Patent Application
20120089347
1. Field of the Invention
Embodiments described herein generally relate to fatigue analysis. More particularly, embodiments described herein relate to performing fatigue analysis of a Floating Production Storage and Offloading (FPSO) unit's process and utility piping.
2. Description of the Related Art
Floating Production Storage and Offloading (FPSO) units have been designed to receive, process, and store hydrocarbons produced from nearby platforms or subsea templates. FPSO units can be constructed by converting an existing floating vessel, for example an oil tanker, or can be constructed as a vessel built specially for the application. In either case, fatigue analysis of process and utility piping located on the FPSO unit is performed as part of the design process to properly size and configure the piping and associated restraints.
To perform fatigue analysis on a FPSO unit, deflections due to the motion of the FPSO unit while at sea are estimated for each pipe restraint location. Typically, the piping is modeled as multiple beam segments, each segment supported at each end. The beam segments are configured within the model to approximate the piping restraint locations for the process and utility piping. The beam segments are excited by simulated motion that approximates the effect of the FPSO unit loading conditions. From the response of each beam segment, the associated deflections for each end of a beam segment or restraint location are estimated.
Such modeling process, however, can be time consuming and error prone. The errors are typically a result of: data entry errors, the floating vessel being designed separately from the piping and restraint locations; the floating vessel design analysis being performed at discrete points about the floating vessel; the discrete points being located at relatively large distances from one another compared to the distances between the piping restraint locations; and the floating vessel design analysis deflection results having little or no obvious relation to the deflections at the various process and utility piping restraint locations.
There is a continuing need, therefore, for improved systems and methods for estimating process and utility piping restraint location deflections, and analyzing fatigue using the estimated deflections while improving the overall analysis efficiency for the design of a FPSO unit process and utility piping.
So that the manner in which the above recited features of the present invention can be understood in detail, a more particular description of the invention, briefly summarized above, may be had by reference to embodiments, some of which are illustrated in the appended drawings. It is to be noted, however, that the appended drawings illustrate only typical embodiments of this invention and are therefore not to be considered limiting of its scope, for the invention may admit to other equally effective embodiments.
A detailed description will now be provided. Each of the appended claims defines a separate invention, which for infringement purposes is recognized as including equivalents to the various elements or limitations specified in the claims. Depending on the context, all references below to the “invention” may in some cases refer to certain specific embodiments only. In other cases it will be recognized that references to the “invention” will refer to subject matter recited in one or more, but not necessarily all, of the claims. Each of the inventions will now be described in greater detail below, including specific embodiments, versions and examples, but the inventions are not limited to these embodiments, versions or examples, which are included to enable a person with ordinary skill in the art to make and use the inventions, when the information in this patent is combined with available information and technology.
As will be described in more detail below, methods for performing a fatigue analysis for process and utility piping on a FPSO unit are provided. In one embodiment, the method can include generating one or more displacement polynomials from bending moments calculated for a floating vessel, generating deflections for process and utility piping restraint locations on the floating vessel using at least the displacement polynomials, and performing fatigue analyses for the process and utility piping using the deflections.
In one or more embodiments, the floating vessel bending moment data set can be provided as at 100 from a naval architect fatigue analysis table associated with the floating vessel. For example, bending moment tables are typically generated by a naval architect during the floating vessel design stage. The bending moment tables can contain the calculated bending moments at various points throughout the floating vessel structure. The points where bending moment or other calculations are performed are typically chosen about the hull longitudinal axis points of deflection.
In one or more embodiments, the floating vessel bending moment data set can be provided from the floating vessel's hull moment of inertia information. For example, a model can be constructed and a first estimate for bending moments can be determined. In one or more embodiments, the model can be a beam with a length equal to the floating vessel length and moments of inertia equal to that of the floating vessel. To estimate the bending moments, the beam can be constrained at a location, for example at the middle of the ship in the amidships region. The constrained beam can be excited simulated forces that approximate the forces from wave action against the floating vessel. Using known techniques, the bending moments can be estimated from the beam's response to the forces.
In one or more embodiments, plotting and curve fitting the bending moment data 110 and polynomial generation 112, 115 can be performed using the provided floating vessel bending moment data 100 and known techniques. For example, a software package with the trade name MICROSOFT EXCEL contains a graphics capability that can generate a polynomial from a plot of data. The software package with the trade name MATLAB can also be used to plot and curve fit the bending moment data set 110 and generate at least one bending moment polynomial 112. For example, for the provided floating vessel bending moment data set 100, MATLAB can be executed to plot, curve fit 110, and determine at least one least-squares fit polynomial 112 from the plot of the provided bending moment data set 100.
In one or more embodiments, to determine bending moment polynomial fit sufficiency as at 113, statistical measures of the polynomial fit can be used. Statistical measures of the polynomial fit accuracy can include the coefficient of determination (R2), the observed ƒ-value for the fit, the observed t-values/confidence intervals for the coefficients, and other known statistical techniques. The values of the statistical measures can be calculated using known techniques and the calculated statistical values can be used to determine fit sufficiency as at 113. For example, the user can decide that an appropriate measure for bending moment polynomial fit sufficiency can be a calculated coefficient of determination equal to or greater than 0.98. The user can calculate the coefficient of determination using known techniques. If the calculated coefficient of determination is not greater than or equal to 0.98, the user can generate a higher order polynomial and/or one or more additional overlapping bending moment polynomials as at 115 until a combination of portions of the one or more bending moment polynomials sufficiently fit the plot of the floating vessel bending moment data set as at 116.
In one or more embodiments, each bending moment polynomial can be a first order, second order, third order, forth order, fifth order, sixth order, seventh order, eighth order, ninth order, tenth order or higher order polynomial. In one or more embodiments, the order of each bending moment polynomial can be selected to achieve a sufficient polynomial fit to at least a portion of the plot of the bending moment data set as at 116. For example, a fourth order bending moment polynomial can be generated and the bending moment polynomial fit sufficiency can be determined using one or more embodiments described herein. If the polynomial fit is not sufficient, a higher order bending moment polynomial can be generated and the fit sufficiency of the higher order bending moment polynomial can be determined as at 113.
In one or more embodiments, if the plot of the bending moment polynomial does not sufficiently fit the plot of the bending moment data set as at 114, in lieu of or in conjunction with raising the polynomial order, one or more overlapping bending moment polynomials can be generated such that a plot of a combination of at least a portion of two or more overlapping bending moment polynomials can represent the plot of the entire bending moment data set. An overlapping bending moment polynomial is defined as a polynomial whose plot overlaps with at least a portion of the plot of at least one other bending moment polynomial. For example, a plot of a combination of a portion of more than one polynomial may be required to properly fit the plot of the entire bending moment data set. For the bending moment data set, the data in the vicinity of the bow, the data in the vicinity of the amidships region, and the data in the vicinity of the aft of the floating vessel can be separately curve fit to produce three distinct bending moment polynomials. In one or more embodiments, at least a portion of the plot of each of the three distinct bending moment polynomials can be combined to describe the entire plot of the bending moment data set. The plot of at least one of the distinct bending moment polynomials can overlap a least a portion of the plot of at least one other distinct bending moment polynomial. The overlapping section can correspond to a solution set over which both polynomials return the same values.
In one or more embodiments, the plot of the bending moment data in the vicinity of an aft region of the floating vessel can include the bending moments associated with the region between the aft of the floating vessel and about 50 meters from the aft of the floating vessel as measured from the aft of the floating vessel along the floating vessel beam toward the bow. The bending moment data in the vicinity of the aft of the floating vessel can include the bending moments associate with the region between the aft of the floating vessel and about 10 meters, about 20 meters, about 30 meters, about 40 meters, about 50 meters, about 60 meters or more from the aft of the floating vessel. The bending moment data in the vicinity of the aft of the floating vessel can include the bending moments associated with the region between the aft of the floating vessel and about 5%, about 10%, about 15%, about 20%, about 25%, about 30%, and about 35% or more of the distance along the full length of the floating vessel beam as measured from the aft of the floating vessel.
In one or more embodiments, the plot of the bending moment data in the vicinity of a bow region of the floating vessel can include the bending moments associated with the region between about 275 meters from the aft of the floating vessel and the bow of the floating vessel. The bending moment data in the vicinity of the bow of the floating vessel can include the bending moments associate with the region between about 250 meters, about 260 meters, about 270 meters, about 280 meters, about 290 meters, and about 300 meters or more from the aft of the floating vessel and the bow of the floating vessel. The bending moment data in the vicinity of the bow of the floating vessel can include the bending moments associate with the region between about 50%, about 60%, about 70% about 80%, and about 90% or more of the distance along the full length of the floating vessel beam as measured from the aft of the floating vessel and the bow of the floating vessel.
The plot of the bending moment data in the vicinity of an amidships region can include the bending moments associated with the region amongst and between the vicinity of the aft of the floating vessel and the vicinity of the bow of the floating vessel. In one or more embodiments, the resultant one or more sufficiently fit bending moment polynomials can be provided for slope and displacement polynomial generation as at 117.
In one or more embodiments, the one or more bending moment polynomials can be integrated using known techniques to generate one or more slope polynomials as at 220. Each of the one or more slope polynomials will contain at least one unknown integration constant as is understood in the mathematical arts. To solve for the one or more unknown integration constants, slope boundary conditions can be chosen and the integration constants determined as at 225. The one or more slope polynomial integration constants can be solved for any specified slope. For example, the one or more slope polynomial integration constants can be solved where the naval architect's fatigue analysis table indicates a location about the floating vessel where the slope is equal to zero.
The solved slope polynomials can be integrated 230 to generate one or more displacement polynomials as at 230. The displacement polynomial integration constants can be solved by selecting the appropriate displacement boundary conditions as at 235. The one or more slope polynomials and the one or more displacement polynomials can be provided for further analyses as at 237.
In one or more embodiments, the plot of two or more slope polynomials can overlap to create one or more slope polynomial overlapping sections as at 344. A point along each slope polynomial overlapping section corresponding to a particular x,y,z location on the floating vessel can be selected as the location on the floating vessel where slope polynomial calculations transition from one slope polynomial to another during slope calculations as at 345. For example, if each slope polynomial contains only one variable “x”, each slope polynomial can be solved for varying “x” values. In the solution set where the values for each overlapping slope polynomial gives the same answer for a subset of “x”, an arbitrary value “a” from the subset of “x” can be chosen as the location on the floating vessel where slope calculations can transition from one slope polynomial to another slope polynomial.
In one or more embodiments, piping restraint location geometry can be provided for determining deflections for restraint locations as at 350. For example, an initial design for the process and utility piping can be provided. The associated piping restraint geometry can be included and/or generated. The various locations for each restraint location can be provided as at 350 and the deflections at each restraint location can be determined as at 355. The deflections at each restraint location can be solved by determining the values from the one or more slope and the one or more displacement polynomials at each of the restraint locations. The complete model of process and utility piping design information, restraint location geometry, and associated deflections can be provided as at 365 and the fatigue analysis for the process and utility piping can be performed as at 370.
The computer 405 can interface with database 477, support computer or processor 475, other databases and/or other processors 479, or the Internet via the interface 480. It should be understood that the term “interface” does not indicate a limitation to interfaces that use only Ethernet connections and refers to all possible external interfaces, wired or wireless. It should be understood that database 477, processor 475, and/or other databases and/or other processors 479 are not limited to interfacing with computer 405 using network interface 480 and can interface with computer 405 in any means sufficient to create a communications path between the computer 405 and database 477, processor 475, and/or other databases and/or other processors 479. For example, in one or more embodiments, database 477 can interface with computer 405 via a USB interface while processor 475 can interface via some other high-speed data bus without using the network interface 480. In one or more embodiments, the computer 405, processor 475, and other processors 479 can be configured as part of a multiprocessor distributed system.
It should be understood that even though the computer 405 is shown as a platform on which the methods described can be performed, the methods described could be performed on any platform. For example, the many and varied embodiments described herein can be used on any device that has computing capability. For example, the computing capability can include the capability to access any communications bus protocols such that the user can interact with the many and varied computers 405, processors 475, and/or other databases and processors 479 that can be distributed or otherwise assembled. These devices can include, but are not limited to and are presented for illustrative purposes only: supercomputers, arrayed server networks, arrayed memory networks, arrayed computer networks, distributed server networks, distributed memory networks, distributed computer networks, desktop personal computers (PCs), tablet PCs, hand held PCs, laptops, devices sold under the trademark names BLACKBERRY™ or PALM™, cellular phones, hand held music players, or any other device or system with computing capabilities.
Still referring to
The memory 420 can store an operating system (OS) 445, a fatigue analysis agent 455, a curve fitting agent 465, and a deflection generation agent 468. The operating system 445 can facilitate control and execution of software using the CPU 410. Any available operating system can be used in this manner including WINDOWS™, LINUX™, Apple OS™, UNIX™ and the like.
In one or more embodiments, the CPU 410 can execute either from a user request or automatically. In one or more embodiments, the CPU 410 can execute the fatigue analysis agent 455, the curve fitting agent 465, and/or the deflection generation agent 468 when a user requests, among other requests, to perform a fatigue analysis on a pipe run or a piping system disposed on the floating vessel. For example, the fatigue analysis can be performed by providing a floating vessel bending moment data set; generating one or more bending moment polynomials from the floating vessel bending moment data set; generating a slope polynomial from each of the bending moment polynomials by integrating each of the one or more bending moment polynomials; generating a displacement polynomial from each of the slope polynomials by integrating each of the one or more slope polynomials; solving the slope polynomial integration constants; solving the displacement polynomial integration constants; providing process and utility piping geometry including piping restraint location geometry; determining the deflections at each restraint location using the slope and displacement polynomials; associating the deflections with each restraint location; providing the deflections for fatigue analysis; and performing the fatigue analysis.
It should be noted that the fatigue analysis agent 455, the curve fitting agent 465, and/or the deflection generation agent 468 can be fully autonomous code sets, partially integrated code sets, or fully integrated code sets and no limitations should be construed from the depiction of the fatigue analysis agent 455, the curve fitting agent 465, and/or the deflection generation agent 468 as separate agents. It should be noted that it is not necessary to execute the fatigue analysis agent 455, the curve fitting agent 465, and/or the deflection generation agent 468 simultaneously nor is it necessary to execute the three agents on the same computer 405. In one or more embodiments, the fatigue analysis agent 455 can be a fatigue analysis computer programs known in the art. Fatigue analysis can be performed by providing the fatigue analysis computer program with the complete model of process and utility piping design information, restraint location geometry, and associated deflections.
Referring to
In one or more embodiments, plotting and curve fitting the bending moment data 110 and polynomial generation 112, 115 can be performed using the provided floating vessel bending moment data 100 and known techniques and/or curve fitting agent 465. For example, the curve fitting agent 465 can execute the software package with the trade name MICROSOFT EXCEL to generate at least one bending moment polynomial 112 that can represent the plot of the bending moment data set 110. In one or more embodiments, the curve fitting agent 465 can use orthogonal polynomial curve fitting techniques to generate at least one associated bending moment polynomial 112. In one or more embodiments, the curve fitting agent 465 can execute the software package with the trade name MATLAB to generate at least one bending moment polynomial 112.
In one or more embodiments, the criteria for polynomial fit sufficiency 113 can be predetermined and set within the curve fitting agent 465. A given value for the coefficient of determination (R2), the observed ƒ-value for the fit, the observed t-values/confidence intervals for the coefficients, or other known fit measures can be set within the curve fitting agent 465 as a threshold by which polynomial fit sufficiency 113 can be determined. The curve fitting agent 465 can iterate on one or more of the embodiments described herein, until the predetermined criteria is met. The curve fitting agent 465 can provide the one or more derived bending moment polynomials that meet the predetermined fit sufficiency criteria to the deflection generation agent 468. For example, if the predetermined criteria for the coefficient of determination (R2) is set to greater than 0.97, the curve fitting agent 465 can generate higher and higher order bending moment polynomials until the criteria is met. In the event that the criteria is not met, the curve fitting agent 465 can generate one or more overlapping bending moment polynomials 115, until the predetermined criteria can be met for an appropriate portion of the provided floating vessel bending moment data set 100.
In one or more embodiments, to determine bending moment polynomial fit sufficiency as at 113, the curve fitting agent 465 can determine the mismatch or residuals of the bending moment polynomial values to the provided floating vessel bending moment data. A plot of the residuals versus the polynomial values can show patterns or trends. The degree of patterns or trends can be a measure of the quality of the fit, where no trend can equal a good fit. In one or more embodiments, a normal-probability plot of the residuals can be used. The fit of the bending moment polynomial values to the provided floating vessel bending moment data can be considered sufficient when the normal-probability plot of the residuals approximates a straight line. The curve fitting agent 465 can iterate on one or more of the embodiments described herein, until there are no trends in the plot of the residual versus the polynomial values. The curve fitting agent 465 can iterate on one or more of the embodiments described herein, until the normal-probability plot of the residual approximates a straight line, within user chosen limits. The curve fitting agent 465 can provide the sufficiently fit one or more bending moment polynomials to the deflection generation agent 468.
In one or more embodiments, to determine bending moment polynomial fit sufficiency, visual comparisons and/or other known techniques can be used. In one or more embodiments, visual comparisons can be performed. For example, when the curve fitting agent 465 generates a first bending moment polynomial, the curve fitting agent 465 can plot the values for the first bending moment polynomial and overlay the plot on a plot of the bending moment data set. The differences between the bending moment polynomial plot and the bending moment data can be visually assessed. In one or more embodiments, the visual assessment can be performed by the user. In one or more embodiments, the visual assessment can be performed by the curve fitting agent 465. It should be understood that the curve fitting agent 465 cannot visualize in the same manner a human visualizes but the curve fitting agent 465 can perform the visual assessment using techniques known in the art, for example using artificial intelligence routines. Based on user judgment or predetermined comparison criteria of the sufficiency of the polynomial fit, the curve fitting agent 465 can generate another polynomial or the user can direct the curve fitting agent 465 to generate another polynomial. In either case, the curve fitting agent 465 can generate a higher order bending moment polynomial and/or an overlapping bending moment polynomial to better approximate the fit of the polynomial values to the bending moment data set. In one or more embodiments, the higher order bending moment polynomials and/or the overlapping bending moment polynomials can be overlaid on the plot of the bending moment data set and reassessed for fit sufficiency as at 113. The analysis cycle can continue until the polynomial fit meet the appropriate criteria. The sufficiently fit bending moment polynomials can be provided to the deflection generation agent 468.
In one or more embodiments, the one or more bending moment polynomials can be provided to the deflection generation agent 468 either automatically or manually. For example, the bending moment polynomials can be provided as at 117 to the deflection generation agent 468 within the memory 420. The bending moment polynomials can be manually provided by the user to the deflection generation agent 468 using the keyboard 430. Other known data entry and programming techniques can be used to provide the bending moment polynomials as at 117 to the deflection generation agent 468.
Referring to
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In one or more embodiments, a coordinate system 520 can be identified for the floating vessel 505. The origin of the coordinate system 520 can be located anywhere about the floating vessel 505. Design calculations for the floating vessel 505, such as bending moments, can be determined at various calculation points about the floating vessel 505. The various calculation points can be located with reference to the origin of the floating vessel coordinate system 520. For example, bending moments can be determined about a point within a first bulkhead and the point can be located relative to the floating vessel coordinate system 520.
The design calculations can be associated with determining the sea-applied bending moments the floating vessel 505 can experience while at sea. During the design of a floating vessel 505, bending moment calculations are typically made at various locations about the vessel 505 for two extremes: when a crest of a wave is in the amidships region 550 and when a trough of a wave is in the amidships region 550. In the first extreme, where the crest of the wave is in the amidships region 550, there is an excess of weight at the floating vessel ends 530, 540 and an excess of buoyancy in the amidships region 550. There is a tendency for the floating vessel ends 530, 540 to droop relative to the amidships region 550. The tendency to droop relative to the amidships region 550 is defined as hogging. In the second extreme, when the trough of the wave is in the amidships region 550, there is an excess of weight in the amidships region 550 and an excess of buoyancy at the floating vessel ends 530, 540. There is a tendency for the amidships region 550 to droop relative to the floating vessel ends 530, 540. This tendency to droop relative to the floating vessel ends 530, 540 is defined as sagging. The calculations associated with the hogging and sagging can be expressed in terms of bending moments, rotational forces, and displacements. The results of the calculations can be documented in naval architect fatigue analysis tables that are known in the art.
The deflections the vessel 505 experiences can include hogging deflections with at least one orthogonal and/or rotational component, sagging deflections with at least one orthogonal and/or rotational component, and/or other defections with at least one orthogonal and/or rotational component. According to one or more embodiments described, determining deflections and entering those deflections into one or more known programs can be performed. For example, using data from the naval architect's fatigue analysis table for the floating vessel 505, one or more curves can be fit to a plot of one or more sagging bending moment data sets. The values of the one or more sagging bending moment data sets can correspond to the sagging bending moments calculated at one or more points about the floating vessel 505. One or more sagging bending moment polynomials can be determined and can represent one or more plots of the one or more sagging bending moment data sets. In one or more embodiments, one or more curves can be fit to a plot of one or more hogging bending moment data sets, values of which can correspond to the hogging bending moments at one or more points about the floating vessel 505. One or more hogging bending moment polynomials can be determined and can represent one or more plots of the one or more hogging bending moment data sets. In one or more embodiments, one or more curves can be fit to a plot of any floating vessel 505 bending moment data set. One or more bending moment polynomials can be determined and can represent the plot of the bending moment data set.
The one or more sagging bending moment polynomials can be integrated and the associated integration constants can be determined to derive one or more sagging slope polynomials for the floating vessel 505. The one or more sagging slope polynomials can be integrated and the associated integration constants can be determined to derive one or more sagging displacement polynomials for the floating vessel 505. The derived sagging slope and displacement polynomials can be used to calculate the sagging deflections at any given point on the FPSO unit 500 using one or more embodiments described herein.
The one or more hogging bending moment polynomials can be integrated and the associated integration constants can be determined to derive one or more hogging slope polynomials for the floating vessel 505. The one or more hogging slope polynomials can be integrated and the associated integration constants can be determined to derive one or more hogging displacement polynomials for the floating vessel 505. The derived hogging slope and displacement polynomials can be used to calculate the hogging deflections at any given point on the FPSO unit 500 using one or more embodiments described herein. The hogging and sagging deflections can be used to perform fatigue analysis using one or more embodiments described herein.
In one or more embodiments, a conversion between the piping system coordinate system 600 and the floating vessel coordinate system 520 can be performed as necessary. For example, the relative location of each restraint location 610 with respect to the origin of the piping system coordinate system 600 can be converted with respect to the origin of the floating vessel coordinate system 520 as appropriate for deflection determinations.
The bending moment data set from the naval architect's fatigue analysis table may have been generated with respect to the floating vessel coordinate system 520. The origin of the floating vessel coordinate system 520 may be different from the origin of the piping system coordinate system 600. The piping restraint locations 610 may be expressed relative to the piping system coordinate system 600. A conversion between the piping system coordinate system 600 and the floating vessel coordinate system 520 may be appropriate prior to calculating the deflections at one or more restraint locations 610 due to the differences in the coordinate system origins. The conversion can be performed using known techniques.
The one or more derived polynomials described above, with reference to
Still referring to
The foregoing discussion can be further described with reference to the following non-limiting, example.
One or more embodiments of the present invention have been trialed on a project for an FPSO unit conversion. In this example, a program with the trade name CAESAR II was used to perform a fatigue analysis on the process and utility piping system design for the FPSO unit conversion. The FPSO unit piping geometry and restraint locations were entered into the CAESAR II program and a fatigue analysis data table or neutral file 700 was generated.
The neutral file 700 contained the piping restraint location geometry 701, 702, 703, 704 in millimeters and stiffness values 705, 706 in units of Newton/millimeter. The geometry and stiffness values were populated in the table 700 using information from an initial process and utility piping design. The blank column 710 corresponds to the static sagging deflections and the blank column 720 corresponds to the static hogging deflections as well as all other deflections to be calculated and entered into the table 700. The displacements 707 and 708 are orthogonal displacements in the x and z-axis directions respectively. Rotation 709 is a slope component of the calculated deflection. The slope 709 is imparted about a given restraint location along the floating vessel y-axis, with x positive from the aft to the fore of the FPSO and where positive z extends into the sky above the FPSO.
The FPSO unit conversion vessel's naval architect bending moment data was entered into a curve fitting agent and a first set of bending moment polynomial curves was fit to the bending moment data. The sufficiency of the fit of the first bending moment polynomial was assessed using both an R2 threshold value of 0.97 and visual inspection of any mismatches between the naval architect bending moment data and the plot of the first bending moment polynomial. Although the R2 threshold was met, the visual inspection indicated mismatches between the naval architect bending moment data and the plot of the first set of bending moment polynomials. The mismatches occurred in the general vicinity of the aft of the FPSO unit and the bow of the FPSO unit.
To alleviate the observed mismatches, three overlapping curves were fit to the naval architect bending moment data. One of the curves was fit to a portion of the plot of the bending moment data associated with the bending moments in the vicinity of the aft of the FPSO unit. The plot of the bending moment data in the vicinity of the aft of the FPSO unit was associated with the bending moments in the region between the aft of the FPSO unit and about 50 meters from the aft of the FPSO unit as measured along the floating vessel beam toward the bow. One of the curves was fit to a portion of the plot of the bending moment data associated with the bending moments in the vicinity between the bow and the aft of the FPSO unit. The plot of the bending moment data between the bow and the aft of the FPSO unit was associated with the bending moments in the region from about 50 meters from the aft of the FPSO unit to about 275 meters from the aft of the FPSO unit. One of the curves was fit to a portion of the plot of the bending moment data associated with the bending moments in the vicinity of the bow of the FPSO unit. The plot of the bending moment data in the vicinity of the bow of the FPSO unit was associated with the bending moments in the region from about 275 meters from the aft of the FPSP unit to the bow of the FPSO unit.
Three tenth-order overlapping bending moment polynomials were generated. The three bending moment polynomials were plotted over the plot of the bending moment data. The fit of the combined portions of the three bending moment polynomials to the bending moment data was assessed using both an R2 threshold value of 0.97 and visual inspection. The fit was determined to be sufficient. The bending moment polynomials were integrated to determine the slope polynomials. The slope polynomials were integrated to determine the displacement polynomials. The slope polynomial and displacement polynomial integration constants were determined using the deflection generation agent and the appropriate boundary conditions for each slope and displacement polynomial.
Referring to
Certain embodiments and features have been described using a set of numerical upper limits and a set of numerical lower limits. It should be appreciated that ranges from any lower limit to any upper limit are contemplated unless otherwise indicated. Certain lower limits, upper limits and ranges appear in one or more claims below. All numerical values are “about” or “approximately” the indicated value, and take into account experimental error and variations that would be expected by a person with ordinary skill in the art.
Various terms have been defined above. To the extent a term used in a claim is not defined above, it should be given the broadest definition persons in the pertinent art have given that term as reflected in at least one printed publication or issued patent. Furthermore, all patents, test procedures, and other documents cited in this application are fully incorporated by reference to the extent such disclosure is not inconsistent with this application and for all jurisdictions in which such incorporation is permitted.
While the foregoing is directed to embodiments of the present invention, other and further embodiments of the invention may be devised without departing from the basic scope thereof, and the scope thereof is determined by the claims that follow.
