Minimum cost design of hybrid cross-ply cylinders with uncertain material properties subject to external pressure

Ocean Engineering 88 (2014) 310–317

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Minimum cost design of hybrid cross-ply cylinders with uncertain material properties subject to external pressure Isaac Sfiso Radebe a, Sarp Adali b,n a b

Department of Mechanical Engineering, Durban University of Technology, Durban, South Africa Discipline of Mechanical Engineering, University of KwaZulu-Natal, Durban, 4041, South Africa

art ic l e i nf o

a b s t r a c t

Article history: Received 2 January 2014 Accepted 22 June 2014 Available online 18 July 2014

Minimum cost design of hybrid cross-ply cylinders is presented which employ high-stiffness and expensive materials in the surface layers and the low-stiffness and inexpensive layers in the middle layers to combine the advantages of the two materials. Hybrid construction takes advantage of the sandwich effect whereby most of the load is carried by the surface layers. The cylinder is subject to external pressure with the material properties displaying uncertain-but-bounded variations around their nominal values. For a given external pressure, the material cost is minimized by minimizing the thickness of the surface layers. Analysis to determine the worst-case combination of material uncertainties makes use of convex modeling to compute the least favorable solution defined here as the minimum buckling pressure. The minimum cost designs are investigated for various problem parameters such as the wall thickness and the level of uncertainty. The relative sensitivities of the buckling pressure to material properties are also studied by defining sensitivity indices. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Hybrid design Material uncertainty Cross-ply cylinder External pressure Minimum cost Sensitivity

1. Introduction Hybridization in composite structures can be used effectively to reduce the material costs by using a more expensive high-stiffness material in the outer layers and a less expensive low-stiffness material in the inner layers. A typical case would be to use a combination of CFRP (Carbon Fiber Reinforced Plastics) and GFRP (Glass Fiber Reinforced Plastics) noting that CFRP is about an order of magnitude more expensive than GFRP. A hybrid layup makes a better use of the expensive material by employing it in the outer layers where it can contribute most to the stiffness of the component (Rahul et al., 2006; Aiello and Ombres, 2007; Abachizadeh and Tahani, 2009; Karakaya and Soykasap, 2011; Montagnier and Hochard, 2013). The cost of the component can be further reduced by minimizing the thickness of the outer layers which is the objective of the present study for a hybrid cylinder subject to external pressure. Previous studies on the optimization of hybrid laminates include Adali and Duffy (1992, 1993), Adali et al. (1995), Adali and Verijenko (1997, 2001), and Aiello and Ombres (1996). Noting that variations in the elastic constants of composite materials are not unusual, the design of the composite cylinder is

n

Corresponding author. Tel.: þ 27 31 2603203; fax: þ 27 31 2603217. E-mail addresses: sfi[email protected] (I. Sfiso Radebe), [email protected] (S. Adali).

http://dx.doi.org/10.1016/j.oceaneng.2014.06.010 0029-8018/& 2014 Elsevier Ltd. All rights reserved.

given subject to uncertainties in the material properties which may arise for a number of reasons such as manufacturing tolerances, fiber misalignment, defects and voids in the fiber/matrix composition, and imperfect bonding between fibers and the matrix. In such situations material properties may be treated as uncertain variables, thereby taking the variations in the elastic constants into account in the design. For a robust design capable of operating under material uncertainty, the design should be able to take the so-called “worst-case” into account. Studies on robust designs include Du and Chen (2002), Lee and Park (2001), Lee et al. (2013), McDowell et al. (2010), Parkinson (2000) and Sandgren and Cameron (2002). Material uncertainties can be studied by employing probabilistic and stochastic models which require statistical data on problem parameters such as probability density functions of random variables. In many cases accurate estimates of probability distributions are fairly difficult to obtain due to lack of sufficient information. On the other hand, information on the bounds on elastic constants could be available leading to bounded-but-uncertain properties. In such cases the problem can be analyzed using convex models of uncertainty which provide an effective alternative to probabilistic models. In the non-probabilistic modeling of uncertainties, variations about the average values of the uncertain variables can be expressed in terms of ellipsoidal sets which lead to the computation of the least favorable solution (worst-case) and consequently to a

I. Sfiso Radebe, S. Adali / Ocean Engineering 88 (2014) 310–317

robust design where the uncertainties are taken into account in a conservative manner (Jiang et al., 2011; Kang et al., 2011; Luo et al., 2009, 2011; Qiu et al., 2009). This approach contrasts with a deterministic analysis where the average values of elastic constants are employed in the calculations, neglecting the variations in the material properties. Recent studies on the application of convex analysis to structural problems include Hu and Qiu (2010), Kang et al. (2011), Qiu (2005) and Qui et al. (2009). Optimization under uncertainty was the subject of the papers by Zhang and Ding (2008), Luo et al. (2009) as well as the review article by Yao et al. (2011). Cylindrical shells subject to external pressure are employed in several branches of engineering and the previous work on the subject include Li and Chen (2002), Messager et al. (2002), Rasheed and Yousif (2001), Ross and Little (2001), Smerdov (2000), Sridharan and Kasagi (1997), Tarn and Wang (2001) and Voce (1969). More recent studies on the subject include Cagdas and Adali (2011), Kardomateas and Simitses (2005), Liang et al. (2003, 2004), Paimushin (2008), Papadakis (2008), Rasheed and Yousif (2005), and Tsukrov and Drach (2010). Optimization of composite cylinders under external pressure with respect to stacking sequence has been studied by Maalawi (2008, 2011). Buckling results given in these references were based on the average values of the elastic constants and the possibility of variations in the material properties was not considered. Results based on deterministic values are, in general, not robust and this situation necessitates taking the data uncertainties into account in a non-deterministic model. Hybrid cylinders under external pressure with uncertain material properties have been studied by Radebe and Adali (2013). In the present study, minimum cost design of a long cylindrical shell made of a hybrid composite material is studied. The load acting on the cylinder is external pressure and its lay-up is defined as symmetrically laminated cross-ply with the material properties taken as uncertain. As the uncertainties only involve the elastic constants, the material remains orthotropic after the uncertainties are implemented. Analysis to determine the worst-case combination of material uncertainties makes use of convex modeling to compute the least favorable solution (minimum buckling pressure). The wall of the cylinder consists of three layers with the outer layers made of a composite with high modulus fibers such as carbon or boron and the middle layer made of a composite with low modulus fibers such as glass. The minimum cost design is achieved by using the minimum amount of the high modulus expensive fibers for a given design pressure. Sensitivity of the buckling pressure to material uncertainty is also investigated by means of sensitivity indices.

311

2. Hybrid cross-ply cylinder The hybrid cylinder under consideration is constructed of layers with different fiber reinforced plastic (FRP) materials. The material for the outer layers is normally of a high modulus composite such as carbon or boron FRP and the material for the middle (inner) layer is of a low modulus composite such as glass FRP. For an anisotropic thin-walled long cylinder with an average radius R, let the axial, tangential and radial coordinates be denoted by ðx; s; zÞ with ds ¼ Rdϕ where ϕ is the circumferential coordinate. The equations governing the buckling of a long cylinder under an external pressure p can be expressed as (Paimushin, 2008; Rasheed and Yousif, 2001, 2005)
 

∂2 M ss ∂N ss ∂w0 ∂w0  v N pR2 þ R   ð1Þ ¼ v ss 0 0 ∂ϕ ∂ϕ ∂ϕ ∂ϕ2   


∂2 M ss ∂ ∂w0 ∂v0 v0  N ss þ p w0 þ ¼ pR2  R N ss þ 2 ∂ϕ ∂ϕ ∂ϕ ∂ϕ

ð2Þ

where M ss and N ss are the moment and force resultants in the tangential direction, and v0 and w0 are displacements in the tangential and radial directions, respectively. For a symmetrically laminated long cylinder the moment and force resultants are given by " # " #" # N ss ε0ss A~ 22 0 ¼ ð3Þ ~ 22 M ss kss 0D where ε0ss is the circumferential mid-plane strain, kss is the ~ 22 are the uncertain circumferential curvature. In Eq. (3), A~ 22 and D extensional and bending stiffness coefficients where the overhead  sign indicates an uncertain quantity. Let the composite material in the outer layers be identified by subscript A and that of the middle (inner) layer by subscript B. Then the stiffness coefficients ~ 22 are given by A~ 22 and D A~ 22 ¼ t A ðQ~ 22 ÞA þt B ðQ~ 22 ÞB ;

D~ 22 ¼ dA ðQ~ 22 ÞA þ dB ðQ~ 22 ÞB

ð4Þ

where dA ¼

1 3 3 ðh  t B Þ; 12

dB ¼

t 3B 12

ð5Þ

are geometrical quantities. Here h is the wall-thickness, t A is the total thicknesses of the outer layers, and t B is the thickness of the middle layer, so that t A þ t B ¼ h as shown in Fig. 1. In Eq. (4), Q~ 22k , where k ¼ A or B is the material index, is the uncertain reduced stiffness modulus and for cross-ply layers, they are

Fig. 1. Cross-ply hybrid cylinder and its cross-section.

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given by: E~ 2k ; Q~ 22k ð01Þ ¼ 1  ν~ 12k ν~ 21k

E~ 1k Q~ 22k ð901Þ ¼ 1  ν~ 12k ν~ 21k

ð6Þ

where E~ 1k and E~ 2k are the uncertain Young's moduli in the material directions, and ν~ 12k and ν~ 21k are uncertain in-plane Poisson's ratios which satisfy the relation: ν~ 21k ¼ ν~ 12k E~ α =E~ 1k

ð7Þ

ε0ss

and kss in terms of displacements and substitutExpressing ing into Eqs. (1)–(3), the coupled differential equations governing the buckling of the long cylinder under external pressure are obtained. The eigenvalue of these equations yields the nondeterministic critical external pressure pcr which is given by (Paimushin, 2008; Rasheed and Yousif, 2001, 2005) ! ~ 22 A~ 22 D pcr ¼ 3 3 ð8Þ R A~ 22 þ RD~ 22 in the case of a symmetrically laminated cross-ply cylinder composed of 01 and 901 plies.

In the present section, a first order uncertainty analysis is implemented to express the buckling pressure in terms of the variations which exist in the uncertain material properties. For this purpose, uncertain elastic constants are defined as follows: E~ 2k ¼ E02k ð1 þ γ 2k Þ;

ν~ 12k ¼ ν012k ð1 þ γ 3k Þ

ð9Þ

where k ¼ A or B, and the superscript }0} denotes the nominal values of the elastic constants. The constants γ ik represent the unknown variations of the elastic constants, can be positive or negative and satisfy the inequalities jγ ik j5 1. They are determined by convex analysis to yield the least favorable buckling load leading to a robust design. In the first order analysis, the uncertain quantities can be linearized in terms of γ ik using the first-order approximation n

2

ð1 7 εÞ ffið1 7 nεÞ þ Oðε Þ

ð10Þ

where the superscript n can take positive or negative values and jεj 5 1. For this purpose we first linearize ν~ 21k ¼ ν~ 12k E~ 2k =E~ 1k in terms of γ ik by noting that ν0 E0 ð1 þ γ 3k Þð1 þ γ 2k Þ ν012k E02k ffi ð1  γ 1k þ γ 2k þ γ 3k Þ ν~ 21k ¼ 12k0 2k 1 þ γ 1k E1k E01k

Next the expressions for the critical external pressures are obtained for cross-ply hybrid composites with stacking sequences ð90oA =0oB =90oA Þ and ð90oA =90oB =90oA Þ. 4.1. Stacking sequence ð90oA =0oB =90oA Þ We first linearize the buckling pressure expression in terms of γ ik for ð90oA =0oB =90oA Þ lamination where the subscripts A and B are the material indices. For this purpose we introduce the notation Q~ A ¼ Q~ A ð901Þ and Q~ B ¼ Q~ B ð01Þ. First, substituting Eqs. (16) and (17) into Eq. (4), stiffness coefficients A~ 22 and D~ 22 are obtained in terms of γ ik . Next they are introduced into Eq. (8) to obtain ! t A dA ðQ~ A Þ2 þ t B dB ðQ~ B Þ2 þ ðt A dB þ t B dA ÞQ~ B Q~ A M pcr ¼ 3 ¼3 N R3 t A Q~ þ R3 t B Q~ þ RdA Q~ þ RdB Q~ A

ð18Þ

B

M ¼ t A dA ðQ~ A Þ2 þ t B dB ðQ~ B Þ2 þ ðt A dB þ t B dA ÞQ~ B Q~ A

ð19Þ

N ¼ R3 t A Q~ A þ R3 t B Q~ B þ RdA Q~ A þRdB Q~ B

ð20Þ

Let s1 ¼ t A dA s3 ¼ t B dB

!2 E011A ; 1  a1A !2 E022B 1  a1B

s2 ¼

ðt A dB þ t B dA ÞE011A E022B ; ð1  a1A Þð1 a1B Þ ð21Þ

Then t A dA ðQ~ A Þ2 ¼ s1 ð1 þ 2ð1  a2A Þγ 1A þ 2a2A γ 2A þ 4a2A γ 3A Þ

ð22Þ

t B dB ðQ~ B Þ2 ¼ s3 ð1  2a2B γ 1B þ 2ð1 þ a2B Þγ 2B þ4a2B γ 3B Þ

ð23Þ

ðt A dB þt B dA ÞQ~ B Q~ A ¼ s2 ð1 þ ð1  a2A Þγ 1A þ a2A γ 2A þ 2a2A γ 3A  a2B γ 1B þ ð1 þ a2B Þγ 2B þ 2a2B γ 3B Þ ð24Þ

ð11Þ and M, given by Eq. (19), can be expressed as 3

ð12Þ

M ¼ f 0 þ ∑ðf i γ iA þ f i þ 3 γ iB Þ

ð25Þ

1

where

where a1k ¼ ν012k ν021k ;

a1k a2k ¼ 1  a1k

ð13Þ

3

f 0 ¼ ∑si ; f 1 ¼ ð1  a2A Þð2s1 þ s2 Þ; 1

f 2 ¼ a2A ð2s1 þ s2 Þ;

Substituting Eqs. (9) and (12) into Eq. (6), we obtain Q~ 22 ð01; γ ik Þ ¼

B

where Q~ A ¼ Q~ A ð901Þ and Q~ B ¼ Q~ B ð01Þ are given by Eqs. (16) and (17). In Eq. (18) M and N denote

Similarly, 1  ν~ 12k ν~ 21k ffi ð1  a1k Þð1 þ a2k γ 1k  a2k γ 2k  2a2k γ 3k Þ

ð17Þ

4. Buckling pressures for hybrid composites

A

3. Uncertainty analysis

E~ 1k ¼ E01k ð1 þ γ 1k Þ;

 E0  Q~ 22 ð901; γ ik Þ ffi 1k 1 þ ð1  a2k Þγ 1k þ a2k γ 2k þ 2a2k γ 3k 1  a1k

E02k 1 þ γ 2k 1  a1k 1 þ a2k γ 1k  a2k γ 2k  2a2k γ 3k

ð14Þ

f 3 ¼ 2a2A ð2s1 þ s2 Þ

f 4 ¼  a2B ðs2 þ 2s3 Þ;

f 5 ¼ ð1 þ a2B Þðs2 þ 2s3 Þ;

ð26Þ f 6 ¼ 2a2B ðs2 þ 2s3 Þ ð27Þ

Similarly, let Q~ 22 ð901; γ ik Þ ¼

E01k 1 þ γ 1k 1  a1k 1 þa2k γ 1k a2k γ 2k 2a2k γ 3k

ð15Þ

Linearizing Eqs. (14) and (15) with the help of Eq. (10) yields Q~ 22 ð01; γ ik Þ ffi

E02k

1  a1k



1  a2k γ 1k þ ð1 þ a2k Þγ 2k þ2a2k γ 3k



s4 ¼

R3 t A E011A ; 1  a1A

s5 ¼

R3 t B E022B ; 1  a1B

s6 ¼

RdA E011A ; 1  a1A

s7 ¼

RdB E022B 1  a1B

ð28Þ

then N, given by Eq. (20), can be expressed as 3

ð16Þ

N ¼ g 0 þ ∑ðg i γ iA þ g i þ 3 γ iB Þ 1

ð29Þ

I. Sfiso Radebe, S. Adali / Ocean Engineering 88 (2014) 310–317

where 7

g 0 ¼ ∑ si ; g 1 ¼ ð1 a2A Þðs4 þ s6 Þ; 4

þ λB

g 2 ¼ a2A ðs4 þ s6 Þ;

g 3 ¼ 2a2A ðs4 þ s6 Þ g 4 ¼  a2B ðs5 þ s7 Þ;

ð30Þ g 5 ¼ ð1 þ a2B Þðs5 þ s7 Þ;

g 6 ¼ 2a2B ðs5 þ s7 Þ ð31Þ

Finally substituting Eqs. (25) and (29) into Eq. (18), and linearizing the resulting expression using Eq. (10), we can express the buckling pressure in terms of γ ik as
 3 3 pcr ffi f 0 þ ∑ðbi γ iA þ bi þ 3 γ iB Þ ð32Þ g0 1

The parameters γ ik and the Lagrange multipliers λA and λB are computed from Eq. (38) as

where bi ¼ g0 f i g f 0 gi .

3 2g 0 βA

4.2. Stacking sequence ð90oA =90oB =90oA Þ

Thus

For a stacking sequence of ð90oA =90oB =90oA Þ, the expression for the buckling pressure can be obtained by the same procedure. The constants which are different from the case of ð90oA =0oB =90oA Þ given above are the following:

γ iA ¼ 8 βA bi

0

s5 ¼

R3 t B E011B ; 1  a1B

s7 ¼

ð1  a1B Þ2

;

RdB E011B 1 a1B

ð33Þ

In the expressions for f i , i ¼ 1; 2; :::6, s2 and s3 have to be replaced by s2 and s3 , and f 4 and f 5 have to be replaced by f 4 ¼ ð1  a2B Þðs2 þ 2s3 Þ;

f 5 ¼ ðs2 þ 2s3 Þa2B

ð34Þ

Similarly, in the expressions for g i , i ¼ 1; 2; :::6, s5 and s7 have to be replaced by s5 and s7 given in Eq. (33), and g 4 and g 5 have to be replaced by g 4 ¼ ð1 a2B Þðs5 þ s7 Þ;

g 5 ¼ a2B ðs5 þ s7 Þ

ð35Þ

With these changes, the expression for pcr given by Eq. (32) remains the same for the ð90oA =90oB =90oA Þ stacking sequence.

ð37Þ

ð38Þ

λA ¼ 7

s3 ¼ t B dB

∑

i¼1

γ 2iB β2B

∂Lðbi ; γ ik Þ ¼0 ∂γ ik

3 bi ; g 0 2λA

ðE011B Þ2

!

3

The minima and the maxima of the Lagrangian are determined from

γ iA ¼ 

ðt dB þ t B dA ÞE011A E011B ; s2 ¼ A ð1  a1A Þð1  a1B Þ

313

γ iB ¼ 

3 bi þ 3 g 0 2λB

!1=2

3

2

∑ bi

i¼1

3

;

λB ¼ 7

ð39Þ 3 2g 0 βB

!  1=2 2

;

∑ bi

i¼1

γ iB ¼ 8 βB bi þ 1

!1=2

3

2

ð40Þ

∑ bi þ 1

i¼1

3

!  1=2 2

∑ bi þ 1

i¼1

for i ¼ 1; 2; 3

; ð41Þ

where the plus and minus signs correspond to the lowest and highest buckling pressures.

6. Minimum cost design The minimum cost design refers to a hybrid design where a minimum amount of the expensive fibers such as carbon or boron is used for a given design pressure denoted by pd , thereby keeping the cost at the minimum level possible. Let αh denote ratio of the thickness t h of the layer made of a high modulus composite to the wall thickness h of the cylinder, i.e., αh ¼

th h

ð42Þ

The minimum cost problem can be stated as ð43Þ

min αh subject to

5. Convex analysis

pd r pcr ðαh ; βA ; βB Þ

In this section the least favorable buckling load corresponding to the worst-case distribution of the uncertainties is computed by convex analysis. For this purpose we introduce constraints on the variations γ ik in the form of ellipsoids, viz. 3

∑ γ 2ik r β2k

ð36Þ

i¼1

where βk o1, k ¼ A or B, determines the level of uncertainty and βk ¼ 0 corresponds to the deterministic case. It is observed that the level of uncertainty could be different for different composites used in the hybrid construction. The buckling pressure pcr depends on the level of uncertainty as measured by βk and a higher βk leads to a smaller buckling pressure as expected. The expression (32) for pcr has to be minimized with respect to the constants γ ik and subject to the constraints (36) to determine the lowest buckling pressure for given levels of uncertainty. For this purpose, a Lagrangian is formulated noting that the constraints given by Eq. (36) yield the lowest pressure when the γ ik values lie on the surface of the ellipsoid, i.e. when ∑3i ¼ 1 γ 2ik  β2k ¼ 0. Thus the Lagrangian takes the following form Lðbi ; γ ik Þ ¼


 3 3 f 0 þ ∑ðbi γ iA þbi þ 3 γ iB Þ þ λA g0 1

3

∑ γ 2iA  β2A

i¼1

!

3

and

∑ γ 2ik  β2k ¼ 0

i¼1

ð44Þ

for a given wall thickness h and for given values of βA and βB . For pd 4 pcr ð1; βA ; βB Þ,, i.e., for t h ¼ h, a feasible design does not exist. Similarly for pd o pcr ð0; βA ; βB Þ, the cylinder can be built of the low cost composite only. Thus the problem yields a feasible solution for pcr ð0; βA ; βB Þ r pd rpcr ð1; βA ; βB Þ where 0 r αh r 1.

7. Sensitivity analysis The sensitivity of the buckling pressure to the variations in the uncertain material properties can be investigated by defining relative sensitivity indices SK ðγ ik Þ given by     γ ik  ∂p ðβ Þ SK ðγ ik Þ ¼  cr k  ð45Þ ∂γ ik pcr ðαh ; 0; 0Þ which is normalized with respect to the deterministic buckling pressure pcr ðαh ; 0; 0Þ. In Eq. (45), the sensitivities of the buckling pressure to the variations in E1k ; E2k and ν12k are denoted by SE1k ðγ 1k Þ, SE2k ðγ 2k Þ and Sν12k ðγ 3k Þ, respectively, so that the subscript K stands for the respective material property, i.e., K ¼ E1k , K ¼ E2k or K ¼ ν12k with k ¼ AorB denoting the material A or B. Eq. (45) indicates that the buckling pressure will have zero sensitivity for γ ik ¼ 0 corresponding to the deterministic case as expected. The

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sensitivities SK ðγ ik Þ can be computed from Eqs. 32 and 45 as jb γ j SK ðγ iA Þ ¼ i iA ; f0

jb γ j SK ðγ iB Þ ¼ i þ 1 iB f0

ð46Þ

noting that pcr ðαh ; 0; 0Þ ¼ 3fg 0 . A recent work on the sensitivity of 0 composite structures using global sensitivity indices is given by Conceição António and Hoffbauer (2013).

8. Numerical results The effect of material uncertainty on the buckling pressure is studied by specifying the high modulus composite as CFRP (Material A) and the low modulus composite as GFRP (Material B). The nominal values of the elastic constants for CFRP are taken as E1 ¼ 181:0GPa, E2 ¼ 10:3 GPa, ν12 ¼ 0:28 and for GFRP as E1 ¼ 57:0GPa, E2 ¼ 14:0GPa, ν12 ¼ 0:277 (see Rasheed and Yousif (2001)). The average radius R of the cylinder is taken as R ¼ 1 m in the computations. The thickness ratio of the outer layers made of CFRP to the total thickness is denoted by αc ¼ t c =h where t c is the thickness of the CFRP layers. We first investigate single material cylinders. Fig. 2 shows the curves of buckling pressure plotted against the thickness parameter α90 ¼ t 90 =h for a GFRP cylinder with a stacking sequence ð90ogl =0ogl =90ogl Þ where t 90 refers to the thickness of the 90o layers and the subscript “gl” to GFRP. The results are given for uncertainty levels β ¼ 0.0, 0.1, 0.2 and 0.3 and indicate that as the thickness of the 90o layers increases, the increase in the buckling pressure tapers off and for α90 Z0:8, this increase is minimal. The

Fig. 2. Buckling pressure vs. thickness ratio for ð90ogl =0ogl =90ogl Þ GFRP cylinders for various uncertainty levels.

Fig. 3. Buckling pressure vs. thickness ratio for ð90oc =0oc =90oc Þ CFRP cylinders for various uncertainty levels.

corresponding results for CFRP are shown in Fig. 3 where it is observed that for α90 ¼ 0:0, the buckling pressure is less that of GFRP due to the lower value of E2 for CFRP as compared to GFRP. The increase in pcr again tapers off for α90 Z0:8. Next we investigate hybrid ð90oc =0ogl =90oc Þ cylinders where the subscript “c” stands for CFRP layers. Fig. 4 shows the curves of pcr plotted against α90 and indicates that after a fairly substantial increase in pcr as the thickness of the 90o CFRP layers increases, this increase tapers off for α90 Z 0:7. The corresponding results for hybrid ð90oc =90ogl =90oc Þ cylinders are shown in Fig. 5 which shows that for low values of α90 , pcr is higher due to GFRP middle layer being 90o , but at higher values of α90 , the increase in pcr tapers off for α90 Z 0:6. In all cases, the buckling pressure decreases as the level of uncertainty increases and the percentage difference between the uncertainty levels remain fairly constant. Next the relation between the wall thickness h of the cylinder and the thickness α90 of the 901outer layers is investigated for ð90oc =90ogl =90oc Þ hybrid cylinders for the deterministic case (Fig. 6a) and the uncertain case with βc ¼ βgl ¼ 0:3 (Fig. 6b). Fig. 6a shows the contour lines of pcr with respect to 0:04m rh r 0:16m and 0:0 r α90 r1:0 for ð90oc =90ogl =90oc Þ cylinders for the deterministic case (βc ¼ βgl ¼ 0:0). The corresponding results for βc ¼ βgl ¼ 0:3 are shown in Fig. 6b. It is observed that as the level of uncertainty increases, a higher thickness is needed for the same value of pcr as expected. Moreover, for a given value of pcr , increasing the wall thickness h constitutes a less expensive option as compared to keeping the thickness constant and increasing the thickness of CFRP layers. This issue is studied in more detail in the next two figures.

Fig. 4. Buckling pressure vs. thickness ratio for hybrid ð90oc =0ogl =90oc Þ cylinders for various uncertainty levels.

Fig. 5. Buckling pressure vs. thickness ratio for hybrid ð90oc =90ogl =90oc Þ cylinders for various uncertainty levels.

I. Sfiso Radebe, S. Adali / Ocean Engineering 88 (2014) 310–317

315

Fig. 6. Contour lines of the buckling pressure (MPa) with respect to wall thickness and thickness ratio αc for a hybrid ð90oc =90ogl =90oc Þ cylinder with a) βc ¼ βgl ¼ 0:0 (deterministic case), b) βc ¼ βgl ¼ 0:3 (uncertain case).

uncertainty levels. Here t c is the thickness of CFRP layers, i.e., outer layers. Fig. 7 shows the results for h ¼ 0:12mand Fig. 8 for h ¼ 0:15m. It is observed that the thickness of the CFRP layers increases sharply for αc Z0:7 as the design pressure increases indicating that after αc Z 0:7, increase in the thickness of CFRP layers is not effective and basically wasteful and it is better to increase the overall thickness to keep αc , i.e., the usage of CFRP, low. Finally sensitivity of the buckling pressure to the uncertainty in Young's moduli E1 of the glass and carbon layers is investigated in Fig. 9 which shows the contour lines of pcr with respect to β ¼ βc ¼ βgl and αc for ð90oc =90ogl =90oc Þ hybrid cylinders. Fig. 9a shows the sensitivity contour lines of pcr for E1c and Fig. 9b for E1g . As expected the buckling pressure is more sensitive to uncertainty in E1c and this sensitivity increases as the thickness of the CFRP layers increases. Fig. 7. Minimum thickness ratio vs. design pressure for a hybrid ð90oc =90ogl =90oc Þ cylinder with h ¼ 0:12m

Fig. 8. Minimum thickness ratio vs. design pressure for a hybrid ð90oc =90ogl =90oc Þ cylinder with h ¼ 0:15m.

The minimum cost results are shown in Figs. 7 and 8 which show the curves of minimum αc ¼ t c =h plotted against the design pressure pd for ð90oc =90ogl =90oc Þ hybrid cylinders for various

9. Conclusions Minimum cost design problem was investigated for hybrid cross-ply cylinders consisting of high-stiffness surface and lowstiffness core layers with the material properties exhibiting variations around their nominal values. The uncertainty in the material properties was taken into account by convex modeling which determines the worst-case combination of material properties to obtain the least favorable solution of the problem. As such the minimum cost designs which are obtained by minimizing the thickness of the expensive, high-stiffness surface layers for a given buckling pressure, are computed for the worst case of uncertainty, yielding a robust design capable of carrying the load under unexpected variations in the elastic constants. Convex modeling leads to a three-dimensional ellipsoid bounding the uncertainties for each material and the method of Lagrange multipliers is implemented to obtain the least favorable solution for a given level of uncertainty. Moreover sensitivity indices are developed to investigate the relative sensitivity of the buckling pressure to the uncertainties in the elastic constants. The numerical results were given for graphite-epoxy/glassepoxy hybrid cylinders. The effects of the level of uncertainty and the relative layer thicknesses on buckling pressure were studied and the increase in the uncertainty was correlated with

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Fig. 9. Contour lines of sensitivity with respect to uncertainty level β ¼ βc ¼ βgl and thickness ratio αc for hybrid ð90oc =90ogl =90oc Þ cylinders a) w.r.t. E1c (SE1c ), b) w.r.t. E1g (SE1g ).

the decrease in the buckling pressure (Figs. 2–5). It was observed that the buckling pressure for hybrid cylinders increases with increasing CFRP layer thickness as expected, but this increase tapers off when about 50% of the cylinder is CFRP, i.e., after a certain thickness of the CFRP layers, their contribution to the buckling pressure becomes less effective leading to an inefficient use of the expensive material (Figs. 4 and 5). Thus, cost effective designs can be obtained with a relatively minor decrease in the buckling pressure using a hybrid construction. On the other hand increasing the overall thickness of the cylinder was found to be more effective in increasing the buckling pressure both for deterministic and uncertain cases (Fig. 6). Minimum thickness ratio for CFRP increases gradually with increasing design pressure, but after a certain point, this ratio increases faster, again indicating that the contribution of the surface layers to the buckling pressure tapers off at higher pressures (Figs. 7 and 8). It was also shown that the buckling pressure was more sensitive to the uncertainties of Young's modulus of the outer layers. It is noted that the present results can be easily extended to non-symmetrical cases by taking the thicknesses of the outer layers different. In this case the expression for the critical buckling pressure should be for the unsymmetrically laminated composite cylinders given by Rasheed and Yousif (2001, 2005).

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