact on the resulting safety index. To further amplify this point, the long-term distribution of the wave loads acting on a ship may be determined by tracing the expected route/s of the ship during its lifetime. Based on ocean wave statistics along the route/s, the long-term (lifetime) wave load probability distribution for the entire history may be determined. In the short-term analysis, extreme load distribution is predicted on the basis of criteria such as one extreme sea storm of a specific encounter probability and duration, or a short-term operation in a specific location under severe sea conditions. It should be noted that there is a fundamental difference between computed results based on these two avenues. In the short-term 80 analysis, the computed probabilities of failure are conditional probabilities given the occurrence of an extreme wave load per selected criterion. Care must be taken in this case in determining the response of the ship to this extreme load since non- linearity will play an important role. In the long-term analysis, the resulting probabilities of failure are associated with the entire history of the expected loads acting on a ship during its lifetime, and are dependent upon the selected route/s likely for the ship. The procedure of ship structural reliability analysis for the short-term analysis is as follows: a) From ship route (if known), obtain ocean wave statistics, and a specified encounter probability (or return period) determine the design storm condition (see section 4.3). b) Calculate the rms value of the wave bending moment in the design sea condition, using either second order strip theory, or towing tank experiment. Calculate also the Stillwater bending moment (see section 4.3). c) Estimate the strength parameters for each failure mode (see Chapter 5) d) Calculate the probability of failure or the safety index for each failure mode. The resulting probabilities are conditional probabilities. They are conditioned on encountering the design storm. In general, the long-term procedure entails the determination of the probability distribution of the maximum load during the lifetime of a ship taking into consideration the wave statistics along the ship route, loading conditions, speed, and heading. The procedure is particularly important for fatigue reliability analysis, where the entire history of loading should be determined. In that fatigue case, the long-term distribution, instead of the maximum load distribution, is required and is usually assumed to be Weibull. Several procedures have been proposed in the literature for determination of the lifetime maximum load distribution. Although their details may vary (sometimes depending on the ship type), most of them have common characteristics as follows: a) Define the mission profile of the ship which includes 1) ship route 2) expected total years of service 3) number of days per year the ship is expected to be at port and underway 4) nominal cruising speed and maximum operating speed in each sea state, and the corresponding fraction of time during operation 5) distribution of ship headings 6) distribution of loading conditions b) From the ship route and available wave statistics, obtain the frequency of occurrence of different sea conditions the ship will encounter in each of the geographic areas (zones). c) On the basis of the above, determine the frequency of encountering different sea conditions, loading conditions, speeds, and headings. d) Determine the wave loads in each sea condition, loading condition, speed, and heading using first- or second-order strip theory. e) Use an extrapolation procedure to determine the distribution of the maximum load in a lifetime. 4.2 Review of computational methods of loads on marine structures This section will present an overview of load computational methods (ISSC, COMMITTEE I.2, 2003). 81 4.2.1 Environmental Loads on Ships With regard to the general loads analysis problem, linear strip theory methods for ship motions and loads predictions, long the workhorse of the industry, are readily being supplanted by three-dimensional methods both linear and non-linear. The flow field around a ship and the resulting motions and loads due to incident waves constitute a three-dimensional non- linear problem. The complete solution of this kind of problem may be accomplished by using computational fluid dynamics methods to solve NavierStokes equations or by using a viscous solution for the near field and an inviscid solution for the far field. The potential flow formulation is a practical alternative approach to a viscous solution. Although strip theory methods continue to be used, three dimension potential methods are rapidly gaining ground. However, the methods currently available do not appear to be sufficiently robust to simulate a full range of hull geometries and operating conditions. Continuing improvements to the computational speeds of computers have made nonlinear time-domain methods more applicable to the design process. However, designers must continue to balance the need for computational accuracy against computational effort. The use of lower level methods as a filter, from which more extensive investigations using higher order methods are made, is still the most pragmatic approach for designers. Ships with flat transom sterns and significant bow flare may be prone to parametric rolling in head seas. Such events can introduce loads on containers and lashing systems that are far higher than those normally predicted by classification rules. Non- linear motion and loads programs can effectively be used to predict whether such events will occur. However, a good estimation of roll damping is required. Since the downward forces of green water offset hydrostatic restoring moment, green water will tend to reduce vertical bending moments. However, green water will significantly increase local structural loads. Wind loads can represent a significant part of the total design load for high-speed ships. 4.2.2 Environmental Loads on Offshore Structure The computation of linear 3D bodies in waves is considered a mature technology. For most structures, this can be done very efficiently on state-of-the-art computers. Remaining challenges are related to the hydroelastic response of very large floating structures. From a practical point of view, it is also important to find ways to incorporate realistic damping estimates (viscous effects) to predict realistic response levels near resonance. This is of particular importance to “mixed type” structures, for example, truss spars. Also, the coupling between CAD (Computer Aided Design) geometry representation for input of body geometry and output pressures and forces in standardized FEM (Finite Element Method) formats would help practical engineering. This is particularly important in time-domain analyses of slamming and extreme load predictions. The topic of trapped modes has received considerable attention. State of the art linear programs seem able to handle the problem, even for very large structures. The understanding of actual resonance frequencies has also matured. It is also recognized that trapped or near trapped waves are found for structures with (almost) axis symmetry, 82 for example, a floating platform with four columns. Computation of the real wave elevation, including the effects of non-linearities associated with trapped waves, remains a challenge. The capability to predict second-order forces on large volume structures has been available for several years. However, it seems to be a challenge to compute local quantities as velocities and wave elevation accurately. This requires careful discretization of the boundary value problem. Perturbation to 3rd-order does not seem to be an accurate way to handle the ringing problem. Fully non- linear potential theory based on computations indicates that the 3rd-order forces are over-predicted in the perturbation scheme, while the fully non- linear formulation gives similar forces as measured. It is assumed that viscous forces play a minor role in the ringing problem. Most programs do not handle breaking waves. It still seems there is a long way to go before the NWT (New wave theory) becomes a practical tool for engineering. Realistic Reynolds numbers are not obtained. However, use of CFD (Computational Fluid Dynamics) techniques is a possible option to estimate viscous damping effects. CFD methods are being used to address VIV (Vortex-induced Vibration)- induced fatigue. However, while the use of such methods is supported by classification societies, it is realized that the general application of CFD to the design process is still aways off. From the fact that several methods for computing the hydroelastic response of a VLFS (Very Large Floating Structure) are used, linear analyses seem to work well. A remaining challenge is obviously non- linear computations to obtain better response estimates close to resonance. The need to explore increasing water depths requires that designers now address issues related to slender bodies and coupled response in the relevant environments. This has led to innovative testing techniques such as hybrid model testing. Such methods provide a means to assess the behavior of new design concepts in a realistic way, but do not provide the designer with insight as to the physical causes of such behavior. In addition to modeling difficulties, the need to operate in ever increasing water depths requires a better understanding of possible areas of operations. For example, current fluctuations may manifest themslves as both a VIV- induced extreme load as well as a fatigue load. Although designers may use a current profile, such profiles typically assume constant direction over depth. A more rigorous handling of such fluctuations, where probabilities of curent speed as a function of depth would be preferable. However, such a scatter diagram will require extensive in-situ measurements. 4.2.3 Hydrodynamic Impact Loads The hydrodynamic impact problem for ships manifests itself in both a global (whipping) and local response. In predicting the global whipping response, it is important that the instantaneous free surface elevation be properly taken into account but local hydroelastic effects can be neglected. However, hydroelasticity may be an issue for local structure. Typically, the importance of hydroelasticity will be a function of natural frequency of both the structure and the duration of the slam load. Thus, the structural designer should take the time to estimate these quantities before suggesting a design load. In addition, when performing model tests, the experiment should be set up such that not only impact pressures but also structural strains are measured. 83 Although 2D methods continue to be used to solve the local hydrodynamic impact problem, 3D methods are beginning to be used. However, full implementation into the design process will require additional computer capacity. Green water and deck impact loads are major design considerations. In estimating design loads, probabilistic estimates are required. Towards that end, exceedence probabilities for waves overtopping the structure appear to follow an exponential curve. When overtopping does occur, the problem becomes similar to the dam break problem with a non- zero initial horizontal velocity. As the presence of structure along the deck will effect velocity magnitude and phases, calculations derived solely from using maximum crest velocity would result in an under-designed structure. Sloshing loads in tanks can be estimated as lo ng as the free surface flows inside the tank are correctly simulated. Both 2D and 3D methods are available for predicting such flows. Although, extensive efforts have been made in developing a variety of numerical techniques for simulating 3D liquid movement in partially- filled tanks they are not sufficient to be successfully used in design practice. The main shortcomings are related to the non- linearity in equations governing motion of liquids, non- linear condition on free surface, flow separation during motion of liquid in tank, considerable solubility of air in real flow in tanks, occurrence of cushions when liquid strikes a tank wall, dynamic flow interaction with flexible tank structure, random occurrence of pressure impulses, necessary work required in preparing data and time-consuming computations and potentially unstable solutions. Due to these difficulties, two-dimensional models are mainly considered. In applying 2D methods, dynamic loads are determined with sufficient accuracy; however, it is difficult to determine impact pressure and pressure impulses. Irrespective of whether a 2D or 3D method is used, frictional forces in the liquid may be neglected, as it is of minor importance in comparison to inertia force occurring in liquid motion in a moving tank. This makes it possible to apply Euler’s equation to describe liquid motion. Sloshing loads in a tank, which are generated by random motion of the ship, are smaller than loads generated by harmonic excitation with the same rms amplitude of excitation. Extreme realistic load and loads having influence on fatigue strength need to be considered in designing the tank structure. There is a need to develop design formulas which would make it possible to estimate sloshing loads during initial stages of design and which could also be used as safety standards. 4.2.4 Probabilistic Methods Except for moderate seas where linear theory is most applicable, there is no one universal method which could be used for the prediction of short-term extreme responses. This shortcoming is due to strong non- linearities in environment and ship’s geometrical properties. In most cases, an iterative approach has to be taken to determine the best fit for the short-term response peaks. The general Gamma distribution, general Pareto distribution, three parameter Weibull distribution and Gumbel distribution can be considered for various applications and successfully applied. 84 Since there exists a range of possible options for looking at short- and long-term load statistics, the application of any one method can be complicated and strongly depends on personal experience of an engineer or scientist, the method used for estimate of distribution parameters and the sample size. The possibility of process standardization, sensitivity of long-term predictions with respect to type of short-term peak probability distribution, and uncertainty in log-term predictions should be further investigated. 4.2.5 Experimental Uncertainty, Verification and Validation of Numerical Codes There has been some progress in the development and practical implementation of experimental and numerical uncertainty procedures. Efforts of experimental and CFD communities are reflected in fact that some sort of validation analysis accompanies more and more publications of measured and computed data. The verification and validations procedure and their practical applications for viscous fluid flow codes (RANS) seem to be better organized than for codes based on potential flow theory. The viscous fluid flows simulations are usually accompanied by detailed quantitative verification and validation procedures, when validation of inviscid / irrotational-based codes mostly means qualitative comparison to published measured data or semi-empirical calculations. Efforts to create a new generation of benchmark data for validation purposes are required. Experiments designed by experimentalists and code developers, carried out according to specified procedure, including measurement of all required parameters with required accuracy are needed. The procedure should consider and include statistical aspects of model handling with respect to course-keeping (manual and/or autopilot) and power delivery (constant speed or revolutions). Standardization of experimental uncertainty procedure: identification of all possible elemental error sources, common methods for calculation of certain properties, and standardization of experimental procedures are required for consistent interpretation of experimental results and uncertainties associated with those experiments. While the process of determining experimental and numerical uncertainty is all well and good, there are some practical concerns. For example, in order to address errors associated with experimental technique, multiple runs require to be performed. In an ideal situation, tests performed within a linear regime should be repeated at least 3 times. These tests should be repeated on not only on different days but with different test crews as well. For tests where non- linearities are expected, a minimum of 5 repeat tests are necessary. Once such data become available the next question is, what is a reasonable level of verification and validation of numerical methods? In theory, a series of convergence studies, followed by comparisons with regular and irregular wave data would suffice. However, in practice this may not be practical. For instance, how many convergence studies should be done? Is one set, performed for one wavelength sufficient, or are more required? And what role should full scale trials play, if any, in the validation process? Then comes the question of hull forms to be studied. Is a validation against one set of model tests for a specific hull form represent validation or just a documented solution? From a pragmatic standpoint, code validation will be limited to what the user is willing to pay for. However, a minimum threshold and some basic guidelines need to be developed. 4.3 Loads 85 4.3.1 Return periods and encounter probabilities Return periods and the associated wave heights are important to determine the probability of the structure encountering a design storm that has a specified return period. This probability of encounter will depend on the lifetime of the structure, i.e., on how long the structure will remain at the location where the return period and the associated wave height are calculated. If the structure life is long, the probability of encounter will be higher. This part will present a procedure for calculating ship/storm encounter probabilities that can be used as a better basis for establishing design criteria. The encounter probabilities involve the life of the structure, as well as wave statistics, in the region of operation. Return periods involve wave statistics only, and do not involve the life of the structure. In the next section, the encounter probability in any ocean zone is developed as a function of the return period of a design wave and the life of the structure. A method of calculating return periods for specific wave heights is described as well. The method is based on extrapolating wave data at the site and depends on the probability distribution of wave heights at that location. 4.3.1.1 Encounter probability The encounter probability is the probability that a particular wave height will be encountered during the portion of a structure’s lifetime spent in a zone. The encounter probability of a specific wave height (or a sea state characterized by a significant wave height), not only depends on its return period R, but also on the life of the structure, L, in years. In this section, the encounter probability in an ocean zone i will be considered. The probability that a wave height x will not be encountered during the portion of a structure’s lifetime Li spent in zone i will be called non-encounter probability Pnei . If the distribution function FYi (x ) of the annual maximum wave heights is available, then from order statistics, the non-encounter probability is: Pnei = P[ no exceedence of x in life Li ] = P[Yi ≤ x ] = [ FYi ( x)] Li (4.3.1) where Yi is maximum wave height during time Li Li is time spent in zone i in years FYi (x ) is distribution function of the annual maximum wave height in zone i The distribution function of the annual maximum wave height can be written in terms of the distribution function of the individual wave heights Fxi ( x) , using, again, order statistics as: FYi ( x ) = [ Fxi ( x )] ki (4.3.2) where k i is the number of wave peaks (cycles) in zone i in one year. Thus equation (4.3.1) can be written Pnei = {[ Fx i ( x)] k i }Li (4.3.3) The return period of a wave height x is defined as the average length of time between exceedence. The waiting period w in years between exceedence in zone i has a probability law given by (Borgman): P[Wi = w ] = FYwi −1 ( x)[1 − FYi ( x )] (4.3.4) 86 and the average waiting period, i.e., the return period, is: Ri = E[W i ] = [1 − FYi ( x )] −1 Ri = [1 − ( Fxi ( x )) ] ki −1 (4.3.5) (4.3.6) The relationship between the non-encounter probability and the return period Ri can be obtained as follows: Pnei = (1 − Ri− 1 ) Li (4.3.7) and the probability of encounter Pei is: Pei = 1 − (1 − Ri−1 ) Li (4.3.8) The return period Ri of a wave height x , in any zone i can be estimated from: n Ri = i y 0 (4.3.9) n0 and n i = [1 − Fxi ( x)] −1 (4.3.10) where n 0 is total number of wave data collected in zone i y 0 is number of years of data collection in zone i n i is expected number of waves in zone i necessary to exceed wave height x n i can be calculated from equation (4.3.10) for any value of design wave x . The procedure for determining the ship/ storm encounter probability in any zone i can be summarized as follows: a) Use wave data in zone i to determine the form and the parameters of the distribution function of wave heights Fxi (x ) , using any method of parameter estimation, e.g., methods of moments, or regression analysis. b) For the prescribed design wave height (or sea state characterized by a significant wave height) predict the number of waves necessary to exceed the design wave height using equation (4.3.10). c) Using equation (4.3.9) to estimate the return period associated with the design wave height or significant wave height. d) Determine the probability of encounter in zone i form equation (4.3.8) and information on how long the ship operates in zone i , i.e., Li . 4.3.1.2 Ship routes and the associated encounter probabilities In order to determine the probability of encountering a wave height (or a specified sea state) along a ship route, the zones and the harbour are considered as members in a series system. A series system is defined as a system in which a state of encounter occurs if an encounter occurs in any of its members. Similarly, the system nonencounter probability Pne can be realized only if mutual non-counter takes place in all zones, i.e., n (4.3.11) Pne = P ∩ Ai i=1 where Ai is the event of no encounter in zone i , P[ Ai ] = Pnei . 87 ∩ is the intersection or mutual occurrence of the events Ai . n is the total number of zones, including harbour. The system (overall) probability of encounter Pe is simply given by: Pe = 1 − Pne If Ai are assumed statistically independent, then: n n i =1 i =1 (4.3.12) Pne = ∏ P( Ai ) = ∏ Pnei (4.3.13) and n Pe = 1 − ∏ Pnei (4.3.14) i =1 On the other hand, if Ai are assumed perfectly correlated, then: Pe = max[ 1 − P( Ai )] = 1 − min P ( Ai ) = 1 − min Pnei i (4.3.15) Thus, the bounds on the system encounter probability Pe are: n 1 − min ( Pnei ) ≤ Pe ≤ 1 − C Pnei i (4.3.16) i =1 These bounds are tight if the non-encounter probability in any of the zones is dominated. If the members of a series system are equally corrected, then an extension of the work by Stuart, summarized in Structural Reliability and Its Application (Thoft-Christensen and Baker, 1982) leads to the following system probability of encounter: ∞ n β + ρt Pe = 1 − ∫ ∏ Φ i (4.3.17) ϕ (t )dt 1 − ρ −∞ i =1 where β i cab be calculated from: β i = −Φ −1 ( Pei ) = −Φ −1 (1 − Pnei ) (4.3.18) Φ and ϕ is the standard Gaussian cumulative distribution and density function, respectively. ρ is the correlation coefficient. 4.3.2 Wave loads and load combinations Estimating wave- induced loads is one of the most important tasks in ship design. Principles of Naval Architecture (Lewis, Ed., 1988-89) suggests that there are four methods by which wave- induced loads can be determined: a) approximate methods b) strain and/ or pressure measurements of full scale ships c) laboratory measures of loads on methods d) direct computation of wave induced fluid loads. The above methods have their own advantages and limitations, respectively. For marine design and marine structural reliability analysis, a simple and relatively accurate method is more important. To calculate wave loads, the Second Order Strip Theory (SOST) is usually used. The following section will describe a simple formulation for determining slightly nonlinear extreme wave loads and load combinations. Then non- linear hogging and sagging 88 bending moments are given. The extreme wave-induced bending moment, M w , is calculated using Second-order Strip Theory. The following will give a simple format of load combination. A simple format was adopted for the load combination: f c = f 1 + Kf2 f1 > f 2 (4.3.19) where f 1 , f 2 are the individual extreme loads and K is a load combination factor defined by: m K = r mc (1 + r 2 + 2 ρ12 r ) 0.5 − 1 (4.3.20) r where ρ12 is the correlation coefficient between the two load components. σ r = 1 is the ratio of the standard deviations of the loads. σ2 [ mr = ln v01T ln v 02T mc = ln v0 cT ln v02T ] v 0i is ratio of zero up-crossing of the load processes, i = 1,2, c . σ i is standard deviation of the loads, i = 1,2 A formulation for calculating the correlation coefficient ρ12 is given in Mansour, 1994. In linear theory, the most probable extreme value of a load peak depends only on the first two moments of the underlying process probability distribution. The mean can be set to zero, without loss of generality, therefore the most probable extreme value depends only on the standard deviation of the process. For ocean loads with Rayleigh distribution peaks, the most probable extreme value is given by f i = σ i 2 ln v0 iT (4.3.21) By introducing a non- linearity parameter δ (see Mansour and Jensen, 1995), equation (4.3.21) becomes: f i = δ iσ i 2 ln v0 iT (4.3.22) where the non-linearity parameter is defined by: α i ( 2 ln v 0iT − 1) γ δ i = k i 1 + + i ( 2 ln v0iT − 3) (4.3.23) (5.8 + 2γ i ) 2 ln v0i T 30 γ i = 1 + 1 .5( β i − 3) − 1 (4.3.24) 2 α i γ i2 + k i = 1 + 0.5 (4.3.25) γ + 3 54 i The difference between sagging and hogging moments manifests itself in the sign of the skewness α , i.e., α is positive for sagging and has the same value but with a negative sign for hogging. An extreme value of a load fη associated with exceedence probability η can be determined by: 89 f η = δσ 2 ln v 0ηT (4.3.26) where v0 (4.3.27) ln( 1 − η) −1 The most probable extreme value is associated with an exceedence probability η = 0.6321. In order to estimate the most probable extreme value of a slightly non-linear load (see equation (4.3.22)), the non-linearity parameter δ needs to be evaluated first. Evaluating δ means determination of the skewness α and kurtosis β . Both of these moments are shown (Mansour and Jensen, 1995) to depend on the significant wave height H S , the zero up-crossing period of waves Tz , ship geometry, ship speed V , and heading angle φ , i.e., α = α ( H S , Tz , shipgeomet ry ,V , φ ) (4.3.28) β = β ( H S , Tz , shipgeomet ry ,V , φ ) (4.3.29) v 0η = m0 m2 m0 , m2 are the sea spectral moments. Furthermore, the skewness and kurtosis have the following relations with H S : α = f (T z , shipgeomet ry, V ,φ ) (4.3.30) Hs β −3 = g (T z , shipgeomet ry, V , φ ) (4.3.31) H s2 The units used for the significant wave he ight H S are metres. The zero up-crossing period, Tz , is expressed in seconds. Charts of the skewness and kurtosis as a function of each parameter are shown in Figures 2.1.11 to 2.1.36 of the literature SSC-398. H S and Tz are not independent and the relation between H S and Tz is given by: where T z = 2π Hs (4.3.32) g The charts for frequently occurring pairs of H S and Tz can be referred to Figures 2.1.38 to 2.1.41 of the literature SSC-398. T z = 11.12 The following is a summary of the nomenclature in Section 4.3: = Coefficients, polynomial series ai = Deck plan area ADK = Waterplane area AW P = Block coefficient cb = Flare coefficient cf ci E[.] = Coefficient, Hermite series = Expected value 90 fi fc fη H i (ω ) H i* (ω ) Hs k K K L mr , mc m n ,i M (t ) M 0 (t ) Ni r Re(.) S x (ω ), S c (ω ) T Tz U (t ) V zf α β γ δ ε σi µ ν 0i ρ ij σ σc σl ϕ ω = Characteristic value of response (stress or deflection) to load component i = Combined response (stress or deflection) = Extreme value associated with exceedence probability η = Frequency response function for load component i (transfer function) = Conjugate complex of H i (ω ) = Significant wave height = Scaling factor = Load combination factor for two correlated load response = Cumulant = Ship length = Ratio referred to equation (4.3.20) = nth spectral moment of component i response = Bending moment process = Normalized bending moment process = Number of peaks associated with load component i σ = Stress ratio 2 σ 1 = Real part of a complex function = Wave and combined response spectra, respectively = Time of exposure = Zero up-crossing period of waves = Standard Gaussian process = Ship speed = Vertical distance between ADK and AW P = skewness = Kurtosis = Refer to equation (4.3.24) = Non-linearity parameter = Band width parameter = Standard deviation = Mean value = Rate of zero up-crossing of load process i = Correlation coefficient between to response components i and j = Non-linear standard deviation = Standard deviation of the combined response = Linear standard deviation = Ship heading angle = Frequency 91 4.3.3 Slamming loads Slamming loads are significant in many types of ocean-going vessels, e.g. those with fine form, low draft, and high speed. The calculation of slam effects (stresses) requires the consideration of hull flexibility. The maximum slam loads typically do not occur when the wave- induced loads are the largest, and such lack of perfect correlation needs to be considered in the calculation of combined load effects. Another characteristic is the marked non- linearity of slam loads with respect to the wave height, resulting in the hull girder response being significantly different for the hogging and sagging parts of the wave cycle. The treatment of slamming in ships is often semi-empirical, relying on insights gained from in-service data and measurements, and includes large uncertainties related to the methods themselves, effect of operational factors and load combinations. The calculations of slamming loads are based on a time-domain approach. The following is a simplified approach to combined slam- and wave-induced loads. In the case of slamming- and wave- induced stresses, the combined stress extreme value f c for a seastate, heading and speed, is given on the basis of the SRSS rule (SSC398, page2-69), by: fc = f w2 + f sl2 where f w and f sl are the individual (wave and slam related) extreme stresses, with the two processes assumed uncorrelated (in terms of frequency, not intensity) because of their typical frequency separation. 4.4 Probabilistic methods The ship design process requires knowledge of extreme lifetime loads. The loads can be obtained from a variety of processes. The non- linear predictions combined with the extreme of random process theory and long-term statistics are the preferable methods. The fully non- linear simulations, though available, are still in development / improvement, verificatio n and validation stages. The partially non- linear and linear codes are still the most popular tools for loads calculations. The 14th ISSC (2000c) VI.1 Extreme Hull Girder Loading Special Task Committee presents very comprehensive overview of approaches for estimate of short-term extreme values and prediction of lifetime design loads. Based on the complexity of the hydrodynamic wave process and length of available time series, different statistical methods can be applied to estimate short- and long-term extremes. In the linear cases, the wave loads in random sea can be calculated from linear potential theory in frequency domain. The loads process follows Gaussian distribution and its peaks are Rayleigh distribution. The Hermite transformation of a standard Gaussian process can be employed to model the slightly non- linear problem. The extreme values can be calculated using formulas similar to linear Rayleigh distribution, which includes parameter-representing non- linearity of the process (skewness and kurtosis). For strong non- linear process, wave- induced loads must be calculated using time-domain methods and statistical analyses are conducted on actual load peaks obtained from load~time histories. Sagging and hogging moment peaks should be treated separately. The threeparameter Weibull distribution is the most successful in modelling the wave- induced 92 load statistics (Wang, 2001). The pdf of the three-parameter Weibull distribution is given by: f ( x ) = cλc ( x − δ )c −1 e − [λ ( x −δ )] Where x=0, c is the shape parameter, ? is scale parameter, and d is location parameter. The advantages of the Weibull model for distribution of vertical bending moment are summarized as follows: The calculation of distribution parameters is relatively simple, It is sufficient for representing non- linearities in sagging and hogging moments for various types of ships and operating conditions, It gives good distribution of peak values around the tail area for reliable estimate of extremes, and It allows for estimate of non- linearities by straightforward comparison to Raleigh distribution. Tawn and Heffernan (2001) analyzed statistical methods for short-term distribution of impact pressure loads from seakeeping model experiments with bulk carriers. The analysis involved vertical pressure loads inserted on the top of the most forward cargo hold during operation in various environments. Two statistical models were considered: the Weibull and generalized Pareto distributions. The quality of fit of both distributions was assessed using P-P (fitted probability vs. sample probability) and Q-Q (fitted quantile vs. sample quantile) diagnostic plots assuming threshold pressure of 5 kPa. In the case of this experiment, the P-P plot shows slightly better fit of Pareto distribution, and Q-Q plot demonstrate superior fit of Pareto distribution. Specifically, the Weibull model overestimates probability of large impact occurrence, which is crucial when extrapolating to obtain desired extremes. The generalized Pareto distribution fit the pressure impact data better that the Weibull distribution for all examined experimental cases. Lu at el. (2002) examines four statistical models for prediction of the most probable extreme value of jackup structure responses due to excitation of random wave motions. The methods are compared in terms of calculated extremes and their sensitivity to structural stiffness, simulation time step and random seed techniques. The analyses were conducted for two typical jackup structures. The extreme value is defined as one with exceedence probability of 10-3 , which reflects a storm of 3 hours persistence. The four applied methods are: drag/inertia parameter method, three parameter Weibull distribution, asymptotic Gumbel distribution and Hermit transformation. In the discussion of results the authors indicate that from the accuracy point, the three last methods provide reasonably close predictions. The Gumbel distribution is described as theoretically most accurate, if enough simulations are generated, and the Hermit transformation as the most robust and efficient. Results of sensitivity analysis are not conclusive and authors indicate need for more research in those areas. c 93 In order to estimate the lifetime extreme value of ship response, a long-term analysis has to be performed. This approach requires that the long-term peak distribution of maxima is obtained as a weighted sum of short term probabilities of exceedence in all possible combinations of mean wave periods, significant wave heights, heading angles and speed. When the linear process is considered the short-term responses can be calculated using the superposition method. To obtain non- linear responses, timedomain simulation or experimental data are required. This leads to an expensive, time consuming and unpractical process. Sagli (2000), Videiro and Moan (1999), and Baarholm and Moan (2001) present the contour line approach and demonstrate that the long-term load extremes for marine structures can be estimated in efficient manner by considering only a few short-term sea states, instead of determining responses for all sea states. The initial step in analysis is to calculate coefficients of contribution CR(si) for linear responses. CR ( si ) = Q R ( R > rD | si , β0 , u0 ) f H S ,TP ( hsi , t si ) wsi∆hi ∆t i QLT ( rD ) where: QR(R>rD|si,ß0 ,u0 ) is the short term cumulative probability distribution, fHs,Tp(his,tsi) is long-term joint probability distribution, wsi is weight function, ? hi, ? ti are grid size, QLT (rD) is long-term probability of exceedence. The values of those coefficients are used, to establish the sea state with the maximum value of CR(si), which is then used in an iterative process of more complicated nonlinear analysis. The sea state with the maximum CR(si) and all sea states with significant coefficient of contribution are established based on non- linear runs. The desired extreme design value can then be calculated. Guedes Soares and Dogliani (2000) investigated the effects, during a double bottom tanker voyage, of loading condition changes on distribution of still-water bending moment at midships. Significant differences with respect to still water loads between departure and arrival conditions were observed, even if the differences in the displacement and mean draft were small. Assuming that the departure and arrival loads can be described by a normal distribution and that the change between departure and arrival is linear, the authors conclude that the still water loads at a random point in time can be obtained from a Gaussian distribution. The mean and standard deviation of that distribution can be calculated as the average of mean and standard deviation at departure and arrival. Wu and Hermundstad (2002) propose a new approach to estimate long-term probability of exceedence of non- linear sagging and hogging moments at midships for the S-175 containership. They applied time-domain simulation code to calculate short-term non94 linear wave-induced loads. The generalized gamma distribution was used to fit shortterm peak distribution and a simplified long-term procedure was used to obtain probability of exceedence of sagging and hogging values given by DNV rules. The applied generalized gamma probability density distribution has the form: c cr cr −1 − ( µy ) c u y e ,0 < y < ∞ Γ( r ) ? (r) is gamma function, and c, r and µ are distribution parameters. The first step is to select an appropriate wave scatter diagram. In the traditional method, the initial number of significant wave height and modal period combinations could be large and require a number of time-consuming non-linear simulations. From the practical point of view, it is desirable to conduct as few simulations as possible. So, the first step of the new process includes simulations for head seas and the most probable mean HS and TP values for each sea state. This process allows for identification of the most critical sea states and then additional runs for variation of headings and TP for the critical sea states. The long-term probability of exceedence for the non- linear process is determined from: c ( µy1 ) c( r −1) e − ( µy1 ) n P( y > y1 ) = ∑ ∑ ∑ P( β ) P( H S , T1 ) Γ( r ) β HS T1 g ( y) = where P(ß) is probability of heading P(H S, T1 ) is joined probability of sea state characterized by significant wave height and mean wave period. Seven sea states were initially considered, and final calculations were conducted for additional seven different wave headings and two modal periods. 4.5 Conclusions In this chapter, loads on marine structures are introduced. For marine structural reliability analysis, the following procedure should be taken for short-time term conditions: 1) From ship route (if known), ocean wave statistics, and a specified encounter probability (or return period) determine the design storm condition. 2) Calculate the rms value of the wave bending moment in the design sea condition using either second order strip theory, or towing tank experiment. Calculate also the stillwater bending moment. 3) Estimate the strength parameters for each failure mode. 4) Calculate the probability of failure or the safety index for each failure mode. For the long-term procedure, reliability analysis entails the determination of the probability distribution of the maximum load during the lifetime of a ship, taking into consideration the wave statistics along the ship route, loading conditions, speed, and heading. The procedure is particularly important for fatigue reliability analysis, where the entire history of loading should be determined. In that fatigue case, the long-term distribution, instead of the maximum load distribution, is required and is usually assumed to be Weibull. The procedure proposed for determination of the lifetime maximum load distribution is as follows: 95 1) Define the mission profile of the ship. 2) From the ship route and available wave statistics, obtain the frequency of occurrence of different sea conditions the ship will encounter in each of the geographic areas (zones). 3) On the basis of the above, determine the frequency of encountering different sea conditions, loading conditions, speeds, and headings. 4) Determine the wave loads in each sea condition, loading condition, speed, and heading using first- or second-order strip theory. 5) Use an extrapolation procedure to determine the distribution of the maximum load in a lifetime. To calculate wave loads, the Second Order Strip Theory (SOST) is usually used. This chapter describes a simple formulation for determining slightly non- linear extreme wave loads and load combinations. Then non-linear hogging and sagging bending moments are given. The extreme wave- induced bending moment is calculated using Second-order Strip Theory. 96