Optimized Design and Reliability of a Hybrid Ship Hull (Procedure and a Demonstration)

Transcription

1 Optimized Design and Reliability o a Hybrid Ship Hull (Proedure and a Demonstration) S. Mushtaq Chevron Shipping Company, Houston USA 1 ABSTRACT The onept o hybrid hull is proposed whih onsists o steel boxes to resist the in-plane loads and omposite sandwih panels to resist the lateral loads. In addition to the redued weight, the sandwih ore in the hybrid hull prevents the lateral bukling o steel until its yield strength is reahed thereby utilizing its ull potential. The sandwih skin separated by the ore resists the inplane load due to hydrostati pressure. This hybrid hull onept is applied to a medium sized existing ship or whih the saety indies are available or omparison. The major loads on any ship struture are estimated rom the seond-order strip theory and the ABS Rules. The sandwih panel subjeted to lateral loads and in-plane loads is optimized with respet to weight to obtain the optimized design and thiknesses o eah skin layer and ore. Reliability proedure is outlined determining the limit state equations or the overall ailure and the loal ailure. The saety indies are ompared with those or the original ship and are ound to be similar. Finally, a proess is laid out or perorming an optimized reliability o a hybrid hull. This paper is part o the PhD thesis (Mushtaq 007) and part o this work has been unded by the Oie o Naval Researh (ONR). INTRODUCTION Polymeri omposite materials have been used in ships, boats, and other marine strutures or over 50 years. Their use has been driven largely by their speial properties and or eonomi reasons. The Preliminary Design o Hybrid Hull setion presents the advantages and disadvantages o the various materials and the appropriate hoie or use in a ship hull. The onept o hybrid hull is explained and a preliminary midship setion is proposed or an already existing medium sized ship. Good design methods and riteria represent the biggest hallenge. While muh o the pratial experiene with FRP, inluding the FRP sandwih, has been built up in a trial-and-error situation with these very high perormane rat (Bergan et al. 199), or larger vessels it is essential to base design and approve on well-doumented properties and riteria or the materials and the strutures. The various loads on a ship hull or the design riteria are determined rom the seond-order strip theory perspetive or the bending moments and the ABS Rules or the hydrostati loads. The optimization o the sandwih panel is perormed by analyzing two prevalent theories or omposite plates, omparing them and hoosing appropriate ormulations or deletion and stresses. A proedure is outlined and is applied to the bottom hybrid hull. For the reliability analysis, the limit-state equations are developed or both overall and loal ailure modes. Appropriate distributions are assigned to the load and strength variables. The irst-order reliability indies obtained or the hybrid hull are ompared to the indies o the original ship. Finally, the proedure is outlined or optimization and reliability analysis or a hybrid hull. 3 PRELIMINARY DESIGN OF HYBRID HULL 3.1 Conept The hybrid hull onsists o steel boxes to resist the in-plane loads and sandwih panels to resist the lateral loads. In addition to the redued weight, the ultimate strength analysis o the hybrid hull (Mushtaq 007) revealed that the sandwih ore in the hybrid hull prevents the lateral bukling o steel until its yield strength is reahed thereby utilizing its ull potential. The sandwih skin separated by the ore resists the in-plane stresses due to lateral loading. IX HSMV Naples 5-7 May 011 1

2 3. Joint between Steel & Composite Material The hybrid hull design o ship has longitudinal steel boxes in the entre and the ends o bottom hull and dek. Suh a struture will bring about the problem o requiring a strong joint between the omposite and the steel omponents. The diiulty with joining omposites to metals is the large dierene in stiness between the adherents, and the large anisotropy o omposites. The stiness mismath generally leads to large stress onentrations and aordingly weak joints (Cao & Grenestedt 00). There has been onsiderable researh in joining omposites to metals lately and is a separate topi in itsel. In this regards, the reent work o Cao & Grenestedt (00) is reerened. The strength o the tested speimens with joints were approximately 90% o that o omposite skin / oam ore reerene beams made o the same materials. 3.3 Choie o Composite Materials Gibson (1993) outlined some key property parameters inluening the seletion o strutural materials or marine use. It is evident that omposites have better harateristis than metals with regard to speii strength. However, in terms o speii stiness, only arbon and aramid omposites outperorm metals. Glass based omposites are more lexible. Apart rom mehanial perormane, strutural materials also have ost impliations. None o the omposites is ompetitive with metals with respet to stiness per volume ost. Furthermore, in strength per volume ost ases, only glass based omposites an ompete with metals. This is the underlying reason or the large usage o glass in large volume appliations suh as ships and oshore strutures. With the researh on joining Glass iber skin sandwih to Steel rom the previous setion (Cao & Grenestedt 00) and due to the above ators, or this paper, glass iber reinored polymer is hosen or the skin. The hoie o Core materials available are PVC oam, balsa wood and honeyomb materials. Expanded losed-ell polyvinyl hloride (PVC) oam has been widely used in many marine appliations. It is available in a range o densities, varying rom 5 to over 50 kg/m 3. PVC oams oer good resistane to water penetration, good thermal and eletrial insulation, and eetive vibration and damping harateristis (Shenoi & Dodkins 000). Balsa wood is one o the most eiient and moderately pried sandwih ore materials. Its main deiieny is suseptibility to water penetration and onsequential swelling, debonding, and rot. Although some suess has been laimed or the balsa ore sandwih onstrution in boats, a number o disastrous instanes o water penetration and subsequent deterioration o balsa ore have also ourred. For these reasons, use o balsa ore in the primary hull and dek struture o ships and boats is not normally advisable (Shenoi & Dodkins 000). Sandwih panels with ultra light honeyomb ores in aluminum developed in many ases or aerospae strutures, are generally very expensive or marine onstrution. They are unlikely to be suitable or the primary hull struture beause o the risk o water penetration and ore skin debonding under impat loads. For the above reasons and using the researh rom Cao & Grenestedt (00) or joining the sandwih struture to steel struture, we hoose PVC Foam Core or this paper. 3. Proposed Midship Setion An existing intermediate ship is seleted or whih the partial saety ators are known so that we may ompare it to the resulting design o the hybrid hull onept. The ship harateristis are shown in Table 1. Table 1: Ship Charateristis or Hybrid Hull Design Length, L pp 59.0 t m. Beam (molded) 55.0 t m. Drat amidship (molded). t. 6.8 m. Displaement (salt water) 9,00 LT metri Ton Trim by Stern -5.0 in m. GMT (orreted).75 t. 0.8 m. LCG at o midship t m. VCG above molded BL 3.0 t m. Roll Gyradius in air.38 t. 6.8 m. Pith Gyradius in air t m. Yaw Gyradius in air t. 0.3 m. IX HSMV Naples 5-7 May 011

3 A preliminary thikness o 0.01 m. or sandwih skin and 0.0 m. or sandwih ore was deided based on simple stress alulations using beam theory. Steel plating was onsidered to have a thikness o in. (0.013 m.) whih is a standard or marine industry. With the above materials, a preliminary design or ross-setion was proposed as shown in igure 1. Figure 1: Proposed Hal Midship Cross-setion LOADS AND STRESS ANALYSIS.1 Seond Order Strip Theory The loads on the hull or suh an intermediate ship are mainly primary loads suh as sagging and hogging o the hull beams and the seondary ores due to the hydrostati pressure. Seond order strip theory is applied to the ship o interest or long-term operating onditions. Though some problems exist with the SOST, it is believed that it an provide insight into the extreme loading problem, partiularly the dierene between sagging and hogging bending moments. Mansour and Jensen (00) derived requeny response untions and standard deviations or the wave indued bending moment amidships or mono-hull. The extreme value predition takes nonlinearities due to bow lare and the eet o whipping into aount. The probability that the individual peak moment values in the long term analysis will exeed a given value is taken as a weighted sum over all sea states and operational proile in the usual manner. peak ombined S,,, S,,, S P M m g m H T V wp H T V dh dt dvd [1] where H S is the signiiant wave height, T is the peak time period, V is the Veloity o the ship and β is the heading angle. The weight ator w is taken as w a wt where, T T 1 1 p H T dh dt a T T S S The mission proile was seleted or the ship in onsideration. The route seleted was mostly North Atlanti with a small portion in the Mediterranean. The same loads are applied to the hybrid hull as in the ase o the original ship to ompare the inal reliability indies. The estimated Long-term Bending Moments are shown in Table and are higher than those omputed using ABS Rules. Table : Long Term Analysis Results Bending Moment (SOST) Long-term Hogging 75,300 Lt.t MN.m Sagging 78,100 Lt.t 8.85 MN.m. Overall Stress Analysis Arbitrary dimensions were given to the steel plates in the preliminary design hosen in the previous setion. Then the primary stress analysis is perormed subjeting the midship hull to the maximum vertial bending moment (both hogging and sagging). For suh an analysis it is required to assume an approximate thikness or the sandwih plate. Due to the high modulus o elastiity o steel ompared to that o the sandwih plate, the ontribution o sandwih plate in sharing the bending moment is very less ompared to that o steel. In the later setion, the plate is also simultaneously subjeted to the lateral pressure determined rom the previous setion to obtain deletion and total stresses. The primary motive was to design a hybridhull ross-setion so that the Steel by itsel would withstand all the primary loads and the sandwih plate should be able to withstand the seondary loads. For the hybrid hull in onsideration, the dimensions determined to satisy this motive are shown in igure and the distribution o stresses in steel or hogging ondition are shown in igure 3. The stress distribution or sagging ondition is IX HSMV Naples 5-7 May 011 3

4 Max Deletion (m.) lower than that o the hogging ondition in this ase. Figure : Proposed Midship Cross-setion Figure 3: Stress Distribution under total Hogging Bending Moment.3 Loal Stress Analysis The Steel plate is subjeted to lateral pressure ombined with the ompressive/tensile stresses rom the bending moment. The stresses obtained rom these two loads are ombined using the appropriate load ombination ators. The hydrostati and hydrodynami pressures are estimated rom the ABS Rules and are ombined with the in-plane stresses rom the SOST to estimate the deletion and stresses or the bottom and dek steel plates. On the bottom plating the water pressure may be onsidered uniorm i the disturbane terms arising rom the presene o ship in waves as well as the small inlination due to roll and pith are negleted. The stresses obtained rom suh loading are heked to be below 80% o the yield stress o steel. 5 OPTIMIATION OF SANDWICH PANEL Sandwih i.e., omposite laminate skin with lightweight ore is treated as plate elements. Two ommonly used laminate plate theories are the lassial and the irst order shear deormation theories. Despite the simpliity and low omputational ost o these theories, they provide suiiently aurate desription o global response or thin to moderately thik laminates, e.g., gross deletions and ritial bukling loads (Reddy 1997). O these theories the irst order shear deormation theory with transverse extensibility appears to provide the best ompromise o solution auray, eonomy and simpliity. The third order theories provide only a slight inrease in auray relative to irst order shear deormation theory (Reddy 1997). For these reasons only the lassial and irst order shear deormation theories and their solutions or deletions and stresses are ompared or various kind o loading in marine appliations and the appropriate one is hosen or optimization analysis. The lassial laminated plate theory (CLPT) is an extension o the lassial plate theory to omposite materials. It is assumed that the Kirho hypothesis holds or this theory. In the irst order shear deormation theory (FSDT), the non-zero transverse shear deormation is taken into aount. The igures, 5 and 6 ompare the two theories or deletion and stresses to see the inluene o the transverse shear Comparison o Deletion in CLPT and FSDT CLPT FSDT No. o terms (m = n =) Figure : Comparison o Maximum Deletion in CLPT and FSDT IX HSMV Naples 5-7 May 011

5 Mas Stress / Yield Stress (aross iber)(transverse) Mas Stress / Yield Stress (along iber)(transverse) Comparison o Stresses in 0 deg ply along iber in CLPT and FSDT 0.16 CLPT FSDT No. o terms (m = n =) Figure 5: Comparison o Maximum Stress in 0 ply in CLPT and FSDT Comparison o Stresses in 90 deg ply aross iber in CLPT and FSDT 0.85 CLPT FSDT No. o terms (m = n =) Figure 6: Comparison o Maximum Stress in 90 ply in CLPT and FSDT It an be seen that the dierene is less than 3%. The thiknesses hosen or the above igures were thiknesses, t 0 = 0.01 m., t 90 = 0.01 m., and t =h=0.5 m. The similar omparison was done or thiknesses o t 0 = 0.01 m., t 90 = m., and h = 0.5 m. and the results diered by less than 3%. The transverse shear eets do not signiiantly inluene the results or deletion and stresses in this ase. CLPT also has muh simpler expressions or deletion and stresses as ompared to FSDT. For these reasons, CLPT theory is used or both optimization and reliability analyses. 5.1 The Optimization Problem Sandwih Panels are well known or providing high stiness with light weight, and are widely used to provide eiient strutures. The design o an eiient sandwih struture entails the speiiation o a number o parameters, inluding the materials, density and thikness o the ore, and thikness o the ae (skin) materials. Beause o the number o design variables, the optimization o the sandwih struture is not trivial, and has been onsidered by a number o authors Objetive Funtion The objetive untion ould be the weight o the sandwih panel or the ost o the sandwih panel. The ost riteria ould be more omplex than the weight riteria as the ost not only depends on the material thiknesses but also on the labor. The more ompliated the design, the more ostly it is, but the weight does not depend upon the omplexity. For example a sandwih ae design o [0 /±5 /90 ] o 0.01 m thikness has the same weight o [0 /90 ] o 0.01 m thikness or the same material, but the ost o the ormer would be higher than the latter as it involves more labour. In this paper the optimization is done on the weight riteria. I the ost untion is dominated mainly by the material osts, then the ost optimization analysis would be similar to the weight optimization analysis. In the design, the ae material is restrited to glass iber reinored polymer and the material o the ore is restrited to PVC (Divinyell H00) oam as disussed earlier. For a sandwih panel o area A, the weight would be w A t t [] where, t and, t are densities and thiknesses o ae and ore respetively. The design variables are the thiknesses, and t. The maximum hydrostati loads at on the bottom hull plate and the minimum lateral loads at on the dek plate. Thereore the design values or thiknesses will be dierent or the dek and the bottom hull plate Failure Modes For a steel struture, the ailure ould be deined when the steel reahes its yield stress or the ultimate stress or rom the von Mises ailure riterion. Similarly, the ailure o a sandwih struture ould also be deined by various riteria. The riteria whih an be used or deining ailure or orthotropi materials are maximum stress ailure, maximum strain ailure, Tsai-Hill (Hill 1950) ailure (an extension o von Mises yield riterion), Homan ailure (Homan 1967) and t IX HSMV Naples 5-7 May 011 5

6 Tsai-Wu tensor (Tsai & Wu 1971) ailure. Other ailure strength riteria are desribed by Sendekyj (1971). Tennyson, Ma Donald and Nanyaro (1978) addressed the next logial step in a urve itting proedure, namely a third-order polynomial it to ailure data. Though the above ailure riteria have identiied some loading onditions under whih their riterion is neessary to properly desribe the atual ailure behaviour, their omplexity has limited their use. For the purposes o optimization, we will use the maximum stress ailure riterion as its results or 0, 5 and 90 plies mathes very losely to the measured ailure data or Glass-Epoxy laminates (Jones 1998) and it is simpler to use this riterion in the optimization analysis. The ae or skin o a sandwih struture onsists o many layers. Eah layer beause o its dierent orientations has dierent properties in longitudinal and transverse diretions o the panel. Thereore there are three dierent ailure modes or eah layer viz. ailure along the iber (breaking o the iber or iber bukling), ailure aross the iber (matrix ailure or matrix ompression ailure) and shear ailure. The skin ailures are mainly lassiied as irstply-ailure (FPF) mode and the irst-ply-iberailure (FPFF) mode. The ormer mode indiates that the ailure has ourred when the stress has reahed an equivalent to the yield stress in the orresponding diretion (transverse or longitudinal or shear) in any layer. The latter mode indiates that the ailure has ourred in the breaking o a iber in any layer. Obviously, the FPF mode preedes the FPFF mode unless they are the same. In some ases, when only the FPF mode is reahed, meaning the matrix ailure or matrix ompression ailure, the struture is still assumed to have not ailed as the load bearing ibers are still intat even though there is a redution in the strength o the struture. For the optimization analysis, the onservative FPF mode is used to deine ailure onstraints. The typial ailure mode or the ore is the shear ailure. The other ailures ould be the ompression o ore in the in-plane diretion and the out-o-plane diretion. The other ailures ould be the delamination between the ore and the ae or in between the dierent layers o the ae. Consider a [0 /90 /Core/90 /0 ] sandwih panel as shown in igure 7. The irst term o maximum stress 1 or 0 ply, max 0 h 16q0 mn 0 0 t Q Q 1 Term 1 1 D ˆ ˆ 1 D3 D N x N y [3] where D11 D1 D16 D D1 D D6 is D 16 D6 D66 the lexural stiness matrix or the sandwih panel o dimensions a and b; D1 D11, D D, D D D ; m and n are odd integers; m a and n b; q 0 = uniorm pressure; N x and N y are in-plane ores; ( k ) Q11 Q1 Q16 ( k ) Q Q1 Q Q 6 is the stiness matrix, Q16 Q6 Q 66 k is either 0, or 90 layer or the ore. Figure 7: [0 /90 /Core/90 /0 ] sandwih panel The terms D 1, D and D 3 depend on the number o layers and their thiknesses. For a [90 /0 /Core/0 /90 ] sandwih panel, the bending stiness matrix, D Q h 0 Q h t1 h Q h t1 t h t [] It an be seen rom equations [3] and [] or stress that eah term is highly non-linear and thereore the lesser the terms, the more eiient is the optimization. It is neessary to hoose the appropriate number o terms or the optimization analysis as too many would result in an ineiient optimization and too less would result in inauray. The onvergene o deletion and stresses or a sample [90 /0 /Core/0 /90 ] sandwih panel with t 0 = 0.01 m, t 90 = 0.01 m and h = 0.5 m or the size o the sandwih panel o 1.6 m x.88 m with respet to the no. o terms is analyzed. For deletion, and stresses in the 11 1 IX HSMV Naples 5-7 May 011 6

7 transverse diretion, m = n = 1 or 3 gives a good approximation and or shear stresses and stresses in longitudinal diretion, m = n = 9 gives a good approximation. The interlaminar normal stress is muh lower than the yield and hene need not be taken into the design onstraints. But a sandwih skin an onsist o plies in various other angles whih an inrease the number o onstraints. It an be noted rom above that or optimizing even a simple sandwih; one has to deal with many onstraints Optimization Proedure Beause o the ompliated nature o the problem, initially a simple oniguration, [90 /0 /Core/0 /90 ] with the same thikness or both 0 and 90 is hosen. This redues the design onstraints to t and h, thikness o eah ply and the ore respetively. Due to the aspet ratio o the plate and its laminate properties, the irst type o ailure in eah ply an be outlined. For the 0 ply, the irst ailure has to be along the iber (transverse) diretion and or the 90 ply, the irst ailure has to be aross the iber (transverse) diretion. This primarily is due to the width (transverse,.88 m) o the plate being signiiantly smaller ompared to the length (longitudinal, 1.6 m) o the plate. So, the stiness properties in the transverse diretion beome more ritial or pure lateral loading. For the ore, the initial ailure will be in in-plane ompression (transverse) or in shear. Thereore, the onstraints are redued thereby inreasing the auray o the optimization program. Sine the ailure onstraints or stresses are only in transverse diretion, a omparison is done between using m = n = 1 or 3. Figures 8 and 9 show the perentage error in using m = n = 1 and 3 ompared to the asymptote value o the summation series or a range o ombination o thikness o plies and ore. The igures indiate that using one term gives a perentage error o at least 13 % whih is not aeptable or optimization. Using our terms i.e., m = n = 3, the perentage error drops down to a maximum o 1 % (Mushtaq 007). This leads us to onlude that in this ase or optimization analysis at least our terms (m = n = 3) have to be used. Due to the inherent unertainty in the laminate properties and the approximations used in these riteria, 80% o the ritial value is used or eah onstraint. The onstraints that will be onsidered or the optimization program are mentioned below. IX HSMV Naples 5-7 May 011 Figure 8: Perentage Error in Max Deletion using up to our terms Figure 9: Perentage Error in Max Stress in 90 layer (aross iber) using up to our terms Maximum deletion ours at a, b : 16q0 D1 D3 D a a b b 16q D1 D3 D a a b b 16q D1 D3 D a a b b 16q ( % o w D1 D3 D a a b b 0.8 x (% o width) [5] Maximum stress in the 0 layer (along iber), 0 1 ours at a, b : q0 h t mn m n Q 0 Q 0 sin sin D m 1 n 1 1 D3 D Nˆ y max al 11 1 [6] 7

8 90 1 Maximum stress in the 90 layer (aross iber), a, b : ours at Q 11 Q1 m1 n1 D1 D3 D N y 16q h t t mn m n sin sin ˆ [7] max al AMPL (Bell Labs), a omprehensive and powerul algebrai modeling language or linear and nonlinear optimization problems, in disrete or ontinuous variables is used to model the above problem. For nonlinear optimization Minos solver is used. Minos (Murtagh & Saunders) is a solver based on MINOS versions 5.51 that an be used either "stand-alone" or with AMPL to solve linear and nonlinear problems expressed in AMPL. For using Minos, the non-linear untion should be smooth but need not be onvex. For problems with nonlinear onstraints, MINOS uses a sparse SLC algorithm (a projeted Lagrangian method). The onvergene is rapid near a solution. The objetive untion and onstraints are input in AMPL and the optimized values or design variables are obtained. 5. Optimization Results The above desribed problem was solved and the obtained values or the design variables are t = 0.3 m and t s = m. The program is also run or various ixed values o the thikness o ore. Table 3 lists the results along with their ritial onstraint. Table 3: Optimized values o thiknesses or same thikness skin layers Core Skin Skin Weight thikness thikness thikness per Area h (m) 0 t (m) 90 t (m) (Kg/m ) Critial Constraint deletion stress aross iber in 90 stress aross iber in 90 stress aross iber in 90 stress aross iber in 90 It an be noted rom Table 3 that the governing onstraint is the maximum stress aross iber in the 90 ply. So, the next step would be to add another design variable by having dierent thiknesses or 0 and 90 layers. The onstraints beome more ompliated with an addition o one more variable. The obtained values or the design variables are t 1 = m, t = 0 and h = 0.79 m. The program is also run or various ixed values o the thikness o ore. The Table lists the results along with their ritial onstraint. Table : Optimized values o thiknesses or dierent thikness skin layers Core thikness Skin thikness Skin thikness Weight per Area Critial Constraint h (m) 0 t 1 (m) 90 t (m) (Kg/m ) deletion deletion Deletion deletion deletion deletion shear stress in 0 The ritial onstraint or this problem is the maximum deletion. O all the dierent orientation layers, 0 layer is the most eetive or reduing deletion. Thereore, a onlusion an be reahed that the addition o ±5 layers will not result in a better optimum value. Hene the addition o any layers apart rom the 0 layer will not derease the weight o the panel. Absene o 90 layer may be a ause o onern as the panel will not be sti in the longitudinal diretion or ertain loads whih may require at least ertain stiness in the longitudinal diretion. For this reason, based upon the loads, one may hoose to add 90 layer with a thikness equalling 10 % o 0 layer. For suh a onsideration, the results are shown in Table 5. The 80% design onstraint used in optimization will inluene the reliability results as a higher perentage should lead to a lower reliable system and vie-versa. The inluene o this perentage is urther disussed in the next setion. IX HSMV Naples 5-7 May 011 8

9 The highlighted row results in the best optimum value and is hosen or reliability analysis. Table 5: Optimized thiknesses with t = 10 % o t 1 Core thikness Skin thikness Skin thikness Weight per Area h (m) 0 (mm) 90 (mm) (Kg/m ) RELIABILITY ANALYSIS Critial Constraint Stress in 90 ply aross iber Stress in 90 ply aross iber Stress in 90 ply aross iber The initial step o a reliability analysis is the deinition o ailure. Failure is said to our when the predited load or demand exeeds the predited strength. I the load is denoted by L and the strength denoted by S, the ailure ours when SL 0 [8] This limit-state equation is then deined as G S L. Both the strength and the load are onsidered to be random variables. Thereore the probability o ailure is the probability that G 0. Mathematially, P P G 0 [9] The loads are ombined using the load ombination ators (Mansour & Thayamballi 1993). This method assumes that the total ombined load an be written as the sum o the omponent loads, with all but one (the largest) o the loads being modiied by a oeiient to aount or the orrelations between the loads, K K [10] where is the total ombined load, i is the i th omponent load and K i is the i th load ombination ator. The limit state equation is ormulated or the desired problem and appropriate distributions and harateristis are assigned to eah variable inluding the load ombination ators. All o this inormation is ed / input to a omputer program CALREL (Liu et al. 1989) to determine the saety index and the probability o ailure. CALREL employs several methods or simpliying this integral and then numerially estimating the probability o ailure. The method used here is the First-order Reliability Method (FORM) whih involves approximating the higher order ailure surae G x 0 by using hyperplanes that are tangent to the ailure surae at speiied design point in a transormed standard normal spae. The design point (most likely ailure point) is ound by an iterative method and the integral is then evaluated numerially. The strength variables are assumed to be lognormally distributed and their Mean and Coeiient o Variation (COV) are shown in Table 6. Table 6: Mean and COV o Strength Variables Variable Units Mean COV M (Hogging) MNm U M U (Sagging) MNm σ y (along-iber) MN/m σ y (aross-iber) MN/m τ y (layer shear) MN/m σ y (ore) MN/m τ y (ore shear) MN/m The moment variables are assumed to be extreme value distributed and the load ators are assumed to have normal distribution. Their adjusted Mean and Coeiient o Variation (COV) are shown in Table 7. Table 7: Adjusted Mean and COV o Load Variables Variable Units Mean COV M MNm W SW M (Hogging) MNm M (Sagging) MNm W M d MNm Pressure, p 0 kn/m In-plane Stress, MMN/m Load Fator, k w Load Fator, k d Limit State Equations The next step in the analysis ater all the variables are haraterized is to ormulate the limit state equations. There are two important dierenes between the hogging and sagging equations. First, there is the addition o the dynami moment and the orresponding load ombination ator. The dynami moment s sign is suh that it will add to the sagging wave bending moment (Mansour & IX HSMV Naples 5-7 May 011 9

10 Thayamballi 1993). The seond dierene is due to the sign o the still water bending moment. Sine the still water bending moment is a hogging moment, it adds to the hogging loads and subtrats rom the sagging loads. Both these dierenes are aounted or in the limit state equations. Overall Failure Mode: Limit State Equation or Hogging Condition: G MU M SW kwmw [11] Limit State Equation or Sagging Condition: G MU M SW kwmw kd M d [1] where M U is the ultimate bending moment strength, M SW is the still water bending moment load, M W is the wave bending moment load and M d is the dynami bending moment load (Mansour & Thayamballi 1993). Loal Failure Mode: Limit State Equation or Maximum Stress in the 0 layer (aross iber), ours at a, b : q0h t 0 0 mn 0 0 m n 1 max al Q1 Q ˆ D D D N sin sin 0 m 1 n y [13] Limit State Equation or Maximum Stress in the 90 layer (aross iber), a, b : ours at q h t t mn m n sin sin max al Q11 Q1 m1 n1 D1 D3 D 6. Reliability Indies and Comparison [1] Table 8 lists the reliability indies () and the probability o ailures (P ) rom FORM or the hybrid hull ship and Table 9 or the original ship or omparison. Table 8: Reliability Results or the Hybrid Hull Sagging Hogging Failure Mode P P Overall Loal E E E E-3 Table 9: Reliability Results or the Original Ship Sagging Hogging Failure Mode P P Primary Seondary E E E E-5 The Reliability indies o the Loal ailure mode or the Hybrid Hull seem to be signiiantly lower than the indies o the Seondary ailure mode or the Original Ship. In order to inrease the Loal mode reliability, the design optimization onstraints used in the optimization setion are made more onservative. Instead o the 80% o max (1.5) saety margin, 70% o max (1.3) saety margin is onsidered. This would hange the optimized variables rom t 1 (0 ply) t (90 ply) h (Core) Objetive 10 mm 1.0 mm 0.5 m 9.9 kg/m to 11 mm 1.1 mm 0.5 m 97. kg/m resulting in an inreased weight. The Table 10 shows the improved reliability indies or the Loal ailure mode. The shaded ells in grey are the lowest in their respetive ailure modes or both the ships. Table 10: Improved Reliability Results or the Hybrid Hull ater the modiiation o design onstraints Sagging Hogging Failure Mode P P Overall Loal E E E E- Table 11: Perentage Dierenes o Reliability Indies or Hybrid Hull and Original Ship ater the modiiation o design onstraints Failure Sag Hog Sag Hog Mode Overall % 5. 0%.58% 7.3% Loal 3.13%.3% 1.33% 11.06% The new Hybrid Hull reliability indies are only 1.3 to 7.3 % lower than the Original Ship. 7 CONCLUSION A preliminary design was proposed with the same harateristis as that o an existing ship. The CLPT is used or obtaining the stress and deletion onstraints. The omplexity and nonlinearity o the problem o optimizing a sandwih plate with strength and stiness onstraints would require a step-by-step optimization ruling out the unimportant onstraints and variables. The hybrid hull s strength and load parameters are determined and suitable distributions assigned. Separate limit-state equations are determined or overall and loal ailure modes and the reliability indies are alulated. The reliability index or overall ailure mode or hybrid hull was 3.79 ompared to.09 or an existing similar ship. The IX HSMV Naples 5-7 May

11 reliability index or loal ailure mode or hybrid hull was 3.08 ompared to 3.75 or the existing similar ship. In order to improve the reliability o the loal ailure mode, the design onstraint or optimization was hanged to a saety margin o 1.3 rom 1.5. The new optimized sandwih plate thiknesses hanged to t 0 = m., t 90 = m., and h = 0.5 m. and resulted in an inrease o the loal ailure mode reliability index to The omplexity and non-linearity o designing, optimizing and perorming a reliability analysis o any hybrid hull given the ship dimensions ould be perormed by the ollowing steps: 1) Determine the major loads on the ship hull lietime vertial bending moment, maximum lateral pressure, et. ) Design a preliminary hybrid hull with steel struture resisting the vertial bending moment and the lateral pressure. This step would deine the minimum thikness o steel required along with its dimensions. 3) Deine the dimensions o the sandwih panel to onnet the steel strutures and rom bulkhead to bulkhead. Determine the maximum lateral load and the in-plane load. With CLPT, ormulate the design onstraints o deletion and stress with a desired saety margin (rom 1.5 to 1.5). Optimize the panel with either weight or ost objetive or layer orientations and thiknesses. a) Start with only 0 and 90 layers with t 0 = t 90 and onsider m=n=3 terms or all stresses and deletion. Estimate the number o terms needed or reasonable auray and the disard the unimportant onstraints. Obtain an optimized plate or 0 and 90 layer skin with ritial onstraints. b) Perorm the optimization without the onstraint t 0 = t 90. The ritial onstraint rom the optimization results determines the ruial layer in resisting deletion. I the shear stresses are the ritial onstraint then adding ±5 layers would be helpul bearing in mind that the sandwih panel should always be symmetrially balaned. ) Consider omitting the third order terms o thiknesses i need be, as it might redue the non-linearity without a signiiant loss in auray. ) Deine the limit-state equations or the overall and loal ailure modes or the hybrid hull with the optimized panel. Choose appropriate means, ovarianes and distributions or the load and strength variables. Obtain reliability indies by the irst-order and seond-order reliability methods. I the reliability index or any variable is signiiantly less than desired, go to Step 3 and inrease the saety margin or the design onstraints. 8 REFERENCES Bell Labs AMPL A Modeling Language or Mathematial Programming, Bergan P.G., Buene L., Ehtermeyer T., Hayman B. (199) Assessment o FRP Sandwih Strutures or Marine Appliation. Marine Strutures 7; pp: Cao J., Grenestedt J.L. (00) Design and testing o joints or omposite sandwih/steel hybrid ship hulls. In: Composites Part A: Applied Siene and Manuaturing, Volume 35, Issue 9, September 00, Pages Gibson A.G. (1993) Composite Materials in Maritime Strutures, Oean Tehnology Series, Cambridge University Press, Cambridge, UK, vol. II, pp Hill R. (1950) The Mathematial Theory o Plastiity, Oxord University Press, London; p. 318 Homan O. (1967) The Brittle Strength o Orthotropi Materials, Journal o omposite Materials, April 1967; pp Mansour A., Thayamballi A. (1993) Probability-Based Ship Design Loads and Load Combinations, Ship Struture Committee-373 Mansour A. and Jensen J.J. (00) Estimation o Ship Long-term Wave-indued Bending Moment using Closed Form Expressions, Trans RINA, 00; pp Jones R.M. (1998) Mehanis o Composite Materials; nd edition 1998; pp. 107 Murtagh B., Saunders M. Minos 5.5, Stanord Business Sotware, In. Mushtaq S. (007) Optimized Design and Reliability o Hybrid Ship Hulls, Ph.D. Thesis, 007; University o Caliornia, Berkeley Reddy J.N. (1997) Mehanis o Laminated Composite Plates, CRC Press. Sendekyj G.P. (1971) A brie Summary o Empirial Multiaxial Strength Criteria or Composites, Composite Materials: Testing and Design (Seond Conerene), Anaheim, Caliornia, 0-; April 1971 Shenoi R.A., Dodkins A.R. (000) Design o Ships and Marine Strutures Made rom FRP Composite Materials. Comprehensive Composite Materials, Volume 6: 9 9 Tennyson R.C., MaDonald D., Nanyaro A.P. (1978) Evaluation o the Tensor Polynomial Failure Criterion or Composite Materials, Journal o Composite Materials, January 1978, pp Tsai S.W. and Wu. E.M. (1971) A General Theory o Strength or Anisotropi Materials, Journal o Composite Materials, January 1971, pp IX HSMV Naples 5-7 May