Gant Ov Nik 2006

Laminated Composite Materials Structural Design Methods of Composite Optimization Examples

Design and Optimization of LaminatedComposite Materials

Vladimir GantovnikClemson University

November 14, 2006

Vladimir Gantovnik Clemson University

Design and Optimization of Laminated Composite Materials


Outline

1 Laminated Composite Materials

2 Structural Design

3 Methods of Composite Optimization

4 Examples




Composite Materials

Combination of a strong material (fibers) with a weakermaterial (matrix)

Layered structure

Stacking sequence

Directional nature of the material - Anisotropy




Composite Components in an Airbus A-320




SEM Micrographs of Carbon Fiber Composite




Carbon Fiber Composite Fuselage Section




Composite Components in Helicopter




Composite Components in Military Aircraft




SpaceShipOne

SpaceShipOne is the first operational space vehicle made entirelyof carbon composite!




Composite Pieces in an Vehicles




Composite Bicycle




Structural Design and Optimization

Galileo Galilei (1638): Optimal cantilever problem (parabolicheight function produces the minimum weight design for atip-loaded, constant width cantilever).




Optimization Problem Formulation

Standard Form:

Minimize f (x)

subject to

gj(x) ≤ 0, j ∈ {1, . . . , q}and

(xi )min ≤ xi ≤ (xi )max ,

i ∈ {1, . . . ,m}.

Linear (LP) and Nonlinear (NL) Programming Problems;Integer Programming Problems (IP);Mixed-Integer Programming Problems (MIP).




Optimization Methods

Mathematical programming techniques

For composites: Kirch (1981), Vanderplaats (1984), Rozvany(1989), Arora (1990), Haftka (1990).

Evolutionary methodsFor composites: Callahan & Weeks (1992), Le Riche &Haftka (1993), Hajela (1993), Nagendra et al. (1993), Ball etal. (1993), Gurdal et al. (1994), Kogiso et al. (1994).





Mathematical programming techniquesFor composites: Kirch (1981), Vanderplaats (1984), Rozvany(1989), Arora (1990), Haftka (1990).







Evolutionary methods

For composites: Callahan & Weeks (1992), Le Riche &Haftka (1993), Hajela (1993), Nagendra et al. (1993), Ball etal. (1993), Gurdal et al. (1994), Kogiso et al. (1994).




Stacking Sequence

90

45

0

45

90

45

h

z=h/2

z=-h/2

z=0

Possible angles: 0◦, ±45◦,90◦

Genetic code: 1,4,7

Laminate Code:[90/±45/0/±45/90/±45]

Integer Design Variable:(7, 4, 1, 4, 7, 4)




Typical Optimization Problems

Design of laminates with required stiffness

Optimization for maximum strength

Design for maximum buckling loads

Thermal effects \ uniform or variable temperature distribution




Wing with Individually Optimized Laminates




Integer Search Space

Number of possible designs:

∑i=1

N i (A),

where A is the integeralphabet; N(A) is the length ofthe alphabet A; ` is the lengthof chromosome, or number ofplies in a laminate.

∑i=1

3i = 3, ` = 1

1 4 7

∑i=1

3i = 12, ` = 2

11 14 17 1041 44 47 4071 74 77 70




Integer Search Space

`N(A)

2 3 4

1 2 3 42 6 12 203 14 39 844 30 120 3405 62 363 13646 126 1092 54607 254 3279 218448 510 9840 873809 1022 29523 34952410 2046 88572 139810020 2097150 5230176600 1466015503700




Selection of Variables

Material related variables

Configuration related variables

Geometry related variables

Decision variables

Design variables




Material Related Variables

Decision variables:

Fiber materialFiber pattern

Continuous fibers (unidirectional, biaxial, woven fabric)Discontinuous fibers (randomly oriented, preferred orientation)

Matrix material

PolymerMetalCarbonCeramic

Design variables:

Fiber volume contentConcentration of fibers with respect to location





Decision variables:

Fiber material

Fiber pattern


Matrix material


Design variables:






Decision variables:

Fiber materialFiber pattern


Matrix material


Design variables:





Configuration Related Variables

Decision variables:Selection of the type of lamination:

Non-hybrid laminateHybrid laminateSandwich structure

Design variables:

Fiber orientationStacking sequenceLayer thicknesses





Decision variables:

Selection of the type of lamination:


Design variables:






Decision variables:Selection of the type of lamination:


Design variables:





Geometry Related Variables

Decision variables:

Location and type of jointsForm of the centerline or the mid-surface (e.g., cylindrical,spherical or paraboloid shells, etc.)Cross-sectional shape (e.g., I-beam, L-beam, etc.)

Design variables:

Cross-sectional dimensionsLocations of supportsCurvature in the case of shell structures




Fuselage Panel




Stiffener




Composite Materials with Different Forms of Constituents




Fiber Arrangement Patterns in the Layer




Best Aircraft?




Design/Optimization Issues

Design Complexity

Modelling complexity:the level of complexity employed in a modelling of a structure(laminate, stiffened plate, segmented plate, grid shell, etc.)Analysis complexity:the constitutive and geometrical properties used in thestructural analysis (linear elastic analysis, plastic, nonlinear andprobabilistic effects, material, loading, and geometricaluncertainties)Optimization complexity:the type of optimization problem (continuous design variables,integer design variables, mixed-integer design variables,convexity of design space, etc.)





Design Complexity

Modelling complexity:the level of complexity employed in a modelling of a structure(laminate, stiffened plate, segmented plate, grid shell, etc.)

Analysis complexity:the constitutive and geometrical properties used in thestructural analysis (linear elastic analysis, plastic, nonlinear andprobabilistic effects, material, loading, and geometricaluncertainties)Optimization complexity:the type of optimization problem (continuous design variables,integer design variables, mixed-integer design variables,convexity of design space, etc.)





Design Complexity

Modelling complexity:the level of complexity employed in a modelling of a structure(laminate, stiffened plate, segmented plate, grid shell, etc.)Analysis complexity:the constitutive and geometrical properties used in thestructural analysis (linear elastic analysis, plastic, nonlinear andprobabilistic effects, material, loading, and geometricaluncertainties)

Optimization complexity:the type of optimization problem (continuous design variables,integer design variables, mixed-integer design variables,convexity of design space, etc.)





Design Complexity

Modelling complexity:the level of complexity employed in a modelling of a structure(laminate, stiffened plate, segmented plate, grid shell, etc.)Analysis complexity:the constitutive and geometrical properties used in thestructural analysis (linear elastic analysis, plastic, nonlinear andprobabilistic effects, material, loading, and geometricaluncertainties)Optimization complexity:the type of optimization problem (continuous design variables,integer design variables, mixed-integer design variables,convexity of design space, etc.)




Laminate under Transversal Loading




Laminated Plate Theory

The strain-displacement relationships:

εxx =∂u

∂x,

εyy =∂v

∂y,

εxy =∂u

∂y+

∂v

∂x.





The strains in terms of middle surface displacements:

εxx =∂u0

∂x− z

∂2w

∂x2,

εyy =∂v0

∂y− z

∂2w

∂y2,

εxy =∂u0

∂y+

∂v0

∂x− 2z

∂2w

∂x∂y.





The strains in terms of middle surface displacements:εxx

εyy

εxy

=

ε0xx

ε0yy

ε0xy

+ z

κxx

κyy

κxy

,

whereε0xx

ε0yy

ε0xy

=

∂u0

∂x∂v0

∂y∂u0

∂y + ∂v0

∂x

,

κxx

κyy

κxy

= −

∂2w∂x2

∂2w∂y2

2 ∂2w∂x∂y

.





The stresses areσxx

σyy

σxy

k

=

Q11 Q12 Q16

Q12 Q22 Q26

Q16 Q26 Q66

k

ε0xx

ε0yy

ε0xy

+ z

κxx

κyy

κxy

,

where Qij are the plane stress-reduced stiffnesses.





The forces and moments for a n-ply laminate:Nxx

Nyy

Nxy

=

∫ h/2

−h/2

σxx

σyy

σxy

dz =n∑

k=1

∫ zk

zk−1

σxx

σyy

σxy

k

dz ,

Mxx

Myy

Mxy

=

∫ h/2

−h/2

σxx

σyy

σxy

z dz =n∑

k=1

∫ zk

zk−1

σxx

σyy

σxy

k

z dz .





The forces for a n-ply laminate:Nxx

Nyy

Nxy

=

A11 A12 A16

A12 A22 A26

A16 A26 A66

ε0xx

ε0yy

ε0xy

+

B11 B12 B16

B12 B22 B26

B16 B26 B66

κxx

κyy

κxy

.





The moments for a n-ply laminate:Mxx

Myy

Mxy

=

B11 B12 B16

B12 B22 B26

B16 B26 B66

ε0xx

ε0yy

ε0xy

+

D11 D12 D16

D12 D22 D26

D16 D26 D66

κxx

κyy

κxy

.





The extensional stiffness (Aij), the coupling stiffness (Bij), and thebending stiffness (Dij) are defined as

Aij =n∑

k=1

(Qij)k(zk − zk−1),

Bij =1

2

n∑k=1

(Qij)k(z2k − z2

k−1),

Dij =1

3

n∑k=1

(Qij)k(z3k − z3

k−1),

where Qij is the reduced stiffness matrix.





The reduced stiffness matrix is

Q11 = Q11c4 + 2(Q12 + 2Q66)s

2c2 + Q22s4,

Q12 = (Q11 + Q22 − 4Q66)c2s2 + Q12(s

4 + c4),

Q22 = Q11s4 + 2(Q12 + 2Q66)s

2c2 + Q22c4,

Q16 = (Q11 − Q12 − 2Q66)sc3 + (Q12 − Q22 + 2Q66)s

3c ,

Q26 = (Q11 − Q12 − 2Q66)s3c + (Q12 − Q22 + 2Q66)sc

3,

Q66 = (Q11 + Q22 − 2Q12 − 2Q66)s2c2 + Q66(s

3 + c3),

where s = sin(θ), c = cos(θ),

Q11,22 =E1,2

1− ν12ν21, Q12 =

ν12E2

1− ν12ν21, Q66 = G12.





Equilibrium equations in terms of forces and moments:

∂Nxx

∂x+

∂Nxy

∂y= 0,

∂Nxy

∂x+

∂Nyy

∂y= 0,

∂2Mxx

∂x2+ 2

∂2Mxy

∂x∂y+

∂2Myy

∂y2+ Nxx

∂2w

∂x2+ Nyy

∂2w

∂y2+

2Nxy∂2w

∂x∂y= 0.




Example

The equation governing the out-of-plane displacement w of asymmetric and balanced laminate subjected to a pressure loading qis

D11∂4w

∂x4+ 4D16

∂4w

∂x3∂y+ 2(D12 + 2D66)

∂4w

∂x2∂y2+

4D26∂4w

∂x∂y3+ D22

∂4w

∂y4= q(x , y).

If we know displacement field of every point it means that we knoweverything about the structure!




Example (out-of-plane displacement)

For a simply supported plate under sinusoidally varying pressure,

q(x , y) = q0 sin(πx

a

)sin

(πy

b

),

the solution is

w =a4q0 sin(πx/a) sin(πy/b)

π4[D11 + 2(D12 + 2D66)(a/b)2 + D22(a/b)4].




Example (out-of-plane displacement)

For a uniform pressure distribution,

q(x , y) = q0,

the solution is

w =16q0

π6

∑m=1,3,...

∑n=1,3,...

wmn sin(mπx

a

)sin

(nπy

b

),

where

wmn =1

mn

[D11

(m

a

)4+ 2(D12 + 2D66)

(mn

ab

)2+ D22

(n

b

)4]−1

.




Example (transverse vibration)

For the transverse vibrations of a laminated plate, we replace qwith the inertia load

q(x , y) = −ρh∂2w

∂t2.

The natural vibration frequencies are given as

ωmn =π2

√ρh

√D11

(m

a

)4+ 2(D12 + 2D66)

(mn

ab

)2+ D22

(n

b

)4,

where m and n are the number of half waves in the x and ydirections, respectively.




Example (buckling)

If the plate is loaded such that Nx = −λNx0 and Ny = −λNy0,then the critical value of λ corresponding to a buckling load isdetermined as

λcr (m, n) =π2

[D11

(ma

)4+ 2(D12 + 2D66)

(mnab

)2+

(nb

)4]

(m/a)2Nx0 + (n/b)2Ny0.

The buckling load multiplier is obtained by finding the lowest valueof λcr for all combinations of m and n.




Future Research

Remember 1466015503700 possible designs (20 layers and 4possible orientations only!)

Single ply thickness is 0.1429 mm

Wall thickness in submarine is 100 mm

Laminate consists of 700 layers!!!

How many possible combinations?



Gant Ov Nik 2006

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