Evaluating the safety of floating structure under the design sea condition

d under its design sea conditions, using environmental statistical data and spectral theory of ship hydrodynamic in irregular waves. The long-term distribution of wave bending moment is approximated to the Weibull distribution based on the results of short-term analyses. For analyzing the longitudinal strength, the calculation is taken for different wave propagation directions and in different sea states corresponding to the wave statistical data. As a result, the study will give a conclusion about the safety of the structure in terms of longitudinal strength. This paper also introduces a reliability based approach for accessing structure’s strength to predict the working safety of floating structures under the real sea conditions. Keywords: shear forces, bending moments, floating structure, longitudinal strength, spectral theory, reliability based approach. 1. Introduction During the lifetime of ships and floating structures, besides loads from structure’s weight, cargo, etc the structures have to work under loads induced by surrounding environment, for example sea waves, wind and current. Wave induced motions and loads on structures are the common topics that are being highly concerned in recent years. In the assessment process of longitudinal strength, the most important load is vertical wave bending moment amidships that depends on many factors, for example the loading conditions, the weight distribution, the angle of incoming waves, etc. These factors are mostly random and must be taken into account in calculations. Because of the randomness, many studies often based on probabilistic theory and spectral theory using wave statistic data of the navigating area. So, the researched structures are then evaluated the safety factor by calculating probability of exceeding extreme values. A.P. Teixeira & C. Guedes Soares [5] presented the reliability based approach to determine the design loads for the remaining lifetime of ships. In their study, the probability distribution of the wave induced loads was obtained by weighting the conditional Rayleigh distribution according to the probability of occurrence of the various sea states in the ship route, such as significant wave high Hs, zero up-crossing frequency T0, the ship heading θ, ship speed v and loading condition c, etc. The exceedance probability of the vertical bending moment (VBM) was approximated to the Weibull distribution. Using these methods, authors evaluated the longitudinal strength of the studied ship in the period of 20 years, predicted the remaining time of the ship. This calculation method was also applied by C. Guedes Soares in the report for “Probabilistic Models for Load Effects in Ship Structures” [6]. In addition, to ensure the safety during working time, the strength of ships are often evaluated through either ships ultimate strength performance or fatigue and fracture analyses. For example, J.K. Paik et al [9], in 2009, used ALPS/ULSAP code for ultimate strength calculations of stiffened plate structures and ALPS/HULL code for progressive hull collapse analysis. The structure in their study is a Suezmax tanker. Z. Shu & T. Moan [10] also presented a study for the “assessment of the hull girder ultimate strength of a bulk carrier using nonlinear finite element analysis”. Regarding the fatigue and fracture analyses, ship structures are often studied in more details, at specific positions which are predicted to occur the fatigue or fracture damages, such as window and door corners of ship structure studied by Mika Bọckstrửm & Seppo Kivimaa [11], fillet welds at doubler plates and lap joints studied by O. Feltz & W. Fricke [12], hatch corners studied by Hubertus von Selle et al [13] and hatch cover bearing pad by Kukkanen T. and Mikkola T. P. J. [15]. 194 This paper presents an analysis of the longitudinal strength of the multi-purpose floating structure built by Quang Trung Mechanical Enterprise. The structure has the main task as a transhipment terminal of containers for container ships in Vietnam, and as a floating dock for building new ship and ship repairing. Because the structure is newly designed with the dimensions exceeding the current upper limit values of Vietnam Register (VR), all the analyses of the structure safety are strictly considered, especially long-term analysis of longitudinal strength. The structure is designed to work along the North coast of Vietnam, between Hon Dau and Hon Ngu islands with the sea data are shown in the section 3 later. The Response Amplitude Operator (RAO) of shear forces and bending moments (SF/BM) of the structure will be calculated, combined with wave spectra data to get the output spectra of SF/BM. From these calculations, the life time of the researched structure will be predicted. 2. Theory background 2.1 Wave load on ships and offshore structures During the working time, there are a number of forces impacting on the structures. Generally, these forces include static loads, low-frequency dynamic loads and high-frequency dynamic loads. Static loads are influenced by weights of ship and her contents, static buoyancy of the ship at rest or moving, thermal loads resulting from nonlinear temperature gradients within the hull, etc. Low – frequency dynamic loads include following components: wave-induced hull pressure variations, hull pressure variations caused by oscillatory ship motions, inertial reactions resulting from the acceleration of the mass of the ship and its contents. High-Frequency dynamic loads are generally generated by propulsive devices on the hull or appendages, reciprocating or unbalanced rotating machinery, interaction of appendages with the flow past the ship, short waves induced loads and termed springing. In fact, gathering all aforementioned loads in one study requires much effort and time. Kukkanen T et al [16] gave a summary report of “Nonlinear wave loads of ships”, in which the these wave loads were detailed by using their own numerical calculations and model test results. Generally, depending on the purpose of particular research, one or several loads are often neglected and the calculations will be easier and faster. Similarly, this paper will focus on the first type of the aforementioned loads: static loads and low – frequency dynamic loads. The low-frequency dynamic loads, loads on ship when neglect dynamic stress amplification are called wave-induced loads. The calculation of these loads requires a previous determination of ship motions induced by waves. This is based on the assumptions of linear theory which both waves and ship motion amplitudes are small. In addition, the viscous forces are considered as a relatively unimportant forces in vertical loads calculations. Thus, the external hydrodynamic force and moment with respect to the neutral axis of a ship are [1]: ⃗( , ) = (− ⃗) (1) ⃗ ( , ) = ⃗ − ⃗ (− ⃗) where w is the wave frequency, Sx is the wetted surface partition from stern to the cross-section, p is the summation of the hydrostatic and total hydrodynamic pressures, vector n is the normal vector of the wetted surface pointing towards the fluid field and x0 is the location of considered intersection point X0 on the neutral axis. 195 Figure 1 Bending Moment, Shear Force and Neutral axis The gravitational force and moment with respect to the intersection point X0 are [1]: ⃗( , ) = 0,0, − − ⃗ (0,0, − ) (2) ⃗( , ) = ( ⃗ − ⃗) 0,0, − − ⃗ (0,0, − ) The inertial force and moment with respect to the intersection point X0 are [1]: ⃗( , ) = ( , , ) (3) ⃗( , ) = ⃗ − ⃗ , , + ( , , )[ ] where is the motion response at the centre of mass of the j-th section, is the moment of inertia of the j-th section. The summations of all load components in equation (1) and (2) are the total shear force( , and, bending) moment on ships. The maximum value of shear force and[ bending] moment RAO among all of the calculated wave frequency points at a particular section is called as SF/BM RAO at that section. 2.2 Short-term analysis for longitudinal structure’s strength The short term analysis is based on the spectral analysis approach developed by Rice (1944) and Wiener-Khintchine theorem that allows us to switch from the time domain to frequency and probability domains. Because of stochastic representation, ocean waves are considered to be a Gaussian random process (Rudnick, 1951) so that the wave ordinate follows the normal Gaussian distribution and the wave amplitude follows Rayleigh distribution. Using seakeeping program, we can obtain the Response Amplitude Operators (RAO) of the structure motion parameters and forces. Thus, the spectra of output response is evaluated by [2]: (4) where is the structure’s response( ) = [spectra,( )] . (is wave) spectra, and RAO ( ) is the response amplitude operators corresponding to the output data that we need for analysis. Subsequently, the spectra of( the) shear forces and bending moments (SF/BM)( ) on structures will be calculated from wave spectra following equation: (5) / / where is the spectrum( ) of the= shear( forces) or bending. ( )moments, is the RAO of shear forces, bending moments, respectively. ( ) / ( ) / 196 Generally, to study the motions and loads on floating structures or ships, the wave frequency is often considered in the range from 0.2 rad/s to 2.5 rad/s. Regarding the sea spectra, we can describe the sea state as a stationary random process. This means that we can observe the sea at a particular position within a limited time period, from 0.5 to 3 hours. This is the short-term description of the sea. Two commonly recommended wave spectra are JONSWAP and Pierson-Moskowitz. The JONSWAP spectrum is recommended by 17th ITTC for limited fetch [3]: ( ) = 155 exp (3.3) ( ) where (6) . . . = exp − √ . and . = 0.07 ≤ . = 0.09 > T1 is the mean wave period defined as: where = 2 / (7) H1/3 is defined as: = ∫ ( ) (8) The Pierson-Moskowitz spectrum is a special/ = 4 case for fully developed long crested sea. The spectral ordinate at a frequency (in rad/s) is [3]: (9) ( ) 0.11 = −0.44 2 2 2 where / (10) T=1 =2 1.086.( )T2 T0 = 1.408.T2 Equation (6) satisfied equation (8) is only true for a narrow-banded spectrum and when the instantaneous value of the wave elevation is Gaussian distributed. Following IACS Recommendation No.34, [4] with the assumption that the process is narrow banded, amplitudes of the vertical wave bending moment (MW) in short-term sea state follows Rayleigh distribution. Thus, the probability function for the maxima (peak values) MW can be obtained following equation: 197 , = − (11) Where process variance is calculated as area below response spectrum: = ∫ | , (12) Where Psh is the probability that wave induced bending moment on ship exceeding the given peak value of the bending moment MW; SR is the spectral of response. 2.3 Long-term analysis for longitudinal structure’s strength Long-term probabilities of the vertical wave induced bending moment exceeding given values are calculated by combining the short-term probabilities with the probabilities of sea state and other factors such as ship headings, ships speeds and loading conditions. Long-term distribution is given in [5]: ∆ , ( ) = ∑ ∑ , , , , , , , . , , . ( , , ) (13) where , , is the relative number of response cycles in each short-term sea state, p(H1/3,T2) is the probability of occurrence of sea state. In long-term analysis, the probability distribution of VBM exceeding the given value MW follows the Weibull distribution FVBM(MW) [6]: ( ) = 1 − − (14) Subsequently, probability of the VBM exceeding the given value Mw is calculated as following function: ( ) = − (15) Where k and w are parameters calculated from the fitting of ( ) to 1 − ( ). 3. Long-term safety evaluation of the multi-purpose floating structure 3.1 Parameters of the structure To evaluate long-term safety of the investigated floating structure, the details of the structure parameters as well as structure’s working environment must be provided. Table 1 shows the summarization of the structure’s parameters. Because the structure is newly designed, then all load conditions data during the life time of the structure are still unknown. So, in this study, the load conditions of the structure are supposed to include three main cases: Full load, Ballast load and Partial load, with the corresponding time consuming proportions are 0.4, 0.4 and 0.2 (of the structure’s working time), respectively. 198 Table 1 Main parameters of the structure Main parameters Lpp [m] 171.000 D [m] 12.000 B [m] 25.000 CB [m] 0.951 T [m] 4.500 Elastic section modulus 4.476 Neutral axis [m] 4.600 amidships [m3] Parameters in particular load conditions Full load Partial load Ballast load Draft amidships [m] 4.500 3.830 2.830 xG [m] -0.804 -0.404 1.261 yG [m] 0.000 0.000 0.000 zG [m] 5.621 4.565 3.949 Displacement [kg] 19173605 16102481 11711206 3.2 Working area data The structure is designed to work along the North coast of Vietnam, between Hon Dau and Hon Ngu islands shown on Figure 2. Figure 2 The design working area of the structure In 2014, Supott Thammasittirong (AIT), Sutat Weesakul (AIT), Ali Dastgheib (UNESCO-IHE) and Roshanka Ranasinghe (UNESCO-IHE) [14] presented a report of their study on “Climate Change Driven Variations in the Wave Climate along the Coast of Vietnam”. This report included statistical wave data of sea area along the North coast of Vietnam, between Hon Dau and Hon Ngu islands. The report also shown that during the period of time from 1981 to 2000, the mean wave height at the above sea area ranged from 0.4 m to 1.0 m; the mean wave period ranged from 4.0 s to 6.5 s and the main wave direction was South-East. These data are well fitted with the data from the National Centre for Hydro - Meteorological Forecasting [7] which are given on the Table 2 below. Table 2 gives basic data for the short-term calculations for the structure strength. With the results gained from the short-term calculations, strength of structure will be evaluated in given periods of time, such as 1 years, 5 years, 10 years, and 20 years. 199 Table 2 Annual statistical data of the design sea area SUM 559 315 101 20 4 1 0 0 1000 >9.5 -- -- -- -- -- -- -- -- 0 8.5 -- -- -- -- -- -- -- -- 0 7.5 -- -- -- -- -- -- -- -- 0 6.5 -- -- -- -- -- -- -- -- 0 5.5 -- -- -- -- -- -- -- -- 0 4.5 -- -- -- -- -- -- -- -- 0 3.5 -- 1 1 -- -- -- -- -- 2 2.5 6 11 5 2 1 -- -- -- 25 1.5 73 85 38 9 2 -- -- -- 207 Significant Wave Height [m]: Hi [m]: Height Wave Significant 0.5 480 218 57 9 1 1 -- -- 766 T0 [s] 3.5 4.5 5.5 6.5 7.5 8.5 9.5 >10.5 SUM T0- mean [s] 4.10 3.3 Results and discussion The structure is supposed to work in its design sea wave environment with different wave frequencies and propagation directions. These frequencies range from 0.2 rad/s to 2.5 rad/s. The wave propagation angles range from -180o to +180o ( = 30o) with the corresponding probability of each is 1/12, [3]. We also suppose that the weight distribution in each load condition is fixed, the liquid’s sloshing in tanks are neglected, and the structure will generally have 1 month docking for small renovation each year, 3 ữ6 months docking for big renovation every 5 years of working. Figure 3 shows examples of the calculated bending moment RAO at mid-section of the structure in different wave propagation directions. 200 Figure 3 RAO of BM in the full load condition at different wave propagation directions The results of bending moment RAO calculations can be combined with wave spectra data to obtain the spectra of bending moment following equation (5). The values of are calculated following equation (12), and the short-term probability (Psh) of bending moment on the structure exceeding the given values are calculated following equation (11), depending on sea state probability, wave propagation directions, wave frequencies. For example, Table 3 shows a result of the full load condition calculation. It shows the probabilities of which the VBM exceed the permissible bending moment of the structure amidships (M0) in the full load condition. Here, the value of the moment M0 is determined by the formula following IACS Recommendation 1989/Rev.6 [8] ≤ [ ] [MN/m2] (16) where [ ] is the permissible stress of the structure steel, and [ ] = 175 [MN/m2] for ordinary hull structure steel. is the structure’s elastic section modulus, and = 4.476 [m3] as shown in the Table 1. From the condition shown in the formula (16), we can obtained the limited value of bending moment of the structure as follows = . [ ] = 787.494 [MN.m] (17) Table 3 Corresponding Probability of BM exceeding the limited value M0 in full load condition SUM 8.5E-175 8.45E-21 3.63E-14 5.16E-22 1.2E-23 0 3.63E-14 >9.5 -- -- -- -- -- -- -- -- 0 8.5 -- -- -- -- -- -- -- -- 0 7.5 -- -- -- -- -- -- -- -- 0 6.5 -- -- -- -- -- -- -- -- 0 5.5 -- -- -- -- -- -- -- -- 0 4.5 -- -- -- -- -- -- -- -- 0 3.5 -- 8.45E-21 3.63E-14 -- -- -- -- -- 3.63E-14 2.5 8.5E-175 2.45E-35 1.12E-22 5.16E-22 1.2E-23 -- -- -- 6.4E-22 1.5 0 5.4E-91 3.15E-55 6.64E-53 1.41E-57 -- -- -- 6.68E-53 Significant Wave Height [m]: Hi [m]: Height Wave Significant 0.5 0 0 0 0 0 0 -- -- 0 T0 [s] 3.5 4.5 5.5 6.5 7.5 8.5 9.5 >10.5 201 It is clearly seen that, in the full load case, the total probability of which the VBM amidships exceed the permissible bending moment of the structure M0 is 3.63E-14. By assuming a range of exceeded VBM MW, we can calculate the corresponding probabilities shown in the Table 4 for all load conditions. Table 4 Short-term Probabilities of the bending moments exceeding the given values MW Probability of exceeding Mw Probability of exceeding Mw MW MW Full Load Partial Ballast Full Load Partial Ballast (kN.m) Psh Psh Psh (kN.m) Psh Psh Psh 10 4.07E-01 3.44E-01 3.39E-01 145 1.43E-03 1.15E-03 1.11E-03 25 1.38E-01 1.16E-01 1.12E-01 160 8.81E-04 7.15E-04 6.86E-04 40 6.77E-02 5.46E-02 5.25E-02 175 5.48E-04 4.48E-04 4.26E-04 55 3.79E-02 2.96E-02 2.90E-02 190 3.43E-04 2.83E-04 2.66E-04 70 2.14E-02 1.66E-02 1.64E-02 300 1.07E-05 9.90E-06 8.63E-06 85 1.20E-02 9.36E-03 9.27E-03 400 5.13E-07 4.79E-07 4.28E-07 100 6.79E-03 5.37E-03 5.29E-03 500 1.95E-08 1.59E-08 1.46E-08 115 3.94E-03 3.15E-03 3.08E-03 600 3.85E-10 2.80E-10 2.54E-10 130 2.34E-03 1.89E-03 1.83E-03 787.494 3.63E-14 2.08E-14 1.78E-14 where Psh = , is the short-term probability of VBM exceeding the given value MW in each load condition. For example, when MW = 115 [kN.m], the total probability of VBM exceeding this MW in the Full load, Partial load and Ballast load conditions are 3.94E-03, 3.15E-03 and 3.08E-03, respectively. By fitting the values of Psh in each load condition with the values of P(MW) given in equation (15), we can find the Weibull parameters k and w shown on Table 5. Table 5 Long-term safety assessment results of the structure Load condition Full load Partial load Ballast load Weibull k 0.7924 0.7244 0.724 parameters w 11.24 9.05 8.876 SSE: 0.0003943 SSE: 0.000131 SSE: 0.0001535 R-sq: 0.9975 R-square: 0.9988 R-square: 0.9986 Goodness of fit Adjusted R-sq: 0.9973 Adjusted R-sq: 0.9988 Adjusted R-sq: 0.9985 RMSE: 0.005507 RMSE: 0.003175 RMSE: 0.003436 T (years) PT PF PP PB 1 year 1.417E-06 3.543E-07 3.543E-07 7.087E-07 5 years 2.998E-07 7.496E-08 7.496E-08 1.499E-07 10 years 1.499E-07 3.748E-08 3.748E-08 7.496E-08 20 years 7.496E-08 1.874E-08 1.874E-08 3.748E-08 P(Mw>M0) 2.55027E-13 9.1948E-12 6.71896E-12 PT(Mw>M0) 1.617E-11 Here, PT is the total probability in all load conditions (full load, partial load and ballast load conditions) at which the structure has one time within T years the BM exceeds its limit value. Similarly, PF, PP, PB are the probabilities at which the structure has one time the wave bending moment exceeds its limit value in the time period of survey in the full load, partial load and ballast load conditions, respectively. 202 PT(Mw>M0) is the totally long-term probability occurring one time BM on the structure exceeding the limited value M0. This probability is 1.617E-11, much lower than 20 years probability shown in the Table 5, which is 7.496E-08. It means that in terms of wave bending moment, the structure have a more than 20 years life time working well under this load. This result should be combined with other calculations such as longitudinal strength analysis of the structure working in hogging and sagging conditions, ultimate strength analysis and other analyses to give a final conclusion of the structure’s longitudinal strength. 4. Conclusion The paper presents a method for long-term analysis of the structure strength under wave induced BM, based on spectra analysis and probability theory. The study also analysed partly the longitudinal strength of the structure in both short-term and long-term periods and predicted the age of the structure under the working sea environment. This is the key study that helps to solve the problems and current arguments of the very large structure built by Quang Trung Mechanical Enterprise. However, there are several issues in the study that need to take into account more carefully, such as the assumptions. Regarding the input data, the sea environment data should be updated to the latest figures. The distribution of weight components along the structure is supposed to be fixed at each load condition, but in practice, this weight distribution will vary significantly, depending on different working states. Moreover, the load conditions in the study are only three main types (full load, partial load and ballast load conditions), which the corresponding time consuming proportions are supposed to follow the traditional rate of ship as 0.4, 0.4 and 0.2. The reason is the structure is newly built, leading to a lack of statistical data of structure working information. Besides, the effect of mooring system should be considered, because it has significant influence on the RAO of wave bending moment. Similarly, the probabilities of wave propagation directions need to be obtained from real sea statistics, which may not be equal between different directions as the assumption in this study. Acknowledgements This study is a part of the safety assessment process for the floating structure built by Quang Trung Mechanical Enterprise which have been received essential data from the company. References [1] Newman, J.N. (1978), 'The theory of ship motions', Advances in Applied Mechanics, Vol.18. 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