FPSO HULL STRUCTURES WITH SANDWICH PLATE SYSTEM IN CARGO TANKS

Nowadays, the floating production storage and offload system (FPSO) is one of the most common platform types for offshore oil production. The traditional arrangement of the stiffened panels creates obstacles for automated cleaning and inspections by remote devices. This paper summarizes the results of an initial study for the design and construction of FPSO hulls with SPS in order to overcome this problem. The main goal is to have the walls and bottom of the cargo tanks free of stiffeners. This research is conducted by first designing the hull with a conventional structural arrangement using steel according to the ABS rules as a benchmark. Following that, the equivalent hull structure with sandwich plates is designed in accordance with the guidelines for SPS construction from DNV rules. Finally, this paper provides the results of a finite element analysis to compare the stresses and ultimate strengths of both types of structures. Briefly, the main results are that the SPS design provides a reduction of 2.8% of the total weight and a better overall structural performance by an increase of 26% for the ultimate strength of the hull.


LIST OF TABLES
-7 -Calculations of the weight for each part of the structure for both types of arrangements .. 55 Table 6

. Background and Motivation
The production of oil and gas reservoirs can be separated by the onshore and offshore production. For the second category, there are several types of floating ocean platforms to be used in deep waters such as semi-submersibles, FPSOs, monocolumns, TLPs and SPARs. The development of this master thesis will be focused on one type of structure: the FPSO.
A floating production storage and offloading (FPSO) unit is a floating vessel used by the offshore oil and gas industry for the production and processing of hydrocarbons, and for the storage of the processed oil. A FPSO vessel is designed to receive hydrocarbons produced by itself or from nearby platforms or subsea template, process them, and store oil until it can be offloaded into a shuttle tanker. FPSOs are preferred in frontier offshore regions as they are easy to install, and do not require a local pipeline infrastructure to export oil. FPSOs can be a conversion of an oil tanker (VLCC) or can be a vessel built specially for the application. They have become common tools for offshore oil production in areas worldwide over the past 10 to 20 years. Thus, the FPSO crew needs to first reallocate the liquids from the tank, which will be inspected, to the others from the vessel without jeopardizing the oil and gas production. The oil tank to be inspected may be out of operation for up to 10 or 15 days. The storage of the produced oil in this period shall be rearranged in the remaining tanks. This rearrange can be quite tricky in the earlier years after the beginning of the operation, because it is where the production reaches its peak as can be seen in Figure   1-1. Therefore, the faster the crew can clean it and inspect it, the less time the production capacity will be reduced and, therefore, the more revenue the FPSO will generate. In this context, new technologies are arising to try to overcome or minimize this problem. One of them is the use of sandwich plates in offshore structures. A sandwich plate is a fabricated material that consists of two steel plates joined to either side of a low-density core material or structure [2]. For example, there are concrete sandwich plates as shown in [3] and the Sandwich Plate System (SPS) as shown [1]. It was developed by Intelligent Engineering in the late 1990s at Canada's Carleton University in response to protecting offshore structures in the Beaufort Sea from impact damage due to ice sheet loads. The solution was a steel-elastomer-steel composite panel, with a high strength-to-weight ratio and durability that would enable the offshore structures to withstand the heavy loads, as shown in Figure   1-2. The elastomer, as a two-part liquid, is injected into closed cavities formed by the steel. Generally, this kind of construction has several advantages that make them a suitable option such as [2]: • High stiffness to weight ratio, making them suitable for lightweight design.
• Good buckling resistance compared to thin orthotropic plate structures.
• Large unsupported spans, thereby reducing the requirement for supporting elements and increasing architectural freedom.
• Reduced assembly times via modular approaches to construction.
• Flat and smooth surfaces Although this technology has several uses in civil (buildings and bridges), marine, military and offshore applications, the main market is the marine/offshore one and it made the debut in 1999 with Ro-Ro and

Objectives
In the light of the problem presented in section 1.1, this master thesis will focus on the research for design and construction of hulls of FPSOs with sandwich plate system without stiffeners in the bottom plate and longitudinal bulkheads of cargo tanks. The main goal is to have the walls and bottom of the cargo tanks free of stiffeners to allow remote cleaning and thickness inspection of bottom and bulkhead plates using autonomous equipment. The risk is reduced because people won't need to inspect all areas from the vessel and these devices halve the time required for cleaning and inspection, thus, less harm to the operation of the FPSO. In order to do that, this study is divided into four steps which are: • Literature review about the SPS in the offshore industry • Design of a midship section in traditional naval steel structures to be used as a benchmark • Design of a midship section with sandwich plates in cargo tanks • Ultimate strength analyses of the midship section for both arrangements.

Structure of the Thesis
The thesis is organized into seven chapters and respective appendices. Chapter 1 is the introduction of the topic to be discussed the related background, main goals and structure of the work. Chapter 2 contains the bibliographic research about ultimate strength and the usage of sandwich panels for this topic. Chapter 3 is the FPSO main dimensions and characteristics that will be used in this master thesis and the estimation of the global loads of the FPSO with the auxiliary of the software HydroD from SESAM. Chapter 4 details the design process for the traditional midship section in steel according to FPI from ABS [5]- [8] and DNV rules [9]- [17]. Chapter 5 shows the design of a midship section with the sandwich plates in the cargo tanks according to the classification society rules (DNVGL-CG-0154 and Lloyds guidelines rules [18], [19]) and compare the weight between both arrangements. Chapter 6 focuses on the FEM analysis for the steel and composite model and the ultimate strength analysis of both sections. In the end, chapter 9 indicates the main conclusions and future work of this thesis.

Beam Theory
The hull girder analysis assumes that the structure satisfies simple beam theory (Bernoulli-Euler). It is a simplification of the linear theory of elasticity which provides a means of calculating the load-carrying and deflection characteristics of beams [20]. Therefore, the hull girder analysis deals only with those longitudinally integrated forces and moments that are dealt with in beam theory: vertical shear force, Fz, longitudinal bending moment in the (ship's) vertical and horizontal planes, My and Mz, and longitudinal twisting moment, Mx. This theory provides the means to estimate the still water shear force/bending moment distribution and also the normal and shear stress for any member of the structure and this master thesis will briefly describe the process to obtain each variable from the load distribution and equilibrium equations.

Shear Force and Bending Moments distributions
As stated in Hughes et. al. [21], the vertical shear force and bending moments distributions can be caused mainly by the unequal distributions of weight and buoyancy along the length of the ship, accentuated by waves. While the horizontal component occurs when the ship is in an inclined condition, as a result of rolling. Once knowing this distribution of resultant vertical/horizontal forces (f(x)) from the balance between weight and buoyancy, a free body diagram can be used to calculate for each infinitesimal length dx the shear force (Q(x)) and bending moment (M(x)) by the following equation [22].
A summary of the procedure to find this curve is presented in Figure 2  These curves have some unique characteristics that will be useful to validate the ones obtained in chapter 4 through suite software SESAM [23], [24]. The first one is that both variables must be zero at both ends of the model, because the hull girder is a "free-free" beam. The second one is the shape of the function. In most cases the loading is approximately similar forward and aft of amidships. Under these conditions the shear force is approximately symmetric, passing through zero somewhere near amidships and having maximum values, positive and negative, near the quarter points. Therefore, since the bending moment is the derivative from the shear force, its shape will be, in general, will be a maximum, positive or negative, near amidships. However, if the load is very asymmetrical, this behavior can change.

Shear and Normal Stress
Besides deflection, the beam equation describes forces and moments and can thus be used to describe stresses. This is due to the fact that Hooke's law of deformation is valid for this case. Thus, both the bending moment and the shear force cause different stresses in the beam. The first one is related to the normal stress while the second one with the shear stress. Generally, the stress due to shear force is maximum along the neutral axis of the beam while and the maximum normal stress is at either the top or bottom surfaces. The main formulas for both are described below [22]. These formulas are extremely important because they provide a way to analyze how close or far are the stresses compared to the yield stress, which is the point where plastic deformation begins. They will be extensively used in chapter 5 to design the thickness of the steel plates.
These formulas are valid for beams with only one homogenic material. However, it is common in engineering to have beams with two or more kinds of material as it is the case for this thesis. Therefore, a modification in the formula must be made in order to calculate the stress for each material. The approach used in this master thesis is described in chapter 6.10 from Introduction to Solids Mechanics where n = E2/E1. In the end, the stresses in material 1 are calculated normally while in material 2 it will be the multiplication of this by n. This will be very useful in order to design and predict the stresses for the SPS panels.

Introduction
A good practice in any design of ships and offshore structures is to use one or more limits states as criteria in addition to the yield stress during the total strength assessment. A limit state is a condition under which a particular structural component or an entire structural system fails to perform its designated function [25]. In other words, it is any condition in which a structure or a structural member has become unfit for one of its intended roles because of one or more loads and/or load effects [21].
The main ones are: serviceability limit states (SLS), ultimate limit states (ULS), fatigue limit states (FLS) and accidental limit states (ALS). This master thesis is concerned with ULS.
Ultimate strength is a critical and fundamental part of the strength assessment in the design of a ship or offshore structure [26]. It is defined as a point beyond which the loading exceeds structural capacity and the structure collapses [27]. There are various methods available in the literature to compute the ultimate strength of different structural elements. They can vary from a more simplified to a more sophisticated one. However, determining this property is a challenging task because it depends on three types of uncertainties: geometry, material and numerical. Each of them has factors that can influence on each other and are summarized on  Nowadays, there are three ways to determine the ultimate strength of ships and marine structures: experimental, numerical simulations and analytical modelling. Due to the very large size of some ships, and the fact there is no standardization of ship types, the shipping industry is in a difficult position to make frequently experimental tests. Therefore, the development of numerical and analytical approaches proofs to be a good solution to this problem [25]. Although numerical simulations can provide good predictions without the need to build/test a structure as the experimental one, the complexity of the problem can increase drastically when considering different kinds of uncertainties. This can be quite time-consuming and expensive, because requires good computers [27]. Thus, analytical methods also have their use once they provide simplified and quicker checks to use it. In this light is that classification societies enter. Their main responsibilities are to provide both analytical and numerical approaches, based on the literature and studies, with the recommended safety factors in order to achieve a safe design of structures. Each classification society such as ABS, BV, DNG GL, LR and more has develop guidelines for both approaches for the corresponding limit state based on either the working stress design (WSD) and the load resistance factor design (LRFD) method.

Analytical Methods
Analytical methods proofed to still be worth in times where the development of numerical methods steady increase yearly. The main causes of that are they can provide a rapid evaluation at the early design stages and easily implemented using a spreadsheet environment, thus, not requiring specific software. There are two kinds of approaches to solve it: closed form methods and progressive collapse methods.
Close form methods (CFM) are a set mathematical expression expressed using a finite number of standard operations and are usually based on semi-empirical data. They are highly used in the classification societies such as in DNV [16] and ABS [5], since they are easy to be implemented.
However, they neglect that ultimate failure is a progressive event, which cannot predict the premature collapse of some structural elements [27]. One of the first attempts to evaluate the ultimate strength was proposed by Caldwell [28] in 1965. He applied "Rigid Plastic Mechanism Analysis" to evaluate the ultimate hull girder strength. The influence of buckling was considered by reducing the yield stress of the material in the buckled part [26]. Since then, several related closed forms were proposed to develop the model such as Paik and Mansour (1995)  The progressive method was firstly proposed by Smith (1977) [32] and it is an iterative method.
Curvature is applied to the section, and the strain experienced by each individual unit is then determined by assuming that plane sections remain plane. Based on this strain, the average stress in each unit is estimated taking into account whether it is in tension or compression, the yield strength of the material, and the buckling behavior of the plating and stiffeners in compression. Summing the contributions over the section determines the bending moment. The curvature is incremented and calculations repeated.
The end result is a nonlinear moment-curvature relationship, the peak value of which defines the ultimate strength of the hull.

Numerical Methods
According to the International Ship and Offshore Structures Congress (ISSC-2015) [26], non-linear finite element models have been growing steadily as a way to assess the ultimate strength of a structure. This is due to the continued development of computer technology and software such as ANSYS, ABAQUS, NASTRAN-PATRAN etc. Typically, the analyzes utilize a progressive method inside the solver together with an equilibrium convergence iterator using either the Riks arc length method or modified Newton-Raphson method [33]. However, as stated by [34] in his doctorate thesis, this is still a challenging task because of model size limitations imposed by software and hardware. Generally, the models are more focused on local parts of the hull in order to optimize the computational time required. Therefore, based on reviewed guidelines from different classification societies [5], [10], [35], [36] and several suggestions from different ISSC [25]- [27], [37]- [39], some good practices can be done to overcome or minimize the problems related to Table 2-1 and are displayed below.
• Nonlinear plate/shell and beam element types should be used; first-order elements are normally the easiest to generate meshes.
• Selective refinement of the model is recommended to keep the model size under control, with the smallest element size used in regions where the results must be more accurate.
• Element formulations must allow for geometric and material nonlinearity; large dis-placement and/or rotation formulations are necessary • Equivalent methods to simplify the geometry in order to reduce the computational time. One example is done by [40], which proposes an equivalent thickness method to estimate the ultimate strength for panels submitted to combined biaxial compression and lateral pressure loads • Use of commercial software once they have an extensive library and community to help the user • An appropriate selection of the longitudinal length of the model to the loading and response effects being modelled. For the hull girder strength, there is, for instance, the suggestion from ABS to use the extension of 3 cargo tanks [5] • For an implicit solver, it is necessary to select an appropriate step size and convergence tolerance. Too large a tolerance will lead to inaccurate solutions; too small a step size may cause convergence difficulties.

Ultimate Strength of Structural Elements
This section presents the recent research and developments on the ultimate strength for both stiffened steel and SPS panels.

Stiffened Panels
A stiffened panel is an assembly of plating and stiffeners. It is normally designed so that the buckling of a local plate panel between stiffeners initially takes place and is then followed by overall collapse due to excessive yielding and/or stiffener failure [25]. As stated in [41], there are 6 types of primary failures for a stiffened panel and 4 of them are presented in Figure 2  Copied from [34] The interaction between the plate elements and support members in terms of their geometrical and material properties and other factors such as loading condition and initial imperfections plays an important role in the ultimate strength, buckling, and plastic collapse patterns of stiffened panels [21].
Moreover, Collapse mode I represents the overall collapse after overall buckling. In this mode, the stiffeners buckle with the plating as a unit, and overall buckling often occurs under an elastic regime.

Sandwich Plate System (SPS)
As introduced in chapter 1, the FPSO studied will use a sandwich plate developed by Intelligent Engineering Limited (IE) and patented as the Sandwich Plate System (SPS), which integrates a steelelastomer-steel composite structural laminate in place of conventional stiffened steel plates (Figure 2- The resulting structure has greatly enhanced impact absorption capability compared to conventional steel structures [4]. This is due to the so called "sandwich effect", because the separation of the facings by a lightweight core acts to significantly increase the second moment of area (and hence the bending stiffness) of the material cross-section with only a small increase in weight [2]. Moreover, according to IE, this technology has civil (such as bridges [45]), military, maritime and industrial applications because of key benefits such as • Lightweight • Eliminates stiffenerssimplified structure (reduction of construction costs) • Improved space utilization • Fatigue and corrosion resistance • Impact, blast and fire protection showed that the sandwich plate configuration had a better performance reducing the stresses at the static load case and also increased the ultimate strength of the panels.
In addition, other projects were carried out by SPS Technology in this field of repair and conversion of FPSOs. The first example is in FPSO Independence from ConocoPihillips. Excessive pitting corrosion of the bottom shell plating in the Cargo Oil Tank resulted in the FPSO Independence requiring local steel reinstatement, which was solved with an SPS Cold Work repair. The FPSO Independence repair was carried out on-station. The repair area was localized and required only one tank to be closed, enabling FPSO Independence to maintain 100% operational capacity throughout the project, which took less than two weeks to complete. The cold work version of SPS delivered a permanent repair, reinstating the existing hull and creating a new composite section. SPS Technology has a solution that uses structural adhesives to join the steel components and form airtight cavities into which the elastomer core is injected. Another example is the FPSO Capixaba. The same installation method from FPSO Independence was used to deliver this permanent repair that reinstated the existing deck and created a new composite section. A combination of bolts and structural adhesive were used to fit and join the steel components that formed airtight cavities into which the elastomer core was injected. All steel components were prefabricated before shipping on-board for installation. This reduced the cost of the project, eliminated welding requirements, reduced time offshore for the installation team and simplified project logistics.
In the scope to provide a safe design to shipyards and engineers, DNV-GL (DNVGL-CG-0154 [18]) and Lloyd's Register [19] developed rules for designing this plate. They provide closed form methods in order to obtain the desirable steel and core plate thickness and, also, strength assessments for the buckling and ultimate strength for both new and overlay construction. The first rule also provides a numerical procedure to estimate both properties.
The Transport Industries Under European Commission in 2013 [2] with the help of several classification societies and universities proposed a coordination action on advanced sandwich structures in the transportation industry (SANDCORE). The result was a guide for best practices for sandwich structures, including the SPS, in marine applications with the main objective to provide knowledge of different types of sandwich plates mainly for shipyards in order to help their integration in ship design and fabrication.
The one that stands out for this thesis is the chapter 3. It is focused mainly on the design of plates with the analytical/numerical/experimental analysis of several limit states such as ultimate strength, buckling and fatigue and other types as vibration and crash response.
In the light of the numerical analysis, two studies will be discussed.  [47] presents a comparison between two approaches to model the SPS panels for a FEM. The first one is a mixture of shell and solids elements and the other is only shell elements. The object of study was double-bottom structure and the main conclusions were that the method of the shell elements presented in the paper gives satisfactory results compared to the theoretical models. Also, compared with the result of the mixture of shell and solid elements, the only shell method proposed in the paper is more efficient and easier to be accomplished. This is paper is important, because one of the main discussions in chapter 7 is how to model the SPS panels for the cargo tanks to make it efficient, easy and return accurate results.
The last paper is a master thesis written by Boersma from Delft University [48]. The main goal of this research is to find an optimal (cost-effective) solution for protection against dropped impact loads comparing a Stiffened Steel Plate (SSP) versus Sandwich Plate System (SPS) structures. The thesis presents several SPS and SSP designs first obtained through analytical methods in order to obtain a preliminary design. Further, by simulating the impact loads with FEM, it tries to obtain an optimum design that can support the load conditions without yielding and other associate failure modes. It also carried out research to obtain the maximum load to obtain the ultimate strength of some candidates for an optimum design. This thesis is valuable for this project because the design process is similar.

FPSO for Case Studies
The objective of this section is to provide the main characteristics of the FPSO for the case studies of this master thesis. It includes data such as main dimensions, initial drawings (longitudinal and transverse division of the ship), materials selected for the midship section and so on.
The main dimensions of the FPSO are presented in Table 3  Taking into account these initial dimensions, the next step is to define the shape and the longitudinal division of the hull. One reference for it is the hull of the FPSO P-66 of Petrobras [49], which has an overall length of 288m, a beam of 54m and a depth of 31.5m. It can be seen in Figure 3-1. In addition to that, the longitudinal division of the FPSO is segmented by the aft part, where the machinery, accommodation modulus, pump and so on will be located, followed by 6 cargo tanks with 2 ballast tanks for each side and in the fore most part a forepeak tank. Moreover, it was considered as well a spacing    Some other initial assumptions were made before the design of the midship section according to the classification society rules. This regards the still water bending moment and the material used for the construction. Based on the still water bending moment from similar FPSOs and the formulas proposed from ABS [9] and DNV [9], [13] a still water bending moment of 950,000 tonf*m was assumed as an initial value. This will be recalculated in chapter 4 with the help of the software SESAM. Also, the material used for the traditional steel midship is the high tensile steel grade 36, which has mechanical properties according to [8], [9], [12].
The final assumption regards the plate division and three aspects need to be considered. First, the dimensions (length, width and thickness) of the plate must comply with the market availability, the width varies between 3.0m to 5.0m. Second, the FPI and DNV rules give formulas to calculate the minimum and/or maximum breadth a type of plate can have. Third, according to DNV Rules [17], the welds between plates and the stiffeners with the plates shall not be less than 50 mm, but does not need to be greater than 100 mm apart from each other. Keeping in mind these three factors, the plate division could be made and it is shown in Figure 3

Software Introduction
These sections have the main goal to detail the methodology to estimate the global loads (bending moments and shear forces) that will be considered for the following chapters. It is very important, once the stresses and ultimate strength are directly related to the value and distribution of both curves. In order to accomplish this, it was used two modules from the software SESAM (GeniE [24] and HydroD [23]) to model the structure with the loads/weights and after solve the equilibrium problem. The solution is the cargo/ballast distribution for a certain equilibrium and load condition, thus, having the distribution of all weights and buoyancy, the global loads can be achieved.
This first section will introduce the software SESAM with its capabilities/uses, the tutorials used both in GeniE and HydroD, the general methodology to estimate the global loads and some initial assumptions.
The following part will detail the construction of the model in GeniE and also the loads considered. The last part is the construction of the model in HydroD and the global loads results from different draughts.
SESAM is a software suite for hydrostatic, hydrodynamic and strength analyses of ships and floating offshore structures developed by DNV. This includes barges, FPSOs, semi-submersibles, TLPs, Spar, buoys and gravity-based structures. Two modules of SESAM have been used in this research work: HydroD and GeniE. HydroD is an interactive application for the computation of hydrostatics and stability, wave loads and motion response for ships and offshore structures [23]. The wave loads and motions are computed by Wadam [50] or Wasim in the SESAM suite of programs. It is based on one common model, fully integrated with finite element analysis (FEA), which is the GeniE. It is a tool for modelling structures composed of beams, flat plates and stiffened shells, which can be meshed to a variety of element types. The load modelling includes equipment, explicit loads, wind loads and generation of compartments in floating structures [24].
The sequence to solve the hydrostatics problem of the master thesis FPSO with the Sesam software is described in several tutorials provided by the company [51]- [55] and by the user's manual of GeniE, HydroD and Wadam [23], [24], [50]. The first step is to create the hydrodynamic and mass model in the The chosen hydrodynamic model is the panel method, which is based on potential theory. The theory behind is further developed in [56]- [59] and the potential proposal provides the necessary and adequate bases used to define the wave problem, linearized from the premise that the incident wave amplitudes are reasonably smaller than their respective wavelengths wave. As a consequence of linearization, the classical hydrodynamic problem of a floating unit subjected to wave passage can be treated from the sum of three main components: hydrostatic portion, radiation portion and diffraction portion [60].
In this master thesis, the hydrostatic problem will be the focus once the main goal is just to obtain the compartment distribution for a still water condition. Therefore, in order to take into account this contribution in the computation of the hydrostatic forces, it is introduced to the system stiffness coefficients, which make up a hydrostatic restoration matrix. This matrix is applied to all panels, which will be generated for all element sides below the still water level and above the sea bed. With that, the rigid body equations can be applied for each one and, by solving it, an equilibrium condition can be found. [50] The next step is the creation of the mass model. It covers the creation of the structure of the FPSO, the boundary conditions and loads imposed. The mass is used both in the hydrostatic calculations to report imbalances between weight and buoyancy of the structure and in the hydrodynamic motion analyses.
In addition to the models, the environmental data and loading conditions must be applied and, then, a hydrostatic analysis is performed. The module returns as output different parameters such as the distribution of contents (oil and ballast) in the tanks for a static equilibrium condition, the GZ curve, still water sectional loads (shear force and bending moments) and stability checks against international codes such as the IMO MODU code [61] and the NMD [62].

GeniE Modelling
This section will cover the procedure to model the traditional steel structure of the FPSO that was designed in chapter 4 of this master thesis. The methodology is strongly based on the tutorials [53], [54] and the user's manual [24] provided by the company DNV.

Hydrodynamic Model
The first step is to create the structural properties, which includes the material and thickness of the  [5] and the result was a structure with a total weight around 42,000 tons. This is an important step in order to predict accurately the distribution of bending moments and shear forces at the HydroD software, once the structure's weight plays an essential role in the force equilibrium equations.
The following phase is to create the geometry of the hull. Thus, it is needed to set guide planes alongside the hull to create the external and internal parts of the cargo tank, forepeak and aft part of the FPSO.
From these guide planes, create the surfaces are created with the correct structural property (material and thickness) that compose each part of the hull. After that, a hydrodynamic model itself can be created.
By following the user's manual from HydroD [23] and GeniE [24] guidelines, the appropriate hydrodynamic model for an FPSO hull is the panel one. In order to do this, the next steps should be made as follow and are based on the specific tutorial [53].
a. Select all wet surfaces, which are those subjected to water. In this master thesis, they are all surfaces from the bottom, side and deck of the hull.
b. Add a wet surface property to them providing that all surface normal points outwards. This is an important step, because if any surface points inwards, this will result in a negative contribution for the buoyancy force [53].
c. Mesh all the selected surfaces with the desired element length. As stated in the user's manuals [23], [24], this part must have a balance between the accuracy of the results with the speed of computational time. This is done automatically by the software to find a good combination of both.
d. The last step is to assign a load case with the dummy hydro pressure option turned on for the set, because the elements within the set selected will be the ones used for the hydrodynamic force's calculation.

Structural Model
With the tanks modelled from the previous section, load cases were assigned for each compartment that is going to be used in the hydrostatic analysis of HydroD. These compartments are the ones that will be filled with either oil or ballast with the desired quantity to reach the stability for a certain draught, heel and trim of the FPSO. Thus, it was created a load case for all the ballast and cargo tanks presented in   Table 4-1. Although they can be modeled as line loads in GeniE, the hydrodynamic analysis in HydroD required that they are represented as point masses rather than line loads [53]. Therefore, these loads were modeled as point masses. The first weight is the superstructure/process plant which was uniformly distributed along the centerline of the deck with a height of 47.3m above the baseline. The second one is the riser group, which includes the riser loads and the lower and upper balconies. All of them were placed at the portside of the ship (Y=28.9m) with the respective vertical center of gravity (VCG) and shown in table 3-2. The third load is the hull stern load was placed similar to the superstructure/process plant, because it was at the centerline of the deck but with a height of 41.3m from the baseline. The fourth weight, which the hull items (equipment, pipes, painting, mooring system), has a VCG of 26m from the baseline. However, the software can only identify the point mass if it is joint with an element from a plate or beam. Therefore, the weights were distributed between the deck and bottom centerline to reach the desired weight and VCG. The fifth one is the hull bow (flare tower, hull equipment, offloading system) and was located in the centerline of the deck as well but with a VCG of 52.3m from the baseline.
The last one is the mooring loads and it was divided into 4-point masses located at the region from the yellow arrow from Figure 3-7 at the deck height (Z=32.3m). In all of the cases except for the mooring loads, it was supposed a spacing between the masses of 5.1m, which is the spacing between web frames. A representation of this step can be seen in Figure 3-7.

HydroD Modelling
This section will describe the method to solve the hydrostatic and compartment balance of the FPSO of this master thesis through the HydroD module. The procedure is based on tutorials provided by the company [52], [55] and by the User's manual [23], [50] and it is divided into two parts. First it will be shown the input data such as loading conditions and environmental data. Then, the results of the weights distribution and the cross-sectional loads (shear force and bending moments) in still water will be shown.

Input Data
Before the hydrostatic/hydrodynamic analysis can be done, several information must be provided for the software so it can find a distribution of the tanks for static equilibrium. The main ones can be divided into four groups: GeniE model, environmental data, tanks characteristics and equilibrium condition. The first group is both panel and structural files made in GeniE, which were introduced to the program. The most relevant information from the second group are the water depth and density, which is assumed as 2000m (deep water) with a density of 1025 kg/m3 (salt water). The third group described the equilibrium point the software has as an objective when computing the filling for each ballast/cargo tank. The main ones are the desired draught, heel angle and trim angle. The heel angle is considered as zero, while, the trim angle is not. Generally, the FPSOs have an aft trim angle in order to help the drainage in the process plant. Thus, in this project it will be considered equal to 1%. For example, if the draught in the fore part is 10m, in the aft part will be 13.23m (10m+0.01x323m). The third group is to inform the software which are ballast and cargo tanks and their liquids properties. As seen in Figure 3-7 and 3-9, there are a total of 6 cargos tanks, 12 ballast tanks (2 for each cargo tank) and one forepeak tank, which is divided into two tanks. The cargo tank shall receive oil with a density of 900 kg/m3 while the ballast and forepeak ones shall receive salt water as content with a density of 1,025 kg/m3. Also, it will be calculated 5 draughts: the minimum (7.8m), the maximum (26.8m) and 3 intermediate values (12.5m; 17.5m; 22.5m).
The minimum and maximum draft could be obtained through an iterative process with the compartments balancing taking as initial draught guess the one from the equilibrium between buoyancy and mass (obtained from GeniE) for a trim equal to 0º. This is explained in the following formula below = * * * * = ℎ + ℎ + ℎ In order to calculate the weight of the tanks, it was assumed that a "full" cargo tank would have 95% and an "empty" one 3% of its volume filled. This tank isn't filled 100%, because of the inert gases presented in the tank that occupies some space. Also, it doesn't reach the 0% mark, because there is always a small amount of oil that the pumps cannot drain from the tanks. After all this input data is given, the following model could be generated and it is presented in Figure 3-9. Thus, the next step is to obtain the filling ratio of the compartments of the FPSO for the desired equilibrium point. The software can't balance tanks with different liquids; thus, a methodology was followed to obtain the final distribution. The first step was to fix a certain amount of ballast for all tanks, for example, 10-20% of the ballast tank capacity and obtain a range of how many "full" cargo tanks would be needed to balance the FPSO using the weights equilibrium forces equation. Then, a certain distribution of these "full" tanks was guessed and the software would compute the filling ratio for each ballast tank to obtain a certain draught. The solver of the software has as objective the maximization of the bending moments. An example is described to illustrate the problem in Table 3-3. Moreover, after a solution was found, the model also was checked by some stability criteria imposed by the classification society rules such as the IMO, ABS and NMD. If the model passed the stability checks, the solution was accepted as definitive. The flowchart representing the iterative process to find the filling ratio of each tank is described below (Figure 3-10).

Results
Applying the methodology of Figure 3

TRADITIONAL STEEL STRUCTURE SCANTLING
This chapter has the purpose of detailing the process of dimensioning a midship section of a FPSO according to the FPI from ABS [5]- [8] and DNV rules [9]- [17]. It is divided into three parts. The first section (5.1) will detail how the dimensions of different elements such as plates, longitudinal stiffeners, brackets and so on were obtained by the FPI rules. The second part (4.2) will do the same but following the DNV rules. The last part (4.3) is a strength analysis of the structure. Both rules recommend doing a hull girder strength (global loads and ultimate strength) and a buckling analysis for the different elements. The first criteria will be studied with the help of MARS 2000 software from the Bureau Veritas.
The other criteria will be studied through the method presented by DNV Rules. With both criteria satisfied, the final distribution of thickness from the plates, dimensions of longitudinal stiffeners, the thickness of brackets can be obtained.

Design According to the FPI Rules
As stated in FPI rules [5], "The design criteria contained in Part 5A, Chapter 3 are applied in two phases.
The first phase provides the basic hull design to reflect overall hull girder and local structural component The main characteristics of the steel used for the dimensioning are: • minimum specified yield point of the material in N/cm 2 , taken as 355 MPa for Grade H36 Steel [8] • E modulus of elasticity of the material, may be taken as 2.06 × 10 7 N/cm 2 for steel [5] • Q material conversion factor, taken as 0.72 for Grade H36 Steel [5] • strength reduction factor taken as 0.908 [5], [8] The main characteristics of the hull girder strength used for the dimensioning are: • • the hull girder moment of inertia in cm2-m2, amidships, is to be not less than

Nominal Design Corrosion Values (NDCV)
As stated in section 5A-3-1/1.7 of the FPI rules [5], oil company is to maximize the profitability of its business and one way to do it is by extracting more oil/gas with the least number of interruptions on its operation. One example of this interruption is sending the FPSO to a dry dock for maintenance and repair. Therefore, with these arguments in mind, it was selected the IACS corrosion margin.
According to Chapter 3 in IACS CSR Rules [63], the corrosion margin can be determined by the following equation.  Table 4-1 for one side exposure to that compartment and in this project tc1 equals tc2 and it is called ICM.

Bottom Structure
According to the FPI rules [5], the definition for the term "bottom shell plating" refers to the plating from the keel to the upper turn of the bilge for 0.4L amidships. Therefore, for the case of this project there are three kinds of plates to be specified: keel plate, cargo tank plate and ballast tank plate. The net thickness for these elements is provided by section 5A-3-3/7.3.1 [5] as: • An observation is also done for the keel plate according to the FPI rule. The net thickness of the flat plate keel is to be not less than that required for the bottom shell plating at that location by 5A-3-3/7.3.1 increased by 1.5 mm.    The main results are presented in Table 4-4. Moreover, as stated in the definition, the longitudinal is associated with an effective plating. Its width can be calculated in section 5A-3-3/7 in Figure 6 item a.
The following formula can be used and this results in an effective width of 728 mm. Generally, a T profile is used at this location, because of the high section modulus required. Also, the dimensions for the web vary from 400mm to 550m in length with a net thickness of 10 to 14mm and for the flange, the length varies from 300mm to 450m with a net thickness between 20 to 35 mm. Thus, by calculating the section modulus with this data in mind and the effective plate width and thickness, the final profile dimensions can be obtained and presented below.

Side Structure
As stated in section 5A-3-3/9.1 from the FPI rules [5], the net thickness of the side shell plating cannot be less than the maximum of the following thickness as specified below for the midship 0.4L: • Two observations are also done as well for the shear strake plate according to the ABS rule [7].
The first one is in the aspect of its minimum width, which must be above 1800mm for the ship's thesis. The second one is that the net thickness shall not be less than the thickness of the adjacent side shell plating.
Using as input the system of equations above and the observation for the shear strake plate, the net thickness can be calculated and are presented below (Table 4-5).  By the FPI rules, the longitudinal stiffener to be fitted in the side structure can be calculated by 5A-3-3/9.5. The main parameter is the net section modulus in association with the effective plating to which it is attached. It can't be less than obtained from the following equation. According to similar midship sections [64], generally an L profile is used in this location so that the water doesn't accumulate in the stiffener when the ship has a roll displacement. Also, the dimensions of the web vary from 350 to 650mm in length and a thickness between 11.5 and 13mm. While the flange has a width of 120mm and a thickness between 18 to 35mm in the bigger profiles. Thus, by calculating the section modulus with this data in mind and the effective plate width and thickness, the final profile dimensions can be obtained and presented below.

Table 4-7 -Final design for the side stiffeners
As can be seen in Figure 3-3 and 3-4, there are horizontal plates that connect the side shell plating of the ship with the side longitudinal bulkhead. The FPI rules [5] and Marine Vessel Rules [7] states that they can be modelled with the equations from the side shell plating with some modifications. They are presented below. . This is presented in section 5A-3-3/11.9 from FPI rules [5] • No internal pressure is considered, once the pressure slightly above and slightly below the plate is quite close. Therefore, only the external pressure is considered.   The profile selected for this element was as well the L, because of the same reason for the side longitudinal stiffener. Thus, by calculating the section modulus with this data in mind and the effective plate width and thickness, the final profile dimensions can be obtained and are presented below.

Deck Structure
According to the Marine Vessel Rules [7], the deck structures can be classified as freeboard, bulkhead and strength deck. This section will be focused on dimensioning the plates of the strength deck, which is the one that forms the top of the effective hull girder at any part of its length [7]. The net thickness can be calculated by 5A-3-3/9.5 from FPI rules and is presented below.
Compared to the bottom and side structures, the pressure input is not that great, once there are not any external or and the sloshing effects are minimum according to the FPI rules [5]. Thus, only the internal pressure from the tank is used, which is not that higher for this region.

Longitudinal Bulkhead
The net thickness provided in section 5A-3-3/13.1 in FPI rules [5] is valid both for the center and side longitudinal bulkhead plating. It can't be lower than as specified below.
• For this kind of structure, the sloshing and internal pressure are the relevant ones. Thus, the sloshing pressure is calculated as specified in 5A-3-2/11.5.1 of FPI rules [5].
Using this system of equations and the observation regarding the pressure, the net thickness of the plates for the center and side could be estimated and it is presented below. The lower the number of the plate, the close it is from the bottom region (Table 4-    The main results are presented in Table 4-18. Moreover, an effective width of 728 mm is used as in the bottom structure section. The characteristic profile for this region is also the L profile for the same reason stated in the side structure section. Moreover, the same range of pattern of longitudinal in the side structure was used to determine the most appropriate for both. After some iterations calculating the section modulus of different patterns, the final profile was selected for both bulkheads and the main characteristics are presented below.

Transverse Bulkhead
The transverse bulkhead formulas apply for the watertight ones located at the ends of the cargo tanks.
Its net thickness plating, which is in 5A-3-3/13.3, is to be not less than t, as specified below: ⁄ o 2 = 9.5 • For this kind of structure, the sloshing and internal pressure are the relevant ones. Thus, the sloshing pressure is calculated as specified in 5A-3-2/11.5.1 of FPI rules [5].
Using this system of equations and the observation regarding the pressure, the net thickness of the plates could be estimated and it is presented below. The height of each plate from the transverse bulkhead coincides with the ones from the center longitudinal bulkhead.

Web Frame Structure
The design of the web frame structure consists of three steps. The first one is the dimensions of the plates by section 5A-3-3/11.11 from the FPI rules. The second step is the calculation of its thickness and stiffeners. The last one is the determination of the dimensions of the brackets.
Section 5A-3-3/11.11 states that "In general, webs, girders and transverses are not to be less in depth than specified below, as a percentage of the span". The bottom and deck height enters in the category from 5A-3-3/11.11.1 "for deck transverses without deck girders for ship-type installations with centerline longitudinal bulkhead". This results in a value of 12.5% of lt, which is the breadth of the cargo tank (23.8m), or 2.975m. Once, this structure will have stiffeners to strengthen it, this value was rounded up to the next multiple of 0.85m. Consequently, the height of the bottom and deck transverse is 3.4m.
Another element is in respect of the longitudinal bulkhead region. In line with 5A-3-3/11.11.3, the depth of the web in this region is equivalent to 14% of ls, which is the distance in Figure 4      The main results are presented in Table 4  The last step to finish the dimensioning of the web frame structure regards the four brackets located in  Thus, assuming a width of the flange equals 365 cm and using the formulation in 5A-3-4/7.9.4, the flange thickness is approximately 38mm. Notice that this flange extends for the whole web frame structure as seen in Figure 4

Design According to the DNV Rules
Although the midship section is already designed by the FPI rules, the DNV is one of the most used in the world for different kinds of ships. Therefore, it is interesting to design the solution proposed by this classification society for two purposes. The first one is to check if the values founded by the first set of rules are realistic, once the results of the plates thickness are similar for all these highly used classification societies. The second purpose is to identify if the DNV Rules design gives a better design or not using as criteria weight of the midship section and the stress in the structure.
However, this section will be focused on the net thickness of the plates. This includes the plates from section 3.2.2. to 3.2.4 (bottom, side, deck and longitudinal bulkhead structures). The other elements such as stiffeners, the transverse bulkhead and the web frame structure and parameters such as stiffeners spacing and increased corrosion margin will remain the same.
The process to obtain the net and gross thickness of the different plates is similar to the one in section 3.2. First, check the material properties and their units that are required in the given formulas of the rules [9], [12]. Second, estimate the loads and accelerations for each section [9], [13]. Moreover, the "Environmental Severity Factors" (ESFs) used in FPI rules were considered the same in the design. The third step, is to estimate the hull girder strength design parameters such as section modulus and inertia of the section [9], [14]. With all this information in mind, the last step is to use all of this as input in the different formulas to estimate the thickness [9], [15], take the bigger one and round it up so it can have a practical thickness to be build.
The main characteristics of the steel used for the dimensioning are: • minimum specified yield point of the material in N/mm2, taken as 355 MPa for Grade H36 Steel [9], [12] • E modulus of elasticity of the material, may be taken as 206.000 N/mm2 for steel [9], [12] • 1 material conversion factor, taken as 1.39 for Grade H36 Steel [9], [12] The main characteristics of the hull girder strength used for the dimensioning are: • the hull girder moment of inertia in cm4, amidships, is to be not less than [9], [14]

Bottom Structure
The bottom structure dimensioning is based on Section 6 from [9]. It includes the dimensioning of the keel plate (minimum breadth, net and gross thickness), the cargo tank plate (net and gross thickness) and the ballast tank plate (net and gross thickness). The keel plate will be designed by following the equations C201 and C202 from Section 6. These equations give the minimum breadth and minimum thickness of the plate respectively and are shown below.

Side Structure
This section will cover the dimensioning of the side structures by following the formulae given in Pt3.
Ch1. Section 7 of the DNV Rules [9]. It includes the side plating (plate 5 to 12) and the shear strake plate (plate 13). The first elements will be designed by following the equations C101 and C102. These equations give the required and minimum thickness of the plate respectively and are shown below.
While the second type of plate will be designed by following the equations C201 and C202. The results are shown below (Table 4-27). Table 4-27 -Side plating net and gross thickness estimation according to the DNV rules

Deck Structure
This section will cover the dimensioning of the deck structures by following the formulae given in Pt3.
Ch1. Section 8 of the DNV Rules [9]. The strength deck is divided the same way as the bottom structure: region of the cargo and ballast tank. The results are shown below. Notice that the thickness is way below the ones obtained in section 3.2.3. Therefore, the deck region in this design will be more focused to be assured it passes the strength criteria (

Longitudinal Bulkhead
This section will cover the dimensioning of the deck structures by following the formulae given in Pt3.
Ch1. Section 9 of the DNV Rules [9]. The plates will be designed by following the equations C100, C102 and C104. These equations give the required, a general minimum and specific minimum for longitudinal bulkhead thickness of the plate respectively and the results are shown below (Table 4-29).

Strength Criteria
As specified in the beginning of chapter 5, a total strength assessment (TSA) of the structures must be carried out against the following three failure modes: Buckling, Material Yielding and Ultimate Strength.
All of the criteria will be done based on the rules [5], [9].
Moreover, the load conditions (shear forces and bending moments) to be considered in the buckling analysis and the ultimate check are the ones presented in Table 4

Buckling Analysis
In this project the axial compressive stress applied to the plate panels of the midship section will be studied. The method adopted is briefly described below and it is summarized in the flowchart of Figure   4-5.
• Guess a certain net thickness tn for the desired plate.
• The ideal elastic buckling stress may be taken as: o Where s is the spacing of longitudinal stiffener   Figure 4-5 -Iterative process to determine the final thickness of the plates for the buckling analysis

Yield Stress
The MARS 2000 is a software developed by Bureau Veritas (BV), which allows to input sections, bulkheads and torsion models, and compute the main geometrical properties of the section, the stress in the plates and stiffeners (primary and secondary) and the ultimate bending capacity all according to the rules [65]. By following the user guide [65]- [67], the midship section from the FPI design with its loads (bending moments and shear forces presented in Table 4-30) could be created and the stresses could be obtained. According to the FPI rules [5], the stresses obtained should be compared with the yield stress multiplied by a safety factor, which is 0.6 for the static case and 0.908 for the dynamic case.
According to the user guide [67], the permissible one stress for the static case from MARS 2000 is 243 MPa (a safety factor of 0.68) for the Grade H36 steel, according to the BV rules. Therefore, it will be used the safety factor from FPI since they are more conservative for both cases.

Ultimate Strength
The last check is the ultimate strength, which was done with the section 5A-3-4/5.3.3. from the FPI rule [5]. The check for the ultimate strength was done based on as stated previously by section 5A-3-4/5.3.3.
from the FPI rule [5]. It focuses more on the ultimate strength of a plate between stiffeners. According to the section, the following formulae must be respected: Where, • fLB -Calculated total compressive stress in the longitudinal direction for the plate • fLT -Calculated total in-plane shear stress, • fTB -Calculated total compressive stress in the transverse/vertical direction • fuL -Ultimate strengths with respect to uniaxial • fuT and fuLT -Ultimate strengths with respect to edge shear,

Final Design
In any project, one of the crucial objectives are is to design a product with the lowest cost as low as possible in order to make profit from it. In the shipbuilding industry, this can be achieved by building a ship or a platform without over dimensioning the whole structure. In this master thesis, the optimal design is the one with the lightest structure that satisfies all the strength criteria (buckling, yield and ultimate strength) imposed for both load conditions (static and dynamic) simultaneous. However, this isn't a simple task. For example, in this master section there are 37 plates whose thickness can vary between the minimum value imposed by the rule and a maximum value, which will be considered in this thesis as 32mm. In this aspect alone, finding an optimal solution is complicated. In addition to that, each individual plate interferes directly with the strength criteria from the other plates, because they can modify the neutral axis height and the moment of inertia and, therefore, the normal and shear stress in the structure. Thus, in order to obtain a feasible solution, it was used a solver. The optimization problem used to do this task is described below. The results from the solver can be seen in Table 4-31 and 4-32. As it can be seen, the solution found complies with most of the strength criteria. The exception is the deck plates for the static load case, which are 3%-5% higher than the allowed (The maximum allowed is 355*0.6 = 213 MPa). Despite this exceed, the solution is feasible for two reasons. First, the condition in which this exceedance of the onboard safety factor takes place is not the worst of all. This helps as a reference for dimensioning, but the important thing is that the plates are able to withstand the loads of the worst condition without exceeding the yield stress, which in fact happens. The second reason is that the vast majority of plates meet the buckling and yield requirements for the calm water condition and all of them are within the stipulated limits for the buckling, yield and ultimate strength criteria for the worst condition, which is that where there are loads of calm waters plus waves.
Moreover, most the of values range from 0.4 to 0.5 and 0.75 to 0.85 from the yield stress for the static and dynamic load case respectively. This is a good sign that the achieved solution is in fact a good one because the structure is not over dimensioned.

SPS PLATES SCANTLING
This chapter has the purpose of detailing the dimensioning of the SPS plates according to DNVGL-CG-0154 and Lloyds guidelines rules [18], [19]. The basis of the SPS design principles is that SPS structures should have strength at least equivalent to that of a conventional steel structure that performs the same function. So, the approach in this case is to take the minimum conventional steel scantlings for structural items, calculated from the Class Rules, and then calculate "equivalent" SPS scantlings.
This chapter is divided into three parts. First, it will be discussed the initial assumptions such as the mechanical properties of the steel and core, plate dimensions, loads considered and minimum thicknesses for the regions to be analyzed. The second part will report the method used to achieve the final design of each composite plate. The last one will display and discuss the results.

Initial Assumptions
The SPS plate is a structural composite material composed of steel and a polyurethane elastomer. Thus, it is essential to define the mechanical properties of each material for the design of the panel. According to the DNVGL-CG-0154 [18], the mechanical properties of the steel are the same as from chapter 4 (High steel grade 36 [6]) and it will be considered the same grade of steel, Grade H36. The core characteristics were obtained through the DNVGL-CG-0154 [18], the Lloyd's Guidelines [19], a master thesis from Delft University [48]. They are shown in Table 5-1. Moreover, the bending moments and shear forces to be considered are the ones presented in Table 4-30, which are the static and dynamic load cases, and the final plate should withstand both cases. Also, the plates dimensions will be similar to the ones from the conventional steel structure. It will be a square plate with a length of 3.4m.
As stated in the DNVGL-CG-0154 [18], there is a range of thickness in which the steel plates and core elastomer are submitted. Typically, it is from 3 to 30 mm for the first element and from 15 to 100 mm for the second. In addition to this constrain, there are also a minimum net and gross thickness for the top and bottom steel plate of the composite panel. They are dependent on the region of application (i.e., bottom structure, side, deck and so on) and the corrosion addition. Using the formulae given in the DNVGL-CG-0154 [18], the minimum net and the corrosion margin could be obtained and are given below (

Design Process
As stated in section 3/1.1.1 from DNVGL-CG-0154 [18], "Scantlings of steel sandwich panels as a part of a hull structure shall satisfy buckling, stress and deflection criteria considering local pressures and inplane stresses from all relevant hull responses such as global hull girder bending, primary girder/floor bending, double hull bending etc.". In addition to this, the steel sandwich panels shall also be designed according to the following principles: • to avoid major plastic yielding of steel face plating • to avoid major plastic yielding of core material • to avoid failure of the bond between the core and the steel face plates.
• to be lighter than the traditional steel configuration.
The Lloyd's Register guidelines [19] proposes a flowchart ( Figure 5-1) as a methodology that will be used in this chapter. Moreover, in the Appendix B of DNVGL-CG-0154 [18] there is a support Excel tool developed by DNV for calculating all these strength criteria properties and comparing it with the rules requirements. This tool was used to make sure the classification society method was followed properly.
Also, in summation of the basic principles, it is interesting to keep the thickness of the core as low as possible in order to make the panel cost-effective.

Buckling Criteria
The present buckling criteria used to design the SPS panel is based on the DNVGL-CG-0154 [18], DNV-RU SHIP Pt3. Ch8. [16] and on DNVGL-CG-0128 [36]. It is described in the following items. • Guess a certain net thickness for the core and steel plates. • Determine the eigenvalue Λ , which is dependent on the ratio between the design loads. For example, if the panel is subject to bi-axial compression or compression/tension combinations or to a single in-plane shear load or to bi-axial and in-plane shear loading in combination.
• Determine the buckling load factor Λ • Determine the actual buckling usage and compares it to the allowed one.

Lateral Pressure Criteria
The other strength assessment is related to the maximum lateral pressure that the panel can support.
This is related to different acting stress on the panel such as the maximum equivalent stress in the steel face plate, the maximum shear stress in the core and the maximum interface shear stress. The procedure to determine all this parameter will be briefly described below.
• For the same thickness, bending moments and shear stresses of the buckling section, calculate the lateral pressure for each plate i of the traditional steel net structure.
• Calculate the lateral plate deflection for that design pressure (w) • The last parameter is the interface shear stress that can be calculated the same way as the core shear stress and it is also compared to the allowable load factor.

Results
Therefore, combining the several information in the previous section such as the hypothesis adopted, the goals established, the methodology to follow and the strength criteria to judge, the results to obtain the thickness for the steel and core of the composite plate could be achieved. This is shown in the next tables (Table 5-3 to 5-6). Notice that for the same load, different combinations of thickness pass the several criteria imposed. Therefore, some weights to make the final decision must be set. For this master thesis, the panel selected was the one that had the best combination of least core material used, a lighter structure, lower stresses for each load case. However, it is important to think about how the new neutral axis and moment of inertia will impact the buckling, yielding and ultimate criteria described in the previous chapter for the other plates. Therefore, the final plate thickness (Table 5-3) is the one that could withstand both load cases (static and dynamic) and also provide all steel plates to pass the buckling, yielding and ultimate criteria from FPI ABS [5]. The new design passed both the buckling and yield criteria ( Table 5-5) and the ultimate strength as well (Table 5-6). Thus, the thickness from Table 5-3 were selected as the final ones. Also, this implies new geometric properties for the structure that are shown in Table 5-4.

Hull Weight Estimation
This section has as the main goal to estimate the hull's weight for both steel and SPS arrangement. This will be based on the thickness and dimensions of the structural elements obtained in chapter 5 and 6.
Moreover, since this master thesis didn't design the structural elements for the aft part where the machines and pump would be located. Generally, this part corresponds between 5% to 15% of the cargo tanks weight (longitudinal and transversal) and for this project it is considered as 10%. The methodology to calculate each part of the structure is presented below and the results are in Table 5-7. As can be seen, the final hull with the SPS panels is 2.8% lighter than the traditional steel hull with the bottom structure being the main responsible for reducing the total weight. A reduction of, approximately, 1250 tons. This is due to the fact that it is the area with one of the highest thicknesses for the steel panels and with the stiffeners with the highest weights.  Therefore, the objective of this chapter is to discuss the finite element analysis done in this master thesis for the traditional steel and the SPS structures. The chapter is divided into two parts. The first one will show the steps to build the FEM (finite element model), which includes the geometry, mesh, boundary condition and loads applied. The second part regards the structural response for each model for the different loading conditions. This chapter is based on the guidelines of the FPI rules from Part 5A-3-A4 [5] and on the chapters from the book The Finite Element Model [68]- [73].
Moreover, for an FEA, many commercial software is available to do so, but, for this master thesis, the ANSYS APDL structural package was used with a student license. Once it is a student version, this has some limitations. For example, as specified on the company's website, the newest version 2021 R1 have the capacity of 128K nodes/elements for structural analysis. This was be taken in mind during the modelling in order to not surpass the software limit.

General Guidelines
As stated previously, the FPI rules [5] have some requirements for the finite element model that must be followed. First of all, the strength analysis is based on a "net" ship approach. Therefore, the nominal design corrosion margin must be deducted from the scantling for the FEA. Second, the analysis must be a three-dimensional global model of three cargo tank lengths located at about amidships with two frames fore and aft of the two end bulkheads ( Figure 6-1, 6-2 and 6-3). Thus, all primary load-carrying members are to be modeled as shown in Figure 6-1, which includes: transverse web frames, longitudinal girders, horizontal girders, side stringers, and centerline ring frames, etc. This is due to the fact that the 3-D global FE analysis's purpose is to determine the overall structural response of the hull girder structure, including primary and secondary bending, and also to obtain appropriate boundary conditions for use in the local fine-mesh FE analysis of local structures.

Geometry
According to [40], [42], [74], the equivalent thickness model is an accurate model to predict the ultimate strength and the stress distribution of the structure. This model consists in using thicker plates that have an inertia equal to a plate with longitudinal stiffeners. There are two benefits of it. The first one is the simplification of a complex geometry. The second one is that with fewer nodes and elements, less computational power is needed and it is easier to achieve a converged solution. Therefore, the steel plates used both in the traditional and with the SPS plates structures are shown in Table 6-1. Ltd [75] and others [2], [47], [76]- [78]. Generally, one of the combinations below shall be applied.
• A single layer of layered shell elements through the thickness of the entire sandwich material with isotropic material properties for each layer • (layered) Shell elements for the faces and solid elements for the core with isotropic material properties for both element types.
• Solid elements for both face and core with isotropic material properties.
The difference between them is related to the size of the model applied. According to private conversations with the engineer from SPS Technology and the DNV-CG-0154 [18], for more complex models, such as the one from this thesis, an accurate solution can be obtained by using the single layer of shell elements for the same reasons from the equivalent thickness model described above. Although the use of solid elements can be modelled the geometry to the degree of detail wanted, this implies a very large number of nodes and elements, and hence the solution time will be very long.

Element Type and Mesh
For finite element modeling of a steel plate, it is typically represented by shell (or bending plate) elements and, in general, the plate element mesh is to follow the stiffener system as far as practicable, hence representing the actual plate panels between stiffeners. Some examples of the mesh spacing can be seen in Figure 6-4. Method [71] and FPI guidelines [5], for the steel plates, the best type of element is the shell one.
Furthermore, the most suitable for the scope of this master thesis is the SHELL181. It is an element based on the Mindlin plate theory and has four nodes with six degrees of freedom at each node: translations in the x, y, and z directions, and rotations about the x, y, and z-axes as seen in Figure 6-5.
Moreover, it is well-suited for linear, large rotation, and/or large strain nonlinear applications and may be used for layered applications for modeling composite shells or sandwich construction.
Therefore, for the middle tank, it was used an element size of 0.85m; and for the aft and fore tank, the size was 1.7m. These numbers are in accordance with the FPI guidelines, once it says that for the middle tank the size shall be equal to the distance between the longitudinal stiffeners and a maximum of 2 times these spaces for the other tanks.
In addition to that, it was done a sensitivity analysis in order to check how the results vary with the mesh size ( the size of 0.85m. There are 5 main variables that will be studied: neutral axis height from the baseline, the normal stress from the bottom and deck plates, the spring reaction at the ends of the model (presented in section 6.1.5) and the time required to run the simulation for the static and ultimate load case. The first three parameters of the reference column were obtained through the beam theory using the geometric properties and loads, the spring reference is detailed in section 7.1.5 and the time static/ultimate was considered as reference the mesh proposed by the FPI rules. As it is expected, the refined mesh provided more accurate results compared to the FPI mesh. However, the computational time required to perform it was 72% and 116% more for the static and ultimate load case respectively.
Moreover, as it can be seen from the column Difference FPI Mesh, the results obtained aren't that distance from the reference one. Thus, the best tradeoff between accuracy of results and computational time required is the one from the FPI rules and this is the one that will be used for all models. The model meshed can be seen in Figure 6-6. Table 6-3 -Comparison between the FPI rules and a refined mesh As will be explained in section 6.1.4, it will be used uniaxial springs at the ends of the model as the boundary condition. According to the ANSYS Manual [79]- [81], one suitable element is the COMBIN14, which is a uniaxial spring-damper as seen in Figure 6-5. In order to become a spring only, the user must input as zero the damping coefficient.  Comparison between the FPI and refined mesh

Materials Properties
The material used in this project is the same as the design process, which is high strength steel with a yield stress of 355 MPa. As seen in ABS material properties [6], the ultimate strength for this steel depends on its grade (AH, DH, EH or FH), which is between 490 MPa to 620 MPa. However, the ANSYS requires the values from the true curve of stress-strain to calculate the stress/displacement of the plates above the yield stress. According to [34], [82]- [84] the grade AH and DH has the highest and lowest ultimate strength respectively as can be seen in Figure 6-7. Therefore, for this master thesis, it was assumed that the structure would be constructed with DH steel. The engineering curve could be obtained by using the relation = (1 + ) and = ln (1 + ). Moreover, according to the Ansys Guidelines [79], [80], one suitable way to model this true curve for steel is the MISO model

Boundary and Load Conditions
According to the FPI guidelines [5], the boundary conditions to be applied at the ends of the cargo tank FE model are ground spring elements (i.e., spring elements with one end constrained in all 6 degrees of freedom) with stiffness in global z degree of freedom are to be applied to the grid points along the deck, inner bottom and bottom shell and with stiffness in global y degree of freedom are to be applied to the grid points along the vertical part of the side shells, inner hull longitudinal bulkheads and oil-tight longitudinal bulkheads. This is shown in Figure 6-8. In addition to this, ground springs with stiffness in global x degree of freedom were applied to these points to prevent rigid body motion to the model. The objective of the application of the loads is to replicate as fairly as possible the vertical shear force and vertical bending moment distributions on the three cargo tanks FE model. Therefore, it is applied at the end of the model the bending moment and vertical force corresponding to that location as Figure 6-9 gives as an example. One observation is done for the shear force, it is only applied to the vertical elements to prevent unrealistic deformation in the horizontal plates as [79] states. Also, according to [79], [80], the moments and forces shall be applied as: Where: • σi is the normal stress at the element i

Static Loads
Before the ultimate strength was calculated, the static load from one load condition was applied (draught equal to 26.8m) for each structure. There are two objectives for this step. The first one is to observe if the model is well calibrated. This is done by comparing the results with the ones expected by the Euler-Bernoulli beam theory and by analyzing how much the reaction force at the spring is related to the resultant forces applied. If the model is well calibrated, the stress at the structures should be similar to the beam theory and the reaction forces at the springs should be lower than 0.3%. The second reason for applying the static load first is because the ultimate strength loads will be obtained using as reference the static load. For example, the input for the ultimate strength analyses can be 2 times the load applied at the static load case.
The loads applied in the tanks for each load case are based on ones obtained in chapter 6. In order to facilitate the understanding of the inputs, it will be divided into two types of loads: fixed and variable.   The first result to compare between structures is the vertical displacement in Table 6-5 and Figure 6-11.
As it can be seen, the steel structure has a bigger deflection than the SPS structure, approximately, 19% bigger. This can be explained because the SPS structure has a higher moment of inertia (2391 m4) rather than the steel structure midship section (2191 m4), because as higher the moment of inertia, the greater is to bend it. Table 6-5 -Vertical static displacement in mm for both types of structures in sagging  The second result is a comparison between the FEM and beam theory presented in Table 6-6. The points taken to be analyzed are at the midship section of tank 2, where the shear stress is zero according to the shear force distribution from HydroD, in order to minimize the boundary effects on the forces by the extremities. At first, it can be seen that the model's loads are calibrated because the spring reaction is lower than the reference. Also, the normal stress and neutral axis are similar to the ones expected by the beam theory for both models. The difference between them can be explained by the slight change of the neutral axis and because the beam theory doesn't consider the plate's thickness, which can change the stresses. Also, notice that the normal stress for the SPS model is calculated for the steel plate, in order to obtain the stress at the core, the stress should be multiplied by the ratio of the Young Modulus ( = 750 206,000 ). This results in a stress of 0.78 MPa, which is 3.7% from the yield stress of the core.   Comparison of static load case for both arrangements [21] by a phenomenon called warping and it is related to the distortion of the longitudinal stresses in the cross-section due to the transverse shear or torsion.

Ultimate Loads
According to Paik (2018) [64], "A ship's hull in the intact condition will sustain applied loads smaller than the design loads, and, in normal seagoing and approved cargo loading conditions, it will not suffer any structural damages such as buckling and collapse. However, the loads acting on the ship's hull are uncertain both due to the nature of rough seas and because of possibly unusual loading/unloading of cargo, the latter due to human error. In rare cases, applied loads may hence exceed design loads and the ship's hull may collapse globally". Therefore, when the structural safety of a ship's hull is considered, the ultimate hull girder strength must then be accurately evaluated. This was first done in chapter 4 and 5 as a first check. However, closed form methods, specially from classification societies, tend to underestimate the real ultimate bending capacity of structures in order to achieve a safety design [85].
Thus, the FEA is a powerful tool to estimate the ultimate bending moment of the hull, because an ability to better assess the true margin of safety should also inevitably lead to improvements in regulations and design requirements.
Moreover, as it was described in chapter 2 and section 6.1.5, the ultimate strength load was obtained by increasing the load from the static load gradually until the structure reaches either a stress value near 665 MPa, which is the maximum stress from the true stress-strain curve of DH36 steel, or there was a collapse of the plates due to buckling. This is due to as applied loads increase beyond the design loads, structural members of the vessel's hull girder buckle in compression and yield in tension. The vessel's hull girder can normally carry further loading beyond the onset of limited member buckling or yielding, but the structural effectiveness of any such failed member clearly decreases, and its individual stiffness can even become "negative," with their internal stress being redistributed to adjacent intact members [64].
The FEA showed that the steel structure and the structure with SPS could withstand 1.94x and 2.45x the static load, which is a 26% increase. This can be explained by the moment of inertia, which is inversely proportional to the normal stress. Therefore, the bigger the inertia, the bigger the bending moment needs to be to reach a specific value. And, since the SPS structure has a bigger one than the steel structure (almost 10% bigger), this structure can reach higher values of bending moments. Table   6-7 summarizes this information. Moreover, the moment-curvature graphic was also done to compare the ultimate bending capacity between the classification society rules (ABS/DNV) with the finite element results (Figure 6-14). Also, this graph shows the ULS for both types of arrangement and where each plate starts to buckle as well.
As it can be observed the ULS for the steel and SPS according to the FEA are 11% and 24% higher respectively than the ones proposed to the classification society rules, because they are more conservative about its estimation. Comparing the values obtained with the same comparison made in   [85], they have a similar behavior and values regarding the comparison between the FEM and the CS. The points circled in the graph are where the buckling starts to occur in each plate of the side shell. As can be observed, both reach the ULS because of the buckling of plates, however, the bottom plates, which were in tension, could support even more load, once they didn't reach the value near 665 MPa.
The main difference between both structures is where the buckling happens. For the steel design, the buckling happens in the deck plates. This could be previously visualized in chapter 5, in which the deck plates obtained the highest factor for the buckling and ultimate strength criteria. However, for the SPS design the region where the plates fail first is the side shell. Similar to the steel ones, they were the ones with the highest values as well for the buckling and ultimate strength criteria. Moreover, their thickness is lower than the deck plates, therefore, their resistance for withstanding buckling is lower. This image phenomenon can be seen both in Figure 6-15, 6-16 and 6-17. As it can be seen in Figure 6-15, buckling occurs in the extremities of the side shell for the SPS plates and for the steel one is the deck panels.  The last result is related to the stresses related to the core of the SPS plates. As seen in chapter 6, the panel has several criteria regarding the core yield and shear. ), the yield stress of the core was 1.82 MPa, which is 8% from the yield stress This resulted also in a shear stress of 3.95 MPa and an interface shear stress, which is related to the bond between core and steel plates, of 4.03 MPa, while the allowable ones are 12 MPa and 7.5 MPa respectively. Therefore, the panel will not collapse because of the core.
To summarize, the SPS design obtained a better structural performance. In the static condition, the stress intensity in the SPS is overall lower than the steel, because of its higher moment of inertia.
Consequently, the stress intensity of the bottom and deck plates were reduced by 1.4% and 15% using as reference the steel arrangement. Moreover, in the ultimate bending capacity, the SPS arrangement could withstand 24% more load than the steel one and the main difference between the models is the region where one plate collapses.
However, it is important to mention some limitations regarding the finite element model. For example, it was only analyzed the sagging condition. Also, due to the size/complexity of the model's geometry and the computational power used, some non-linear properties were simplified in order to achieve a solution.
These kinds of limitations can influence the structural behavior of the model and, thus, could be individually analyzed for a deeper understanding of the structure and its ultimate bending strength.

CONCLUSIONS
This work focused on the research for an initial design and construction of hulls of FPSOs with sandwich plates without stiffeners in the bottom plate and longitudinal bulkheads of cargo tanks. The main reason is to have the walls and bottom of the cargo tanks free of stiffeners to allow remote cleaning and thickness gauging of bottom and bulkhead plates using autonomous equipment.
The first step was to estimate the global loads for the case study FPSO in chapter 3. This was done with the help of GeniE and HydroD, in which loads and equilibrium conditions were assumed, and 5 cargo/ballast distributions were calculated. One of them is for the empty cargo condition, three intermediate draughts and one for the full cargo condition (10/12 cargo tanks filled). Therefore, with the balance between loads and buoyancy for each equilibrium condition, the bending moments and shear force distribution could be obtained.
In chapter 4 the initial design of the traditional steel arrangement was made by using the results of chapter 3 and guidelines of the ABS FPI rules [5]. This design was also compared to the DNV one to check possible changes and in the last section, the total strength assessment was made. In general, the design obtained passed all the criteria of yielding (static and dynamic load case), buckling and ultimate strength proposed. The only exception was the deck plates at the static load case, which obtained a factor of 0.62 from the yield stress. This design was considered acceptable because it is close to the maximum safety factor (0.6 for the static load case) and, since the static load case won't get near the yield stress, it is very unlikely that these plates permanently deformity plastically.
In chapter 5, the design of the hull with SPS panels was made by using the steel panels dimensions and properties with the guidelines of DNVGL [18]. The main results were SPS panels that can withstand both static and dynamic load cases as the steel arrangement, but with a lighter structure. Also, it was calculated FPSO's hull weight for both types of arrangement and it was observed that the SPS one was overall 2.8% lighter than the steel one primarily due to the bottom structure. This happens because the bottom structure has the plates with the highest thickness and with the more robust stiffener for the traditional hull structural arrangement.
Chapter 6 focused more on the construction of a finite element model for the analysis of the ultimate strength for both arrangements: traditional steel and SPS panels. After the creation of the model, the static load case was used as a load condition in order to validate the model using as reference the results from beam theory. Both steel and SPS models achieved results similar to the ones predicted.
Following that the ultimate analysis was done and the main result is that the structure with SPS panels could withstand 26% more load rather than the steel one. This could be explained because the first structure has a higher moment of inertia.
Therefore, to summarize it, the SPS design demonstrated a superior structural performance over the traditional steel one. However, further research in the following areas would be interesting: • Perform a local ultimate analysis for the SPS plates. This would provide more insights into the buckling and ultimate strength of the panels for the load cases • Study the building process to investigate how the SPS panels could be efficient build.
• Research about the financial attractiveness of this new panel both during the building and operation step in order to know how the different kind of costs varies.
• Research different fillings materials and methods for the polyurethane in order to have alternatives.
• Research the impact of this new technology during the operational life of the FPSO, converting inspections requirements, maintenance and repair works.

APPENDIX 1
Figure A-9-1 -Lines plan from the hull