Gantovnik2006

Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Design and Optimization of Laminated
Composite Materials
Vladimir Gantovnik
Clemson 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

Composite Materials
Combination of a strong material (ﬁbers) with a weaker
material (matrix)

Composite Materials
material (matrix)
Layered structure

Composite Materials
material (matrix)
Layered structure
Stacking sequence

Composite Materials
material (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 entirely
of carbon composite!

Composite Pieces in an Vehicles

Composite Bicycle

Structural Design and Optimization

Structural Design and Optimization
Galileo Galilei (1638): Optimal cantilever problem (parabolic
height function produces the minimum weight design for a
tip-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}.

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).

(1989), Arora (1990), Haftka (1990).
Evolutionary methods

(1989), Arora (1990), Haftka (1990).
Evolutionary methods
For composites: Callahan & Weeks (1992), Le Riche &
Haftka (1993), Hajela (1993), Nagendra et al. (1993), Ball et
al. (1993), G¨urdal et al. (1994), Kogiso et al. (1994).

Stacking Sequence
90
45
0
45
90
45
h
z=h/2
z=-h/2
z=0

Stacking Sequence
90
45
0
45
90
45
h
z=h/2
z=-h/2
z=0
Possible angles: 0◦, ±45◦,
90◦

Stacking Sequence
90
45
0
45
90
45
h
z=h/2
z=-h/2
z=0
90◦
Genetic code: 1,4,7

Stacking Sequence
90
45
0
45
90
45
h
z=h/2
z=-h/2
z=0
90◦
Genetic code: 1,4,7
Laminate Code:
[90/±45/0/±45/90/±45]

Stacking Sequence
90
45
0
45
90
45
h
z=h/2
z=-h/2
z=0
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
Ni
(A),
where A is the integer
alphabet; N(A) is the length of
the alphabet A; is the length
of chromosome, or number of
plies in a laminate.
i=1
3i
= 3, = 1
1 4 7
i=1
3i
= 12, = 2
11 14 17 10
41 44 47 40
71 74 77 70

Integer Search Space
N(A)
2 3 4
1 2 3 4
2 6 12 20
3 14 39 84
4 30 120 340
5 62 363 1364
6 126 1092 5460
7 254 3279 21844
8 510 9840 87380
9 1022 29523 349524
10 2046 88572 1398100
20 2097150 5230176600 1466015503700

Selection of Variables
Material related variables
Conﬁguration related variables
Geometry related variables

Selection of Variables
Material related variables
Conﬁguration related variables
Geometry related variables
Decision variables
Design variables

Material Related Variables

Decision variables:
Fiber material

Decision variables:
Fiber material
Fiber pattern

Decision variables:
Fiber material
Fiber pattern
Continuous ﬁbers (unidirectional, biaxial, woven fabric)
Discontinuous ﬁbers (randomly oriented, preferred orientation)

Decision variables:
Fiber material
Fiber pattern
Matrix material

Decision variables:
Fiber material
Fiber pattern
Matrix material
Polymer
Metal
Carbon
Ceramic

Decision variables:
Fiber material
Fiber pattern
Matrix material
Polymer
Metal
Carbon
Ceramic
Design variables:
Fiber volume content
Concentration of ﬁbers with respect to location

Conﬁguration Related Variables

Decision variables:

Decision variables:
Selection of the type of lamination:

Decision variables:
Non-hybrid laminate
Hybrid laminate
Sandwich structure

Decision variables:
Non-hybrid laminate
Hybrid laminate
Sandwich structure
Design variables:

Decision variables:
Non-hybrid laminate
Hybrid laminate
Sandwich structure
Design variables:
Fiber orientation
Stacking sequence
Layer thicknesses

Geometry Related Variables

Decision variables:
Location and type of joints
Form 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.)

Decision variables:
Design variables:

Decision variables:
Design variables:
Cross-sectional dimensions
Locations of supports
Curvature 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

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

Design Complexity
Analysis complexity:
the constitutive and geometrical properties used in the
structural analysis (linear elastic analysis, plastic, nonlinear and
probabilistic eﬀects, material, loading, and geometrical
uncertainties)

Design Complexity
Analysis complexity:
the constitutive and geometrical properties used in the
structural analysis (linear elastic analysis, plastic, nonlinear and
probabilistic eﬀects, material, loading, and geometrical
uncertainties)
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



=



ε0
xx
ε0
yy
ε0
xy



+ z



κxx
κyy
κxy



,
where



ε0
xx
ε0
yy
ε0
xy



=



∂u0
∂x
∂v0
∂y
∂u0
∂y + ∂v0
∂x



,



κxx
κyy
κxy



= −



∂2w
∂x2
∂2w
∂y2
2 ∂2w
∂x∂y



.

The stresses are



σxx
σyy
σxy



k
=


¯Q11
¯Q12
¯Q16
¯Q12
¯Q22
¯Q26
¯Q16
¯Q26
¯Q66


k






ε0
xx
ε0
yy
ε0
xy



+ 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





ε0
xx
ε0
yy
ε0
xy



+


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





ε0
xx
ε0
yy
ε0
xy



+


D11 D12 D16
D12 D22 D26
D16 D26 D66





κxx
κyy
κxy



.

The extensional stiffness (Aij ), the coupling stiffness (Bij ), and the
bending stiffness (Dij ) are defined as
Aij =
n
k=1
( ¯Qij )k(zk − zk−1),
Bij =
1
2
n
k=1
( ¯Qij )k(z2
k − z2
k−1),
Dij =
1
3
n
k=1
( ¯Qij )k(z3
k − z3
k−1),
where ¯Qij is the reduced stiffness matrix.

The reduced stiﬀness matrix is
¯Q11 = Q11c4
+ 2(Q12 + 2Q66)s2
c2
+ Q22s4
,
¯Q12 = (Q11 + Q22 − 4Q66)c2
s2
+ Q12(s4
+ c4
),
¯Q22 = Q11s4
+ 2(Q12 + 2Q66)s2
c2
+ Q22c4
,
¯Q16 = (Q11 − Q12 − 2Q66)sc3
+ (Q12 − Q22 + 2Q66)s3
c,
¯Q26 = (Q11 − Q12 − 2Q66)s3
c + (Q12 − Q22 + 2Q66)sc3
,
¯Q66 = (Q11 + Q22 − 2Q12 − 2Q66)s2
c2
+ Q66(s3
+ 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 a
symmetric and balanced laminate subjected to a pressure loading q
is
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 know
everything 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 q
with 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 y
directions, 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 is
determined as
λcr (m, n) =
π2 D11
m
a
4
+ 2(D12 + 2D66) mn
ab
2
+ n
b
4
(m/a)2Nx0 + (n/b)2Ny0
.
The buckling load multiplier is obtained by ﬁnding the lowest value
of λcr for all combinations of m and n.

Future Research

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

Future Research
Single ply thickness is 0.1429 mm

Future Research
Wall thickness in submarine is 100 mm

Future Research
Laminate consists of 700 layers!!!

Future Research
Laminate consists of 700 layers!!!
How many possible combinations?

Gantovnik2006

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