Optimal design of filament wound truncated cones under axial compression

Accepted Manuscript Optimal design of filament wound truncated cones under axial compression Izzet U. Cagdas PII: DOI: Reference:

S0263-8223(16)32006-2 http://dx.doi.org/10.1016/j.compstruct.2017.03.023 COST 8344

To appear in:

Composite Structures

Received Date: Revised Date: Accepted Date:

29 September 2016 2 March 2017 8 March 2017

Please cite this article as: Cagdas, I.U., Optimal design of filament wound truncated cones under axial compression, Composite Structures (2017), doi: http://dx.doi.org/10.1016/j.compstruct.2017.03.023

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

OPTIMAL DESIGN OF FILAMENT WOUND TRUNCATED CONES UNDER AXIAL COMPRESSION Izzet U. Cagdas Civil Engineering Department, Faculty of Engineering, Akdeniz University, 07058, Antalya, Turkey, [email protected] FOOTNOTE: Tel: +90 5323101166, fax: +90 2423106306 Honorary Researcher: School of Mechanical Engineering, University of KwaZulu-Natal Durban, South Africa Abstract In this study, filament wound truncated cones under axial compression are optimized with the objectives of minimizing the total weight and maximizing the failure load, which is defined as the minimum of the buckling and the first-ply failure (FPF) loads. The numerical results are obtained using an axisymmetric degenerated shell element based on a refined first-order shear deformable shell theory and a 2D degenerated shell element is used for verification purposes. It is shown that, FPF is more critical than buckling for thicker cones with lower cone angles. Optimal designs, where FPF and buckling are imposed as design constraints, are presented for filament wound cones using Micro-Genetic Algorithms. The results show that, using more layers having different winding angles has negligible influence on the failure load and the optimal design is not FPF critical for moderate levels of axial compression. The influence of the rotational boundary conditions on the optimal failure load is also demonstrated. Keywords: Cone; buckling; finite elements; genetic algorithms; design optimization 1. INTRODUCTION In this study, optimal designs for axially compressed filament wound truncated cones are presented. The primary objective is to maximize the failure load which is defined as the minimum of the buckling load and the first-ply failure load and the secondary objective is to minimize the total weight. Filament winding is a fabrication technique well suited to automation and several studies exist in the literature on the buckling behavior of filament wound laminated composite cones [1-10]. The main cause of the complexity regarding these structures is that, thickness and ply angles of filament wound cones along the axial direction are not constant, as stated by Morozov [1], Goldfeld and Arbocz [2, 5], Goldfeld et al. [3,4,6], Patel et al. [7], Blom et al. [8], Goldfeld [9,10], and Maleki and Tahani [11]. Optimal design of thin-walled fiber reinforced composite structures under compressive loads have received considerable attention in the literature as these structures are prone to buckling. Design/optimization of filament wound truncated cones under buckling loads has been treated previously by several researchers. Brown and Nachlas [12] presented the optimal ply angles maximizing strength. Goldfeld et al. [3] obtained optimal lamination angles for variable thickness laminated cones using response surface method. Kabir and Shirazi [13] obtained optimal laminate configurations for filament wound laminated cones under axial compression using the penalty function method. Maleki and Tahani [11] considered variable thickness conical shell panels under thermomechanical loads. Naderi et al. [14] studied the influence of fiber paths on buckling loads of laminated cones under axial compression. Shadmehri et al. [15] used Ritz Method to analyze axially compressed filament wound cones and made some recommendations for design. However, FPF has not been considered in these studies. To the authors knowledge, the present study is the first study where a FPF constraint is considered in design/optimization of axially compressed cones. Also, the influence of the

boundary conditions is investigated in the present study and it is revealed that the rotational restraints at the large end has significant influence on the buckling load. A suitable method should be selected for design/optimization because of the several design parameters involved such as material properties, fiber orientations, number of layers, cone geometry. As stated by Gürdal et al. [16], Moita et al. [17], Moita et al. [18], and Correia et al. [19] simulated annealing and genetic algorithms may be selected for problems having a large number of variables. In the present study, “Micro Genetic Algorithms” (MicroGA) is selected as the optimal design tool as it is simpler and computationally less costly than conventional genetic algorithms. The failure load is increased and the total weight is minimized by employing a weighted sum approach defined by Konak et al.[20] and Madeira et al. [21]. A modified mutation strategy, which helps quicker convergence, is also developed and implemented in the code. The structural analyses required are conducted using an axisymmetric degenerated shell element developed by Cagdas [22], which is based on a refined first-order shear deformable shell theory developed by Leissa and Chang [23] and Qatu [24] and was used by Qatu[24, 25] to solve several shell vibration problems. In this theory, the (1 + z R ) term is included in the stress resultant equations and exclusion of the term can only be valid for shallow or very thin shells, where it is close to unity, as stated by Bert [26] and Qatu [24, 25]. The same shell theory was used by Cagdas and Adali [27] for design/optimization of cross-ply laminated composite cylinders under hydrostatic pressure without considering FPF. The axisymmetric shell element developed by Cagdas [22] is modified in the present study to be able to be able consider thickness and ply angle variation. Also, a 2D degenerated shell element developed by Cagdas and Adali [28], which is based on a shell element developed by Mallikarjuna and Kant [29], is used for verification purposes. The optimal stacking sequences and the optimal cone angles for some selected cases are determined and the optimization method used (MicroGA) is found out to be reliable as several independent runs yielded nearly the same results for the optimization problems solved. It is determined whether if using more layers having different winding angles will help increase the failure load and whether if FPF constraint is active or not for the selected cases. The influence of the rotational boundary conditions on the optimal failure load is also demonstrated. 2. STRUCTURAL ANALYSIS The numerical results presented here are obtained using a degenerated, curved, and first order shear deformable axisymmetric shell finite element developed by Cagdas [22]. Also, a 2D element developed by Cagdas and Adali [28] is used here to check the results. Formulation of the axisymmetric element modified here for variable thickness is given next. 2.1. Axisymmetric Finite Element Formulation The axisymmetric element is schematically shown in Fig. 1 and the element is based on the following displacement field;  ur  cn u    θ  m 0    uz  = ∑  0 V  n=0  0   αi  0 V   θ i 

0 sn

0 0

0 0

0 0

cn 0

0 cn

0

0

0

0   u rn    0   uθn    0   u zn   0  ψ αn i    sn  ψ θni 

(1)

where cn , sn , and m denote cos (n θ ) , sin(nθ ) , and the total number of harmonics, respectively. urn ,

uθn , u zn , ψ αn i , and ψ θni denote the radial, circumferential, and axial displacement components and the rotations in the nodal coordinate system corresponding to harmonic n, respectively. Vα i , Vθ i denote the nodal rotations of the transverse normal.

2.1.1. Strain-displacement relations A local coordinate system ( α , θ , z ' ) is defined at a Gauss point (GP) on the mid-surface of the shell as it can be seen from Fig. 1(a) that φGP is the angle between the unit vectors in r and z' directions at the GP and c , and s denote cos(φGP ) , and sin(φGP ) , respectively. The linear strain-displacement relations in the local coordinate system are given below, where rGP denotes the value of r at the GP and Rθ =rGP/cGP.

ε α = ε 0α + z ' χ α ε z' = 0 

1

(2) (3)



(ε 0θ + z ' χ θ ) ε θ =   1 + z ' Rθ     1 1 ε 0αθ + z '  γ αθ = ε 0θα +   1 + z ' Rθ  1 + z ' Rθ  

1

(4)    χαθ + χθα   



(γ 0θz ' ) γ θz ' =   1 + z ' Rθ  γ αz ' = γ 0αz ' = ψ α + w0 ,α

(5) (6) (7)

where,

u 0,α     ε 0α   1 ( s u0 + c w0 + v0,θ ) ε     r  ε 0 =  0θ  =  GP , v0,α ε 0αθ    1 ε 0θα   (u0,θ − s v0 )    rGP  

(8)

ψ α ,α     χα   1 ( s ψ α + ψ θ ,θ ) χ    θ   rGP  χ= = , ψ χ θ ,α  αθ     χθα   1 (ψ − s ψ ) α ,θ θ  r  GP 

(9)

ψ α + w0,α   γ 0αz '    w φ=  =  0,θ − c v0 + ψ  . θ γ 0θz '   r rGP  GP 

The in-plane parts of the non-linear strain-displacement relations are given below;

(10)

[

]

ε 0nlα =

1 (u0,α ) 2 + (v0 ,α ) 2 + ( w0 ,α ) 2 2

ε 0nlθ =

2 2 2 1 (u0,θ ) + (v0,θ ) + ( w0,θ ) + 2 s u0 v0,θ + 2c w0 v0,θ − 2 s v0 u0,θ + K   2 2rGP − 2c v0 w0,θ + s 2u02 + v02 + 2c s u0 w0 + c 2 w02 

(11)

(12)

γ 0nlαθ =

1 (u0,α u0,θ + v0,α v0,θ + w0,α w0,θ ) rGP

(13)

γ 0nlθα =

1 ( s u0v0,α − s v0u0,α − c v0 w0,α + c w0v0,α ) rGP

(14)

2.1.2. The force and moment resultants

The force and moment resultants are given below,  Nα   A11 N    θ   A12  =  Nαθ   A16  Nθα   A16

A12 Aˆ 22 A26 Aˆ

A26 A66

 M α   B11 M    θ   B12  = M αθ   B16  M θα   B16

B12 Bˆ

22

B16 B26

B26 Bˆ 26

B66 B66

26

Qα  5  A55  =   Qθ  6  A45

A16   ε 0α   B11    Aˆ 26   ε 0θ   B12  + A66  ε 0αθ   B16   Aˆ 66  ε 0θα   B16

A16

A66

B16   ε 0α   D11    Bˆ 26   ε 0θ   D12  + B66  ε 0αθ   D16   Bˆ 66  ε 0θα   D16

A45   γ αz '  .  Aˆ 44  γ 0θz ' 

B12 Bˆ 22

B16 B26

B26 Bˆ 26

B66 B66

D12 Dˆ

22

D16 D26

D26 Dˆ 26

D66 D66

B16   χα    Bˆ 26   χθ   , B66  χ αθ   Bˆ 66   χθα  D16   χα    Dˆ 26   χθ   , D66   χ αθ   Dˆ 66   χθα 

(15)

(16)

(17)

The rigidity terms appearing in Equations (15-17) were presented by Leissa and Chang [23], who truncated terms having orders higher than ( z ' Rθ )2 . Qatu [24] have presented more accurate results for these terms which are given below; Aij = Aij − c0 Bij Aˆij = Aij + c0 Bij

(18)

Bij = Bij − c0 Dij

Bˆ ij = Bij + c0 Dij

(19)

Dij = Dij − c0 E ij

Dˆ ij = Dij + c0 Eij

(20)

where, i, j = 1,2,4,5,6 and N

( )k (z'k − z 'k 1 )

Aij = ∑ Qij k =1

Bij =

1 N ∑ Qij 2 k =1

(21)

−

( ) (z ' k

2 2 k − z ' k −1

)

(22)

Dij =

1 N ∑ Qij 3 k =1

( ) (z ' k

3 3 k − z 'k −1

)

(23)

where, z'k , z 'k −1 are the coordinates of the lower and upper surfaces of kth layer in z ' direction and c0 = −1 Rθ . Note that, the lamination angle is taken as the angle between the fiber direction and the local α axis. 2.1.3. The element stiffness matrix

The 3-noded element stiffness matrix is given below,

(

K en = ∫ B Tχ D B χ + B εT B B χ + B φT C Bφ + B εT AB ε + B Tχ B B ε e

)

n

dAe

(24)

A

where Bε , B χ , Bφ are the strain-displacement matrices, and the superscript n stands for the nth harmonic. Note that, the rigidity matrices, A , B , C , and D defined in Equations (18-23) are calculated separately at each Gauss point to take the variable stiffness into account and reduced integration technique is used to prevent shear locking. After integration in circumferential direction, Eq. (24) becomes; 1

(

K en = kπ ∫ BTχ D B χ + BεT BB χ + BφT C Bφ + BTε ABε + BTχ B Bε

)

n

J rGP dξ

(25)

−1

where, k = 2 for n = 0 , and k = 1 for n = 1,K, m and ξ denotes the shape function coordinate. The Jacobian at a GP is given below, J = s r,ξ − c z,ξ

(26)

The mid-surface strains in the local coordinate system are; ε 0 = [B ε1 , B ε 2 , B ε3 ]{δ1 , δ 2 , δ 2 }

T

(27)

where,

0

Bε i

 s   =  s Si   r0 

S i ,α Si r0 s n (− n ) 0

0 Si cn (n ) r0 − s Si sn r0 S i ,α sn

− c S i ,α 0 − c Si r0

sn (− n ) 0

0 0  0 0   0 0  0 0 

(28)

Deflections due to bending in local coordinates are;

[

]

χ = B χ1 , B χ 2 , B χ 3 {δ1 , δ 2 , δ 3 }T

where,

(29)

Bχ i

0  0 =  0  0

0 0 0 0 0 0 0 0

  S i E11i cn (n )  r0  − s Si E11i  sn  r0  S i ,α E11i s n  0

S i ,α cn s Si cn r0 Si sn (− n ) r0 0

(30)

In Eq.(30), E11i = − si s − ci c and si , and ci denote cos(φ i ) , and sin(φ i ) , and φi is the angle between the unit vectors in r and z' directions at node i. Deflections due to shear in local coordinates are;

[

]

φ = B φ1 , B φ 2 , B φ 3 {δ1 , δ 2 , δ 3 }T

(31)

where, 0  c S i ,α cn  − c Si Bφ i = c S i  sn (− n ) sn r0  r0

s Si ,α cn Si c n s Si sn (− n ) 0 r0

  S i E11i sn   0

(32)

2.1.4. The element geometric stiffness matrix The element geometric stiffness matrix, K e,Gn , given below is derived using the non-linear strain displacement relations given in Equations (11-14) considering that the structure is subjected to a prescribed in-plane stress system leading to buckling. 1

K eG,n = kπ ∫ (G T SG ) ( n) J rGP dξ

(33)

−1

where, Gδ e = [G 1 , G 2 , G 3 ]{δ 1 , δ 2 , δ 3 } = {u 0 T

v0

0 − c S i cn  s S i cn  0 Si sn 0   c S i cn 0 s S i cn  0 − c S i ,α c n  s S i ,α cn  0 S i ,α sn 0 Gi =  0 s S i ,α c n  c S i ,α c n  s S s (− n ) 0 c S i s n (n )  i n 0 S i cn (n ) 0   c S s ( − n ) 0 s S s i n (− n )  i n

w0

u 0,α

v 0 ,α

0 0 0 0 0 0  0 0 0 0 ,  0 0 0 0  0 0  0 0

and the in-plane forces are stored in matrix S given below;

w0 ,α

u 0,θ

v 0 ,θ

T w0 ,θ } ,

(34)

(35)

 s 2 Nθ  2  rGP   0    0    0  S= 0    0    0    0    0 

0

2csNθ

Nθ

0

2 rGP

2 rGP

0

2 sNθα rGP

0

0

2 sN θ

− 2 sNθα rGP

0

− 2cN θα rGP

− 2 sNθ

0

0

2cN θα rGP

0

0

2 rGP

2 rGP

0

c 2 Nθ

0

0

Nα

0

0

2 N αθ rGP

0

0

0

0

Nα

0

0

2 N αθ rGP

0

0

0

0

Nα

0

0

0

0

0

0

0

Nθ

0

0

0

0

0

0

0

0

0

0

0

0

0

2 rGP

2 rGP

2cN θ 2 rGP

Nθ 2 rGP

0

   − 2cNθ   2 rGP   0    0   0   2 N αθ  rGP   0    0   Nθ  . 2 rGP  0

(36)

The in-plane forces in Eq.(36) are calculated at each Gauss node using Eq. (15). 2.2. Stability analysis The buckling load parameter, λcr , can be obtained by solving the following eigenvalue problem;

[

det K ncr − λcr K Gncr

]

=0

(37)

where, K n is the global stiffness matrix, and K Gn is the global geometric stiffness matrix corresponding to harmonic ncr . The related eigenvalue problem is solved using the inverse iteration method described by Cook [30].

2.2.1. Verification study for stability analysis A filament wound truncated cone having 4 layers under axial compression is analyzed for stability using both the axisymmetric and the 2D shell elements. The buckling loads N cr (kN/m) for axially compressed variable stiffness conical shells with stacking sequence at the small end taken as [90°/45°/-45°/90°], Material I, boundary conditions SS3-SS4 are presented in Table 1 in comparison with the numerical results presented by Goldfeld and Arbocz [2, 5], Patel et al. [7], and Goldfeld [9]. The geometrical and material parameters for the verification problem solved are given as follows;

L = 0.2m , R1* = 0.1325m , h1 = 1.16mm where L and h1 denote the cone length and the cone thickness at the small end, respectively. Material I:

E1 = 42.6 GPa , E2 = 11.7GPa , G12 = G13 = G23 = 4.8GPa , ν 12 = 0.302 .

The cross-section of the variable stiffness cone and the coordinate systems are shown in Fig 1(a) and the axial compressive load acts in α direction, i.e. skew loading is considered as shown in Fig. 1(c). The ply angle θi and layer thickness hi of the ith layer are calculated using the following equations;

 R1*

 R * cosθ i1 sin θ i1  and h i ( x ) = hi1 1 R cosθ i  R 

θ i = arcsin 

(38)

where, R1* , θ i1 , and hi1 are the radius of the parallel circle, ply angle and layer thickness at the small end of the cone and i = 1, K N . The radius of the parallel circle at the large end is denoted by R2* and φ denotes the cone angle. The volume of a filament wound cone, V∑ , can be calculated using the expression given by Goldfeld et al. [4 ]; N

V∑ = 2π R1* ∑ hi1 cos θ i1  S 22 − S12 sin 2 θ i1 − S1 cos θ i1    i =1

(39)

where S1 = R1* sin φ and S 2 = R2* sin φ . The results presented in Table 1, which are obtained using 100 axisymmetric shell elements, are in good agreement with the reference results especially for higher φ values. Best agreement is with the results presented by Goldfeld and Arbocz [5] who have used a method based on the finite difference method. 2.3. FPF Analysis

The stresses in the local coordinate system are evaluated as given below (see Leissa and Chang [25]); Q11 σα   σ  Q12  θ  σ θz '  =  0  σ   0  αz '   σ αθ (m ) Q16

Q12 Q22 0 0 Q26

0 0 Q44 Q45 0

0 0 Q45 Q55 0

Q16   ε α     Q26   ε θ    0  γ θz '   0  γ αz '    Q66  (m ) γ αθ  (m )

(40)

where, the reduced stiffness coefficients for mth layer, Qij , are calculated separately at each Gauss point and the FPF loads at the top and bottom of each layer are calculated using the maximum stress criterion which is given by; FI = F1MS σ 1 + F2MS σ 2 + 2 F12MS σ 1σ 2 + F11MS σ 12 + F22MS σ 22 + F44MS σ 42 + F55MS σ 52 + F66MS σ 62 ≥ 1

(41)

where, FI denotes the failure index, ( σ1 , σ 2 ) are normal stress components in the fiber coordinate system, ( σ 4 , σ 5 , σ 6 ) are shear stress components in the fiber coordinate system, and the F MS terms were defined by Reddy and Pandey [31] and Ochoa and Reddy [32]. Note that, the stresses calculated at the Gauss points are extrapolated from the Gauss points to the element nodes as stress concentrations may occur at the boundaries; see Hinton and Owen [33] 2.2.1. Verification study for FPF analysis

FPF and buckling loads, N FPF and N cr (kN/m), for simply supported (SS) cones made up of material T300/5208 and having R1* =100 mm, h1 = 2 mm, L=400 mm are calculated and presented in Table 3. It is assumed that the axial compressive load acts vertically as shown in Fig 1(b) and the stacking

sequence at the small end of the cone is taken as [45°/-45°/90°/90°/45°/-45°]. The properties of T300/5208 graphite/epoxy pre-preg are listed in Table 2. It was observed during the preliminary study that, restraining both nodal rotations at the ends of the cone causes numerical problems in the 2D shell model. Therefore, both nodal rotations are taken as free in both 2D shell and axisymmetric shell models to be able to make a comparison. It can be observed from Table 3 that, the FPF and buckling loads calculated using the 2D (20×20) shell model and the axisymmetric model composed of 100 axisymmetric elements are very close and the axisymmetric element yields more accurate results. Also, N FPF < N cr for cone angle φ =15° and N FPF > N cr for φ =30°, φ =45°, and φ =60°. These results show that FPF may occur before global buckling for lower cone angles and therefore it is obvious that there is a need to consider FPF in design/analysis. These results further show that, increasing the cone angle causes the buckling load to decrease sharply while the decrease in the FPF load is milder. A similar conclusion for buckling was reached by Shadmehri et al. [15]. The variations σ α and σ θ at the top of each layer for a SS cone having φ = 60°, R1* =100 mm, h1 = 2 mm, L=400 mm SS: [45°/-45°/90°/90°/45°/-45°] and for P =-1 kN/m are presented in Figures 3(a) and 3(b). The distributions of Nα , and Nθ and M α , and M θ are also shown in Figures 4(a) and 4(b). The FI distribution is also shown in Figures 5(a) and 5(b) for the same case but for the FPF load; i.e. N z =-127.359 kN/m. These figures show that stress concentrations occur due to boundary conditions and FPF occurs at the smaller end as, for the given geometry, the perimeter at the smaller end is lower comparing with the perimeter at the larger end. Therefore, FPF occurs at the smaller end even though the thickness is higher there. The buckling mode shape for the same case (the last row of Table 3) obtained using the 20×20 2D shell model and the axisymmetric model are shown in Figures 6(a) and 6(b), respectively. It can also be observed from these figures and Table 3 that the numerical buckling loads obtained by the two models are in good agreement but the results obtained using the axisymmetric model are more accurate as they are lower. 3. DESIGN OPTIMIZATION

The geometry of the filament wound cone considered in this study and the downward (z-dir.) acting axial compressive loading are shown in Figures 1(a) and (b). The cones are constructed of n angle-ply variable thickness layers. The material is selected as T300/5208, properties of which are listed in Table 2. 3.1. Problem Statement

The design objectives are minimizing the total volume, V∑ , and maximizing the failure load, N f , of a filament wound truncated cone by optimizing the filament winding angles at the smaller end, θ i1 . Here N f denotes the maximum axial compressive load the cone can carry without buckling or first-ply failure, which is defined as; N f = min ( N cr , N FPF ) .

The design problem is defined as follows:

(42)

Determine the ply angles θ i1 , i = 1,2,..., n of an angle-ply laminate of small end thickness h1 such that N f will be maximized, and V∑ will be minimized, viz; 10 ≤ θ i1 ≤ 80

such that

max DI 1 θi

and

FI < 1 .

(43)

The design index, DI, is defined as;

DI = N f − β

N f ,max ⋅ V∑

(44)

V∑1

where, β is a constant which is taken as equal to 0.1 in the present study and V∑1 denotes the volume of a uniform thickness cone with thickness equal to h1 . N f ,max denotes the failure load of the cone having the highest N f value obtained during optimization. Thus, the multi-objective optimization problem is converted into a single objective optimization problem as defined by Konak et al. [20]. Note that, a higher β value may be used for a lighter design but this in the end will decrease the failure load of the optimal design. 3.2. Optimization Procedure

The optimization problem is solved using Micro-GA stages of which are described as follows;

i.

Randomly generate an initial population entries of which are stored in matrix I p as shown below.

[I ] p

 θ 1,1 θ1,2 ... θ1,nθ  = M M O M θ NC ,1 θ NC , 2 ... θ NC ,n  θ

    

(45)

where, 10 ≤ θ i , j ≤ 80 . The initial population consists of NC different stacking sequences (or total number of chromosomes) each containing nθ unknown ply angles (genes). In this study, NC is taken as equal to 5. For these values, the ith stacking sequence StS i(nθ ) corresponding to the ith chromosome will become;

{

{StS( ) } = {θ nθ

i

i ,1

− θ i,1 θi ,2 − θi ,2 θi ,3 − θ i,3 ... θi ,nθ

− θi ,nθ

}

}

(46)

ii.

Structural analysis-Fitness evaluation: Calculate N f value for each chromosome.

iii.

Selection: Generate separate vectors which represent gene pools for each θ i, j in the stacking

iv.

sequence and put the lamination angles according to their DI or fitness values into these pools. The better genes having higher DI will have higher fitness values and they will be represented more in the related gene pools and the genes of the stacking sequences with lower fitness values will have less contribution to the related gene pools. The worst stacking sequence is discarded and does not contribute to the gene pools. Then lamination angles for the new population are randomly selected from these pools. Reproduction rate is selected as 90% in this study. Memory: The best member of the population is preserved.

v. vi.

Mutation: 40% of the genes of the offsprings, except for the best member, are changed randomly by ±5°. Go to step ii until convergence or a prescribed maximum number of iterations, niter , is reached. The last obtained maximum failure load is denoted by N f ,opt .

After the optimal winding angles are obtained, refinement may be achieved by re-starting the optimization process after inserting the last obtained best stacking sequence into the initial population. 3.3. Optimization Results and Discussion

Filament wound cones are generally used as adapters between cylinders having different radii and in this section it is aimed to determine the cone angle which yields the best design. Adapters considered having 4 different cone semi-vertex angles are schematically shown in Fig. 7 and optimization results are presented for both ends simply supported (SS) axially compressed filament wound cones for 3 different cases in the following sub-sections where for Case I: R2* R1* = 4 , R2* = 0.4m , for Case II:

R2* R1* = 2 , R2* = 0.4m , and for Case III: R2* R1* = 4 , R2* = 0.2m . For the SS case, both rotations are set free at both ends. The material is selected as T300/5208. It should also be determined whether if having more layers will help increase the failure load. Thus, two different stacking sequences designated as StS (4 ) and StS (8 ) which respectively have 4 and 8 unknown lamination angles are selected for optimization using MicroGA. Another objective is to check whether if FPF constraint is active or not for the selected cases. Finally, some representative numerical results are presented for an example problem with both ends clamped (CC), both ends simply supported (SS) and small end simply supported-large end clamped (SC) boundary conditions to examine the influence of the rotational boundary conditions. 3.3.1. Case I: R2* R1* = 4 , R2* = 0.4 m

Optimization results are presented in Fig. 8(a). for nθ=4, R1* = 0.1 m , R2* = 0.4 m and for cone semivertex angles φ = 15°, 30°, 45°, and 60°, where P∑ ,opt denotes the optimal axial compressive load for given weight. Note that, FPF constraint was not active for the cases presented in Fig. 8(a) and can only be active for very high axial compressive load values not covered here. It can be observed from Fig.8(a) that, P∑ ,opt increases quadratically with increasing V∑ and the lowest weight design is obtained for φ =45° and the heaviest design is obtained for φ =15°, which is somewhat unexpected because the critical buckling load decreases with increasing cone semi-vertex angle. The reason of this unexpected result is that, for given weight, the length is higher and the thickness is lower for φ =15° comparing with they are for higher φ values. It can also be observed from Fig. 8(a) that the optimal weights for φ =45° are less than the ones for φ =60°. Therefore, it is obvious that increasing the cone angle does not necessarily increase the design efficiency and for the selected material and geometry, 45° is the best semi-vertex angle out of the four different φ values considered. However, Goldfeld et al. [4], keeping the length L and the small end radius R1* of the cones constant, have reported that the buckling load is lower for cones having higher cone semi-vertex angles. The reason of this contradiction is that, when L and R1* are kept constant R2* will take higher values for higher φ . In the present study, as stated before, R1* and R2* are kept constant and L varies and therefore the conclusions reached are different.

The influence of the total number of different winding angles on the optimal results is investigated next. Sample optimization results for φ=45°, R1* =100 mm, R2* =400 mm, h1 = 4 mm, L=424.264 mm, niter = 30 are given in Table 4(a) for 4 unknown winding angles, i.e. nθ=4. N f ,opt = 86.378 kN/m with

ncr=13 and V∑,opt =1037.907 cm3 for this case. The same optimization problem is solved again for nθ =8 and the results are presented in Table 4(b) and N f ,opt = 86.485 kN/m with ncr=13 and V∑,opt =1041.392 cm3 for nθ=8. The best stacking sequences are [±25°/±10°(3)] and [±25°/±20°/±10°(6)] for nθ=4 and 8 respectively, which are in good agreement with the optimal winding angles proposed by Goldfeld et al. [4], and Kabir et al. [13]. It can be observed from Tables 4(a) and 4(b) that the differences between the optimal values are negligible and therefore, for the given geometry and the material properties, there seems to be no need to increase nθ as production may be more complicated for higher nθ. In a related study, Goldfeld et al. [4] have shown that two ply angles are sufficient for the optimum configuration for maximum buckling load of a laminated conical shell. Considering this, the optimal winding angles for the same problem are obtained again for nθ =2 to check whether if the results differ considerably. The use of MicroGA is not required for nθ =2 as the design space is narrow; i.e. there exists only 225 different stacking sequences. Thus, the best stacking sequence for nθ =2 is found out to be [±10°/±10°] and N f ,opt = 86.035 kN/m with ncr=13 and V∑,opt =1054.090 cm3. This result is important as it means that increasing nθ does not significantly improve the failure load for axially compressed filament wound cones with R2* R1* = 4 and R2* = 0.4m . The stacking sequence yielding the best design for nθ=2, for all φ and for all of the small end thickness values considered is found out to be [±10°/±10°]. 3.3.2. Case II: R2* R1* = 2 and R2* = 0.4 m.

Optimization results are presented in Fig. 8(b) for nθ=2, R1* = 0.2 m, R2* = 0.4 m. Comparison with the results presented in Fig. 8(a) shows that, the P∑ values are slightly higher than the ones for the case

R1* = 0.1 m, R2* = 0.4 m. As was observed for Case I, the lowest weight designs are obtained for φ=45°, and 60° and the heaviest designs are obtained for φ=15°. The stacking sequence yielding the best design for φ=30°, 45°, and 60° is [±25°/±25°] and for φ=15° it is [±30°/±30°] for all of the small end thickness values considered. Comparison of these results with the results obtained for Case I shows that, for all of the φ values considered, the optimal winding angles increase with increasing R1* when R2* is kept constant. FPF is again not observed for all of the small end thickness values considered. 3.3.3. Case III: R2* R1* = 2 and R2* = 0.2 m.

Next it is investigated whether if similar conclusions reached for the case R2* R1* = 2 can be reached for R2* R1* taken equal to 2 again but for a lower R2* value, i.e. for R2* = 0.2 m. Optimization results are presented in Fig. 8(c) for nθ=2, R1* = 0.1 m, R2* = 0.2 m and for cone angles φ = 15°, 30°, 45°, and 60°. Note that, the FPF constraint is again not active for this case and using more layers having different winding angles does not significantly increase the failure load. Also, the stacking sequences yielding the best design are similar to the ones given previously for Case II.

Comparison of Figures 8(b) and 8(c) show that the behavior is similar; i.e. the lowest weight designs are obtained for φ =45°, and 60° and the worst are obtained for φ =15°. For given P∑ , the optimal volumes are found out to be much lower for Case III than they are for Case II. For example, for P∑ =100 kN, V∑,opt ≈ 1400cm3 for Case II, and V∑,opt ≈ 200 cm3 for Case III. 3.3.4. The effect of the rotational boundary conditions

Next, some additional results showing the effect of the rotational boundary conditions are presented in Table 5 for R2* R1* = 4 and R2* R1* = 2 , R2* = 0.4m , φ=45°, nθ = 1, β =0, stacking sequence

StS (1) = [± θ1 / ± θ1 ], and for SS, CC and SC boundary conditions. Here CC denotes that the end rotations are fixed at both ends. SC denotes that the larger end is restrained against rotation and the zdirection displacement component is free for all B.C. It can be observed from Table 5 that, clamping the small end of the cone has negligible influence on the buckling load and the optimal winding angles differ only marginally and the optimal failure loads for the CC and SC boundary conditions, P∑CC ,opt and P∑SC,opt , are up to about 45% higher than the ones for the SS boundary conditions, ie. P∑SS,opt , and this increase is more pronounced for thicker cones. Interestingly, for some thickness values, the failure loads for the SC B.C. are found out to be very slightly higher than the ones for CC B.C., which probably is due to the additional stresses caused by the end bending moments. The optimal winding angles are found out to be similar to the Cases I and II presented above. It can be observed from Table 5 that the optimal winding angle increases with decreasing R2* R1* for constant R2* .

4. CONCLUSIONS

Optimal designs for filament wound truncated cones under axial compression are presented. The first objective is to maximize the failure load which is defined as the minimum of the buckling load the first-ply faiure (FPF) loads and the second objective is to minimize the weight. The optimal winding angles, where FPF and buckling are imposed as design constraints, are obtained using Micro-Genetic Algorithms. The reliability of the optimization technique used is demonstrated and the optimization method used is found out to be reliable as several independent runs yielded nearly the same results for the optimizations problems solved. The numerical results are obtained using an axisymmetric degenerated shell element based on a refined first-order shear deformable shell theory and a 2D degenerated shell element is used for verification purposes. The verification problems solved show that, for the selected cases, the FPF constraint may be active for thick cones with lower cone angles. A thorough parametric study is not conducted as the conclusions reached for given material and boundary conditions will differ. However, some recommendations for design, which may be valuable when similar material is used, are made using the results presented here. The results obtained here for axial compressive loading show that, the highest failure index occurs at points closer to the small end and FPF is generally not critical as the optimal designs are found out to be buckling critical for all of the small end thickness values and the cone geometries considered. The results obtained in this study show that the FPF constraint may be active for very high axial compressive force levels and/or poorly designed cones. Also, it is estimated that, for a combination of axial compression and external pressure, the FPF constraint may be active.

It is revealed that, for R2* R1* = 2, and 4, the highest design efficiencies may be achieved for φ=45°, and 60°, respectively and similar design efficiencies may be obtained for these two cone angles for both cases. φ=15° yields the lowest design efficiency, as the cone will be longer and thinner for given weight, R1* , and R2* . Filament winding process may be simpler for φ=45° rather than it is for φ=60° and therefore φ=45° may be the best cone semi-vertex angle for an adapter yielding the highest design efficiency. The results also show that the optimal winding angles considerably differ for different R2* values. Another finding is that, the failure load and the design efficiency drastically increases with decreasing R2* . The results obtained for all of the cases considered in this study show that using more layers having different winding angles has negligible influence on the failure load. This is one of the most important outcomes of this study as it is easier to produce filament wound cones having lower number of different winding angles. The results obtained for representative geometries, R2* R1* = 2, 4, and R2* = 400 mm and the cone semi vertex angle φ=45°, show that the failure loads for clamped cones are up to about 45% higher than the ones for the simply supported cones and the optimal winding angles differ marginally for given R2* R1* ratio. The results obtained also show that clamping the large end is much more effective than clamping the small end. This may be a good motivation to increase the torsional rigidity of the cone end stiffener element at the large end, if exists. REFERENCES

1. Morozov EV. Theoretical and experimental analysis of the deformability of filament wound composite shells under axial compressive loading. Composite structures. 2001 Dec 31;54(2):255-60. 2. Goldfeld Y., Arbocz J., ‘Buckling of laminated conical shells taking into account the variation of the stiffness coefficients’, AIAA J; 2004; 42(3): 642-649. 3. Goldfeld Y., Sheinman I., Baruch M., ‘Imperfection sensitivity of conical shell’, AIAA J, 2003; 4(3):517-24. 4. Goldfeld Y., Arbocz J., Rothwell A. Design and optimization of laminated conical shells for buckling. Thin-Walled Structures. 2005 Jan 31;43(1):107-33. 5. Goldfeld Y., Arbocz J. Elastic buckling of laminated conical shells using a hierarchical high-fidelity analysis procedure’, ASCE J Eng Mech 2006; 132(12):1335-1344. 6. Goldfeld Y., Vervenne K., Arbocz J., Van Keulen F. Multi-fidelity optimization of laminated conical shells for buckling. Structural and Multidisciplinary Optimization. 2005 Aug 1;30(2):128-41. 7. Patel BP., Singh S., Nath Y. Postbuckling characteristics of angle-ply laminated truncated circular conical shells. Communications in Nonlinear Science and Numerical Simulation. 2008 Sep 30;13(7):1411-30. 8. Blom AW., Tatting BF., Hol JM., Gürdal Z. Fiber path definitions for elastically tailored conical shells. Composites part B: engineering. 2009 Jan 31;40(1):77-84. 9.

Goldfeld Y. Imperfection sensitivity of laminated conical shells. International journal of solids and structures. 2007 Feb 28;44(3):1221-41.

10. Goldfeld Y. An alternative formulation in linear bifurcation analysis of laminated shells. Thin-Walled Structures. 2009 Jan 31;47(1):44-52. 11. Maleki S., Tahani M. Non-linear analysis of fiber-reinforced open conical shell panels considering variation of thickness and fiber orientation under thermo-mechanical loadings. Composites Part B: Engineering. 2013 Sep 30;52:245-61. 12. Brown RT., Nachlas JA., ‘Structural optimization of laminated conical shells’, AIAA J, 1985; 23(5):781-7 13. Kabir MZ., Shirazi AR. Optimum design of filament-wound laminated conical shells for buckling using the penalty function. Iranian Aerospace Society. 2008 Jan 1;5(3):115-21. 14. Naderi AA., Rahimi GH., Arefi M. Influence of fiber paths on buckling load of tailored conical shells. Steel and Composite Structures. 2014 Mar 1;16(4):375-87. 15. Shadmehri F., Hoa SV., Hojjati M. Buckling of conical composite shells. Composite Structures. 2012 Jan 31;94(2):787-92. 16. Gürdal Z., Haftka R.T., Hajela P., Design and Optimization of Laminated Composite Materials, John Wiley & Sons, NY, USA, 1999. 17. Moita JM, Correia VM, Martins PG, Soares CM, Soares CA. Optimal design in vibration control of adaptive structures using a simulated annealing algorithm. Composite Structures. 2006 Sep 30;75(1):79-87. 18. Moita JS, Martins PG, Soares CM, Soares CA. Optimal dynamic control of laminated adaptive structures using a higher order model and a genetic algorithm. Computers & structures. 2008 Feb 29;86(3):198-206. 19. Correia VM, Soares CM, Soares CA. Buckling optimization of composite laminated adaptive structures. Composite Structures. 2003 Dec 31;62(3):315-21. 20. Konak A., Coit DW., Smith AE. Multi-objective optimization using genetic algorithms: A tutorial. Reliability Engineering & System Safety. 2006 Sep 30;91(9):992-1007. 21. Madeira JF, Araújo AL, Soares CM, Soares CM, Ferreira AJ. Multiobjective design of viscoelastic laminated composite sandwich panels. Composites Part B: Engineering. 2015 Aug 31;77:391-401. 22. Cagdas IU. Stability analysis of cross-ply shells of revolution using a curved axisymmetric shell finite element. Thin-Walled Structures 2011; 49: 732-742. 23. Leissa, A.W., Chang, J. Elastic deformation of thick, laminated composite shallow shells. Composite Structures. 1996; 35: 605-616. 24. Qatu MS. Accurate equations for laminated composite deep thick shells. International Journal of Solids and Structures. 1999 Jul 1;36(19):2917-41. 25. Qatu MS. Vibration of laminated plates and shells. Elsevier, 2004. 26. Bert CW. Structural theory or laminated anisotropic elastic shells. Journal of Composite Materials 1967;1,414-423. 27. Cagdas IU., Adali S. Buckling of cross-ply cylinders under hydrostatic pressure considering pressure stiffness. Ocean Engineering 2011; 38: 559-569.

28. Cagdas IU., Adali S. Effect of Fiber Orientation on Buckling and First-Ply Failures of Cylindrical Shear-Deformable Laminates. Journal of Engineering Mechanics. 2012 Nov 5;139(8):967-78. 29. Mallikarjuna, Kant T. A general fibre-reinforced composite shell element based on a refined shear deformation theory. Comput Struct 1992; 42(3):381-388. 30. Cook RD. Concepts and applications of finite element analysis: a treatment of the finite element method as used for the analysis of displacement, strain, and stress. John Wiley & Sons, 1974. 31. Reddy JN., Pandey AK. A first-ply failure analysis of composite laminates. Computers & structures. 1987 Dec 31;25(3):371-93. 32. Ochoa OO., Reddy JN. Finite element analysis of composite laminates. Springer Netherlands, 1992. 33. Hinton E., Owen D.R.J. Finite Element Programming. London, Academic Press, 1977. 34. Baruch M., Arbocz J., Zhang G.Q. Laminated conical shells-considerations for the variations of the stiffness coefficients. Delft University of Technology, Faculty of Aerospace Engineering; 1992.

Tables and Figures

Table 1. Buckling loads (kN/m) for axially compressed variable stiffness conical shells, skew loading, stacking sequence at the small end: [90°/45°/-45°/90°], Material I, B.C. SS3-SS4. Table 2. Material properties of T300/5208 graphite/epoxy pre-preg; Ochoa and Reddy (1992) *

Table 3. FPF PFPF and buckling Pcr loads (kN/m) for R1 =100 mm, h1 = 2 mm, L=400 mm, simply supported boundaries (both rotations are free at the ends), material T300/5208. Un-skew loading, stacking sequence at the small end: [45°/-45°/90°/90°/45°/-45°]. Table 4(a). Micro-GA results for φ=45°, R1* =100 mm, R2* =400 mm, h1 = 4 mm, L=424.264 mm, SS boundaries, material T300/5208, niter =30, nθ = 4. Table 4(b). Micro-GA results for φ=45°, R1* =100 mm, R2* =400 mm, h1 = 4 mm, L=424.264 mm, SS boundaries, material T300/5208, niter =30, nθ = 8. *

Table 5. The influence of the boundary conditions for φ=45°, R1 =200 mm, R2* =400 mm, material T300/5208, β = 0.1 nθ = 2. _______________________________________________________________________________ Figure 1.

(a) Cross-section of the conical shell and the coordinate systems. (b) The application of the z-dir. axial compressive load. (c) The skew-support and the skew application of the load

Figure 2. Variation of FI for a SS cone having φ = 60°, R1* =100 mm, h1 = 2 mm, L=400 mm Stacking seq.: [45°/-45°/90°/90°/45°/-45°] using (a) 2D model (the darker the color the higher the FI ) (b) Axisymmetric model for N z =-127.359 kN/m Figure 3. Variation of (a) σ α (b) σ θ at the top of each layer for a SS cone having φ = 60°,

R1* =100 mm, h1 = 2 mm, L=400 mm, Stacking seq.: [45°/-45°/90°/90°/45°/-45°] and for N z =-1 kN/m *

Figure 4. Variations of (a) Nα , and Nθ and (b) M α , and M θ for a SS cone having φ = 60°, R1 =100 mm, h1 = 2 mm, L=400 mm, Stacking seq.: [45°/-45°/90°/90°/45°/-45°] and for N z =-1 kN/m. Figure 5. Variation of FI for a SS cone having φ = 60°, R1* =100 mm, h1 = 2 mm, L=400 mm Stacking seq.: [45°/-45°/90°/90°/45°/-45°] using (a) 2D model (the darker the color the higher the FI ) (b) Axisymmetric model for N z =-127.359 kN/m *

Figure 6. Buckling mode shape for simply supported cone having φ = 60°, R1 =100 mm, h1 = 2 mm, L=400 mm, SS: [45°/-45°/90°/90°/45°/-45°] using (a) 2D shell model (b) axisymmetric model

Figure 7. Geometry of the conical adapter Figure 8. (a)

R2*

Variation of optimal P∑ with weight for,

= 0.4m , R2* R1* = 4 , (b) R2* = 0.4m , R2* R1* = 2 (c) R2* = 0.2m , R2* R1* = 2

Figure 1. Cross section of the curved axisymmetric element and the displacement components at a Gauss point.

(a)

(b)

(c) Figure 2. (a) Cross-section of the conical shell and the coordinate systems. (b) The application of the z-dir. axial compressive load. (c) The skew-support and the skew application of the load

0.00 -0.25 φ=60o h1=2 mm R1*=100mm L=400 mm SS: [45o/-45o/90o/90o/45o/-45o]

-0.50

σα (top) (N/mm2)

-0.75 -1.00

Layer 1 Layer 2 Layer 3 Layer 4 Layer 5 Layer 6

-1.25 -1.50 -1.75 0

50

100

150

200

250

300

350

400

α (mm)

(a)

1.00 Layer 1 Layer 2 Layer 3 Layer 4 Layer 5 Layer 6

0.75 0.50

σθ (top) (N/mm2)

0.25 0.00 -0.25

φ=60o h1=2 mm R1*=100mm L=400 mm SS: [45o/-45o/90o/90o/45o/-45o]

-0.50 -0.75 -1.00 0

50

100

150

200

250

300

350

400

α (mm)

(b)

Figure 3. Variation of (a) σ α (b) σ θ at the top of each layer for a SS cone having φ = 60°,

R1* =100 mm, h1 = 2 mm, L=400 mm, Stacking seq.: [45°/-45°/90°/90°/45°/-45°] and for N z =-1 kN/m

0.00

-0.50

N (N/mm) -1.00 φ=60o h1=2 mm R1*=100mm L=400 mm SS: [45o/-45o/90o/90o/45o/-45o]

-1.50

Nα Nθ

-2.00 0

50

100

150

200

250

300

350

400

α (mm) 0.16

(a)

φ=60o h1=2 mm R1*=100mm L=400 mm SS: [45o/-45o/90o/90o/45o/-45o]

0.12

M (Nmm/mm)0.08

Mα

0.04

Mθ

0.00 0

50

100

150

200

250

300

350

400

α (mm)

(b) * 1 =100

Figure 4. Variations of (a) Nα , and Nθ and (b) M α , and Mθ for a SS cone having φ = 60°, R mm, h1 = 2 mm, L=400 mm, Stacking seq.: [45°/-45°/90°/90°/45°/-45°] and for N z =-1 kN/m.

(a)

1.00

0.75

FI

Layer 1 Layer 2 Layer 3 Layer 4 Layer 5 Layer 6

φ=60o h1=2 mm R1*=100mm L=400 mm SS: [45o/-45o/90o/90o/45o/-45o]

0.50

0.25

0.00 0

50

100

150

200

250

300

α (mm)

350

400 (b)

* 1 =100

Figure 5. Variation of FI for a SS cone having φ = 60°, R mm, h1 = 2 mm, L=400 mm Stacking seq.: [45°/-45°/90°/90°/45°/-45°] using (a) 2D model (the darker the color the higher the FI) (b) Axisymmetric model for N z =-127.359 kN/m

(a)

250 cone cross-section buckling mode shape

200

150

z (mm) 100

50

φ=60o h1=2 mm R1*=100 mm L=400 mm SS: [45o/-45o/90o/90o/45o/-45o]

0 100

150

200

250

300

350

400

450

r (mm)

(b) *

Figure 6. Buckling mode shape for simply supported cone having φ = 60°, R1 =100 mm, h1 = 2 mm, L=400 mm, SS: [45°/-45°/90°/90°/45°/-45°] using (a) 2D shell model (b) axisymmetric model

Figure 7. Geometry of the conical adapter

300 R1*=100 mm R2*=400 mm

250

φ=15o φ=30o φ=45o φ=60o

200

PΣ 150 (kN) 100 50 0 0

500

1000

1500

2000

2500

VΣ (cm3)

(a)

300 R1*=200 mm R2*=400 mm

250

φ=15o φ=30o φ=45o φ=60o

200

PΣ 150 (kN) 100 50 0 0

500

1000

1500

2000

2500

VΣ (cm3)

(b)

300 250 200

PΣ 150 (kN)

R1*=100 mm R2*=200 mm

100

φ=15o φ=30o φ=45o φ=60o

50 0 0

150

300

450

600

VΣ (cm3)

Figure 8.

750

(c)

Variation of optimal P∑ with weight for,

(a) R2* = 0.4m , R2* R1* = 4 , (b) R2* = 0.4m , R2* R1* = 2 (c) R2* = 0.2m , R2* R1* = 2

Table 1. Buckling loads (kN/m) for axially compressed variable stiffness conical shells, skew loading, stacking sequence at the small end: [90°/45°/-45°/90°], Material I, B.C. SS3-SS4. φ Pcr (ncr) Pcr (ncr) Pcr Pcr (ncr) Pcr (ncr) Goldfeld and Arbocz Goldfeld and Arbocz Patel et al. Goldfeld Current Study [5] [2] [7] [9] (100 axis. elem.) 10° 75.976 (11) 74.54 (11) 75.10 75.42 (11) 78.327 (11) 20° 59.704 (12) 59.18 (12) 58.97 59.12 (12) 60.628 (12) 30° 48.894 (13) 48.13 (13) 48.24 48.30 (13) 49.175 (13) 45° 36.144 (13) 35.38 (13) 35.63 35.61 (13) 36.030 (13) 55° 28.417 (12) 27.83 (12) 27.99 27.95 (12) 28.245 (12) 65° 20.835 (11) 20.42 (10) 20.55 20.44 (11) 20.613 (11) 75° 13.092 (8) 12.66 (8) 12.82 12.80 (8) 12.913 (8) 4.837 (5) 85° 4.097 90° 0.510 (0) 0.508 (0) 0.508 0.508 (0) 0.510 (0)

Table 2. Material properties of T300/5208 graphite/epoxy pre-preg; Ochoa and Reddy [26] 132379.37 E1 XT 1513.40 Mpa MPa E2 XC 10755.82 MPa 1696.11 MPa E3 10755.82 MPa YT = ZT 43.78 MPa G12 = G13 YC = Z C 5653.70 MPa 43.78 MPa G 23 3378.43 MPa 67.57 MPa R ν 12 = ν 13 0.24 86.87 MPa S =T ν 23 0.49

_______________________________________________________________________________

Table 3. FPF PFPF and buckling Pcr loads (kN/m) for R1* =100 mm, h1 = 2 mm, L=400 mm, simply supported boundaries (both rotations are free at the ends), material T300/5208. Un-skew loading, stacking sequence at the small end: [45°/-45°/90°/90°/45°/-45°]. φ PFPF PFPF Pcr ( ncr ) Pcr (100 Axis.el.) 2D (20×20) 2D (20×20) (100 Axis mesh mesh el.) 15° 217.614 220.702 330.871 (11) 363.401 30° 203.651 207.645 164.129 (13) 183.897 45° 173.078 174.674 88.630 (13) 98.230 60° 127.359 130.688 40.583 (12) 42.416

Table 4(a). Micro-GA results for φ=45°, R1* =100 mm, R2* =400 mm, h1 = 4 mm, L=424.264 mm, SS boundaries, material T300/5208, niter =30, β = 0.1 , nθ = 4, StS (4) = [± θ1 / ± θ 2 / ± θ 3 / ± θ 4 ] , Member

Initial population (Chromosomes)

StS1(4 )

[±25°/±40°/±50°/±10°]

StS 2(4 )

[±60°/±55°/±10°/±45°]

StS 3(4 )

[±35°/±35°/±65°/±60 °]

StS 4(4 )

[±30°/±60°/±55°/±55°]

StS5(4 )

[±75°/±70°/±20°/±35°]

N f (ncr ) (kN/m) Vol (cm3) 68.062 (14) 914.979 54.361 (15) 788.061 43.033 (17) 738.757 42.647 (17) 730.666 41.347 (15) 673.913

Final population (Chromosomes)

[±25°/±10°/±10°/±10°] [±25°/±10°/±15°/±10°] [±30°/±10°/±15°/±10°] [±25°/±10°/±20°/±10°] [±20°/±15°/±15°/±15°]

N f (ncr ) (kN/m) Vol (cm3) 86.378 (13) 1037.907 85.757 (13) 1034.080 85.490 (13) 1025.495 84.875 (13) 1028.694 84.803 (13) 1033.394

_______________________________________________________________________________

Table 4(b). Micro-GA results for φ=45°, R1* =100 mm, R2* =400 mm, h1 = 4 mm, L=424.264 mm, SS boundaries, material T300/5208, niter =30, β = 0.1 , nθ = 8. Member Initial population N f (n cr ) Final population N f (n cr ) (Chromosomes)

StS1(8 ) (8 )

[±75°/±45°/±45°/±60°/±20°/±55°/±45°/±50°]

StS 2

[±45°/±65°/±65°/±50°/±55°/±35°/±15°/±65°]

StS 3(8 )

[±70°/±60°/±65°/±50°/±55°/±35°/±15°/±65°]

(8 )

StS 4

[±25°/±35°/±30°/±65°/±25°/±15°/±35°/±25°]

StS5(8 )

[±70°/±25°/±30°/±45°/±25°/±10°/±50°/±25°]

(Chromosomes)

(kN/m) Vol (cm3 ) 45.687 (16) 727.785 43.590 (16) 720.147 61.940 (14) 850.539 70.139 (14) 913.970 64.064 (15) 871.264

[±25°/±20°/±10°/±10°/±10°/±10°/±10°/±10°] [±20°/±20°/±10°/±10°/±10°/±10°/±10°/±10°] [±15°/±20°/±10°/±15°/±10°/±10°/±10°/±15°] [±15°/±15°/±15°/±10°/±10°/±15°/±20°/±10°] [±25°/±15°/±20°/±15°/±10°/±25°/±10°/±10°]

(kN/m) Vol (cm3) 86.485 (13) 1041.392 86.383 (13) 1044.877 85.746 (13) 1043.742 84.985 (13) 1041.828 84.691 (13) 1029.473

Table 5. The influence of the boundary conditions for φ=45°. R2* =400 mm. material T300/5208.

β = 0 . nθ = 1. StS (1) = [± θ1 / ± θ1 ].

R1* (m m)

t (m m) 1 2 3 4

100 5 6 7 8 1 2 3 4 200 5 6 7 8

SS

θ opt P∑ ,opt (kN) 10 3.600 ° 10 12.56 2 ° 10 29.42 0 ° 10 54.11 9 ° 10 86.99 1 ° 10 128.4 85 ° 10 179.1 68 ° 10 239.4 18 ° 25 7.521 ° 25 31.73 ° 1 25 74.54 ° 6 25 137.1 ° 57 25 221.0 ° 55 25 331.9 ° 11 25 458.4 ° 17 25 610.7 ° 40

CC

V∑,opt (cm3) 263.5 23 527.0 45 790.5 68 1054. 090 1317. 613 1581. 135 1844. 658 2108. 181 337.7 63 675.5 25 1013. 288 1351. 051 1688. 814 2026. 576 2364. 339 2702. 102

θ opt P∑ ,opt (kN) 10 4.037 ° 10 15.50 ° 6 10 37.42 ° 5 10 70.33 ° 6 10 114.8 ° 76 10 172.2 ° 90 10 243.6 ° 50 10 328.8 ° 01 30 9.278 ° 25 41.29 ° 9 25 100.2 ° 88 25 189.4 ° 25 25 310.1 ° 00 25 465.3 ° 81 20 655.8 ° 07 20 893.6 ° 80

V∑,opt (cm3) 263.5 23 527.0 45 790.5 68 1054. 090 1317. 613 1581. 135 1844. 658 2108. 181 329.5 02 675.5 25 1013. 288 1351. 051 1688. 814 2026. 576 2410. 086 2754. 384

SC (large end clamped) θ opt P∑ ,opt V∑,opt (kN) (cm3) 10 4.037 263.5 ° 23 10 15.50 527.0 ° 6 45 10 37.42 790.5 ° 4 68 10 70.33 1054. ° 6 090 10 114.8 1317. ° 79 613 10 172.2 1581. ° 95 135 10 243.6 1844. ° 30 658 10 328.7 2108. ° 31 181 25 329.50 9.277 ° 2 25 41.29 675.52 ° 3 5 25 100.0 1013.2 ° 65 88 25 188.9 1351.0 ° 28 51 25 310.0 1688.8 ° 18 14 25 467.0 2026.5 ° 83 76 25 658.3 2364.3 ° 78 39 25 892.5 2702.1 ° 70 02

SC P∑CC ,opt P∑ ,opt

P∑SS,opt P∑SS,opt

1.12 1 1.23 4 1.27 2 1.30 0 1.32 1 1.34 1 1.36 0 1.37 3

1.12 1 1.23 4 1.27 2 1.30 0 1.32 1 1.34 1 1.36 0 1.37 3

1.23 4 1.30 2 1.34 5 1.38 1 1.40 3 1.40 2 1.43 1 1.46 3

1.23 4 1.30 1 1.34 2 1.37 7 1.40 2 1.40 7 1.43 6 1.46 1