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Transcript of 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
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Outline
1 Laminated Composite Materials
2 Structural Design
3 Methods of Composite Optimization
4 Examples
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Composite Materials
Combination of a strong material (fibers) with a weakermaterial (matrix)
Layered structure
Stacking sequence
Directional nature of the material - Anisotropy
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Composite Materials
Combination of a strong material (fibers) with a weakermaterial (matrix)
Layered structure
Stacking sequence
Directional nature of the material - Anisotropy
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Composite Materials
Combination of a strong material (fibers) with a weakermaterial (matrix)
Layered structure
Stacking sequence
Directional nature of the material - Anisotropy
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Composite Materials
Combination of a strong material (fibers) with a weakermaterial (matrix)
Layered structure
Stacking sequence
Directional nature of the material - Anisotropy
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Composite Materials
Combination of a strong material (fibers) with a weakermaterial (matrix)
Layered structure
Stacking sequence
Directional nature of the material - Anisotropy
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Composite Components in an Airbus A-320
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
SEM Micrographs of Carbon Fiber Composite
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Carbon Fiber Composite Fuselage Section
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Composite Components in Helicopter
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Composite Components in Military Aircraft
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
SpaceShipOne
SpaceShipOne is the first operational space vehicle made entirelyof carbon composite!
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Composite Pieces in an Vehicles
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Composite Pieces in an Vehicles
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Composite Bicycle
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Structural Design and Optimization
Galileo Galilei (1638): Optimal cantilever problem (parabolicheight function produces the minimum weight design for atip-loaded, constant width cantilever).
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Structural Design and Optimization
Galileo Galilei (1638): Optimal cantilever problem (parabolicheight function produces the minimum weight design for atip-loaded, constant width cantilever).
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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).
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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).
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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).
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Optimization Methods
Mathematical programming techniquesFor 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).
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Optimization Methods
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).
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Optimization Methods
Mathematical programming techniquesFor 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).
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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)
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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)
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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)
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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)
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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)
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Wing with Individually Optimized Laminates
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Selection of Variables
Material related variables
Configuration related variables
Geometry related variables
Decision variables
Design variables
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Selection of Variables
Material related variables
Configuration related variables
Geometry related variables
Decision variables
Design variables
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Material Related Variables
Decision variables:
Fiber material
Fiber 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
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Configuration Related Variables
Decision variables:Selection of the type of lamination:
Non-hybrid laminateHybrid laminateSandwich structure
Design variables:
Fiber orientationStacking sequenceLayer thicknesses
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Configuration Related Variables
Decision variables:
Selection of the type of lamination:
Non-hybrid laminateHybrid laminateSandwich structure
Design variables:
Fiber orientationStacking sequenceLayer thicknesses
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Configuration Related Variables
Decision variables:Selection of the type of lamination:
Non-hybrid laminateHybrid laminateSandwich structure
Design variables:
Fiber orientationStacking sequenceLayer thicknesses
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Configuration Related Variables
Decision variables:Selection of the type of lamination:
Non-hybrid laminateHybrid laminateSandwich structure
Design variables:
Fiber orientationStacking sequenceLayer thicknesses
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Configuration Related Variables
Decision variables:Selection of the type of lamination:
Non-hybrid laminateHybrid laminateSandwich structure
Design variables:
Fiber orientationStacking sequenceLayer thicknesses
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Configuration Related Variables
Decision variables:Selection of the type of lamination:
Non-hybrid laminateHybrid laminateSandwich structure
Design variables:
Fiber orientationStacking sequenceLayer thicknesses
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Fuselage Panel
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Stiffener
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Composite Materials with Different Forms of Constituents
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Fiber Arrangement Patterns in the Layer
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Best Aircraft?
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Best Aircraft?
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Best Aircraft?
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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.)
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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.)
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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.)
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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.)
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Laminate under Transversal Loading
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Laminated Plate Theory
The strain-displacement relationships:
εxx =∂u
∂x,
εyy =∂v
∂y,
εxy =∂u
∂y+
∂v
∂x.
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Laminated Plate Theory
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.
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Laminated Plate Theory
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
.
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Laminated Plate Theory
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.
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Laminated Plate Theory
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 .
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Laminated Plate Theory
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
.
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Laminated Plate Theory
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
.
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Laminated Plate Theory
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.
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Laminated Plate Theory
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.
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
Laminated Plate Theory
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.
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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!
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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].
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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
.
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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.
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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.
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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?
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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?
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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?
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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?
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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?
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials
Laminated Composite Materials Structural Design Methods of Composite Optimization Examples
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?
Vladimir Gantovnik Clemson University
Design and Optimization of Laminated Composite Materials