Blake, J.I.R.

Shenoi, R.A.

Sobey, A.

English

Southampton (e-Prints Soton)

Comparative Optimisation of First Principles Design Methods withClassiﬁcation Society Rules for Boat Hull StructuresAdam Sobey, James Blake, Ajit Shenoi, University of Southampton11 IntroductionBoat production works closely with classiﬁcation society rules as conforming to these rules hasa certain implicit recognition in port legislation. It is therefore important to make sure that eitherclassiﬁcation society rules are used, relevant to the country sold in, or that ﬁrst principle methods aredetermined safe by those same societies. Classiﬁcation society rules use safety factors to ensure thatthe boats produced will not suﬀer failure during use. These safety factors, while ensuring a safe boatand fast design cycle, often mean that the topologies developed are heavier than can be calculatedfrom ﬁrst principle methods. ISO 12215-5 for scantling determination has recognised this fact and hastried to reduce these safety factors, Curry (2004). However, boats can be cheaper and more eﬃcientif designed by ﬁrst principles methods. First principle methods can be passed through classiﬁcationsocieties but the process can be expensive as all calculations must be carefully checked and this processincurs added cost. Therefore, these methods must oﬀer large increases in either cost eﬀectiveness orboat eﬃciency in order to be used.2 Structural AnalysisThe structural analysis used is based upon Navier method grillage theory, Vedeler (1945). Maneepanet al. (2007) showed that the Navier method grillage analysis closely approximates the exact answerwhile being computationally more eﬃcient than other grillage methods. The Navier method is basedon the determination of deﬂections at the intersecting points found between longitudinal girders (indexg) and the transverse beams (index b) as seen in Fig.1. It is then possible to determine the momentsin the stiﬀeners, allowing an investigation of the stresses and strains. This method has been used formany years for structural analysis of isotropic plates. Datoo (1991) developed a method based onelastic stress analysis that allows orthotropic materials to be analysed. This method is described inmore detail in section 2.2.Figure 1: Plate and stiﬀener arrangement1Ship Science, Southampton SO17 1BJ, UK, R.A.Shenoi@soton.ac.ukSchiﬀstechnik Bd.54 - 2007/Ship Technology Research Vol. 54 - 2007 1The elastic equivalent properties are developed by splitting each stiﬀener into four separate ele-ments, i, labeled in Fig.2. Each of these elements can have diﬀerent properties and each ply of eachelement, k, can also be analysed using diﬀerent materials and ply angles.Figure 2: Stiﬀener element names and numbers2.1 Grillage analysisThe grillage analysis uses the Navier summations of points within the grillage to develop thedeﬂection of the stiﬀeners. The values of the wave numbers m and n were kept at 17, as highernumbers extended computational time with only small increases in accuracy. The equation givingdeﬂection of the stiﬀened plate, w(x,y), is a double summation dependent on the wave numbers givenas well as the length and breadth of the panel, L and B, and the longitudinal and transverse positionalong the panel, x and y, given beloww(x,y) =∞ Xm=1∞ Xn=1amn sinmπxLsinnπyB(1)The coeﬃcient amn depends on the ﬂexural rigidities, Dg and Db, found from Eq.(13) as part of theelastic equivalent properties as well as the pressure, P:amn =16PLBπ6mnm4(g + 1)DgL3 + n4(b + 1)DbB3 (2)The coeﬃcient amn is found based on the assumption that the change in potential energy from thedeﬂection will be a minimum. From the deﬂection curve of the qth beam and pth girder, where x ory are kept constant, giving xq = qL/(b + 1) and yp = pB/(g + 1), to investigate the deﬂections alongthe speciﬁed beam, it is possible to derive the strain energy V :V =L Z0Dg2 ∂2w∂x2!2y=ypdx +B Z0Db2 ∂2w∂y2!2x=xqdy (3)The work done on the grillage is:L Z0B Z0P∞ Xm=1∞ Xn=1amn sinmπxLsinnπyBdx dy (4)Minimising the potential energy (∂V/∂amn) and equating it to the work done, it is then possible toﬁnd amn in Eq.(2). The moments can be found in the girders or beams (Mg, Mb):Mg = −Dg∂2w∂x2 Mb = −Db∂2w∂x2 (5)2 Schiﬀstechnik Bd.54 - 2007/Ship Technology Research Vol. 54 - 2007Finally using the maximum moments in the grillage the maximum stress σmax can be determined:σmax =Es(i)MsZsDs(6)Es(i) is the Young’s modulus of the element of a stiﬀener (girder or beam), Ms is the moment createdin the stiﬀener, Zs is the vertical distance of the centroid of an element to the neutral axis and Ds isthe structural rigidity of a stiﬀener:2.2 Elastic equivalent propertiesIt is possible to determine the reduced stiﬀness terms, Qij, from the elastic properties in each plyof each element:Q11 =E11 − ν12ν21, Q22 =E21 − ν12ν21, Q12 =ν21E11 − ν12ν21, Q66 = G12 (7)E1, E2, ν12, ν21 and G12 are the properties of the material in each element i. These values allowcalculating the transformed reduced stiﬀness terms ¯ Qij for each ply, depending on the angle of the plyspeciﬁed:¯ Q11 = cos4θQ11 + sin4θQ22 + 2cos2θsin2θQ12 + 4cos4θsin2θQ66 (8)¯ Q12 = cos2θsin2θQ11 + cos2θsin2θQ22 + (cos4θ + sin4θ)Q12 − 4cos2θsin2θQ66 (9)¯ Q22 = sin4θQ11 + cos4θQ22 + 2cos2θsin2θQ12 + 4cos4θsin2θQ66 (10)θ is the angle of each ply of each element. The laminate stiﬀness terms for each element can then befound by totalling the transformed reduced stiﬀness terms for each of the plies:Aij =N Xk=1tk( ¯ Qij)k (11)tk is the thickness of each ply of each element. The Young’s modulus of the material for each elementof the stiﬀener is then:Ei =A11A22 − A212A22 · t(12)The ﬂexural rigidity of the stiﬀener in either the longitudinal or transverse directions follows from:Dg =Ng Xi=1Eg(i)Ig(i) Db =Nb Xi=1Eb(i)Ib(i) (13)Finally it is also possible to ﬁnd the second moment of area for each element of the stiﬀener:I(i) = Icx(i) + a(i)d2na(i) (14)Icx(i) is the moment of inertia of each element about its own neutral axis, a(i) the area of each elementand dna(i) the distance of the element’s cross section to the beam or girders neutral axis. The ﬂexuralrigidity found using stress analysis can then be used to determine the stresses in the stiﬀeners usingthe Navier grillage method.2.3 Higher-Order Shear Deformation TheoryThe grillage method used ﬁnds the maximum stresses in the stiﬀeners by assuming that all ofthe load is passed through to the stiﬀening members. It is also important that the plates of thehull must be thick enough to withstand the expected loads. For single plies, this can be veriﬁedeasily using classical laminate plate theory and ﬁrst-order shear deformation theory. For multi-layeredstructures it is necessary to use higher-order shear deformation theories which are computationallymore expensive. The plate analysis has been based on third-order shear deformation theory (TSDT),Schiﬀstechnik Bd.54 - 2007/Ship Technology Research Vol. 54 - 2007 3developed by Reddy (2004), to determine the properties required for the failure criteria, as this allowsthe full beneﬁts of using diﬀerent layups in the material to be used.The boundary conditions for a plate can be deﬁned from the following equations.Where displacement in the x direction is given by:u0(x,y) =∞ Xn=1∞ Xm=1Umn cosαxsinβy (15)Displacement in the y direction is given by:v0(x,y) =∞ Xn=1∞ Xm=1Vmn sinαxcosβy (16)Displacement in the z direction is given by:w0(x,y) =∞ Xn=1∞ Xm=1Wmn sinαxsinβy (17)Initial rotation about the y axis direction is given by:φx(x,y) =∞ Xn=1∞ Xm=1Xmn cosαxsinβy (18)Initial rotation about the x axis direction is given by:φy(x,y) =∞ Xn=1∞ Xm=1Ymn sinαxcosβy (19)Each value (Umn, Vmn, Wmn, Xmn and Ymn) is a coeﬃcient that must be determined from Eq.(22),α = πm/L and β = πn/B. The forces at each point on the plate q(x,y) are determined from:q(x,y) =∞ Xn=1∞ Xm=1Qmn sinαxsinβy (20)where Qmn is the lateral loading on the plate:Qmn(z) =4LBL Z0B Z0q(x,y)sinmπxLsinnπyBdx dy (21)It is then possible to ﬁnd the coeﬃcients of the boundary conditions using the stiﬀness matrix [C] bysubstituting Eqs.(15) to (21) into the equations of motion.[C]          UmnVmnWmnXmnYmn          =          00Qmn00          (22)Qmn = −16q0/(π2mn) and q0 is the load on the plate. The stiﬀness matrix [C] can be used to showthe relation between the stress resultants and the strains:4 Schiﬀstechnik Bd.54 - 2007/Ship Technology Research Vol. 54 - 2007  {N}{M}{P}  =[A] [B] [E][B] [D] [F][E] [F] [H]  ε(0)ε(1)ε(2)  (23)({Q}{R})="[A] [D][D] [F]#(γ(0)γ(2))(24)where N,M,P,Q and R are the stress resultants and ε(0),ε(1),ε(2), γ(0) and γ(2) are the strains in theplate. The values relating to this matrix [C] can be found from, (i,j = 1,2,6) and (m,n = 1,2,4,6):(Amn,Bij,Dmn,Eij,Fmn,Hij) = ( ¯ Qij, ¯ Qmn)(1,z,z2,z3,z4,z6) dz (25)It is then possible to determine the values of the strains from the displacement relations.  εxxεyyεxy  =              ∂u0∂x+12∂w0∂x2∂v0∂y+12∂w0∂y2∂u0∂y+∂v0∂x+∂w0∂x∂w0∂y              + z            ∂φx∂x∂φy∂y∂φx∂y+∂φy∂x            + z3                  −c1 ∂φx∂x+∂2w0∂x2!−c1 ∂φy∂y+∂2w0∂y2!−c1 ∂φx∂y+∂y∂x+ 2∂2w0∂x∂y!                  (26)(εyzεxz)=    φy +∂w0∂yφx +∂w0∂x    + z2    −c2φy +∂w0∂y−c2φx +∂w0∂x    (27)These strains can then be used with Eqs.(26) and (27) to determine the stresses in the plate. Thestresses and strains then allow the use of failure mechanisms to determine whether a given thicknessof plate will fail. In this presented method the maximum stress criteria is being used and the ultimatestresses are found using the material data from Lloyd’s Register rules.2.4 Classiﬁcation Society RulesThe structural design was also investigated following classiﬁcation society rules, using Lloyd’sRegister rules for Special Service Craft. The classiﬁcation society rules are based on developing generalrules, with safety factors, to determine the geometry of the panel and stiﬀeners based on the expectedconditions. These were taken from Part 5 Chapter 2 Section 4. For a fair comparison between thetwo models, we assumed that the panel was a side panel and that the horizontal position chosen foranalysis was at the point of maximum pressure. This allows the plate to have a standard thicknessthroughout its dimensions. The pressure used for both models came from that developed in Lloyd’srules. The panels were 12 m long and 0.5 m wide for both approaches. The geometries were developedfrom Part 8 Chapter 3, assuming the craft as a mono-hull. The values for dimensions without aspeciﬁed minimum thickness in Lloyd’s Register rules were generated using a genetic algorithm. Thesame cost models and genetic algorithms were used in each comparison, but Lloyd’s rules were usedto determine failure for the classiﬁcation society model.3 Validation of approachThe ﬁrst principles structural analysis method was validated to ensure that elastic stress theorywould create reasonable results with grillage analysis. The results from the grillage method werecompared to those found in Clarkson (1965) for a panel with a length and width of 3180 mm. Thepanel consisted of four beams and girders with dimensions 254 mm deep 127 mm wide with 18.288mm thick ﬂanges and 9.144 mm thick webs. A pressure of 137.9 kPa was applied to each panel. Theresults are presented in Table I. Similar values were found, and therefore the grillage method was usedfor the stiﬀener modeling.Schiﬀstechnik Bd.54 - 2007/Ship Technology Research Vol. 54 - 2007 5Table I: Validation of Navier method grillage analysisClarkson Grillage ModelDeﬂection 10.95 mm 10.9 mmStress 183.27 MPa 170.13 MPaThe elastic equivalent properties were compared to Datoo (1991) using lamina properties E1 = 140kN/mm2, E2 = 10 kN/mm2, G12 = 5 kN/mm2, ν12 = 0.3 and a ply thickness of 0.125 mm for each ofthe 8 plies, all having a 0◦ ply angle, where the result was identical to Datoo’s value of 140 GPa.For the validation of third-order shear deformation theory, a layup of [0/90/90/0] was used. Thelength to width ratio of the plate was L/B = 1, the length to thickness ratio L/t = 100. Thematerial properties were E1 = 175 GPa, E2 = 7 GPa, G12 = G13 = 3.5 GPa, G23 = 1.4 GPa, andν12 = ν13 = 0.25. The load acting on the plate was q0 = 50 kPa. Although only a wave numberof one was used, there was only a 4% diﬀerence between Reddy’s (2004) non-dimensionalised centerdeﬂection, ¯ w=w0(E2h3q0a4 ), of 0.00434 result and the result from calculations of 0.00415.4 Multi-objective OptimisationGenetic algorithms, Coley (1999), allow global optimisation but often lead to results that are onlyclose to the global optimum. Eﬃciency and the ﬁnal result depend on assorted parameters, particularlychosen mutation rate, cross-over rate and selection rate. The optimisation used embedded geneticalgorithms to ﬁrst search for the optimum stiﬀener geometry, for a given stiﬀener spacing, then for theoptimum stiﬀener spacing. The objectives used were mass and cost and each of these had an equalweighting.The application was a side panel, with pressure 24 kPa, with length 12 m and width 0.5 m.Parameter studies revealed that for the application at hand the best results were obtained for mutationrate of 0.02 and a cross-over rate of 0.8, Sobey et al. (2008). For these parameters, the ﬁrst principleapproach gave a 4% lower cost and a 29% lower weight than the rule-based approach. Table II andIII give the associated plate and stiﬀener scantlings.Table II: Plate geometry after optimizationlong. stiﬀener spacing transv. stiﬀener plate thicknessrule-based 425 mm 500 mm 6.1 mmﬁrst principles 962 mm 500 mm 2.0 mmTable III: Stiﬀener scantlings; rule-based (RB) and ﬁrst principles (FP)crown web basewidth thickness thickness height widthlongitudinal (RB) 55 mm 1 mm 4.1 mm 55 mm 100 mmlongitudinal (FP) 126 mm 0.5 mm 0.5 mm 70 mm 192 mmtransverse (RB) 85 mm 1 mm 4.7 mm 70 mm 85 mmtransverse (FP) 55 mm 0.5 mm 0.5 mm 69 mm 100 mmThe crown and web thicknesses are very low compared to the expected results. This could becaused through the use of simpliﬁed failure criteria. The ﬂexural rigidity will also be an importantfactor for a box beam. This has not yet been included in the model and therefore the addition ofshear stresses and buckling will further constrain the thicknesses of the stiﬀeners.The panel thickness is much smaller in the ﬁrst principles approach. The geometries of the stiﬀenersare similar in most cases; however the ﬁrst principles approach gives a larger longitudinal stiﬀener and6 Schiﬀstechnik Bd.54 - 2007/Ship Technology Research Vol. 54 - 2007a smaller transverse stiﬀener. This could also be veriﬁed and improved in the future by using acomparison with a ﬁnite element analysis package to determine correct results. However, the initialresults show consistently that the minimum criteria approach of classiﬁcation societies rules appearsto be too stringent and further development of these models will help to show whether this is true.5 ConclusionsThe method gives a fast result for the optimum solution and can be further expanded to increasethe diﬀerent variables used as inputs. The method should be easily implementable into a concurrentengineering environment giving rapid assessment of the diﬀerences between the boat hull designedusing classiﬁcation society rules and those built using ﬁrst principles methods.Further development will be required to properly constrain the stiﬀeners using the ﬁrst principlesmethod. Development of a better cost model will allow accurate comparison between the resultsfor each method. Better cost modeling will also allow the geometry of the stiﬀeners to be betterdetermined, as cost constraints will stop some shapes of stiﬀeners from being developed. Use of morecomplex failure criteria should help to constrain the geometries of the stiﬀeners and panels. Furtherwork on the algorithm to optimise diﬀerent layups and orientations should also be taken into accountas this will reduce the size that is required for the ﬁrst principles approach. The model should alsotake into account more production factors, as this will more eﬀectively constrain the results. Furtherdevelopment of cost models will also allow diﬀerent production techniques to be compared givingproduction designers an idea as to which method of production could be the cheapest. Finally forindustrial applications it will be important to use direct methods at the end of the genetic algorithmfurther increasing the accuracy of the method and also decreasing the runtime required to gainoptimum results.ReferencesCLARKSON, J. (1965), The elastic analysis of ﬂat grillages, Cambridge University PressCOLEY, D.A. (1999), An introduction to genetic algorithms for scientists and engineers, World ScientiﬁcCURRY, R. (2004), An assessment of ISO 12215 small craft hull construction with classiﬁcation society rules,Trans. RINA Part B: Int. J. Small Craft Technology 147DATOO, M.H. (1991), Mechanics of Fibrous Composites, Elsevier Science Publ.MANEEPAN, K.; SHENOI, R.A.; BLAKE, J.I.R. (2007), Genetic algorithms (GAs) based optimisation of FRPcomposite plated grillages in ship structures, Trans. RINA Part A: Int. J. Maritime Engineering 149, pp.1-19REDDY, J.N. (2004), Mechanics of Laminated Composite Plates and Shells, CRC Press Inc.SOBEY, A.J.; BLAKE, J.I.R.; SHENOI, R.A. (2008), Optimization of composite boat hull structures, 7th Conf.Computer and IT Applications in the Maritime Industries (COMPIT), Liege, pp.502-515VEDELER, G. (1945), Grillage beams in ships and similar structures, Grondahl & SonSchiﬀstechnik Bd.54 - 2007/Ship Technology Research Vol. 54 - 2007 7

Genetic algorithms (GAs) based optimisation of FRP composite plated grillages in ship structures,

Grillage beams in ships and similar structures,

Optimization of composite boat hull structures, 7th Conf.

The elastic analysis of ﬂat grillages,

Comparative optimisation of first principles design methods with classification society rules for boat hull structures

Comparative optimisation of first principles design methods with classification society rules for boat hull structures

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