Ultimate Strength and Optimization of Aluminum Extrusions M.D. Collette(AM), X. Wang(M), J. Li(M) Recent large aluminum high-speed vessels have made use of custom extrusions to efficiently construct large flat structures including internal decks, wet decks, and side shell components. In this paper an efficient method for designing and optimizing such extrusions to minimize structural weight is presented. Strength methods for extrusions under in-plane and out-of-plane loads are briefly reviewed and shortcomings in existing aluminum strength prediction methods for marine design are discussed. A multi-objective optimizer using a genetic algorithm approach is presented; this optimizer was designed to quickly generate Pareto frontiers linking designs of minimum weight for a wide range of strength levels. The method was used to develop strength vs. weight Pareto frontiers for extruded panels for a main vehicle deck and a strength deck location on a nominal high-speed vessel.

KEY WORDS: aluminum; ultimate strength; optimization; extrusion; genetic algorithm

INTRODUCTION The current commercial and military interest in large high-speed vessels has resulted in the development of monohull, catamaran, and trimaran designs between 70m and 130m in length for both transportation and combat roles. In this design space, deadweight is restricted, and the vessel operates under a constant trade-off between cargo capacity, achievable speed, and achievable range, quite unlike conventional displacement vessels. Given these restrictions minimization of lightship weight, and hence structural weight, is of great significance in the design of the vessels. Most vessels in this category have been constructed out of aluminum to reduce structural weight. In addition to being a lighter material than steel, aluminum is marked by its ability to be extruded very economically into custom profiles. This ability gives the designer the freedom to replace conventional plate and welded-stiffener panels with extrusions where the plate thickness may be varied, or where the plate and stiffener construction may be replaced by a sandwich-type structure. Such extrusions can be used economically on large flat deck structures such as cargo and passenger decks, cross-decks for multi-hull vessels including the wet deck forming the lower connection between the hulls on a multihull vessel, and the side shell above the waterline. Such extrusions offer the possibility of weight savings, along with easier welding and reduced complexity of the resulting structure. A conventional panel and various types of extruded panels are shown in Fig. 1. The conventional panel, constructed by welding stiffeners to a large, flat plate, is shown in the upper left-hand corner. On the upper right, an extruded panel is shown where the stiffener and attached plate is extruded as a single unit. Such extrusions are then joined by butt-welds to form a panel. Other types of panels that have found favor include a hat-type stiffener, shown on the lower left of Fig. 1, and a sandwich-type extrusion, shown on the lower right. To optimize the design of such structures, ultimate limit-state design is the preferred approach. Using limit-state design to calculate the loads at which the structure will actually fail in service, a more rational risk assessment and comparison of

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alternatives can be made in the optimization process. At the present time, ultimate strength methods are only available for conventional plate and stiffener panels. The more complex, yet potentially more efficient, designs made possible by extruded aluminum cannot easily be considered. This lack of tools and assessment techniques means that designers are restricted in the types of structures they can consider. Additionally, robust methods for performing such optimization are required if optimization is going to become a practical tool for use in design offices. For such design applications, optimization output would ideally be available quickly using only standard PC computers.

Fig. 1: Type of Aluminum Extrusions Considered This paper presents a brief survey of existing ultimate strength techniques for plate and panel components that could be used to predict the strength of novel extrusions. The shortcomings in the current state-of-the-art are also noted. An optimization approach is then developed using closed-form strength formulas and extrusion weight as conflicting objectives. A sample application of this approach is made to a nominal high-speed vessel, considering extrusions subjected to both in-plane and lateral loadings, and conclusions on the weight benefit of different extrusions and the suitability of the proposed approach are discussed. This paper is based on a larger body of work recently sponsored by the Ship Structure Committee and published as report SSC-454 (Collette et al. 2008.)

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STRENGTH METHODS To be able to accurately compare different structural designs, it is desirable to use a limit-state-based approach in place of allowable-stress design. Paik and Thayamballi (2003) have outlined reasons for preferring limit-state design over allowable stress design for steel marine structures, and their reasons are equally valid for aluminum structures. When considering limitstate formulations, it is important to note that aluminum as a material has many differences from steel. Marine structures are normally assembled from one of two series of aluminum alloys the 5xxx-series which is a work-hardened alloy series, and the 6xxx-series which is a heat-treated alloy series. These two alloy series also have important differences for limit-state design. For structural limit state formulation, three of the most significant material differences are: 1. Aluminum has roughly 1/3rd the elastic modulus as structural steel, making it more susceptible to compressive instability than steel for similar structural profiles. Many limit-state formulations explicitly include the material’s elastic modulus, but others which use geometric properties directly, such as breath-to-thickness ratios, can not be moved from steel to aluminum without further adjustment. 2. The shape of the aluminum stress-strain curve is generally more rounded than that of steel. Typically, no defined yield point can be identified in the material stress-strain curve. Thus, a 0.2% offset proof stress is used in place of the yield stress. The 0.2% offset proof stress is defined as the stress where the plastic component of the strain is 0.2%. The 5xxx-series alloys have a particularly rounded stress-strain curve, and their local tangent modulus may fall significantly below the elastic modulus before the proof stress is reached. This indicates that these alloys may be more prone to buckling in the inelastic regime than equivalent steel or 6xxx-series alloy structures. As the 5xxx-series alloys are strain hardened, the proof stress is often higher in tension than compression, a fact often overlooked in marine structural analysis. Most extrusions utilize 6xxxmaterials for reasons of producability and cost, though 5xxx extrusions can be found in limited applications. 3. Both 5xxx and 6xxx series alloys become weaker in a local region near the weld when welded by fusion welding. This local weak region is known as the heataffected zone (HAZ). For 5xxx-series alloys, the HAZ material is typically similar to annealed material. For the 6xxx-series, the HAZ is typically an over-aged region in terms of the alloys’ precipitation hardening. This means that while the proof stress is reduced for both the 5xxx and 6xxx HAZ regions, the 6xxx series suffers a larger loss of ultimate tensile strength in the HAZ than the 5xxx-series alloys. An additional consideration when moving from standard rolled profiles used in steel vessel construction to custom extrusions is that careful consideration of the all potential failure modes of the extrusions is required. Some failure modes – such as local web buckling – may have been designed out of standard steel rolled

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shapes; thus, traditional steel-based strength approaches may not address them. However, when designing custom extrusions via numerical optimization, it is important that all failure modes be included. While nonlinear finite element (FE) analysis has proven to be a useful tool for analyzing limit states in aluminum structures (Rigo 2003, Paik and Duran 2004), the model set-up - particularly initial imperfections and residual stresses - and the computational requirements make nonlinear finite element methods difficult to apply with optimization techniques outside of research settings. For that reason, the current study focuses on simplified limit state formulations that are available in closed-form or simple iterative expressions. Methods focused on predicting both the strength of individual plate components and assemblies of plate components have been reviewed. For both types of methods, approaches to predict in-plane strength and strength from lateral loads are reviewed. After reviewing individual plate elements and assemblies of plate elements, a strength prediction approach is outlined for use with the optimizer later is this paper, including simplified load combination relationships.

Plate Elements Uniaxial compression Failure by instability in compression is one of the dominant limit states for most ship structures. A key component of such failure is the instability failure of individual plate elements – typically, the plate between stiffeners in conventional vessel construction. While such collapse can occur under combined loads, in many large vessel applications, uniaxial compression from primary hull-girder bending is the most significant load source. A useful parameter in classifying the susceptibility of plate elements to compression instability is the plate slenderness factor, β, defined as shown in Eq. 1: β =

b

σ 02

t

E

(1)

Where b is the plate width, perpendicular to the applied loading, t is the plate thickness, σ02 is the yield or proof stress of the plate material and E is the elastic modulus of the plate material. β does not include any effect of the actual aspect ratio of the plate, although for ship structures the conservatism resulting from assuming infinite aspect ratios is typically small. Simplified methods were examined that could predict failure under uniaxial compressive loading, and these were compared to two data sets available in the open literature. The first is a data set of primarily aerospace alloys tested by Anderson and Anderson (1956). This data set contains 58 plates in aerospace alloys 2024, 2014, and 7075 in the T3 and T6 tempers. The plates tested by Anderson and Anderson were made of thin sheet material, normally 1.59mm thick, and were long enough for five buckling waves or more to form over the length of the plates. The test program covered b/t ratios between 14.6 and 58.2, and β values between 1.1 and 4.84. None of these plates were welded. Thus, the Anderson and Anderson data set represents useful data on heat-treatable aluminum alloys in general, but not with the HAZ

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regions around welds, residual stress, and imperfections typical of welded marine construction. The second plate data set was reported by Mofflin (1983), and consists of a series of 76 plates in the civil and marine alloys 5083 and 6082 tested in the United Kingdom in the O, F, and TF (roughly corresponding to the modern T6) tempers. The test results by Mofflin are generally similar to plates commonly encountered in aluminum vessels. These plates were all approximately 6mm thick and were tested with a length-to-width ratio of four, with compressive displacements applied along the short edges of the plate. Simple support boundary conditions were approximated as closely as possible during the test program, with unloaded edges free to pull in so large membrane stress would not develop. The unloaded edges were not constrained to remain straight in-plane. Two levels of initial out-of-plane deformations were introduced into the plates, with maximum values of roughly 0.001 times the plate width for small deformations, and 0.005 times the plate width for large deformations. Mofflin simulated the effects of welding on the plates by making TIG passes along the long, unloaded edges of certain plates without depositing weld metal. Two levels of welding were used in the study, defined as “light” and “heavy”, with heat inputs roughly corresponding to MIG fillet welds of 3mm and 4mm leg lengths, respectively. Of the total of 76 plates tested by Mofflin, 66 were either un-welded or had welds simulated in this fashion. A further ten plates had MIG welds made in the middle of the plate, perpendicular to the applied loading. However, these plates were not investigated in the current study. The test program covered b/t ratios between 20 and 85, and non-dimensional slenderness ratio, β, values between 0.93 and 5.41. A noted advantage of the Mofflin test program is that compressive, as opposed to tensile, material properties were recorded. This is important for strain-hardened aluminum alloys, such as those in the 5xxx-series, where tensile and compressive proof stresses may differ substantially. A key weakness of both data sets is that neither included plates with welds across the shorter, loaded edges. For stockier plates with low β values, Kristensen (2001) showed via numerical analysis that these welds are especially damaging to plate strength. A range of simplified ultimate strength methods from both marine and civil engineering codes were tested against these plate test results. The first is a group of methods that predict the uniaxial compressive strength of plate element via variations on the equation shown in Eq. 2a below: σU C1 C 2 = − 2 (2a) σ 02 β β Where σU is the ultimate compressive strength of the plate and C1 and C2 are constants to be determined. The equation is typically further limited so that the ultimate strength can not exceed the material yield or proof strength. The U.S. Navy (Naval Sea Systems Command 1982) has historically used a version of this expression with C1 = 2.25 and C2= 1.25 in their design data sheet DDS 100-4 for steel naval vessels. Faulkner (1975) proposed a slightly more conservative version of this equation that has been widely accepted with C1 = 2.0 and C2= 1.0 for steel vessels. More recently, Wang et al. (2005) extended this method for aluminum

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plates, using Faulkner’s values for C1 and C2, and adjusting the value of β used to an effective β that accounts for the impact of HAZ around welds in the plate. These results were based on a systematic series of finite element analysis. The Eurocode 9 (CEN 1998) civil engineering code strength formulation also follows the general form of Eq. 2a, with the constants further modified to reflect boundary conditions, welding, and alloy types. Furthermore, the β factor in the bottom has been modified so that elastic modulus does not figure in the equation, but alloy strength does. For an aluminum-only code where the elastic modulus is roughly constant between alloys, this achieves the same purpose as the full β expression from Eq. 1 above. The Eurocode 9 approach is shown in Eq. 2b below. σU C1 C2 σ 02

=

b

σ 02

t

250 MPa

−

b t 

  250 MPa  σ 02

2

(2b)

The Eurocode 9 formulation also includes some additional checks for welds which may further reduce the strength of plate. Three other methods were also reviewed, which use different formulations for predicting strength. The first was a group of regression equations proposed by Paik and Duran (2004). These were based on a series of nonlinear finite element analyses carried out on 5383-alloy plates, with welds on all four plate edges and initial imperfections with amplitude of 0.009 times the plate width. The impact of welds on the plate strength was accounted for by modifying the plate’s β value, based on a volume-average proof strength including HAZ and base material regions. Kristensen (2001) similarly proposed a series of regression equations based on finite element analysis of aluminum plates made of several different alloys. Kristensen fitted these equations to the lower band of FE results for alloy type, aspect ratio, and welded strength. Kristensen’s regression equation used in this paper assumed a HAZ width of 25mm, and welding was present on the loaded edges of the plate. These assumptions are most likely conservative for the current data sets. Kristensen also addressed the case of transverse compression in the plate elements, where the loaded and unloaded edges are interchanged. The final method is the U.S. Aluminum Association Aluminum Design Manual (The Aluminum Association 2005). The Aluminum Design Manual divides compressive plate failure into three regions based on the slenderness of the plate, a squash region where stocky plates fail plastically, an intermediate region where inelastic buckling occurs, and a slender region where elastic buckling occurs followed by further post-buckling strength. The transition between regions and failure performance in the buckling regions is predicted using expressions with alloyspecific constants, thus incorperating different material stressstrain curves. Welds at the perimeter of the plate are accounted for through a series of calculation adjustments. Calculations for plates with weld in the direction of the applied load use an areaaveraged strength similar in concept to Paik and Duran’s approach, but differing in implementation. All seven methods were applied to the Anderson and Anderson and Mofflin data sets. For the civil engineering code methods,

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any code-specified partial safety factors were set equal to 1.0, as all the other methods attempt to predict mean strength. Thus, any remaining conservatism should be inherent in the strength formulation itself. The results are shown in Table 1, where the bias is defined as the mean of the predicted strength divided by the experimentally observed strength, and the coefficient of variation (COV) of the prediction is also listed, defined at the standard deviation of the bias values divided by the mean bias. Table 1 indicates that the plate methods generally gave very good results with predicted bias falling close to 1.0 and most COV between 5% and 10%. The DDS 100-4 method is slightly nonconservative, while the Faulkner coefficients for Eq. 2 gave better predictions. There was some tendency in all the predictions to have slightly more variability for the stockier plates. In this inelastic failure region, the differences between the assumed welding (many methods assumed welding on all four plate boundaries), varying experimental residual stress ratios, as well as more complex physical failure processes may impact the results. Table 1. Performance of simplified plate strength methods Mofflin Anderson Overall Method Bias

COV

Bias

COV

Bias

COV

DDS 100-4

1.12

8%

1.05

8%

1.09

9%

Faulkner

1.05

7%

0.97

6%

1.02

8%

Wang et al.

0.96

12%

0.97

6%

0.97

10%

Eurocode 9

0.97

6%

0.95

5%

0.96

6%

Paik & Duran

0.96

10%

0.92

14%

0.94

12%

Kristensen

0.94

11%

1.05

13%

0.99

12%

Alum. Assoc.

1.01

8%

1.0

4%

1.00

6%

Lateral loading In addition to carrying in-plane loads, plate elements are often loaded laterally, either by sea pressures acting on the outside of the hull or by internal loads from vehicles and accommodation spaces. Unlike compressive collapse, where a plate ceases to be able to support increases in in-plane loads, true collapse of plate components from lateral loads is difficult to achieve and often occurs only after very large out-of-plane deformations have taken place. Thus, in formulating limit states for plates under lateral loading, it is customary to specify a level of out-of-plane deformation that will be considered a serviceability failure. Another common approach, though not strictly a limit state approach, is to limit the working stress in the plate to a certain percentage of the yield stress. Both types of approach were reviewed and presented here. Unfortunately, experimental results for marine aluminum plates undergoing lateral loads are current scarce, and only one study from Denmark dealing with patch loads (Abildgaard, Hansen and Simonsen 2001) has been found, so formal validation of these approaches is not currently possible. Among the permanent set approaches, yield-line theory (Kmieck 1995) is often explored, though Hughes (1998) notes that for stockier ship structures, it can be nonconservative in terms of load estimation. Hughes (1998) presents an alternative semi-empirical approach for steel structures with better performance for stocky plates. For allowable stress approaches, Sielski (2008) updated a U.S. Navy allowable stress

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equation with material constants for modern marine alloys, including 6082-T6. This equation has three sets of coefficients, allowing it to be used for locations where no set, some set, and more set would be allowed. A similar equation without the variable levels of allowable set forms the basis of the recent ABS Guide for High-Speed Naval Craft (ABS 2008.) As no experimental data was available for validation, a brief comparative study between these methods was made for two different plate thicknesses, assuming 6082-T6 material with a proof strength of 260 MPa and a plate breadth of 300mm. The results are shown in Table 2. For the yield line theory and Hughes’ method, a deformation of 0.5t was allowed. For the ABS HSNC guide, an allowable stress of 60% of the material’s yield strength was assumed, corresponding to general deck loading. As can be seen from the table, the methods span a wide range of allowable pressures. The ABS HSNC guide and the U.S. Navy approach with Sielski’s coefficients for no set gave similar pressures, which are considerably below those based on allowable set. None of these methods have been developed to consider the weak regions near welds in aluminum; neither do they explicitly consider different welding patterns, such as welds in the midregion of the plate as well as welds at the plate boundaries, so care is required when applying them to aluminum. In general, unlike compressive stresses, the current engineering approaches for limit state design of marine aluminum plates under lateral loading requires further research. Table 2. Comparative prediction under lateral load Method t=5mm t=8mm Yield Line Theory 533 kPa 1360 kPa Hughes Theory 435 kPa 1050 kPa ABS HSNC Guide 87 kPa 222 kPa Sielski – No set 65 kPa 167 kPa Sielski – Some set 190 kPa 485 kPa Sielski – More set 310 kPa 795 kPa

Stiffened Plate Elements In addition to individual plate elements, many failure modes such as plastic deformation under lateral loads or column-type buckling occur at the assembly of plate elements level. Several authors have proposed methods for investigating stiffened panels under combined in-plane and lateral loads. Most of these methods idealize a single “unit” of a stiffened panel, consisting of a single stiffener and attached plating, acting as an independent beam-column spanning the distance between transverse frames. For nonconventional extruded panels, the basic repeating shape of the extrusion can be idealized as the beam-column “unit”. In this section, two different approaches to this beam-column formulation will be reviewed: first, a method developed by Hughes (1998) for conventional steel panels; and second, an adaptation of the U.S. Aluminum Association (2005) design code for beam-columns for conventional panels and extrusions. These methods will be compared to the test data from the recent SSC451 study (Paik 2008) for uniaxial compression. Unfortunately, no test data was found for panels loaded by in-plane loads and

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lateral loads simultaneously, nor has a comprehensive test program for aluminum marine extrusions been carried out and reported in open literature. Hughes’ method Hughes (1998) developed a collapse methodology for conventional steel stiffened panels using the beam-column approach. In this approach, three explicit failure modes are checked and the lowest failure mode is taken as limiting. The three failure modes were defined as failure induced by stiffener flange yielding in compression (Mode I), failure induced by compressive collapse in the plate (Mode II), and failure induced by a combination of tensile yield in the stiffener flange and compressive collapse in the plate (Mode III), which can occur under certain high values of lateral load. For all three failure modes, Hughes adapts the basic elastic beam-column formula as a starting point: M y σ A (δ + ∆ ) y σ =σ + + φ (3) 0

a

I

a

0

I

Failure is assumed to occur when the resulting stress, σ, reaches a pre-defined value, typically the yield or proof stress of the material. σa is the applied in-plane compressive loading, y is the distance from the neutral axis to the location of interest, M0 is the applied bending moment, A and I are the cross-sectional area and moment of inertia of the beam-column, δ0 is the deflection from the lateral loading, ∆ is the initial column-type imperfection, and φ is magnification factor on the deflection from the applied lateral loads. For Mode I, the case of tensile failure in the stiffener flange, this formula can be applied as-is, as the amount of inelastic response before failure is small. However, for Modes II and III, the buckling of the plate requires a more advanced approach, and Hughes proposes an effective width approach based on empirical relationships for steel plates. Details of the approach to Mode II and III can be found in Ship Structural Design (Hughes 1998). The Hughes method was applied to panels in SSC-451. As no lateral loading was present in these panels, only Modes I and II were calculated. However, the Hughes effective width treatment of the plate is based on empirical relationships developed for steel, which was seen as a potential weakness in applying the method to aluminum. Two variants of the method were carried out where: • The Hughes plate strength equation was replaced by the Aluminum Association plate strength formula reviewed in the previous section. • The Hughes plate strength equation was replaced by the Aluminum Association plate strength formula and furthermore, the effective width formula for the plate was replaced by Faulkner’s (1975) effective width formula. Adaption of Aluminum Association approach The U.S. Aluminum Association has published formulas for beam and column action in its Specification for Aluminum Structures as part of the Aluminum Design Manual (The Aluminum Association, 2005 – referred to as the Specification). While this section does

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not explicitly address the modeling of stiffened panels as beamcolumns, the individual strength formulations within the specification can be assembled into a formulation capable of addressing the principal failure modes of such panels in a manner similar to Hughes’ approach discussed previously. This has the advantage that such formulations can be extended to extruded shapes as well as conventional plate-and-stiffener panels. The approach developed considers beam and column action separately, and uses an interaction equation to combine the two sources of loading. It is important to note that this approach is the authors’ own adaption of the design rules in the Specification for the purposes of this project, and has not been reviewed or validated beyond what is presented in this paper. Further validation of this approach against experiments or finite element analysis to ensure all failure modes are included is recommended. For column behavior, the governing strength was taken as the least of the following three strengths: 1. The strength of the overall combination as a column. This was calculated following the formulas in Section 3.4.7 of the Specification. For cases where the panel consists of both 5xxx and 6xxx series alloys, volumeaveraged material properties were used in these calculations. 2. The area-averaged local plate buckling strength of all plate elements in the column cross section with each plate strength calculated in accordance with the individual plate element strengths reviewed previously. This is defined in Section 4.7.2 of the Specification. 3. Local-overall buckling interaction as defined in Section 4.7.4 of the Specification. This was assumed to occur only when the elastic buckling strength of a plate element in the column is less than the elastic buckling strength over the overall column. Limiting stresses were reduced to include the impact of welds when the weld cross-sectional area was more that 15% of the panel gross area for column buckling. This was done by using the weighted-average stress approach presented in Section 7.2 of the Specification, and is shown in Eq. 4. A σ U _ W = σ NW − W (σ NW − σ AW ) (4) A Where σU_W is the final strength of the component considering the weld, σNW is the strength of the component as if it had no welds, σAW is the strength of the component as if it were entirely composed of welded material, AW is the weld cross-sectional area, and A is the cross-sectional area of the entire panel. While lateral-load responses were not required to model the SSC451 dataset, to be considered for the optimizer in this project, an extension for lateral loads was sought. Again adapting components of the Specification for this purpose, the following checks were added, assuming a beam with simply-supported end conditions with the lateral pressure converted to an equivalent distributed load: 1. Calculating the applied moment that would result in tensile yielding of the extrusion, in accordance with Section 3.4.2 of the Specification.

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2.

Calculating the applied moment that would cause compressive collapse of each flat plate element in the cross section, in accordance with Section 3.4.15/16 of the Specification. 3. Calculating the applied moment that would cause compressive collapse of the vertical plate elements (webs) in the cross section that are bending in their own plane, in accordance with Section 3.4.18 of the Specification. 4. Calculating the lowest shear buckling capability of the vertical plate elements (webs). This was taken as the limiting shear stress, calculated in accordance with Section 3.4.20 of the Specification. A corresponding applied moment was calculated assuming the shear force was uniformly distributed over the web elements of the extrusion. This is a simplification, but fairly accurate, given that details of the end connections of the panel were not included in this study. The impact of welding was considered in a similar manner to the columns. For the cases where lateral pressure and in-plane loads coexist, interaction equations were used to determine the limiting loads. First, two interaction equations from Section 4.1.1 of the Specification were checked, which deal with overall combined bending and in-plane loads. These equations were simplified as there is only bending in one direction: fa Cfb + ≤ 1.0 fa  FA  FB  1 −  (5)  FE 

fa

+

fb

≤ 1.0 FAVG FB Where fa is the applied in-plane compressive stress, FA is the limiting in-plane compressive stress with no other loads acting, fb is the applied bending compressive stress and FB is the limiting compressive bending stress. FE is the elastic buckling stress of the beam-column unit as a column, C is an interaction coefficient, and FAVG is the limiting column stress calculated from the areaaverage compressive strength of each element of the column only. In addition to the combined bending and compression, the webs of the beam-column unit were checked for combined shear and compression using the formula given in Section 4.4 of the Specification. 2

fa

 fs   ≤ 1.0  FS 

+

(6) FA Where fs is the applied shear stress, FS is the limiting shear stress with no other loads acting, and all other variables are as above. Comparison of results The two methods were compared to the 78 panel tests reported in SSC-451, which cover a range of conventional plate-and-stiffener geometries in 5083, 5383, and 6082 alloys. These panels were tested in uniaxial compression without lateral loads. The results of the comparison are shown in Table 3 below, where the bias and COV definitions are as Table 1 above. All partial safety factors were again set equal to 1.0 for the Aluminum Association formulations. All of the methods had

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a slight positive (nonconservative) bias. This may be a result of the lack of tripping failure modes in either of the strength methods. The panels tested in SSC-451 included a number of flat-bar panels that failed by tripping. Several steel-based tripping formulations are available and would make a good first approximation for aluminum and could be used to enhance these results. Additionally, only tensile material test data were reported in SSC-451; for the 5083 and 5383 alloys, these values were used as-is without any correction for potentially lower compressive strengths. The modifications to the Hughes method to include more aluminum-specific plate behavior did improve the method’s performance slightly. Compared to the estimates of plate response, the prediction bias and COV of the bias were larger, but this is not surprising given the more complex failure modes of an assembly of plate components compared to the individual plate components. Table 3. Comparison between methods and SSC-451 data Method Mean Bias COV Hughes method 1.09 17% Hughes w/A.A. plate 1.05 18% Hughes w/A.A. plate + 1.07 19% Faulkner effective width Adaption of A.A. method 1.20 16%

Simplified Strength Approach for Optimizer Given its ability to handle a wide range of potential extrusion cross sections, the adaptation of the Aluminum Association’s strength formulas was selected as the basis for the simplified strength approach for the optimizer. Two additional features were added to this approach to address failure modes still missing that the optimizer might exploit. First, for all individual plate elements identified as carrying lateral load in the extrusions cross section, the allowable lateral load was calculated in accordance with the ABS HSNC Guide formula presented in the previous section on plate element strength. For the overall extrusions, the lowest lateral load from any plate element or from the overall bending response of the extrusion cross section was taken as limiting. No interaction between in-plane compression and lateral load was included for the individual plate elements. Secondly, as many aluminum vessels are multi-hull, it was desirable to be able to include the impact of bi-axial compression in certain plate elements, such as shell plate or deck plate. This was included mainly to ensure that the optimizer did not drive the solution space in such a way that very little bi-axial compression capability was achieved, and the implementation is somewhat approximate. Bi-axial compression on a plate element level was handled through a simple interaction equation proposed by Stonor et al. (1983,) which seemed to fall close to the lower bound of a series of interaction finite element studies carried out by Kristensen (2001.) The strength of the plates in the transverse direction was estimated by the regression equations proposed by Kristensen (2001.) A more detailed review and comparison of these methods is presented in Collette et al. (2008.) The final strength method was implemented in a simple series of C++

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routines suitable for rapid analysis in conjunction with the optimization method.

OPTIMIZATION METHOD Designing ship structures is fundamentally a multi-objective (MO) optimization problem. Objectives such as low weight, minimum cost, and highest strength are naturally in conflict and, as such, a multi-objective optimization approach is required. MO optimization provides the information about the different possible alternative solutions that can be achieved for a given set of objectives. By analyzing the spectrum of solutions, the most appropriate solution can be selected. In this study, we use a genetic algorithm-based approach to find a set of alternative solutions considering only two objectives, weight and structural strength. The details of MO optimization are summarized in the SSC report corresponding to this work (Collette et al. 2008) and further information is available in many of the standard textbooks on optimization, such as that of Deb (2001.) When dealing with MO problems, a method for ranking potential design variants is needed; this is done through the concept of domination. We say that solution x1 dominates solution x2 if: (1) x1 is no worse than x2 in all objectives; and (2) x1 is strictly better than x2 in at least one objective. This concept can be expanded into Pareto optimality by the definition that x* is Pareto optimal if there does not exist another feasible solution in the entire design space that would decrease some objectives without causing a simultaneous increase in at least one other objective. Unfortunately, this concept almost always gives not a single solution but, rather, a set of solutions called the Pareto optimal set. The solutions included in the Pareto optimal set are called non-dominated. The plot of the objective functions whose non-dominated vectors are in the Pareto optimal set is called the Pareto front. For the relatively simple two-objective optimization approach considered here, the Pareto front can be easily visualized as a curve on a plot with an axis of strength and weight, dividing the domain into feasible and non-feasible strength-to-weight ratios for the given optimization problem. In the last 10 to 15 years, evolution-based algorithms have become more and more mature and popular for both single and multi-objective optimization, where they can be used in place of methods that convert MO problems to single-objective problems. Evolution-based algorithms use a different search strategy; instead of point-by-point search, they conduct searches in multiple points simultaneously, using operators inspired by evolution such as crossover and mutation to produce offspring, and by applying selection pressure to evolve multiple search points for optimal solutions. Evolution-based algorithms have also been used elsewhere to investigate the design of steel sandwich panels for marine applications (Romanoff and Klanac, 2007.) There exist many different evolution-based algorithms that have proved capable of tackling a wide variety of real world problems. In this study, we use a version of a multi-objective genetic algorithm (MOGA) that implements elitism strategy and nondominated sorting, the so-called NSGA-II approach (Deb 2001.)

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There are two main advantages with MOGA. First of all, MOGA deals simultaneously with a set of possible solutions (the so-called population) that generally results in discovering multiple members of the Pareto optimal set in a single optimizer run. Secondly, MOGA can easily handle discontinuous and/or nonconvex Pareto fronts. To implement MOGA, a coding strategy must be selected to encode the design variables in a certain structure (e.g., vector or matrix) that forms what is called a chromosome or individual. Two coding strategies are widely used; one is the classical binary coding, the other is real-value-based coding. The binary or real elements in a chromosome are usually called genes. In most cases, a predefined number of individuals are randomly generated to constitute the population, and the population number is kept constant throughout the entire search process. The duration of the search process is defined by the number of generations of the population that the method creates. Three types of genetic operators are usually used to generate new search points and form new generations: crossover, mutation and selection. Crossover is used to produce two offspring individuals from two randomly selected parents. Mutation is used to alter one or more randomly selected genes of a chromosome. A selection operator is used to pick individuals in a population with higher fitness values (e.g., objective values) to create the next generation. To briefly illustrate how MOGA works, a flow chart is provided in Fig. 2. A certain number of chromosomes is predetermined and initially randomly generated to form the current population. For each chromosome, a fitness evaluation is performed to assign a fitness value as a base to carry out ranking among individuals. Fitness evaluation in MOGA involves computing every objective value given an individual. Crossover and mutation, defined further below, then operate on the current population to produce offspring. The elitist non-dominated sorting algorithm in Deb (2001) is used as the selection procedure to create the next population. The key issues with MOGA are to make sure that the selection pressure from the fitness and elitist sorting methods causes the solution to approach the true Pareto front while, at the same time, maintaining sufficient diversity in the population so the approach does not become trapped in local optima or converge to a few sparsely scattered solutions. The elitist nondominated sorting algorithm has proven quite robust (refer to Deb (2001) for details). A real-coded MOGA was implemented in C++ for this study. The real value coding strategy was selected since most engineering problems involve continuous variables. The use of real-coded genes allows us to achieve arbitrary precision in the design space and also to avoid the Hamming cliffs effects associated with binary-coded genetic algorithms. Hamming cliffs arise due to significant change in real value by altering just a single bit (such as 10000 to 00000 by flipping the left-most bit from 1 to 0) in a binary coded chromosome, which hinders a smooth search in continuous variable space.

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The selection procedure follows the Crowded Tournament Selection operation described in Deb (2001), except that the constraint violations of individuals are taken into accord when performing selection. First, non-dominated sorting is conducted for a pair of individuals to determine their domination status. Then, each individual is assigned a rank. The ones within the best non-dominated set are ranked 0, the ones within the next-best set are ranked 1, and so on. In so doing, the individuals with the same ranks are grouped into the same fronts, which divide the population into non-overlapping sets. Second, a randomly picked pair of individuals takes part in the tournament selection to decide who will win the tournament and survive into the next generation. In this step, when the two individuals are in the same front, the crowding distance metric is used so that individuals in the less crowded regions are given a better chance to enter the next round of search. This has proven to be critical in obtaining more distributed Pareto front at convergence of the optimizer.

Code Design Variables Current Population Strength Estimate Fitness Evaluation Weight Estimate Cross-Over/Mutation Non- Dominated Sorting Next Population

Terminate?

Fig. 2: Flowchart of MOGA Approach Two specific genetic operators were developed for the real-valued chromosome, arithmetic crossover and delta mutation (Li 1997). Suppose the chromosome has the form of

C = ( x1 , ⋯ , xi , ⋯ , xn )

The steps in generating an offspring population, Q, from a given parent population, P, in the NSGA-II algorithm are listed below. This assumes that the population size, N, is greater than 0. These steps are:

(7)

Given two parent chromosomes Cu and Cv, the arithmetic crossover is defined as:

Cu = λ Cu + (1 − λ )Cv , Cv = (1 − λ )Cu + λ Cv '

'

(8)

1. 2.

Where: 3.

t /T (9) λ = r + (1 − r ) r ∈ [0,1] is a random value drawn from a uniform distribution, t is the current generation and T is the maximum number of generations.

4.

The delta mutation is defined as follows: suppose the i-th gene is selected for mutation; the resulting gene will be:

5.

xi = '

i

+ ∆ , if a random number is 0

i

− ∆ , if a random number is 1

x  x

(10)

Where:

( x ∆=  ( x

u

i

− xi ) ⋅ (1 − r − xi ) ⋅ (1 − r l

i

(1 − t T )

2

(1 − t T )

2

), if a random value is 0

The optimization method was implemented in C++ routines which were then linked with the strength evaluation routines described previously. A simple volume calculation was used to determine panel weight, providing the second objective evaluation for members of the population.

(11) ), if a random value is 1

r ∈ [0,1] is a random value drawn from a uniform distribution, t is the current generation and T is the maximum number of generations. It is clear from Eq. 9 and Eq. 11 that λ and ∆ are functions of t/T, which implies that, as the search approaches the maximum number of generations, the changes made to the selected solutions through these genetic operators get smaller and smaller. It is generally true that in later generations the nondominated set of solutions is getting closer to the true Pareto front and we do not want to introduce significant alterations to the population that could slow down convergence. Results have shown these two operators are very effective.

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Given Pt, generate Qt using the crowded tournament selection, arithmetic crossover, and delta mutation. Combine Pt and Qt and perform non-dominated sorting to group individuals into different fronts based on their ranks. Generate Pt+1 by combining the first r fronts identified in (2) until encountering the r+1-th front where adding all the individuals in this front will cause the overall population in Pt+1 to exceed N. Compute crowding distances for individuals within front r+1 and perform the crowding distance tournament to fill the rest in Pt+1 while not exceeding N. If termination criteria are not met, repeat (1).

Collette

OPTIMIZATION EXAMPLE Problem Overview The multi-objective optimization scheme described in the section above was applied to the optimization of two different stiffened panels on a nominal high-speed car ferry, a typical use of aluminum extrusions. A nominal high-speed vessel was selected based on previous Ship Structure Committee work in SSC-438, Structural Optimization for Conversion of Aluminum Car Ferry to Support Military Vehicle Payload (Kramer et al. 2005). SSC-438 studied the conversion of a 122m LOA catamaran commercial car/truck ferry to handle military cargoes. This vessel was constructed out of aluminum alloy with three main decks, a main vehicle deck, and upper vehicle deck, and a strength deck. The frame spacing is 1,200mm. It was decided to use the centerline

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stiffened panels on the strength deck and main vehicle deck as the target of optimization. These panels extend 3,375mm off centerline, ending on the first longitudinal girder. In addition to the vessel’s frame spacing of 1,200mm, it was decided to include an additional panel with a hypothetical length of twice the actual frame spacing, or 2,400mm. All panels were assumed to have three simultaneous loads components acting on them, an out-ofplane load or lateral pressure applied normal to the plate, and transverse and longitudinal stress acting in the plane of the plating. However, for each of the locations different load components were selected to be the dominate loading and maximizing the ability to resist this load component was the objective of the optimization. For the main vehicle deck panels, lateral loads from vehicles (here assumed as uniform lateral pressure) was selected as the strength optimization objective and longitudinal stress applied in the plane of the plate was the strength objective for the strength deck panels. This is summarized in Table 4. The lateral load on the strength deck was estimated from the ABS HSNC Guide (ABS 2008), while the transverse and longitudinal in-plane components were taken as 10% of the material proof stress. Extrusions were assumed to be made from a 6082-T6 aluminum alloy, with a base material proof stress of 262 MPa and a proof stress of 138 MPa in the HAZ near welds. The elastic modulus was taken as 70000 MPa. Table 4. Load cases Load Type

Main Strength Deck Deck Longitudinal compression 26 MPa Maximize Transverse compression 26 MPa 26 MPa Lateral load Maximize 4.9 kPa Two constraints were placed on the extrusion cross sections to ensure producability: first, the minimum wall thickness was set to 2mm; and second, the ratio of thicknesses of adjacent plate elements could not exceed 2:1. More advanced constraints, such as total cross-sectional area constraints or constraints related to the size and arrangement of the hollow shape could easily be included in the optimization method if required by a specific aluminum extrusion press. Three different styles of extrusions were examined for each panel location, as shown in Sub-Figures a-c of Fig. 3.

Fig. 3: Three types of extrusions considered For the extruded stiffener construction of Figure 3a, the plate elements were idealized as shown in Fig. 4, where heavy dots indicate the edges of individual elements. Span and thickness limitation were taken as shown in Table 5. Plates 3 and 4 of Fig. 4 were restricted to be mirror images of each other. Once the number of stiffeners was known, the width of plate 1 in Fig. 4 was

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selected by the requirement that the panel span the 3,375mm from centerline to the first longitudinal girder. 3

4

2

y 1

x

Fig. 4: Stiffener extrusion plate elements Table 5. Extrusion parameters for stiffener extrusion Design Variable Lower Upper Bound Bound Plate thickness (plate 1) 2mm 14mm Web thickness (plate 2) 2mm 14mm Web height (plate 2) 20mm 150mm Flange thickness (plate 3&4) 2mm 14mm Flange width (plate 3&4) 10mm 100mm Number of stiffeners 1 22 HAZ regions with extent three times the plate thickness for butt welds were included in plate 1 of Fig. 4. In cases where the total width of the extrusion was less than 150mm, it was assumed that the two plate/stiffener combinations could be extruded together in a single die, and the amount of welding was proportionally reduced. This corresponds to a limiting overall extrusion diameter of 300mm, which is reasonable for most mills. The sandwich-type and hat-type extrusions were broken down into basic repeating sections with plate elements as shown in Fig. 5-6, and Tables 6-7. 3

2

y x

1

Fig. 5: Sandwich-type extrusion plate elements Table 6. Extrusion parameters for sandwich extrusion Design Variable Lower Upper Bound Bound Plate thickness top & bottom 2mm 14mm (plate 1 and 3) Web thickness (plate 2) 2mm 14mm Web height (plate 2) 15mm 150mm Number of webs 1 70 For the sandwich and hat extrusions, weld joints were assumed to be at the edge of the extrusion, with HAZ widths of three times the thickness of the plate element being joined. Similar to the stiffener extrusion, in cases where the total width of the repeating section was less than 150mm, it was assumed that the two or more

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sections could be extruded together in a single die, and the amount of welding was proportionally reduced. 5 y

3 x

1

4 2

Fig. 6: Hat-type extrusion plate elements Table 7. Extrusion parameters for hat-type extrusion Design Variable Lower Upper Bound Bound Plate thickness- plate 1 2mm 14mm Plate thickness- plate 2 2mm 14mm Plate thickness- plate 3 and 4 2mm 14mm Hat “top” thickness (plate 5) 2mm 14mm Hat height 20mm 150mm Hat “top” width as % of bottom 50% 100% width (plate 5) Hat “bottom” width (plate 2) 30mm 350mm Plate 1 width 20mm None Number of stiffeners 1 22

Optimization and Results The MOGA optimizer was applied to three different extrusion types for each of the two panel locations. In general, it was found that a stable Pareto frontier was generated after processing 300 generations of 40 individuals for the sandwich and hat stiffener, and 200 generations of 40 individuals for the extruded stiffener panels. Run times of the optimizer were on the order of one to five minutes on a standard desktop PC for development of the entire Pareto front. Pareto frontiers showing strength vs. weight for the strength deck panel are shown in Fig. 7 and Fig. 8 for the two panel lengths assumed. In each figure, all three types of extrusions are plotted together to allow the relative efficiency of each extrusion to be judged. Weight is plotted on the y-axis, and allowable in-plane load is plotted on the x-axis, including the reduction for the transverse compression and lateral load present on these panels. Both Pareto fronts show a fairly sharp corner, where weight increases rapidly as the panel strength approaches the proof strength of the material – the upper strength bound for the strength method used. Below roughly 200 MPa of strength, the relationship between strength and weight is roughly linear for all types of extrusions. For the 1,200mm long panel shown in Fig. 7, all three types of extrusions perform roughly equally well, with the sandwich panel being perhaps slightly less weightefficient than the other two types of panels. This trend is extended for the 2,400mm-long panel that is shown in Fig. 8, where the sandwich panel is noticeably heavier than the other two panels. Interestingly, the hat-type stiffener panel appears to be roughly as weight-efficient as the conventional stiffener extrusion. This was unexpected, as the hat-type appears to use more material than a conventional stiffener. While the hat-type

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panel does have the advantage that stiffener tripping is likely to be precluded by stiffener shape, stiffener tripping was not included as an explicit failure mode in the current strength routine, and so this benefit should not be apparent in the current results. The genetic optimization approach does not allow an explicit range of panel weights and strengths to be specified ahead of time. While all Pareto fronts are fairly well covered, it was seen in the 2,400mm-long panel that the optimizer focused on lower weight and lower strength panels as compared to the 1,200mm-long panel. The same process was repeated for the main deck panels; however, the upper plate thickness bounds of 14mm proved too high for the lateral load problem, as the bulk of the resulting Pareto front focused on panels whose strength were far in excess of practical design loads. The optimization problem was re-run with lower bounds on thickness. A maximum plate element thickness of 6mm was permitted for both the extruded stiffener and hat-shaped stiffener panels, and a maximum plate element thickness of 5mm was permitted for the sandwich panels. These limits produced more designs with lower allowable lateral pressures, as shown in Fig. 9. Fig. 9 confirms that, for lateral pressures less than about 0.08 MPa, the sandwich panel became less weight-efficient, while the extruded stiffener and hat-stiffener panels were roughly equal in performance. The higher weight of the sandwich panel is likely a result of the 2mm minimum thickness requirement as the sandwich panel must have both a top and bottom surface – the panel cannot go below an “average” material thickness of 4mm. In practical design problems, local concentrated loads or class society minimum thicknesses may force the use of thicker plate elements than the 2mm allowed here, but such requirements are likely to shift all three Pareto fronts upward by the same amount in the lower lateral pressure region. Because the loading and constraint assumptions used in this project differ from those used in the actual vessel design, it is difficult to directly compare the results of this optimization problem to the as-built vessel structure. However, the structures generated by the optimizer were similar in scantlings and slightly more weight efficient for the current study’s objectives than the as-built structure (Collette et al. 2008.)

CONCLUSIONS An optimization approach for designing aluminum extrusions was developed and successfully employed on a high-speed vessel example problem. Simplified strength methods for aluminum extrusions were reviewed. While compressive collapse under uniaxial loading appears to be well covered in the existing literature, method for predicting the permanent set of plate elements under lateral load and load combination methods still appear to be lacking. This is especially true when considering different HAZ patterns from welding – few methods addressed the impact of the HAZ on lateral strength or load combination. A multi-objective genetic algorithm optimizer was developed to investigate trade-offs between strength and panel weight. The approach used an implementation of the NSGA-II algorithm to progressively determine a Pareto frontier for a given multi-

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objective function. Using a large high-speed vehicle ferry investigated previously, two hypothetical optimization problems were developed, one for a strength deck panel primarily loaded in-plane, and one for a vehicle deck panel primarily loaded outof-plane. In general, the optimization approach proved robust, and Pareto frontiers could be determined in a matter of minutes on a standard desktop PC. The performance of the three extrusions types was very similar for each application, without any clear favorites for improved strength-to-weight ratio, though the sandwich panel did slightly lag the other two types of extrusions for low-load/low-weight applications. This finding suggests that, for these applications, it may be possible to select the extrusion type based on considerations other than primary strength – such as fatigue strength or ease of construction, and that a weightefficient panel can then be optimized from any of the three types of extrusions investigated. Overall, the combination of the robust multi-objective genetic algorithm optimizer and the closed-form strength equations proved a powerful tool for optimizing the design of aluminum extrusions.

ACKNOWLEDGEMENTS This work is funded by the Ship Structure Committee as project SR-1457. The authors would like to express their appreciation to the SSC and to the SSC Technical Representative Mr. H. Paul Cojeen, and to all member of the projects’ PTC for their encouragement, support, and guidance in the execution of this project. The authors wish to additionally recognize Dr. Robert Sielski for his review of the final report and helpful comments and suggestions which improved the quality of this study. The contributions of Tin-Guen Yen and Jessica Walters at SAIC to this work are also acknowledged.

REFERENCES Abildgaard, P., P. Hansen, and B. Simonsen. 2001. Ultimate Strength of Welded Aluminium Structures. In HIPER 2001, 4-18. Hamburg, Germany. American Bureau of Shipping. 2008. Guide for Building and Classing High-Speed Naval Craft. Houston, TX: American Bureau of Shipping. Anderson, R., and M. Anderson. 1956. Correlation of crippling strength of plate structures with material properties. Washington, DC: NACA Technical Note 3600, January 1956. CEN (European Committee for Standardization). 1998. Eurocode 9: Design of Aluminium Structures. Brussels: European Committee for Standardization (CEN). ENV 1999-1-1: 1998 E. Collette, M. et al. 2008. Ultimate Strength and Optimization of Aluminum Extrusions. Washington, DC: Ship Structure Committee, Report SSC-454. October 2008. Deb, K. 2001. Multi-Objective Optimization using Evolutionary Algorithms. Chichester, England: John Wiley & Sons, Ltd. Faulkner, D. 1975. A. Review of Effective Plating for Use in the Analysis of Stiffened Plating in Bending and

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Compression. Journal of Ship Research 19, no. 1 (March): 1-17. Hughes, O. 1988. Ship Structural Design. Jersey City, NJ: The Society of Naval Architects and Marine Engineers. Kmiecik, Marian. 1995. Usefulness of the yield line theory in design of ship plating. Marine Structures 8, no. 1: 67-79. doi:10.1016/0951-8339(95)90665-H. Kramer, R., et al. 2005. Structural Optimization for Conversion of Aluminum Car Ferry to Support Military Vehicle Payload. Washington, DC: Ship Structure Committee Report SSC-438. February, 2005. Kristensen, Odd. 2001. Ultimate Capacity of Aluminium Plates under Multiple Loads, Considering HAZ Properties. PhD Thesis, Norwegian University of Science and Technology. Li. J. 1997. Oil Tanker Market Model, Analysis and Forecasting using Neural Network, Fuzzy Logic and Genetic Algorithms. Ph.D. Thesis, University of Michigan, Ann Arbor. Mofflin, David. 1983. Plate Buckling in Steel and Aluminium. PhD Thesis, Trinity College, University of Cambridge, August. Naval Sea Systems Command. 1982. Design Data Sheet 100-4 – Strength of Structural Members Revised 15 November 1982. Paik, Jeom Kee. 2008. Mechanical Collapse Testing on Aluminum Stiffened Panels for Marine Applications. Washington, DC: Ship Structure Committee, February 2008. Paik, J.K. and A. Duran. 2004. Ultimate Strength of Aluminum Plates and Stiffened Panels for Marine Applications. Marine Technology 41, no. 3 (July): 108-121. Paik, J.K., and A.K. Thayamballi. 2003. Ultimate limit state design of steel plated structures. Chichester, England: John Wiley and Sons. Rigo, P. et al. 2003. Sensitivity analysis on ultimate strength of aluminium stiffened panels. Marine Structures 16, no. 6 (August): 437-468. doi:10.1016/j.marstruc.2003.09.002. Romanoff, J. and A. Klanac. 2007. Design Optimization of Steel Sandwich Hoistable Car-Decks Applying Homogenized Plate Theory. In PRADS 2007. Houston, Texas, September 30 – October 5. Sielski, R. 2008. Aluminum Marine Structure Design and Fabrication Guide. Washington, DC: Ship Structure Committee, February. Stonor, R et al. 1983. Test on Plates under Biaxial Compression. Report CUED/D-Struct/TR98, Cambridge: Cambridge University, Engineering Department. The Aluminum Association, Aluminum Design Manual: Specification for Aluminum Structures - Load and Resistance Factor Design Specification, 8th ed., Washington, D.C. Aluminum Association, 2005. Wang, X. et al. 2005. Buckling and Ultimate Strength of Aluminum Plates and Stiffened Panels in Marine Structures. In The Fifth International Forum on Aluminum Ships. Tokyo, Japan, October.

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350

300

Hat Sandwich

Weight (kg)

250

Conventional

200

150

100

50

0 0

50

100

150

200

250

300

Longitudinal Compressive Stress (MPa)

Fig. 7: Pareto Frontier – Strength Deck 1200mm Panel Length 700

600

Hat Sandwich Conventional

Weight (kg)

500

400

300

200

100

0 0

50

100

150

200

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Longitudinal Compressive Stress (MPa)

Fig. 8: Pareto Frontier – Strength Deck 2400mm Panel Length

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400 350

Hat (6mm Max Thickness) Conventional (6mm Max Thickness)

300

Sandwich (5mm Max Thickness)

Weight (kg)

250 200 150 100 50 0 0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Governing Pressure (MPa)

Fig. 9: Pareto Frontier – Main Deck 2400mm Panel Length – Maximum Thicknesses Reduced

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