RPJ_Laan_MSc_Thesis.

RPJ_Laan_MSc_Thesis.
Influence of stiffeners on a
variable stiffness laminate
optimisation
Master of Science Thesis
Faculty of Aerospace Structures and Computational Mechanics
R.P.J. Laan
I NFLUENCE OF STIFFENERS ON A VARIABLE
STIFFNESS LAMINATE OPTIMISATION
M ASTER OF S CIENCE T HESIS
by
R.P.J. Laan
in partial fulfillment of the requirements for the degree of
Master of Science
in Aerospace Engineering
at the Delft University of Technology,
to be defended publicly on Monday March 21, 2016 at 2:00 PM.
Thesis committee::
Dr. M. M. Abdalla,
Dr. S. R. Turteltaub,
Dr. S. Teixeira De Freitas,
Ir. D. M. J. Peeters,
TU Delft
TU Delft
TU Delft
TU Delft
This thesis is confidential and cannot be made public until March 21, 2016.
An electronic version of this thesis is available at http://repository.tudelft.nl/.
Copyright © Aerospace Structures & Computational Mechanics
All rights reserved.
D ELFT U NIVERSITY OF T ECHNOLOGY
D EPARTMENT OF
A EROSPACE S TRUCTURES & C OMPUTATIONAL M ECHANICS
The undersigned hereby certify that they have read and recommend to the Faculty
of Aerospace Engineering for acceptance a thesis entitled
I NFLUENCE OF STIFFENERS ON A VARIABLE STIFFNESS LAMINATE OPTIMISATION
by
R.P.J.L AAN
in partial fulfillment of the requirements for the degree of
M ASTER OF S CIENCE
Dated: March 21, 2016
Committee chairman:
dr. M. M. Abdalla
Committee members:
dr. S. R. Turteltaub
dr. S. Teixeira De Freitas
ir. D. M. J. Peeters
Acknowledgements
In this thesis I would like to thank a lot of people who supported me during my study and this thesis.
First, I would like to thank Mr. Mostafa Abdalla for giving me this project and for his support during
the project.
Special thanks go to Mr. Daniël Peeters who helped me by guiding me through the entire project.
Without his support and time he invested in me, the thesis would have been a lot harder to complete.
Thanks for answering all the questions I had, at literally every time of the day.
I would like to thank my family and friends who mentally supported me when times were difficult.
They made my study a lot easier.
I would particular like to thank my parents Jan Laan and Margriet Laan-Dam. They supported me
in every decision I made.
Last but not least, I would like to thank the Thesis committee for grading my work.
iv
Summary
Engineers are still looking for ways to improve the performance of materials used in aircraft. A major
step was going from metal parts to laminates in order to save weight significantly. Those laminates
existed of fibres with a constant orientation, and became laminates with changing fibre directions.
Variable stiffness laminates (VSL) are used for its possibility of exploiting the anisotropic properties
of composites. In order to improve the buckling load of a panel, VSL are used. Those laminates steer
the loads through the laminate. This is beneficial for maximising the buckling load of a panel.
An optimiser is built by the Delft University of Technology to optimise VSL by changing its fibre
orientations throughout the entire panel. A manufacturing constraint is added to the optimiser to
prevent the fibre paths to have a turn radius of smaller than 0.333[m]. Plate designs with fibres paths
having a smaller turn radius than this constraint, are assumed to be unmanufacturable. The software
optimises fibre paths of VSL against buckling and keeps the stiffness of the entire plate equal or higher
than the quasi isotropic design.
Since the aircraft of nowadays are stiffener dominated, an extra functionality is added to the optimiser. The optimiser is further developed to be capable of optimising stiffened plates. Stiffeners are
taken into account during the optimisation. Extra functionalities such as integrated stiffeners are
introduced. These are stiffeners sharing layers with the plate.
The extended optimiser is verified by comparing the internal loads from five cases with the values
calculated with ABAQUS. After having the optimiser verified, it is used to calculate results for twelve
different cases. Those cases range from square unstiffened plates to rectangular plates with two stiffeners. The results show the influence of stiffeners on the optimised panel with changed fibre paths.
Those cases are compared to the plate designs with the stiffeners excluded from optimisation. Performance increases of up to 174% are obtained for balanced symmetric laminates. The results show
clearly that when stiffeners are added to a panel, the area in-between the stiffeners can be treated as a
simply supported plate. Integrated stiffeners also show a buckling load increase of 113% with respect
to the quasi isotropic case.
The fibres of the buckling optimised plates show a relating behaviour. Fibre orientations going towards [45,-45] are present in buckling critical areas. Near stiffeners and near edges, [0,0] areas are
present in order to introduce the load through the structure. A relation is found between having
more [0,0] orientations in the inner plies of the laminate and significantly more [45,-45] orientations
at the outer plies of the laminate. The outer plies are more critical for buckling, which requires
more fibres being close to a [45,-45] in order to higher the buckling load. The inner areas are less
vulnerable for buckling, which creates the opportunity of creating stiffness in the inside of the laminate.
v
Contents
Acknowledgements
iv
Summary
v
List of Figures
viii
List of Tables
xi
List of Abbreviations
xiii
Introduction
1 Optimal plate design
1.1 Variable stiffness laminates
1.2 Manufacturing constraint .
1.3 Integrated stiffeners . . . .
1.4 Optimisation . . . . . . . .
1.4.1 Objective function .
1.4.2 Ranking weights . .
1.4.3 State variables . . .
1.4.4 Design variables . .
1.4.5 Constraints . . . . .
1
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2
2
3
3
4
4
4
5
5
5
2 The initial optimiser
2.1 About the software . . . . . . . .
2.2 Inputs and outputs . . . . . . . .
2.2.1 Option inputs . . . . . . .
2.2.2 Expected results . . . . .
2.3 Optimiser application . . . . . .
2.3.1 Mesh generation . . . . .
2.3.2 Model creation . . . . . .
2.3.3 Symmetry matrix . . . . .
2.3.4 Properties . . . . . . . . .
2.3.5 Response approximations
2.3.6 Sensitivities . . . . . . . .
2.3.7 Stopping criteria . . . . .
2.4 Developing the optimiser . . . . .
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6
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7
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12
13
13
13
3 Stiffened panel
3.1 Single stiffener . . . . . . . .
3.1.1 Stiffener options . . .
3.1.2 Addition to the model
3.1.3 Optimisation . . . . .
3.2 Two or multiple stiffeners . .
3.3 Stiffener integration . . . . .
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14
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14
15
16
16
17
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vi
CONTENTS
3.3.1
3.3.2
CONTENTS
Fibre orientations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Verification
4.1 Input values . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Model constraints . . . . . . . . . . . . . . . . . . . . . .
4.3 Model verification . . . . . . . . . . . . . . . . . . . . .
4.3.1 Unstiffened plate . . . . . . . . . . . . . . . . . .
4.3.2 Single stiffened plate (center located) . . . . . . .
4.3.3 Single stiffened plate (off-center located) . . . . .
4.3.4 Double stiffened plate (Symmetrically located) .
4.3.5 Double stiffened plate (Asymmetrically located)
4.3.6 Crippling . . . . . . . . . . . . . . . . . . . . . .
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . .
5 Optimisation results
5.1 Initial values . . . . . . . . . . . . . . . . .
5.2 Results . . . . . . . . . . . . . . . . . . . . .
5.2.1 Unstiffened plate . . . . . . . . . . .
5.2.2 Single stiffened plate (center) . . . .
5.2.3 Single stiffened plate (off-center) . .
5.2.4 Double stiffened plate (symmetric) .
5.2.5 Double stiffened plate (asymmetric)
5.2.6 Rectangular plates . . . . . . . . . .
5.2.7 Shared layers . . . . . . . . . . . . .
5.3 Conclusion . . . . . . . . . . . . . . . . . .
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17
18
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19
21
21
21
22
24
25
26
27
28
28
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29
29
30
30
31
32
33
33
34
38
40
Conclusion & recommendations
42
Bibliography
43
Appendices
45
A Laminate properties
A.1 Engineering constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2 Material Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.3 Lamination Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
46
46
46
B ABD-matrix
47
C Frobenius inner product
48
D FEM
49
E Verification plots
E.1 Unstiffened plate . . . . . . . . . .
E.2 Single stiffened plate (Center) . . .
E.3 Single stiffened plate (Off-center) .
E.4 Double stiffened plate (Symmetric)
51
51
54
57
60
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vii
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CONTENTS
CONTENTS
E.5 Double stiffened plate (Asymmetric) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
viii
63
List of Figures
1.1
1.2
Schematic drawing of a laminate variable stiffness laminate . . . . . . . . . . . . . . .
A laminate with a stiffener integration . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
Schematic drawing of a balanced laminate layer with steered fibres (left), and the fibre
directions per node (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Schematic drawing of a Mesh with 17x17 nodes (left) and 11x11 nodes plus element
symmetry in two directions (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Software Flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Positive (Counter Clockwise) element definition . . . . . . . . . . . . . . . . . . . . . .
Standard constraint on the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The distances between the center of the laminate and the ply borders[19] . . . . . . .
Placing the fibres under an angle with respect to the loading[19] . . . . . . . . . . . .
2.2
2.3
2.4
2.5
2.6
2.7
3.1
3.2
3.3
3.4
3.5
3.6
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
A schematic drawing of a plate with stiffener model . . . . . . . . . . . . .
The model definition with a Main part and Sub parts . . . . . . . . . . . .
Element definition, ’226 - 242 - 227’ . . . . . . . . . . . . . . . . . . . . . .
A basic model with multiple stiffeners . . . . . . . . . . . . . . . . . . . . .
A visualisation of an integrated stiffener . . . . . . . . . . . . . . . . . . . .
A schematic drawing of two integrated stiffeners showing one layer with the
fibre directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
7
8
9
9
10
11
. .
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as
. .
14
15
16
16
17
The 4 verification cases with 6 models in total . . . . . . . . . . . . . . . . . . . . . . .
Nxx of the Optimiser (Left) and Nxx of ABAQUS (Right) . . . . . . . . . . . . . . . .
Nyy of the Optimiser (Left) and Nyy of ABAQUS (Right) . . . . . . . . . . . . . . . .
Absolute error between both models, unstiffened plate Nxx (Left) Nyy (Right) . . . . .
Relative error between both models, unstiffened plate Nxx (Left) Nyy (Right) . . . . .
The absolute error between both models, single stiffened plate center Nxx (Left) Nyy
(Right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The relative error between both models, single stiffened plate center Nxx (Left) Nyy
(Right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The absolute error between both models, single stiffened plate off-center Nxx (Left) Nyy
(Right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The relative error between both models, single stiffened plate off-center Nxx (Left) Nyy
(Right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The absolute error between both models, double symmetric stiffened plate Nxx (Left)
Nyy (Right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The relative error between both models, double symmetric stiffened plate Nxx (Left)
Nyy (Right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The absolute error between both models, double asymmetric stiffened plate Nxx (Left)
Nyy (Right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The relative error between both models, double asymmetric stiffened plate Nxx (Left)
Nyy (Right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
22
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23
ix
. . . .
. . . .
. . . .
. . . .
. . . .
arrows
. . . .
2
4
17
24
24
25
25
26
26
27
27
LIST OF FIGURES
LIST OF FIGURES
4.14 Buckling of model one at Pcr = 5.6e05[N ] with a stiffener height of 0.05[m] (left)
Crippling of the stiffener from model two at Pcr = 2.79e05[N ] with a stiffener height of
0.15[m] (Right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
Fibre direction per node (Left) fibre paths of a laminate layer (Right) . . . . . . . . .
Optimised fibre paths of an unstiffened plate. From the top layer (Left up) to the
laminate center(Right below) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optimised Fibre paths of a center stiffened plate. From the top layer (Left up) to the
laminate center (Right below). The red line shows the stiffener location . . . . . . . .
Optimised fibre paths of an off-center stiffened plate. From the top layer (left up) to
the laminate center laminate (right below). The red line shows the stiffener location .
Optimised fibre paths of a symmetric double stiffened plate. From the top layer (Left
up) to the laminate center (Right below). The red line shows the stiffener location . .
Optimised fibre paths of an asymmetric double stiffened plate. From the top layer (left
up) to the laminate center (right below). The red line shows the stiffener location . . .
Unstiffened panel (left) Stiffened panel (right) both 0.5[m] by 1[m] . . . . . . . . . . .
Optimised fibre paths of an unstiffened rectangular plate of 0.5[m] by 1[m]. From the
top layer (Left) to the laminate center (Right) . . . . . . . . . . . . . . . . . . . . . .
Optimised fibre paths of a double stiffened rectangular plate with 4 plies per stiffener
of 0.5[m] by 1[m]. From the top layer (Left) to the laminate center (Right). The red
line shows the location of the stiffeners . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optimised fibre paths of a double stiffened rectangular plate with 8 plies per stiffener
of 0.5[m] by 1[m]. From the top layer (Left) to the laminate center (Right). The red
line shows the location of the stiffeners . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optimised fibre paths of a double stiffened rectangular plate with 12 plies per stiffener
of 0.5[m] by 1[m]. From the top layer (Left) to the laminate center (Right). The red
line shows the location of the stiffeners . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optimised fibre paths of an double stiffened rectangular plate with 16 plies per stiffener
of 0.5[m] by 1[m]. From the top layer (Left) to the laminate center (Right). The red
line shows the location of the stiffeners . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optimised fibre paths of an double stiffened square plate with 16 plies per stiffener of
0.5[m] by 1[m]. From the top layer (Left top) to the laminate center (Right bottom).
The red and green lines show the location of the stiffeners. Green also means that the
plies go into the stiffener and the fibre orientations at that location will be the fibre
orientation of the stiffener layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Optimised fibre paths of an double stiffened rectangular plate with 16 plies per stiffener
of 0.5[m] by 1[m]. From the top layer (Left) to the laminate center (Right). The red
and green lines show the location of the stiffeners. Green also means that the plies go
into the stiffener and the fibre orientations at that location will be the fibre orientation
of the stiffener layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
D.1 A drawing of master and slave nodes with their adjacent nodes (Left) and a drawing of
the forces going from adjacent nodes to the master and slave nodes . . . . . . . . . . .
E.1 The displacements of the unstiffened plate in all three spatial directions (Optimiser),
U1 on top, U2 in the middle, and U3 at the bottom . . . . . . . . . . . . . . . . . . .
E.2 The displacements of the unstiffened plate in all three spatial directions (ABAQUS),
U1 on top, U2 in the middle, and U3 at the bottom . . . . . . . . . . . . . . . . . . .
x
28
29
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31
32
33
34
34
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35
36
37
37
38
39
50
51
51
LIST OF FIGURES
LIST OF FIGURES
E.3 The section forces of the unstiffened plate in all three spatial directions (Optimiser),
Nx on top, Ny in the middle, Nz at the bottom . . . . . . . . . . . . . . . . . . . . . .
E.4 The section forces of the unstiffened plate in all three spatial directions (ABAQUS), Nx
on top, Ny in the middle, Nz at the bottom . . . . . . . . . . . . . . . . . . . . . . . .
E.5 The absolute error of the unstiffened plate section forces . . . . . . . . . . . . . . . . .
E.6 The relative error of the unstiffened plate section forces . . . . . . . . . . . . . . . . .
E.7 The displacements of the center stiffened plate in all three spatial directions (Optimiser),
U1 on top, U2 in the middle, and U3 at the bottom . . . . . . . . . . . . . . . . . . .
E.8 The displacements of the center stiffened plate in all three spatial directions (ABAQUS),
U1 on top, U2 in the middle, and U3 at the bottom . . . . . . . . . . . . . . . . . . .
E.9 The section forces of the center stiffened plate in all three spatial directions (Optimiser),
Nx on top, Ny in the middle, Nz at the bottom . . . . . . . . . . . . . . . . . . . . . .
E.10 The section forces of the center stiffened plate in all three spatial directions (ABAQUS),
Nx on top, Ny in the middle, Nz at the bottom . . . . . . . . . . . . . . . . . . . . . .
E.11 The absolute error of the center stiffened plate section forces . . . . . . . . . . . . . .
E.12 The relative error of the center stiffened plate section forces . . . . . . . . . . . . . . .
E.13 The displacements of the off-center stiffened plate in all three spatial directions (Optimiser), U1 on top, U2 in the middle, and U3 at the bottom . . . . . . . . . . . . . . .
E.14 The displacements of the off-center stiffened plate in all three spatial directions (ABAQUS),
U1 on top, U2 in the middle, and U3 at the bottom . . . . . . . . . . . . . . . . . . .
E.15 The section forces of the off-center stiffened plate in all three spatial directions (Optimiser), Nx on top, Ny in the middle, Nz at the bottom . . . . . . . . . . . . . . . . . .
E.16 The section forces of the off-center stiffened plate in all three spatial directions (ABAQUS),
Nx on top, Ny in the middle, Nz at the bottom . . . . . . . . . . . . . . . . . . . . . .
E.17 The absolute error of the off-center stiffened plate section forces . . . . . . . . . . . . .
E.18 The relative error of the off-center stiffened plate section forces . . . . . . . . . . . . .
E.19 The displacements of the unstiffened plate in all three spatial directions (Optimiser),
U1 on top, U2 in the middle, and U3 at the bottom . . . . . . . . . . . . . . . . . . .
E.20 The displacements of the unstiffened plate in all three spatial directions (ABAQUS),
U1 on top, U2 in the middle, and U3 at the bottom . . . . . . . . . . . . . . . . . . .
E.21 The section forces of the unstiffened plate in all three spatial directions (Optimiser),
Nx on top, Ny in the middle, Nz at the bottom . . . . . . . . . . . . . . . . . . . . . .
E.22 The section forces of the unstiffened plate in all three spatial directions (ABAQUS), Nx
on top, Ny in the middle, Nz at the bottom . . . . . . . . . . . . . . . . . . . . . . . .
E.23 The absolute arror of the unstiffened plate section forces . . . . . . . . . . . . . . . . .
E.24 The relative arror of the unstiffened plate section forces . . . . . . . . . . . . . . . . .
E.25 The displacements of the unstiffened plate in all three spatial directions (Optimiser),
U1 on top, U2 in the middle, and U3 at the bottom . . . . . . . . . . . . . . . . . . .
E.26 The displacements of the double asymmetric stiffened plate in all three spatial directions
(ABAQUS), U1 on top, U2 in the middle, and U3 at the bottom . . . . . . . . . . . .
E.27 The section forces of the unstiffened plate in all three spatial directions (Optimiser),
Nx on top, Ny in the middle, Nz at the bottom . . . . . . . . . . . . . . . . . . . . . .
E.28 The section forces of the unstiffened plate in all three spatial directions (ABAQUS), Nx
on top, Ny in the middle, Nz at the bottom . . . . . . . . . . . . . . . . . . . . . . . .
E.29 The absolute arror of the unstiffened plate section forces . . . . . . . . . . . . . . . . .
E.30 The relative arror of the unstiffened plate section forces . . . . . . . . . . . . . . . . .
xi
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54
55
55
56
56
57
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58
58
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60
61
61
62
62
63
63
64
64
65
65
List of Tables
2.1
Material properties used throughout the entire Thesis . . . . . . . . . . . . . . . . . .
7
4.1
4.2
The used material properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ply orientations of the plate and stiffener(s) . . . . . . . . . . . . . . . . . . . . . . . .
21
21
5.1
5.2
Initial ply orientations of the plate and stiffener(s) . . . . . . . . . . . . . . . . . . . .
The improvements of each square design compared to: 1. Stiffeners excluded from
optimisation and added afterwards 2. The QI design . . . . . . . . . . . . . . . . . . .
The improvements of each rectangular design compared to: 1. Stiffeners excluded from
optimisation and added afterwards 2. The QI design . . . . . . . . . . . . . . . . . . .
30
5.3
xii
40
41
List of Abbreviations
AFP
CTS
econn
FEA
g
H
MPC
QI
VSL
Automated Fibre Placement
Continuous Tow Steering
Element connectivity matrix
Finite Element Analysis
Gradient
Hessian
Multi Point Contraints
Quasi Isotropic
Variable Stiffness Laminates
xiii
Introduction
Designing a light weight aircraft is still one of the design objectives of aircraft engineers. The first
composite-dominated planes were the B-787 and the A400M. Those were optimised by replacing metal
parts by laminates. The further development of composite structures has become a trending topic.
Although much progress has already been made[2], many areas are still not studied regarding composite
optimisation. Many different ways of optimising composite laminates are done[3][4]. A significant
weight decrease could be achieved when the fibres are having variable directions throughout the entire
laminate [5]. Using those Variable Stiffness Laminates (VSL) increases the freedom in design and gives
control to fine-tune materials such that local design requirements can be met [6].
The Delft Technical University developed an optimiser to optimise the fibre paths of a variable
stiffness laminate. The optimiser is developed to optimise unstiffened plates against buckling. Since
aircraft of nowadays are stiffener dominated, the capability of adding stiffeners to the optimisation
cycle will be implemented. It is expected that when the stiffeners are also taken into account during
the optimisation of the plate, its design is improved even more.
This report will discuss the ’Influence of stiffeners on a variable stiffness laminate optimisation’.
In order to study this influence, the existing developed optimiser will be extended. A stiffener will be
added to the input of the optimiser. A new way of defining the properties will be created. Without
touching the optimisation algorithm, the optimiser will be capable of showing the results of a plate
optimised against buckling under influence of a stiffener.
Chapter 1 discusses VSL, manufacturing constraints, and integrated stiffeners in order to make the
reader familiar with those concepts. Chapter 2 discusses the initial state of the optimiser used for the
research to the influence of stiffeners on variable stiffness optimisations. Chapter 3 shows the steps
taken in order to implement the functionality of stiffener addition to the optimiser. The verification
will be done in chapter 4. After the verification the results will be shown in chapter 5 and a conclusion
will be drawn. In the end, the overall conclusion and recommendations will be discussed.
1
1
Optimal plate design
The use of laminate materials is a very trending topic in the industry of aircraft design. Variable
stiffness laminates (VSL) were discovered as a result from optimising laminate material. This is
discussed in the first section. The Delft technical university developed an optimiser in order to
optimise VSL by changing fibre paths. To keep VSL designs practical, a manufacturing constraint is
introduced. This will be discussed in section two. The third section discusses another way to improve
aircraft designs by integrating stiffeners. The fourth section discusses the general optimisation problem
applied on designing a VSL.
1.1
Variable stiffness laminates
Variable stiffness materials are, as the name already reveals, materials that do not have the same
stiffness throughout the entire material. Variable stiffness materials can be controlled or uncontrolled.
An example of an uncontrolled variable stiffness material is wood. Wood can be from the same tree
which does not guarantee that the piece of material has the same properties throughout the entire
wooden structure. Usually those stiffness changes are not immense. VSL are a type of controlled
variable stiffness material. Controlled variable stiffness materials can be made of fibres and resin.
The reason why VSL are used is the possibility of exploiting the anisotropic properties of composite
materials to a larger extent than was previously possible[7]. The variations of fibre directions can be
used to give beneficial load and stiffness distribution patterns. For buckling of composite panels it is
proven that VSL can be very effective[8]. An example of a VSL can be found in figure 1.1, which is
a VSL plate with a hole optimised against buckling. It can be noticed that the different areas of the
laminate contain fibres in different directions ranging from -5 to -89 degrees. The buckling load has
increased by 126% compared to a quasi isotropic panel [5].
Fig. 1.1: Schematic drawing of a laminate variable stiffness laminate
2
CHAPTER 1. OPTIMAL PLATE DESIGN
1.2
1.2. MANUFACTURING CONSTRAINT
Manufacturing constraint
Although the laminate is optimised against buckling (figure 1.1), it can be concluded that the design is
not really practical because of its orientation jumps from one area to another. Those borders between
two orientation areas are extremely critical for crack growth. It is thereby strongly recommended
to use continuous fibres in variable stiffness laminates. A disadvantage in this case is the maximum
angle the continuous fibre can make on a certain distance. So it is impossible to go from -89 degrees
to -67 degrees on such a small distance as has been shown in the figure. Using continuous fibres for
the already shown example would therefore not be possible. So when optimisers are optimising a
VSL, its manufacturability needs to be taken into account[9]. Described by Peeters (et al.)[10], a local
and global steering constraint is required for designing a continuous fibre laminate. The optimiser is
thereby bounded by the constraint of obeying a minimum radius of curvature for the fibre paths.
The Automated Fibre Placement (AFP) technique described by Kim (et. al)[11], makes it possible
to lay fibres in any direction of choice. Inefficiencies like tow gaps and tow overlap can be prevented
using Continuous Tow Steering(CTS). With both methods, the fibres need to be continuous[12]. As
long as the fibres are continuous and stay within the steering constraint, it will be accepted as manufacturable design. The maximum steering angle can be represented by ζU . The following constraint
can be set up:
ζ 2 − ζU2 ≤ 0
(1.1)
The squared form is taken because the angles can be negative as well. Changing this to an optimisation
problem, together with equation 1.1 it will become:
Minimize
Such that
max (f1 , f2 , ..., fn )
fn+1 , ..., fm ≤ 0
ζ 2 − ζU2 ≤ 0
Where f1 to fn are representing the structural responses that are optimised and fn+1 to fm are
constraints.
1.3
Integrated stiffeners
Laminate structures do not only exist of flat plates and basic stiffeners. According to Renton (et al.),
in 2023 most composite structures are expected to be fully integrated with minimal assembly parts.
The same goes for using stiffeners in aircraft. Those stiffeners will be integrated where possible. In
this case, stiffeners could be integrating with the plate by using shared layers between the plate and
the stiffener. Such an integrated stiffener is shown in figure 1.2. The gray layers are the layers which
only continue in the plate, the yellow and pink layers are shared between the plate and stiffener, and
the blue layers are only from the stiffener.
The advantages of stiffener integration are[13]:
• Integrated parts do no have to be assembled which means a reduce of assembly time/costs
• The structure is stronger with respect to the cases of assembling a stiffener with bolts
One of the main disadvantages is that the design is harder to fabricate which means that it costs
more money. So a trade-off must be made between short term costs and long term costs. Weight
optimisation applies to a integrated structure and less maintenance is required. Although as has been
said, the production costs are rising with integrated stiffeners.
3
1.4. OPTIMISATION
CHAPTER 1. OPTIMAL PLATE DESIGN
Fig. 1.2: A laminate with a stiffener integration
1.4
Optimisation
It needs to be clear that during the thesis, the optimisation algorithm stays untouched. But in order
to understand the implementation of adding a stiffener to the optimiser, this section is written. In
this thesis a plate will be optimised by changing the fibre paths in order to change the stiffness of the
plate. The manufacturability is assured by adding the manufacturing constraint. Those inputs will
be explained in this section.
Optimisation can be described as "finding an optimal solution by changing the parameters". It has
to be taken into account that there are requirements which needs to be satisfied. Those requirements
can exist of "finding a maximum or minimum", and it can consist of constraints which limit the design
space of the solution. This can for example be that a certain parameter is not allowed to be bigger
than another certain parameter. In general an optimisation problem is defined to be a mathematical
problem[14]. The design in a mathematical form is shown in equation 1.2. This equation is used to
compare different designs in order to rank them[15, 16, 17].
hj =
n
X
wi · fi (x, y)
(1.2)
i=1
Where:
fi (x, y)
wi
x
y
1.4.1
Is
Is
Is
Is
called the objective function
the weight given for one of the ranking criteria.
a vector of all state variables (which are constant inputs)
a vector of all the changeable design variables (change during optimization)
Objective function
The outcome of the objective function (f1 (x, y)) represents the score of the variables for a certain
design aspect. The objective function gives a score for how the design variables (x) together with
the state variables (y) are suitable for this design. The objective functions of the optimiser discussed
in this thesis is the buckling load for the first two buckling modes. The critical load for buckling is
maximised during the optimisation.
1.4.2
Ranking weights
The weights (wi ) of a function to rank the designs (hj ) are determined by the user of the optimization
algorithm. In this thesis the weights are only given to the buckling objective. If multiple objectives
would have been chosen, multiple weights should be given according to the importance of a given
objective.
4
CHAPTER 1. OPTIMAL PLATE DESIGN
1.4.3
1.4. OPTIMISATION
State variables
The state variables (x) can be easily described as variables that initialize the environment. Those
variables are all fixed during the optimisation. Those state variables are called such because they
determine the state of the system before changes are applied. Of course those variables are allowed to
be changed, but that means that the optimization process must be done again because what is optimal
in one environment does not have to be optimal in another. The state variables in the optimiser are
the material properties of the fibres, design dimensions such as plate width, applied loads, and the
boundary conditions. When the stiffeners are added, all those input variables will be added to the
state variables as well. They will not be changing during the optimisation but only be influencing the
end result.
1.4.4
Design variables
An optimisation problem exists of multiple types of design aspects. As mentioned above, the problem
contains design variables (y). The numerical values of those variables will be used to get to the optimal
solution. Those values are changed in order to find a better solution. In this optimisation problem,
the design variables are the fibre angles. By changing the fibre angles, the design will score higher
or lower on the objective function. The design variables define the design space which exist of all
possible combinations of the design variables. Depending on the type of constraints, the design space
is preferably inside the limitations set by the constraints. When strict limitations are set, the design
is not allowed to get behind those limitations, but the user can also apply a ranking penalty when a
constraint is not met.
1.4.5
Constraints
Constraints are representing the limitations of the design. When a potential solution does not fall
into the limited design space, two options can be chosen as a user of the optimization algorithm. The
first is that every design which falls outside the design space surrounded by the constraints, is invalid
so the ranking score for this design is equal to zero. The second option is to give this design such a
penalty that potential designs which do not fall behind this constraint, have an advantage regarding
to its position in the ranking sequence.
Constraints exists in two different forms. Equality constraints and inequality constraints. The
equality constraints requires a function or variable to be equal to another variable, function or constant.
Mostly written as (=). The inequality constraints requires a function or variable to be unequal to
another variable, function or constant. This can be in three ways, those are often represented by the
mathematical symbols of unequal (6=), bigger than (>) and smaller than (<). Of course combinations
can be made as well such as, equal to or bigger than (≥) and equal to or smaller than (≤). In some
cases, those formulae are written separately in an equality constraint and an inequality constraint.
The already discussed manufacturing constraint is a combination of both. Another constraint in this
thesis is the optimised panel design needs to be at least as stiff as a QI design. So:
Edesign ≥ EQI
5
(1.3)
2
The initial optimiser
This chapter will describe the initial state of the optimiser. Discussing the modules of the optimiser
will give a clear insight in the software. Some aspects must be known first in order to understand the
further development of the optimiser which is going to be discussed in chapter 3. To start with section
2.1, the purpose of the software and its history will be discussed. Section 2.2 will go into a little more
depth regarding the possibilities of the software and explains the inputs and expected outputs. After
that, the different steps will be shown in a schematic flow chart and each of the important modules
will be explained in the sub-sections 2.3.1-2.3.6.
2.1
About the software
It is difficult to retrieve the exact start date when the first parts of code were written, but since some
modules are dated from 2008 it can be assumed the first part are written at least 7 years ago. This
software package exists of multiple modules which will be discussed in Section 2.3. It is developed on
the TU Delft in order to create an optimisation tool for optimising VSL against buckling by changing
the angles of the fibres in the different layers.
Fig. 2.1: Schematic drawing of a balanced laminate layer with steered fibres (left), and
the fibre directions per node (right)
The software is capable of changing the angle of the fibres inside the layer on multiple locations.
In other words, the same layer can contain fibres with multiple directions throughout the length of
the composite. An example is given in figure 2.1a. As can be seen, the fibres are steered throughout
the laminate. In order to determine those angle changes, the laminate is divided in elements and
nodes. On every node the angle of the fibres can be different. In figure 2.1b, the different fibre angles
per node are visualised with arrows. In between nodes, the average angle change is taken in order to
meet the angle of the fibres which are required on the next node. The continuity of the fibres and the
maximum steering angle must be taken into account since the fibres can not turn an infinite angle in
an infinite small distance [10]. The multiple possibilities of the software will be explained in the next
sections.
2.2
Inputs and outputs
This section discusses the multiple options that the user has in order to get the wanted outputs. As
has been discussed in chapter 1, state variables and design variables need to be defined. Those belong
to the options of the user.
6
CHAPTER 2. THE INITIAL OPTIMISER
2.2.1
2.2. INPUTS AND OUTPUTS
Option inputs
At first, the user is deciding on what exactly needs to be optimised. Things like how accurate the
final answer must be, and which limitations the design has, are put in the software. The accuracy
of the calculations is related to the density of the mesh. Together with the symmetry properties of
the elements, a mesh is generated. In figure 2.2a, an example can be found for a dens mesh of 17
by 17 nodes. Figure 2.2b shows a wider mesh and with two symmetry axis for the elements. Every
node contains a vector of all the layer orientations on that location. Throughout the optimisation,
the angles of the different layers change but the nodes and elements stay on the same place. Only the
properties of those elements and nodes are changing during the optimisation.
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
1
-1
1
0.8
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
0.6
1
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
Fig. 2.2: Schematic drawing of a Mesh with 17x17 nodes (left) and 11x11 nodes plus
element symmetry in two directions (right)
The next choice for the user is about the number of layers the design will have. Each of these
layers get an initial angle which is constant along the entire layer at the start of the optimisation.
This means that all the nodes are having the same layer orientations at the start. After the first
optimisation iteration, this will already be changed. At every node, the fibre angles will be changing
towards an optimal design. The user specifies which layers are allowed to change in order to give the
optimiser its design space. Other things determined by the user are the design options like having a
balanced laminate, the lengths in x-axis and in y-axis, and the steering constraints. It has to be kept
in mind that the laminates designed with the optimiser will always be symmetric. The final things
determined by the user are the parameters for the fibre material and the thickness of each layer. The
standard properties of the material can be found in table 2.1. Those properties are used throughout
the entire thesis. All verification steps are done with those values.
Table 2.1: Material properties used throughout the entire Thesis
Material:
E1
E2
G12
ν12
t
AS4D/TC3500
1.54e11 [Pa]
1.08e10 [Pa]
4.02e9 [Pa]
0.317[-]
3e−4 [m]
7
2.3. OPTIMISER APPLICATION
2.2.2
CHAPTER 2. THE INITIAL OPTIMISER
Expected results
The expected output of the optimisation software is an optimised
laminate with the number of layers kept the same. The angles of
the fibres are optimised on every node and the angle differences
between the nodes do not exceed the manufacturing constraint.
The visual output for one layer has already been shown in figure
2.1. The output can be different for every layer. Since it is an
optimisation process which does not cover all possible solutions,
it is not assured that the best solution is found but an optimal solution. If more information is wanted about optimisation
processes and algorithms, the reader is referred to the literature
study[18].
The next section will discuss how to get from the inputs to
the outputs.
2.3
Optimiser application
This section will mainly be used to show and describe the working of the optimiser. A flow-chart is shown in order to make it
easier for the reader to understand the process that the software
is dealing with (see figure 2.3). As can be seen, multiple steps
of the optimiser are visualised in the flow chart. Starting with
the user inputs, and ending with the saved data containing the
results. It can be noticed there is a feedback loop going from
the stopping criteria to the property determination. This is a
part of the optimisation phase. For every improved design, the
optimiser needs to calculate the new properties, response and
sensitivities. Those new values are used for the new optimisation iteration.
2.3.1
Mesh generation
The mesh generation is done by dividing the square plate into
triangles. The defined number of nodes and symmetry properties are determining the layout of the mesh. All lengths of the
elements on each axis are equally divided. All those nodes are
stored in a coordinate matrix and an element connectivity matrix. This element connectivity matrix contains all data about
which nodes belong to which element. The elements must be
saved such that the orientation of the element does not change. Fig. 2.3: Software Flowchart
All elements are stored in a positive (counter clockwise) reference system. This is very important for determining the properties of a specific element. As can be
seen in figure 2.4, element 1 is the triangular element containing the three nodes shown in the figure.
The element is oriented in a counter clockwise direction, which means that the element is made of
’Node 1’-’Node 17’-’Node 2’ (in that exact order).
8
CHAPTER 2. THE INITIAL OPTIMISER
2.3. OPTIMISER APPLICATION
Fig. 2.4: Positive (Counter Clockwise) element definition
2.3.2
Model creation
After the generation of the mesh, the model is created. The model exists of a design, a load case and
the constraints. Since the design is a square/rectangle, the design has four sides to constrain. In figure
2.5 the standard model is shown. As can be seen, the loaded edge, the edge along the X-axis and the
one on the other side of the plate (parallel to the X axis at y = 0, y = L) are simply supported. In
other words, at those edges:
z=0
(2.1)
The side parallel to the Y axis is constraint in all spatial directions. So x = y = z = 0.
Fig. 2.5: Standard constraint on the model
The load will be exerted on the side which is parallel to the Y axis. A constant strain load will
be put on the entire edge using Multi Point Constraints (MPC). Those multi point constraints are
also used for the other edge constraints. A MPC uses one master node and multiple slave nodes. The
slave nodes are having the exact same displacement in x and y direction as the master node is having.
Since the z direction is already constraint, all spatial directions of the slaves are equal to those of the
master node. In this way, the load will be equally spread at all times during the optimisation.
9
2.3. OPTIMISER APPLICATION
2.3.3
CHAPTER 2. THE INITIAL OPTIMISER
Symmetry matrix
A symmetry matrix is a matrix which relates symmetric model parts. For a plate having symmetric
loadings, symmetric properties and symmetric boundary conditions, all the elements on both sides of
the symmetry axis can be related to each other. Using symmetry properties, only half of the plate
needs to be calculated. Knowing this, the computational effort will be reduced significantly. Only the
response for half of the plate is determined and its symmetric counterpart will have exactly the same
response. So for example, the plate of figure 2.5 could be treated as such. The symmetry line could be
drawn at Y = 0.5Ly (where Ly is the length of the plate in Y-direction). Then only the computations
should be done on one side of the plate. The symmetry matrix creates a relation between symmetric
parts of the model. Using half of the computational effort, the entire plate is analysed.
The symmetry matrix is created in order to lower the computational effort before it goes into
the finite element analysis. The symmetry matrix is created on a spatial point of view. The nodes
are compared and processed searching for symmetry. The optimiser assumes the plate to be entirely
symmetric with a symmetric load case. So the system assumes the output to be symmetric as well.
Although this might save computational effort, it has to be noted that the assumption is not entirely
correct! For example the laminate is symmetric regarding to the layer orientation (so the upper layers
and lower layers are symmetric), and the laminate is balanced. At first sight, one would think that
the load case is entirely symmetric so the response approximation would be as well. This is not true!
As can be seen in figure 2.6, the distance of every layer to the middle of the laminate is different. This
means that if a balanced symmetric laminate is made; for example [10, −10, 45, −45]S , the 10 laminate
has a larger distance to the laminate center which requires this layer to carry slightly more load. So
the plate will twist a little due to the bending phenomenon. So using the advantage of requiring less
computational effort, has its price in accuracy. In this case, the accuracy is only affected by a very
small amount.
Fig. 2.6: The distances between the center of the laminate and the ply borders[19]
2.3.4
Properties
The properties of the laminate are determined for every single node of the mesh. Every node can have
different values due to different fibre angles per node. The response approximations will be done on
the elements which will be discussed later. The A and D matrices are calculated for the membrane and
bending properties using classical laminate theories[19]. Transforming elastic constants to engineering
10
CHAPTER 2. THE INITIAL OPTIMISER
2.3. OPTIMISER APPLICATION
constants is part of the laminate theories:
Qxx =
Ex
1 − νxy · νyx
(2.2)
Qyy =
Ey
1 − νxy · νyx
(2.3)
Where ν is the Poisson’s ratio, the Qyy is the stiffness of fibres in the direction of the force, and Qxx
is the stiffness perpendicular to the force. The two other engineering constants are:
Qxy =
νxy Ey
νyx Ex
=
1 − νxy · νyx
1 − νxy · νyx
(2.4)
Qss = Gxy
(2.5)
These equations are all made for the case that the load is in the 1-direction. But for fibres, the force
can also be under an angle (see figure 2.7). To go from those engineering constants to the right angle,
Fig. 2.7: Placing the fibres under an angle with respect to the loading[19]
the next equations are used (1 is the direction of the applied force):
Q11 = m4 Qxx + n4 Qyy + 2m2 n2 Qxy + 4m2 n2 Qss
(2.6)
Where n = sin (θ) and m = cos (θ). The θ is the angle between the direction of the force and the
direction of the fibre. The other 5 equations can be found in the Appendix A.1. In order to simplify
the optimisation, the engineering constants are written as Material Invariants in a linking Matrix:
U1 =
(3Qxx + 3Qyy + 2Qxy + 4Qss )
8
U1 U2
U4
0

0
0

U =
U1 −U2

0
0
U5
0

0
0
U2
2
0
U2
2
0
h
U3
0
−U3
0 

0
U3 


U3
0 

0
−U3 
−U3
0

V = 1 cos (2θn,l ) sin (2θn,l ) cos (4θn,l ) sin (4θn,l )
11
(2.7)
(2.8)
iT
(2.9)
2.3. OPTIMISER APPLICATION
CHAPTER 2. THE INITIAL OPTIMISER
Where θn,l is the node number and layer number respectively. The remaining U terms can be found
in Appendix A.2. Calculating A and D matrices per node in the standard way:
A = 2 · [U ][V ]
(2.10)
2 +z ·z
2
zi+1
i
i+1 + zi
· [U ][V ]
(2.11)
3
In order to predict the property changes in A and D for a changing angle at a node, the derivatives
with respect to the angles are calculated as well:
D =2·
dA
d[V ]
= 2 · [U ]
dθ
dθ
(2.12)
z 2 + zi · zi+1 + zi2
dD
d[V ]
= 2 · i+1
· [U ]
dθ
3
dθ
(2.13)
d2 A
d2 [V ]
=
2
·
[U
]
dθ2
dθ2
(2.14)
2 +z ·z
2
zi+1
d2 D
d2 [V ]
i
i+1 + zi
=
2
·
·
[U
]
(2.15)
dθ2
3
dθ2
are the distances between the ply heights and the center of the laminate (see figure
Where zi and zi+1
2.6).
In case of constant fibre paths per layer, all those calculations were made for each layer and not
for each node. But since every node can now have different fibre angles, all those properties are
calculated for every single node. After the properties of each node are determined, it goes to the
response approximations module.
2.3.5
Response approximations
The FEA takes a lot of computational effort. In order to save computational effort, approximations
are made for the response. Those approximations are using less computational effort. With those
approximations, a well defined direction for the optimiser can be computed in a very short time. If
for every possible design the FEA needs to be done, it will take a lot more time. The general form of
an approximated response looks like[10]:
F (1) ≈
X
φm : A−1 + φb : D−1 + ψm : A + ψb : D + C
(2.16)
n
Where A and D are the in-plane and out-of-plane stiffness matrices (found in Appendix B), ’:’ is
called the Frobenius inner product (Appenddix C), the φ and ψ are linear approximation terms
calculated from sensitivities[20], and m and b denotes the membrane and bending part respectively.
This function runs over all the nodes n. In order to perform the FEA, the global membrane/bending
stiffness matrix must be known first. This matrix is determined using the A matrix, the D matrix,
the model coordinates, and the model element data. The slave and master nodes along with the MPC
play also a big role again (discussed in section 2.3.2). The force is divided along the edge of the plate.
The MPC requires the entire edge to have the same spatial displacements.
The entire calculation of the FEM can be found in Appendix D.
12
CHAPTER 2. THE INITIAL OPTIMISER
2.3.6
2.4. DEVELOPING THE OPTIMISER
Sensitivities
The sensitivity values are determined per node. Those values are used to indicate how large the effect
will be on the properties of the laminate if the specific node undergoes some changes[20]. In this way
the optimiser will be more efficient. The sensitivity of a node can be determined by calculating the
second level approximation. The general form of the second level approximation is[10]:
(1)
F (2) ≈ F0
+ g · ∆θ + ∆θT · H · ∆θ
(2.17)
Where g and H are the gradient and Hessian respectively. The gradient can be defined as:
∂F
∂θi
∂F ∂A ∂F ∂D
=
+
·
·
∂A ∂θi ∂D ∂θi
∂A
∂D
= φm ·
+ φb
∂θi
∂θi
(2.18)
∂2F
∂θi ∂θj
∂
=
·g
∂θ
∂2A
∂2D
∂φm ∂A ∂φb ∂D
= φm ·
+ φb
+
·
+
∂θi ∂θj
∂θi ∂θj
∂θj ∂θi
∂θj ∂θi
(2.19)
gi =
The Hessian is calculated as followed:
H=
In order to be sure the Hessian is convex, the blue part of the equation is left out. Since the equation
is already an approximation, and the function and gradient are equal at the approximation point, it
is allowed to omit the blue part of the Hessian equation. This will give:
H=
2.3.7
∂φm ∂A ∂φb ∂D
·
+
∂θj ∂θi
∂θj ∂θi
(2.20)
Stopping criteria
The optimiser is trying to optimise until one of the stopping criteria is met. Two stopping criteria are
active for the optimiser. The first stopping criterion is: after at least 5 iterations, the new solution is
not improved more than 0.1% (with respect to the critical load). The second stopping criterion is: a
maximum of 100 iterations is reached. In both cases, the optimisation process will be ended. In every
iteration, the process starts to determine the new properties and sensitivities again. It continues until
at least one of the stopping criteria is met.
2.4
Developing the optimiser
The optimiser is capable of optimising a VSL by changing the fibre paths in the laminate. Further developing the optimiser will make the software capable of optimising a VSL under influence of stiffeners.
This chapter has shown the initial state of the optimiser. The next chapter will go into further detail
of implementing one or multiple stiffeners to the optimiser. The knowledge gained in this chapter is
used to further develop the optimiser with those additional features.
13
3
Stiffened panel
As has been mentioned earlier, aircraft structures contain many stiffeners. Optimising composite
structures with taking stiffeners into account as well, will result in even better optimised structures.
The optimiser will be extended with the capability of adding a stiffer to the design, and take this
into account during optimisation. This chapter discusses this new capability. Section one discusses
how the implementation of a single stiffener on a plate in the software works. After having shown
this implementation of a single stiffener to the optimiser, the second section will show the addition of
multiple stiffeners to the optimiser. The third section discusses one step further; the implementation
of the integrated stiffener functionality.
3.1
Single stiffener
The first step is to design the optimiser for adding a single stiffener. Three topics regarding the
implementation of a stiffener to the optimiser are discussed. The first topic is about the stiffener
options. The second topic is how to add the stiffener to the model. The third topic is about how it is
implemented in the optimisation.
3.1.1
Stiffener options
The stiffener options are defined by the user. Starting with the number of stiffeners which need to be
added. If this number is larger than zero, the remaining options will be defined. If the option of zero
stiffeners will be chosen, the optimiser will work exactly the same as it would have when the optimiser
was untouched. Which results in an optimised plate design without stiffeners.
Many of the defined options of the unstiffened plate, are needed to be defined for the stiffener as
well. The stiffener can be balanced, but is always symmetric. The spatial options are the dimensions
of the stiffener, and the location of the stiffener. The mesh density in the z-direction is free to change,
but the mesh density in the x-direction, is fixed to the mesh density of the plate. The stiffener will
always be placed on a row of nodes of the plate (see figure 3.1). Invalid locations will be round off to
the nearest row of nodes. The remaining options are the symmetry properties of the elements which
are similar to the options of the plate. The dimensions are taken such that the length of the stiffener is
equal to the width of the plate. This prevents having loose hanging parts of the stiffener. The height
Fig. 3.1: A schematic drawing of a plate with stiffener model
is taken to be 10% of the stiffener length (this can be changed by the user). So in the case of figure
3.1, the stiffener length is 0.5[m] in x-direction, zero in y-direction and 0.05[m] in z-direction.
The material properties, ply thickness and volume fraction of the stiffener are defined to be equal to
the plate. They are chosen by default.
14
CHAPTER 3. STIFFENED PANEL
3.1.2
3.1. SINGLE STIFFENER
Addition to the model
As has been discussed in section 2.3.2, a model of the plate is created. The stiffener will be added
to this model. The plate is defined to be the main part, and the stiffener is going to be a sub-part.
Together they are the model. This is similar to an object oriented environment where a class is created
for the separate parts[21]. The plate is the main object and the stiffener is its sub-object. While every
individual part is an instance of the class ’Model’, the stiffener could not ’survive’ without its main
part: the plate. A visualisation of how the Main-Sub-model works can be found in figure 3.2. It shows
that the stiffeners are a sub-part of the plate. They are all together the entire model.
The reason for why this set up has been chosen is straight forward. The stiffener is in fact the
same as an unstiffened plate. The optimiser treats stiffeners as separate plates now. Defining the
properties and sensitivities for the stiffener goes similar to the ones defined for the plate. Reusing
existing functions in the software saves the optimiser computational effort during optimisation and
storage space since a plate concept is already known to the optimiser. The only differences between
a stiffener and a plate are the boundary conditions on the edges. The small edges of the stiffener are
having the same boundary conditions as the plate: simply supported. The stiffener edge touching the
plate is clamped to the plate. The upper edge is free to move.
Fig. 3.2: The model definition with a Main part and Sub parts
The mesh of the stiffener is also build up of elements. Similar to the plate elements, the properties
for the stiffener elements are calculated as well. In order to do FEA on the model, the elements of
the plate and the elements of the stiffener must be treated as one model. The entire model with its
elements must be merged to one part. The nodes of the stiffener must integrate with the nodes of
the plate at the same location. Those merged nodes are called "connection nodes". A second model
is created with all the elements of the stiffener and plate. Those elements are then connected via the
connection nodes. This new model is called the "Fused model". In the fused model, all duplicates
of nodes with the same coordinates are taken out. This model can now be used for the FEA using
the properties gained from the initial model with main and sub parts. So the difference between the
initial model and the fused model is that in the fused model it is merged to one piece while in the
initial model there is still a difference between a plate and a stiffener.
As has been discussed for the plate as well, the element definitions are very important. The plate
elements discussed in section 2.3.1 are oriented counter clockwise. The stiffener elements need be
15
3.2. TWO OR MULTIPLE STIFFENERS
CHAPTER 3. STIFFENED PANEL
defined in the same way. In figure 3.3 it is shown that three nodes of an element are numbered. Since
the Y -axis is going left, the X-axis is going right, and the Z-axis is going up, the counter clockwise
orientation of the nodes are in the sequence of "226-242-227".
Fig. 3.3: Element definition, ’226 - 242 - 227’
3.1.3
Optimisation
The default case of optimising a panel with a stiffener, is adding a stiffener with a predefined stacking
sequence. This stacking sequence remains unchanged during the entire optimisation. Depending on
its location, size and layers, the stiffener influences the buckling load significantly. The stiffness of the
stiffener is directly put in the optimisation as being a support for the plate. For the buckling modes,
the fused model (with the stiffener included) is analysed for its eigenvalues. The fused model can be
analysed for buckling of the plate and for crippling of the stiffener. Since the plate and stiffener are
merged to one model, for the optimiser there is no difference between both.
3.2
Two or multiple stiffeners
In the case of two or multiple stiffeners, the model can be simply extended with extra stiffeners. The
working of this process is similar to the single stiffener. All the stiffener options must be given for
every additional stiffener. Those can be different from each other. As can be seen in figure 3.4, a plate
with two stiffeners is shown. This is one of the ways, the stiffeners could be placed.
Fig. 3.4: A basic model with multiple stiffeners
16
CHAPTER 3. STIFFENED PANEL
3.3
3.3. STIFFENER INTEGRATION
Stiffener integration
The next step for extending the optimiser is developing the stiffener integration capability. As has
been discussed in chapter 1, those are stiffeners with shared layers from the plate to the stiffener.
Figure 3.5 shows an example of a integrated stiffener. The two brown top layers of the plate, go
into the stiffener. The blue stiffener layers are having a predefined stacking sequence. First the fibre
orientations of the shared layers will be discussed. Then how this is implemented in the optimiser will
be explained.
Fig. 3.5: A visualisation of an integrated stiffener
3.3.1
Fibre orientations
Sharing layers with the plate is partially restricting the stacking sequence of the stiffener. The stiffener
is restricted to have the same stacking sequence as the plate has, at the connection nodes. Only the
shared layers are restricted, the blue layers from figure 3.5 are not shared so those are not restricted.
Those are predefined by the user. Figure 3.6 shows a composite layer of a plate with two stiffeners
which are integrated. As can be seen, the fibre orientations are visualised by arrows. The blue arrows
represent the fibre orientations of the plate. They have each its own orientation along the plate. The
red arrows are visualising the fibre orientations of the stiffener. As can be seen, the red arrows above
each other are having the same orientation as the blue arrow below them. The plate layer with a
certain orientation, goes up and those fibre orientations remain the same as the last fibre orientation
of the plate at the connection node.
Fig. 3.6: A schematic drawing of two integrated stiffeners showing one layer with the
arrows as fibre directions
17
3.3. STIFFENER INTEGRATION
3.3.2
CHAPTER 3. STIFFENED PANEL
Optimisation
The optimiser will be slightly adapted due to this extra functionality. The entire stacking sequence of
the stiffener is no longer constant through the optimisation. The fibre orientations of the shared layers
in the stiffener change when the underlying nodes of the plate are changing (figure 3.6). This means
that the influence of those connection nodes on the model response is increased. When one of the fibre
orientations at the connected nodes is changed, the entire column of all stiffener nodes above it are
changing as well. Therefore the sensitivities of those connection nodes will be significantly increased.
This is done by adding the gradient of the above laying nodes to the gradient of the connection node.
This is the same gradient as has been discussed in section 2.3.6.
gi = φ m ·
n
∂A
∂D X
∂D
∂A
+ φb
+
+ φb
+φm ·
∂θi
∂θi j=1
∂θj
∂θj
Where ’i’ is the plate node and ’j=1,2,...,n’ are the nodes above the plate node.
18
(3.1)
4
Verification
This chapter shows the verification process of the developed optimiser with its added features. Four
main cases are verified. Some of those cases have two different configurations:
1. Unstiffened plate
2. Stiffened plate with one stiffener (at two locations)
3. Stiffened plate with two stiffeners (symmetrically placed and assymmetrically placed)
4. Crippling of a single stiffened plate
The first three cases will be verified by comparing the displacement and section forces from the
optimiser model with the ABAQUS model using the exact same mesh and same forces. The fourth
case is verified by showing that the optimiser is taking crippling into account when it is expected to
occur. It will be shown that for a specific case, crippling occurs at a lower load than plate buckling.
The four cases with the six models can be found in figure 4.1.
The determination of the Displacement is done during the FEA, and the section forces can be
calculated using:




Nxx
h i εxx




N = Nyy  = A εyy 
(4.1)
Nxy
εxy
Where ε is the strain and A is the A-matrix discussed earlier. Normally the stresses are used in order
to verify such models. In this case the section forces are used since the stresses are changing from
layer to layer in a laminate. The section forces are taken from all the layers together.
A mean error of the models is calculated with the mean error of the vector lengths:
n
X
1
|NA |−|NM |
= ·
n i=1,..,n
|NA |
δmean
q
n
X
1

= ·
n i=1,..,n
NA · NTA −
q
i
q
NM · NTM
NA · NTA

(4.2)

i
Where n is the number of elements, A means ABAQUS model, and M means MATLAB model.
The input values are similar for each case and are discussed first. In the second section, plots will be
shown in order to make the different cases comparable. All the results of the computed displacements
and sectional forces can be found in Appendix E. Starting with the unstiffened plate, its sectional
forces Nxx and Nyy will be discussed. After that, the absolute error and the relative error of the same
two sectional forces are discussed. Since the absolute error and relative error of the sectional forces
are of most interest, only those will be discussed per design case. For more plots, the reader will be
referred to the appendix where the remaining sectional forces and the displacements of each model
can be found. In the end of this chapter a conclusion will be drawn about the verification.
19
CHAPTER 4. VERIFICATION
Fig. 4.1: The 4 verification cases with 6 models in total
20
CHAPTER 4. VERIFICATION
4.1
4.1. INPUT VALUES
Input values
The mesh for each model is similar. The plate has a mesh of 21x21 nodes and the stiffener(s) have
a mesh of 3x21 nodes. The only exception is the model for verifying whether the optimiser is taking
crippling into account. This stiffener mesh is 7x21 nodes since this stiffener has another height than
the stiffeners in the other cases. The mesh is equally spread in every model. The dimensions of the
plates are 0.5 [m] by 0.5 [m]. The Stiffeners are all 0.05 [m] except for the stiffener of the crippling
verification which is 0.15[m]. The material properties can be found in table 4.1. Every case in this
entire thesis uses a constant thickness for plies.
Table 4.1: The used material properties
E1
1.54e11[Pa]
E2
1.08e10[Pa]
G12
4.02e9[Pa]
ν12
0.317[-]
tply
3e-4[m]
The layups of the plate and stiffener laminates can be found in table 4.2. Those lay-ups are chosen
semi-randomly, it is preferred to have at least three layer orientations at least at 15 degrees difference.
So although the choice of fibre orientations is random, they were not allowed to all point in the same
direction. It has to be kept in mind that in this verification process, no fibre angle optimisations
are done. Those constant ply orientations are predefined and non-changing throughout the entire
verification. The fibre orientations are all with respect to the X-axis.
Table 4.2: Ply orientations of the plate and stiffener(s)
Plate Layup
[10,-10, 20,-20, 80,-80, 60,-60, 30,-30, 70,-70]S
4.2
Stiffener Layup
[45,-45, 30,-30, 60,-60, 0, 0]S
Model constraints
Every model has simply supported edges (as has been discussed in chapter 2). A force of 1[N] is
exerted along the edge at x = L. A MPC is added for this edge in order to keep the constant strain
along the edge. The short edge of the stiffener is also included in the MPC. The force will also be
applied on the stiffener.
4.3
Model verification
The six models discussed in the beginning of this chapter are compared in this section. The verification
is done by comparing the sectional forces and the displacements during the load case. Those data
are visualised in plots. The section force plots and the displacements plots can be found in Appendix
E. The error between the optimiser model and the Abaqus model are of more interest, so those plots
are shown and discussed in this chapter as well. The displacement plots will not be discussed in this
chapter. By studying the response and section forces of the six different cases, a conclusion about the
additional stiffener concept can be drawn.
21
4.3. MODEL VERIFICATION
4.3.1
CHAPTER 4. VERIFICATION
Unstiffened plate
The first model to be verified is the initial unstiffened plate. Starting with the section forces. Figure
4.2 shows the Nxx of the optimiser and the ABAQUS calculations. Figure 4.3 shows the Nyy of the
optimiser and the ABAQUS calculations. Only for the plate, the section forces are shown. Since the
errors between both models is of more interest, showing the sectional forces has only be done for the
unstiffened plate. For the other cases, only the errors are shown in this chapter.
Fig. 4.2: Nxx of the Optimiser (Left) and Nxx of ABAQUS (Right)
Fig. 4.3: Nyy of the Optimiser (Left) and Nyy of ABAQUS (Right)
The error between the optimiser and the Abaqus model is shown in figure 4.4, the relative error
is shown in figure 4.5. As can be notices, the relative errors for the Nxx are very small. At most
of the area, it is even within the 0.1% error. At some outliers, the error reaches 8% max. For the
Nyy , the relative errors are significantly higher, although the absolute errors are not immense high.
Especially the corners where probably force concentrations occur are having deviating values. Most
of the errors are still within the 10% range. For the Nxy (which is shown in Appendix E.1) the errors
are even worse. But this is also because of the relative forces. The forces calculated for Nxy are so
small, that an error of 0.01 can result in a relative error of 1500%. A good example can be a vector
of [0.01, 0.01, 1] with an error of [0.01, 0.01, 0.01]. In this example the error in the first two directions
22
CHAPTER 4. VERIFICATION
4.3. MODEL VERIFICATION
are 100% while the error in the third direction is only 1%. The total error is very small because of
the large third component.
Since the direction of interest is the direction of the force (x-direction), the errors for Nxx can
mainly be used to draw a conclusion. The mean error (equation 4.2) compensates for this occurrence
by dividing the absolute error by the length of every [Nxx ,Nyy ,Nxy ]-vector. The so called mean error
is: δ = 0.08%. This is enormously low because of the contribution of the very low errors of the Nxx .
So this unstiffened plate is verified.
Fig. 4.4: Absolute error between both models, unstiffened plate Nxx (Left) Nyy (Right)
Fig. 4.5: Relative error between both models, unstiffened plate Nxx (Left) Nyy (Right)
23
4.3. MODEL VERIFICATION
4.3.2
CHAPTER 4. VERIFICATION
Single stiffened plate (center located)
The second model is the single stiffened plate with a stiffener placed at the center of the plate. The
error plots can be found in figure 4.6 for the absolute error and figure 4.7 for the relative error. The
stiffener is placed at y = 0.25[m]. The lay-up is equal to the other stiffeners and is discussed in the
beginning of this chapter. The same as for the unstiffened plate, the errors of the Nxx are relatively
low. It can be noticed that they are higher than for the unstiffened plate. The relative errors for Nyy
are also significantly higher. As has been discussed, this is the result of absolute errors in a region
where very small forces are determined. According to the mean error δ = 1.64%, and the responses
(which can be found in Appendix E.2) the single center stiffened plate with its stiffener in the centre
is verified.
Fig. 4.6: The absolute error between both models, single stiffened plate center Nxx (Left)
Nyy (Right)
Fig. 4.7: The relative error between both models, single stiffened plate center Nxx (Left)
Nyy (Right)
24
CHAPTER 4. VERIFICATION
4.3.3
4.3. MODEL VERIFICATION
Single stiffened plate (off-center located)
The off-center located stiffener is placed at 0.75 of the plate length. So at y = 0.375[m]. The error plots
can be found in figure 4.8 for the absolute error en figure 4.9 for the relative error. The calculated
mean error δ = 1.63%, is remarkably similar to the error calculated for the single center stiffened
plate. The errors for Nxx and Nyy are also similar. The same conclusion for this case can be drawn
as for the single center stiffened plate. The single off-center stiffened is verified as well according to
the responses (Appendix E.3) and the mean calculated error.
Fig. 4.8: The absolute error between both models, single stiffened plate off-center Nxx
(Left) Nyy (Right)
Fig. 4.9: The relative error between both models, single stiffened plate off-center Nxx
(Left) Nyy (Right)
25
4.3. MODEL VERIFICATION
4.3.4
CHAPTER 4. VERIFICATION
Double stiffened plate (Symmetrically located)
The double stiffened plate has stiffeners located at 0.25 and 0.75 of the plate length. So at y = 0.125[m]
and y = 0.375[m] respectively. The error plots can be found in figure 4.10 for the absolute error en
figure 4.11 for the relative error. As can be seen, especially on the edges, the error is relatively higher
than towards the center of the panel. Again the Nxx has relatively low errors, but also the relative
errors are bigger than for the single stiffened plates. The mean error is calculated to be: δ = 2.62%.
The errors are still within an acceptable range. The double symmetrically stiffened plate is verified
according to the mean calculated error and its responses which can be found in Appendix E.4.
Fig. 4.10: The absolute error between both models, double symmetric stiffened plate Nxx
(Left) Nyy (Right)
Fig. 4.11: The relative error between both models, double symmetric stiffened plate Nxx
(Left) Nyy (Right)
26
CHAPTER 4. VERIFICATION
4.3.5
4.3. MODEL VERIFICATION
Double stiffened plate (Asymmetrically located)
The asymmetric double stiffened plate has stiffeners located at 0.5 and 0.75 of the plate length. So at
y = 0.25[m] and y = 0.375[m] respectively. The error plots can be found in figure 4.12 for the absolute
error en figure 4.13 for the relative error. The mean error is calculated to be: δ = 2.59%. The same
as has been concluded for the case of a symmetrically double stiffened plate, this case is verified as
well according to the responses (Appendix E.5) and the calculated mean error.
Fig. 4.12: The absolute error between both models, double asymmetric stiffened plate
Nxx (Left) Nyy (Right)
Fig. 4.13: The relative error between both models, double asymmetric stiffened plate
Nxx (Left) Nyy (Right)
27
4.4. CONCLUSION
4.3.6
CHAPTER 4. VERIFICATION
Crippling
The crippling verification is only meant to verify the fact that the optimiser software should account
for crippling. The method chosen is creating two models with both almost identical dimensions. Both
models are created in the optimiser. No ABAQUS model is required to verify the crippling capabilities
of the optimiser. Both models have a plate of 0.5[m] by 0.5[m] and a stiffener placed in the centre of
the plate at y = 0.25[m]. The height of the stiffener is the only difference between the two models. The
first model has a stiffener height of 0.05[m] and the second model has a stiffener height of 0.15[m]. The
stiffener from the second case, has an increased height to intentionally let the stiffener cripple earlier
than the plate will buckle. So in order to verify the crippling functionality of the optimiser, the second
model should buckle later because of the increased stiffness of the plate plus stiffener. But it should
cripple even earlier than it should buckle. Using eigenvalue analysis on the stiffness matrices, the first
has a critical load of Pcr = 5.6e + 05 and the second case has a critical load of Pcr = 2.79e + 05. From
those two values it can already be concluded that the crippling is occurring since a stiffer stiffener is
added to the second case but the critical load has decreased. After having the displacements plotted
in figure 4.14, it can be concluded that those expectations were right. The displacements are shown
larger than they really are, in order to visualise the crippling of the model in the figure.
Fig. 4.14: Buckling of model one at Pcr = 5.6e05[N ] with a stiffener height of 0.05[m] (left)
Crippling of the stiffener from model two at Pcr = 2.79e05[N ] with a stiffener height of
0.15[m] (Right)
4.4
Conclusion
This chapter shows that each of the cases is verified. It can be concluded that those models have
all some outliers in the different section forces but the mean calculated errors are within a range of
3%. The relatively high errors for Nyy and Nxy can be assumed normal because those sectional force
values are really small as well. This makes a small absolute error, relatively big while this is not the
case. The mean errors are compensating for this effect. Together with analysing the relative values
of the biggest sectional force component (which is in those cases Nxx ), and the responses (which can
be found in Appendix E) a valid conclusion can be drawn. All the errors of the main sectional force
(Nxx ) stay within the 10% to -10% range while most of the errors are even within the 1% to -1%
range.
For crippling it is shown that the optimiser takes this into account. With the stiffener of an
0.05[m] height it clearly shows that the plate is buckling. With the case of a stiffer height of 0.15[m],
the stiffener is crippling before the entire model buckles. Its critical load is lower in the second case
since the stiffener starts to cripple earlier. So therefore, the optimiser is verified for adding one or
multiple stiffeners to the plate.
28
5
Optimisation results
After verification of the renewed optimiser, the optimiser is ready to obtain results. This chapter will
discuss the results of the optimiser. Five main cases will be studied and discussed. Those cases are
listed:
1. Unstiffened panel
2. Single stiffened panel (centered and off-centered)
3. Double stiffened panel (symmetric and asymmatric)
4. Double stiffened panel with changing stiffener layers
5. Integrated stiffeners (square and rectangular plate)
The unstiffened case is used to show the influence of stiffeners by comparing the initial unstiffened case
with the cases with stiffeners. The case with double stiffeners has multiple variations. The number of
plies varies between the 0 and 16 plies in order to show the influence of adding and removing stiffener
plies. Two cases for the integrated stiffeners will be studied. Integrated stiffeners mean that plies from
the plate go into the stiffener as has been discussed in chapter 1.3. This will all be discussed in this
chapter. The results can be shown in two ways (see figure 5.1); 1. the fibre directions at the nodes
are shown with an arrow 2. the fibre paths through the laminate layer. The second way of showing
the results is chosen because those are easier to imagine for the reader. It has to be kept in mind that
all the results are showing two layers which are balanced as can be seen in the figure. All the designs
are optimised for the first two buckling modes and being as stiff as the Quasi Isotropic configuration.
In this case Quasi Isotropic is determined by putting all lamination parameters equal to zero.
Fig. 5.1: Fibre direction per node (Left) fibre paths of a laminate layer (Right)
5.1
Initial values
The same model constraints are valid as for the verification phase. The mesh is different from the
verification phase and will be noted for every case. The local steering constraint is 333[mm], which
means that the radius of curvature of the fibres needs to be at least 333 [mm]. The initial guess for
ply orientations can be found in table 5.1. The initial values such as dimensions and stiffener locations
29
5.2. RESULTS
CHAPTER 5. OPTIMISATION RESULTS
will be specified for each case in the next section. All the stiffeners are the same size: 0.5[m] by
0.05[m]. The mesh density does change from case to case and will be mentioned per case as well. For
the stiffener(s) there are always 3 nodes in the direction perpendicular to the plate and the number
of nodes in the length direction are always equal to the number of plate nodes in the same direction.
Table 5.1: Initial ply orientations of the plate and stiffener(s)
Plate Layup
[30, −30, 30,-30, 30,-30, 30,-30, 30,-30, 30,-30]S
5.2
Stiffener Layup
[45,-45, 30,-30, 60,-60, 0, 0]S
Results
This section shows the result of optimised plates coming out of the optimiser. Since every laminated
plate in this chapter is symmetric and balanced, only six pictures with layups are shown. Each
laminate contains 24 plies. Every picture is showing a balanced layer (two plies at once), and the
symmetric counterpart of the laminate is now shown since those layers are exactly the same. Only
the top 6 balanced layers are shown and in the order of the top layer first (left up), then going to
the right for the second layer and ending with the most inside balanced layer on the right bottom of
each figure. Next to the layups, a small 3D visualisation of the model is shown in order to show the
locations of the stiffeners. For every case, the buckling load will be given in order to draw conclusions
on the increase or decrease of performance of the specified case.
5.2.1
Unstiffened plate
The unstiffened plate contains 19x19 equally spread nodes on an area of 0.5[m] by 0.5[m]. The results
can be found in figure 5.2. As can be seen in the figure, the layers do look similar but are not identical.
It can be noticed that every layer has a large area of the layer which goes towards [45,-45].It is known
from buckling analysis that [45,-45] layers are wanted in a design optimised against buckling[22].
Fig. 5.2: Optimised fibre paths of an unstiffened plate. From the top layer (Left up) to
the laminate center(Right below)
Since the outer plies have more influence on the buckling mode, the outer layers are more optimised
30
CHAPTER 5. OPTIMISATION RESULTS
5.2. RESULTS
towards a [45,-45] laminate than the more inner layers. As can be seen, the inner layers have areas
going towards [45,-45] as well but have more fibre directions going towards [0,0] since this is optimal for
creating stiffness which is an optimising requirement. As has been discussed, the plate must be at least
as stiff as the quasi isotropic plate. The buckling load of the quasi isotropic plate is: Pcr = 1.51e05[N ].
The buckling load of this optimise plate is: Pcr = 2.36e05[N ]. As can be noticed, the critical load has
increased by 56.3% with respect to the QI case. This unstiffened plate will be taken as standard result
in order to compare the other results with. The influence of stiffeners on this optimisation process
will be discussed with respect to this standard case.
5.2.2
Single stiffened plate (center)
The single stiffened plate with a stiffener located at the center of the plate is optimised for a mesh
with 17x17 nodes on an area of 0.5[m] by 0.5[m]. The results can be found in figure 5.3. The red line
in the figure shows the location of the stiffener. As can be seen, the first two plies are even closer to
a [45,-45] orientation compared to the unstiffened plate (counting the plies from top of the laminate
to the center of the laminate). The fourth, fifth and sixth ply are clearly showing the influence of the
stiffener. Towards the location of the stiffener, the fibres are going towards 0 degrees (parallel to the
stiffener) in order to introduce the direct loads. The areas between the stiffener and the outside edges
are treated more or less as separate plates with each area optimised against buckling. The plies closer
to the laminate center contain more fibre orientations closer towards [0,0]. Stiffness is mainly created
in the center of the laminate and the [45,-45] degrees fibres are more wanted on the outside of the
laminate. This is because the outside is more sensitive for buckling than the center of the laminate.
Fig. 5.3: Optimised Fibre paths of a center stiffened plate. From the top layer (Left up)
to the laminate center (Right below). The red line shows the stiffener location
The single center stiffened plate has a critical load of Pcr = 7.9773e05[N ]. It would be useless to
compare an unstiffened plate with a stiffened plate. Adding the exact same stiffener to the optimised
unstiffened plate would give a good insight in the optimised performance. This critical load is: Pcr =
6.5413e05[N ]. This is an increase of 22.0%. In fact that is the difference between adding the stiffener
to the optimisation procedure or only added after the optimisation has been done. The QI model has
a Pcr = 6.3244e05[N ], which means an increase of 26.1%.
31
5.2. RESULTS
5.2.3
CHAPTER 5. OPTIMISATION RESULTS
Single stiffened plate (off-center)
The single stiffened plate with a stiffener on a quarter length of the plate is optimised for a mesh of
17x17 nodes on an area of 0.5[m] by 0.5[m]. The results can be found in figure 5.4. Those plies show
clearly the influence of the stiffener regarding the separation of two areas. The stiffener is preventing
the plate from buckling in the area where the stiffener is placed. This automatically creates two
areas with non-identical properties. The larger area is still critical for buckling modes. This can be
concluded from the results as well, the larger area consists of more [45,-45] area than the smaller area
above the stiffener which contains more fibre orientations going towards [0,0]. Comparing the fourth,
fifth and sixth layer (the three layers closest to the laminate center) with the same layers of the center
stiffened plate, the larger area of the off-center stiffened plate shows more [45,-45]. This probably
results from the fact that the buckling mode of the smaller area above the stiffener is far from critical
regarding the buckling modes which makes it more optimal to design this area to create stiffness. This
can be concluded from the plots as well.
Fig. 5.4: Optimised fibre paths of an off-center stiffened plate. From the top layer (left
up) to the laminate center laminate (right below). The red line shows the stiffener
location
The critical load for this single off-center stiffened plate is: Pcr = 7.5919e05[N ]. Adding the same
stiffener at the unstiffened plate gives: Pcr = 4.8112e05[N ]. So, adding the stiffener to the optimising
procedure gives an increase of 57.8% in the critical load. Compared to the critical load of the QI case
having a Pcr = 3.7320e05[N ], which is an increase of 103%.
32
CHAPTER 5. OPTIMISATION RESULTS
5.2.4
5.2. RESULTS
Double stiffened plate (symmetric)
The double symmetric stiffened plate is optimised for a mesh of 17x17 nodes on an area of 0.5[m] by
0.5[m]. The results can be found in figure 5.5. It can be seen that also in this double stiffened case, the
outer layer (layer one) is dominated by fibre going towards [45,-45] orientations. The remaining layers
contain fibres which are going towards a [45,-45] orientation in the middle area, and going towards
a [0,0] orientation on the outer two areas. As could already been seen in the figure of the single
off-centred stiffened plate, the small areas between the edges and the stiffeners contain many fibres
with an orientation going towards a [0,0] orientation because those areas are less sensitive for buckling
modes. The results of this double symmetric stiffened plate will be used to compare to a larger panel
discussed in section 5.2.6.
Fig. 5.5: Optimised fibre paths of a symmetric double stiffened plate. From the top layer
(Left up) to the laminate center (Right below). The red line shows the stiffener location
The critical load of this double stiffened plate is: Pcr = 1.7666e06[N ]. Adding the stiffeners after
the optimisation gives an critical load of: Pcr = 1.5804e06[N ]. That is an increase of 11.8%. The
critical load for the QI case is: Pcr = 8.6987e05[N ]. This is an increase of 103%.
5.2.5
Double stiffened plate (asymmetric)
The double asymmetric stiffened plate is optimised for a mesh of 17x17 nodes on an area of 0.5[m] by
0.5[m]. The results can be found in figure 5.6.
As could be expected, the result of this double asymmetric stiffened plate is a combination of the
single stiffened plates. The first layer (counting from top layer towards the laminate center) contain
areas mainly orientated as [45,-45] and the second to the sixth layer the fibre orientations are more
going to [0,0] orientations in the two upper areas in order to create stiffness. The smaller areas are
again less vulnerable for buckling. The bigger area is mainly optimised against buckling. The critical
load for this double asymmetric stiffened panel is: Pcr = 1.6765e06[N ]. Adding those stiffeners to the
unstiffened plate gives: Pcr = 7.6551e05[N ]. This is an increase of 119%. The QI case has a critical
load of Pcr = 7.8407e05[N ], which is an increase of 114%.
33
5.2. RESULTS
CHAPTER 5. OPTIMISATION RESULTS
Fig. 5.6: Optimised fibre paths of an asymmetric double stiffened plate. From the top
layer (left up) to the laminate center (right below). The red line shows the stiffener
location
5.2.6
Rectangular plates
In order to study the optimising behaviour of the fibres under influence of stiffeners, the area inbetween two stiffeners is studied more close as well. The area between two stiffeners is often assumed
to be a plate which is simply supported. In this section, a rectangular plate with two stiffeners is
discussed. The area in-between the two stiffeners contains the same dimensions as the unstiffened
square plate at the beginning of this chapter. In this way, the influence of the stiffener can be studied
and the assumption of having a simply supported plate in-between the two stiffeners is discussed. A
rectangular plate with 17x33 nodes and an area of 0.5[m] by 1[m] is optimised for five different cases.
The only difference between the five cases is the stiffener design, having zero, four, eight, twelve, or
sixteen plies. In this way, the influence of the stiffener can be studied in more detail by changing its
stiffness. The stiffeners are located at 0.25 and 0.75 of the panel length. Starting with the unstiffened
(zero plies) rectangular plate (see figure 5.7).
Fig. 5.7: Unstiffened panel (left) Stiffened panel (right) both 0.5[m] by 1[m]
34
CHAPTER 5. OPTIMISATION RESULTS
5.2. RESULTS
Unstiffened
This unstiffened rectangular plate has fibres in slightly other directions than expected (see figure 5.8).
The upper layer exists of an inner area with [0,0] and contains more [45,-45] areas on the sides. From
this it can be concluded that due to the rectangular shape, the buckling phenomenon is less critical.
In order to satisfy the constraint of having a stiff panel which is at least being as stiff as the QI panel,
in all six layers the areas with [0,0] oriented fibres can be found. Going towards the center of the
laminate, more [0,0] directions go to the upper and lower edge of the panel, and the directions going
towards [45,-45] are going towards the center where buckling is more critical. The critical load of
Fig. 5.8: Optimised fibre paths of an unstiffened rectangular plate of 0.5[m] by 1[m].
From the top layer (Left) to the laminate center (Right)
the unstiffened rectangular plate is: Pcr = 2.3229e05[N ]. The QI rectangular plate with the same
dimensions has a critical load of: Pcr = 1.1883e05[N ]. That is an increase of 95%.
Double stiffened (four plies)
This double stiffened rectangular plate has two stiffeners with each four plies of [45,-45,-45,45]. Comparing those six layers with the case of no stiffeners, the design contains less areas with [0,0] oriented
fibres (see figure 5.9).
Fig. 5.9: Optimised fibre paths of a double stiffened rectangular plate with 4 plies per
stiffener of 0.5[m] by 1[m]. From the top layer (Left) to the laminate center (Right). The
red line shows the location of the stiffeners
35
5.2. RESULTS
CHAPTER 5. OPTIMISATION RESULTS
This means that the stiffeners add a rather large contribution to the stiffness of the plate which
causes the layers to be optimised especially against buckling. Comparing those plies with the double
stiffened square plate, it becomes clear that the stiffeners in this rectangular design are not preventing
the entire panel from buckling. In this case the panel is already divided in three separate buckling
areas. As the number of stiffener plies increase, it is expected that this separation of buckling areas
will occur again but will be even more visible.
The critical load of this rectangular plate with 2 stiffeners of both 4 plies thick is: Pcr = 7.1653e05[N ].
Adding the stiffeners after optimisation of the plate leads to a critical load of: Pcr = 4.1964e05[N ].
This is an increase of 70.7%. The QI case has a critical load of Pcr = 2.6177e05[N ], that is an increase
of 174%.
Double stiffened (eight plies)
The results of the double stiffened rectangular plate with two stiffeners of each 8 plies thick can be
found in figure 5.10. The stiffener plies exist of [45,-45,30,-30]S . It can be seen that those stiffeners
have a slightly bigger influence than the one from the previous case. The areas on the plate are partly
dividing again like what happened with the square plate. The middle area becomes more critical
with respect to buckling. The outer areas are going towards the [0,0] design again in order to create
stiffness.
Fig. 5.10: Optimised fibre paths of a double stiffened rectangular plate with 8 plies per
stiffener of 0.5[m] by 1[m]. From the top layer (Left) to the laminate center (Right). The
red line shows the location of the stiffeners
The critical load of this double stiffened rectangular plate is: Pcr = 8.3771e05[N ]. Doing the
optimisation for the unstiffened case and adding the stiffeners afterwards again, leads to: Pcr =
4.3368e05[N ]. This is an performance increase of 93.2%. The QI case has a critical load of Pcr =
3.5804e05[N ], that is an increase of 134%.
Double stiffened (twelve plies)
The double stiffened plate with two stiffeners with each having twelve plies of [45,-45,30,-30,60,-60]S
is optimised. The results of the optimisation can be found in figure 5.11. The same as for the
square plate, the inner area of this design contains fibres close to the [45,-45] orientation and the two
outer areas contain fibres going towards a [0,0] orientation. Remarkable is that the area between the
stiffeners go more and more to an unstiffened square plate when the number of plies per stiffeners
is increasing. This verifies the assumption of the middle area can be treated as a simply supported
36
CHAPTER 5. OPTIMISATION RESULTS
5.2. RESULTS
square plate. But only when the stiffeners are stiff enough. Stiffer stiffeners create a clear separation
between the three areas of the laminate.
Fig. 5.11: Optimised fibre paths of a double stiffened rectangular plate with 12 plies per
stiffener of 0.5[m] by 1[m]. From the top layer (Left) to the laminate center (Right). The
red line shows the location of the stiffeners
The critical load of this optimised rectangular plate is: Pcr = 8.5300e05[N ]. When the stiffeners
would be added after optimisation: Pcr = 4.3909e05[N ]. This is an increase of 94.3%. The QI case
has a critical load of Pcr = 3.7131e05[N ], that is an increase of 130%.
Double stiffened (sixteen plies)
The double stiffened plate with two stiffeners with each having sixteen plies of [45,-45,30,-30,60,60,0,0]S is optimised. The results of this optimisation can be found in figure 5.12. It is clear that due
Fig. 5.12: Optimised fibre paths of an double stiffened rectangular plate with 16 plies
per stiffener of 0.5[m] by 1[m]. From the top layer (Left) to the laminate center (Right).
The red line shows the location of the stiffeners
to the increased stiffness of the stiffeners the panel can be divided in three areas again. The middle
area comes really close to the results of an unstiffened plate with the same dimensions. As has been
discussed, the middle area can be treated as a simply supported square plate when the stiffeners are
stiff enough. The other smaller areas are mainly used to create stiffness for the stiffness requirement.
The critical load for this stiffened rectangular plate is: Pcr = 8.8838e05[N ]. The critical load for
adding the same stiffeners afterwards: Pcr = 4.5581e05[N ]. This is an increase of 94.9%. The QI case
has a critical load of Pcr = 3.8242e05[N ], that is an increase of 132%.
37
5.2. RESULTS
5.2.7
CHAPTER 5. OPTIMISATION RESULTS
Shared layers
This section discusses the results of the integrated stiffeners functionality which has already been
discussed in section 1.3. Two cases are shown, the square plate and the rectangular plate both with
two fully integrated stiffeners. Those results will be compared to the constant non-integrated stiffened
plates. The advantages of integrated stiffeners has already been discussed in section 1.3. In this
section, fully integrated stiffeners are discussed. This means that every layer from the plate goes into
the stiffener.
Square plate
Looking at the results of the square plate with two integrated stiffeners (see figure 5.13), it looks
quite similar to the double stiffened square plate with non-shared stiffeners. In the figure it can be
noticed that the first four balanced layers contain green lines. This means that the layers are going
up into the stiffeners with the same fibre orientations. These are the integrated layers of the stiffener
(see section 1.3 for information about integrated stiffeners). Since the layers of the stiffeners have
the same fibre orientations as the plate fibres at that same location, the design has become a little
different. Where the square plate without integrated stiffeners had much [0,0] fibre oriented areas,
the plate with the integrated layers does not have it. Which means that integrated layers can have a
significant influence on the design. The reason is that if the plate layers below the stiffeners go to [0,0]
oriented fibres, the stiffeners are changing its fibre orientation such as well. The critical load for this
plate is: Pcr = 1.4208e06[N ]. Comparing to the square plate with predefined stiffener layups gives:
Pcr = 1.5804e06[N ]. This is a decrease in performance of 10.1%. The critical load for the QI case is:
Pcr = 8.6987e05[N ]. This is an increase of 81.7%.
Fig. 5.13: Optimised fibre paths of an double stiffened square plate with 16 plies per
stiffener of 0.5[m] by 1[m]. From the top layer (Left top) to the laminate center (Right
bottom). The red and green lines show the location of the stiffeners. Green also means
that the plies go into the stiffener and the fibre orientations at that location will be the
fibre orientation of the stiffener layer
38
CHAPTER 5. OPTIMISATION RESULTS
5.2. RESULTS
Rectangular plate
The influence of integrated stiffeners on the rectangular plate can be found in figure 5.14. In this
figure the fibre orientations of the stiffeners are not fixed. The plies go from the plate into the stiffener
(as has been discussed in section 1.3). The same as in the previous section, the green lines represent
the layers going up from the plate into the stiffener. From the results it can be concluded that the
optimised plate does not change significantly compared to the rectangular plate with 16-ply-stiffeners
which are having a constant fibre orientation. From this it can be concluded that especially the
thickness of the stiffener influences the design significantly instead of the fibre orientation. This could
have been expected since the geometric stiffness of the stiffener provides stiffness against buckling.
This prevents the plate to buckle on smaller loadings. Comparing the results of the plate with sixteen
shared layers of the stiffener and the rectangular plate with sixteen constant layers, can be used to
verify the extra functionality of stiffener integration. The initial optimiser discussed in chapter 2 was
not able to account for integrated stiffeners.
Fig. 5.14: Optimised fibre paths of an double stiffened rectangular plate with 16 plies
per stiffener of 0.5[m] by 1[m]. From the top layer (Left) to the laminate center (Right).
The red and green lines show the location of the stiffeners. Green also means that the
plies go into the stiffener and the fibre orientations at that location will be the fibre
orientation of the stiffener layer
The critical load of the rectangular plate with integrated stiffeners is: Pcr = 8.6279e05[N ]. The
critical load of the optimised rectangular plate with no integrated stiffeners (discussed in section 5.2.6)
is: Pcr = 8.8838e05[N ]. This is a decrease of only 2.89%. This verifies the fact that integrated stiffeners
can be used without affecting the critical loads too much. Although this percentage looks very small,
it can not be used to represent the decrease of performance in each possible case. This time it is
compared to a case with constant stiffeners of [45,-45, 30,-30, 60,-60, 0, 0]S , an extra study must be
done to know the influence for the worst cases. This is also the reason why only a fully integrated
stiffener is discussed in this thesis, from sem-integrated stiffeners with half non-integrated and half
integrated no conclusions can be drawn either. The critical load of the QI plate with two stiffeners is:
Pcr = 4.0563e05[N ]. This means that the rectangular design with the integrated stiffeners does still
have a performance increase of 113% with respect to the QI case.
39
5.3. CONCLUSION
5.3
CHAPTER 5. OPTIMISATION RESULTS
Conclusion
Looking at the results of this chapter, the designs have all improved regarding the critical loads.
Fibre orientations going towards [0,0] are beneficial for creating stiffness, and fibre orientations going
towards [45,-45] are used to optimise the buckling critical areas. Inner plies overall contain more fibre
paths in the direction of the load than the outer layers where buckling modes are more critical. This
is because of the distance from those layers to the center of the laminate.
The influence of the stiffeners is clearly visible in the fibre path optimisations. The stiffer the
stiffeners are, the more the optimiser treats the panel as three separate areas with simply supported
properties. A great example is the comparison between the middle area of the rectangular plate with
stiff stiffeners, and the unstiffened square plate. It shows that the middle area can be taken as a
separate plate with simply supported boundary conditions as long as the stiffeners are stiff enough.
Those designs get close to each other regarding the fibre paths.
The rectangular panel with shared layer stiffeners looks very similar to the panel of stiffeners
with constant fibre orientations. This verifies the optimiser with the functionality of using integrated
stiffeners. The integrated stiffener designs are a better than the QI designs: 81.7% for the square
and 113% for the rectangular plate. From comparing a predefined non-changing stiffener with a
constantly changing one, no conclusions could be drawn. Multiple designs should be studied before a
valid conclusion can be drawn about the performance of integrated stiffener designs with respect to
non-integrated stiffener designs.
The rate of improvement of the square and rectangular plate with predefined stiffener stacking
sequences, can be found in table 5.2 and table 5.3. In both tables, the first column shows the design
which is optimised. The second column shows the rate of improvement with respect to the same design
but the stiffeners excluded from the optimiser. The third column shows the rate of improvement
compared to the same design with a QI material. According to the calculated critical loads, all the
non-integrated stiffeners are having a positive influence on the optimised design. Ranging from a
critical load increase of 11.8% to 119% for the squared plate and 70.7% to 94.9% for the rectangular
plate. This means that when the stiffeners are not included in the optimisation cycle, the design could
still have been improved by the rate shown in the second column. The real rate of improvement is
shown in the third column when it is compared to the QI case. It is shown that it is worth optimising
with stiffeners included.
Table 5.2: The improvements of each square design compared to: 1. Stiffeners excluded
from optimisation and added afterwards 2. The QI design
Unstiffened Plate
Single Stiffened (center)
Single Stiffened (off-center)
Double Stiffened (symmetric)
Double Stiffened (asymmetric)
40
Excl. Stiffeners
22.0%
57.8%
11.8%
119%
QI
56.3%
26.1%
103%
103%
114%
CHAPTER 5. OPTIMISATION RESULTS
5.3. CONCLUSION
Table 5.3: The improvements of each rectangular design compared to: 1. Stiffeners
excluded from optimisation and added afterwards 2. The QI design
Unstiffened
Stiffeners of 4 plies
Stiffeners of 8 plies
Stiffeners of 12 plies
Stiffeners of 16 plies
Excl. Stiffeners
70.7%
93.2%
94.3%
94.9%
41
QI
95%
174%
134%
130%
132%
Conclusion & recommendations
Variable stiffness laminates (VSL) are proven to be an excellent substitution for constant fibre laminates in aerospace structures. The optimiser developed at the Delft University of Technology is perfect
for optimising VSL against buckling. The software includes a manufacturing constraint in order to
model manufacturable designs. But the optimiser was short on the capability of including stiffeners
in the optimisation.
After further development of the optimiser, it is now capable of adding stiffeners to the optimisation.
This implementation is realised by merging stiffeners to the plate with an object oriented programming approach. Using main- and sub-parts, computational effort is saved by reusing already existing
calculation functionalities.
The addition of stiffeners is verified by comparing displacements and sectional forces from the optimiser with the calculated values of ABAQUS. The overall mean errors, stayed under the 3%, despite
of the somewhat enormous outliers in the verification process. The optimiser accounts for crippling of
stiffeners which has been verified as well.
The optimiser has optimised 12 simply supported models for the first two buckling modes while the
models were constrained to be at least as stiff as their quasi isotropic designs. Looking at the results of the optimiser; the influence of stiffeners with a predefined stacking sequence gives a buckling
load increase of 11.8% to 119%. Stiffeners separate the areas on the plate. The area in-between two
stiffeners shows similar properties to a simply supported plate. The larger areas are more sensitive
to buckling than the smaller areas. This results in optimal fibre paths going to a [45,-45] fibre orientation. The other areas are creating stiffness by changing its fibre orientations closer to a [0,0]
design. The outer layers of a laminate are found to be more critical to buckling than the inner layers.
The outer layers contain more fibres going towards a [45,-45] direction. And since the inner layers
are less sensitive to buckling, stiffness is created here by orienting the fibres closer to a [0,0] orientation.
The integrated stiffener designs are shown to be better than the quasi isotropic design but can be less
performing compared to the stiffeners with a fixed layup. No valid conclusions can be drawn about
using shared layers with respect to its performance increase. More research needs to be done to draw
more conclusions about comparing integrated stiffeners with non-integrated stiffeners.
Recommendations can be done regarding this thesis and for future work:
• The optimiser could be optimised even more to reduce computational effort and storage space.
• As already been mentioned, more study can be done to integrated stiffeners. If the stiffeners
become VSL as well, a structure is expected to be optimised even further. This requires the
stiffener layers to be included in the optimisation as well.
• The optimiser could be developed in an entire object oriented environment. This will cost a lot
of time to develop but will save a lot of time during analysis. It will also save a lot of time if
other types of structures are going to be implemented. For example, plates with holes.
• A further study could be done to different load cases.
42
Bibliography
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March 6, 2016]
[2] Gurdal, Z., Haftka, R.T., Hajela, P., 1999. Design and Optimization of Laminated Composite
Materials, New York: John Wiley & Sons
[3] Callahan, J.K., Weeks, G.E., 1992. Optimum Design of Composite Laminates Using Genetic Algorithms. Composites Engineering, 2(3), pp. 149-160
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Part I: Constant stiffness design, Composite Structures, 90, pp.1-11
[5] Hyer, M.W., Lee, H.H., 1991. The Use of Curvilinear Fiber Format to Improve Buckling Resistance
of Composite Plates with Central Circular Holes, Composite Structures, 18, pp. 239-261
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structures, Composite Structures, 72, pp.311-320
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distributions for optimal variable stiffness laminates, Composites, B 43, pp. 354-360
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maximum buckling load using lamination parameters, AAIA, 48(1), pp. 134-143
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[10] Peeters, D.M.J., Hesse, S., Abdalla, M.M., 2015. Stacking sequence optimization of variable
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angle tow composites, Composites Part A, 43, pp. 1347-1356
[12] Kim, B.C., Hazra, K., Weaver, P., Potter, K., 2011. Limitations of fibre placement techniques for
variable angle tow composites and their process-induced defects, Proceedings of the 18th international conference on composite materials
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vehicle structures (2002-2023), J Aircraft ,41(5), pp.986-98.
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[15] Stewart, J., 2008. Calculus: early transcendentals, Boston: Cengage Learning.
[16] Okechi, C and Onwubiko., 2000. Introduction To Engineering Design Optimization, New Jersey:
Prentice-Hall
[17] Polak, E., 1997. Optimization; algorithms and consistent approximations, Berlin: Springer
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BIBLIOGRAPHY
[18] Laan, R.P.J., Optimization of Variable Stiffness Laminates, Internal Publication, Delft: Technical
University
[19] Kassapoglou, C., 2013. Design and Analysis of Composite Structures, New York: John Wiley &
Sons
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integration with numerical optimization techniques for structural design, Computers & Structures,
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44
Appendices
45
A
Laminate properties
A.1
Engineering constants
Q11 = m4 Qxx + n4 Qyy + 2m2 n2 Qxy + 4m2 n2 Qss
(A.1)
Q22 = n4 Qxx + m4 Qyy + 2m2 n2 Qxy + 4m2 n2 Qss
(A.2)
Q12 = m2 n2 Qxx + m2 n2 Qyy + m4 + n4 Qxy − 4m2 n2 Qss
Q66 = m2 n2 Qxx + m2 n2 Qyy − 2m2 n2 Qxy + m2 − n2
2
(A.3)
Qss
(A.5)
(A.6)
Q16 = m3 nQxx − mn3 Qyy + mn3 − m3 n Qxy + 2 mn3 − m3 n Qss
Q26 = mn3 Qxx − m3 nQyy + m3 n − mn3 Qxy + 2 m3 n − mn3 Qss
A.2
Material Invariants
3Qxx + 3Qyy + 2Qxy + 4Qss
8
Qxx − Qyy
U2 =
2
Qxx + Qyy − 2Qxy − 4Qss
U3 =
8
Qxx + Qyy + 6Qxy − 4Qss
U4 =
8
Qxx + Qyy − 2Qxy + 4Qss
U5 =
8
U1 =
A.3
(A.4)
(A.7)
(A.8)
(A.9)
(A.10)
(A.11)
Lamination Parameters
A
V0,1,2,3,4
B
V0,1,2,3,4
1
=
2
Z1
[1, cos 2θ(z), cos 4θ(z), sin 2θ(z), sin 4θ(z)] dz
(A.12)
[1, cos 2θ(z), cos 4θ(z), sin 2θ(z), sin 4θ(z)] · zdz
(A.13)
[1, cos 2θ(z), cos 4θ(z), sin 2θ(z), sin 4θ(z)] · z 2 dz
(A.14)
−1
Z1
=
−1
D
V0,1,2,3,4
3
=
2
Z1
−1
46
B
ABD-matrix

V0A V1A
V2A
0
0
A11
A
A
A

V
−V
V
0
0 
A22 
1
2
 0




A
A
A 
0
−V2 V0
0 
 0

 12 

A=
 = h
 0
0
−V2A 0 V0A 
A66 




V3A


A
A16 
V
0
0
 0

4
2
A
V3
A
A26
−V4
0
0
0
2




0 V1B
V2B
B11
0 −V B
V2B
B22 
1




B 
0
0
−V2B
h2 
 12 

B=
=
0
0
−V2B
B66 
4 



V3B

B16 
V4B
0
2
B
V3
B26
−V4B
0
2


0
0
0
0
0
0

0


0

(B.2)

0 0

0 0

V0D V1D
V2D
0
0
D11
D
D
D

−V1
V2
0
0 
D22 

V0




D
D
D 
3
0
0
−V
V
0


2
0
h 
 12 

D
D
D=
=

0
0
−V
0
V
2
0 
D66 
12


D


V3


D
D16 
V
0
0

 0
4
2
V3D
D
D26
0
−V4
0
0
2

(B.1)


(B.3)
!
ABDmatrix =
47
A B
B D
(B.4)
C
Frobenius inner product
The Frobenius inner product is written as A : B. It is the component-wise inner product of the A
and B matrices as though they are vectors. It can be calculated as follows:
A:B=
X
Aij Bij = vec (A)T vec (B)
i,j
Where ’vec’ stands for vectorization. This can also be written in the better known form of ’traces’ of
matrices:
vec (A)T vec (B) = tr AT B
48
D
FEM
(from section 2.3.5): The general problem which needs to be solved numerically is:
Km · U = F
The real problem is more difficult than it looks like. It needs to be taken into account that the model
has free master degrees of freedom, free degrees of freedom and slave degrees of freedom. In the
software, the stiffness matrix of the free degrees of freedom is called K11 , the free master degree of
freedom K22 , and the stiffness between the free and free master degrees of freedom K12 . The internal
forces exerted on the free nodes is in equilibrium with the structure:
K11 u1 = f1
It has to be noticed that this equation is only valid for a complete structure which only exists of free
nodes. The influence of the master and slave nodes must be taken into account as well. So the real
equation becomes:
K11 u1 = f1 − f21
Where f21 is the internal force exerted by the adjacent nodes of the master and slave nodes (see figure
D.1):
f21 = K21 · u2
Writing this out gives:
K11 · u1 = f1 − K12 · u2
Using Cholesky decomposition for K11 :
R0 · R · u1 = f1 − K12 · u2
Where R0 is the lower triangle, and R the upper triangle. The working of the Cholesky decomposition
can be found in Appendix C. Rewriting the equation results in:
u1 = R\ R0 \f1 − R0 \K12 · u2
Using A = R0 \f1 and b = R0 \K12 gives:
u1 = R\(A − b · u2 )
In order to write out u2 , the same principle is valid as well:
K22 · u2 = f2 − f12
Where:
f12 = K12 · u1
Which results in:
K22 · u2 = f2 − K12 · u1
Substituting u1 :
K22 · u2 = f2 − K12 · [R\(b − A · u2 )]
49
(D.1)
APPENDIX D. FEM
Rewriting this:
f2 = K22 · u2 + K12 · [R\(b − A · u2 )]
f2 = K22 · u2 + [R\(K12 · b − K12 · A · u2 )]
f2 = R\(K12 · b − K12 · A · u2 + R · K22 · u2 )
R · f2 = K12 · b + (R · K22 − K12 · A) · u2
Bringing u2 to the left side:
(R · K22 − K12 · A) · u2 = R · f2 − K12 · b − R · K12
(R · K22 − K12 · A) · u2 = R · f2 − K12 · b − R · K12
u2 = (R · K22 − K12 · A) \(R · f2 − K12 · b)
In order to write it easier, A0 = R\K12 :
u2 = K22 − A0 · A \ f2 − A0 · b
(D.2)
Those u1 and u2 can fill up the displacement matrix ’U ’, only the MPC-matrix must be multiplied
Fig. D.1: A drawing of master and slave nodes with their adjacent nodes (Left) and a
drawing of the forces going from adjacent nodes to the master and slave nodes
with the u2 in order to calculate the displacements of the slave nodes as well. For calculating the
force in every degree of freedom, equation D can be easily used. Where U is the displacement matrix
containing all the displacements for every degree of freedom of the model.
50
E
Verification plots
E.1
Unstiffened plate
Fig. E.1: The displacements of the
unstiffened plate in all three spatial
directions (Optimiser), U1 on top,
U2 in the middle, and U3 at the
bottom
Fig. E.2: The displacements of the
unstiffened plate in all three spatial
directions (ABAQUS), U1 on top,
U2 in the middle, and U3 at the
bottom
51
E.1. UNSTIFFENED PLATE
APPENDIX E. VERIFICATION PLOTS
Fig. E.3: The section forces of the
unstiffened plate in all three spatial
directions (Optimiser), Nx on top,
Ny in the middle, Nz at the bottom
Fig. E.4: The section forces of the
unstiffened plate in all three spatial
directions (ABAQUS), Nx on top,
Ny in the middle, Nz at the bottom
52
APPENDIX E. VERIFICATION PLOTS
E.1. UNSTIFFENED PLATE
Fig. E.6: The relative error of the
unstiffened plate section forces
Fig. E.5: The absolute error of the
unstiffened plate section forces
53
E.2. SINGLE STIFFENED PLATE (CENTER)
E.2
APPENDIX E. VERIFICATION PLOTS
Single stiffened plate (Center)
Fig. E.8: The displacements of the
center stiffened plate in all three
spatial directions (ABAQUS), U1
on top, U2 in the middle, and U3
at the bottom
Fig. E.7: The displacements of the
center stiffened plate in all three
spatial directions (Optimiser), U1
on top, U2 in the middle, and U3
at the bottom
54
APPENDIX E. VERIFICATION PLOTS
E.2. SINGLE STIFFENED PLATE (CENTER)
Fig. E.9: The section forces of the
center stiffened plate in all three
spatial directions (Optimiser), Nx
on top, Ny in the middle, Nz at the
bottom
Fig. E.10: The section forces of the
center stiffened plate in all three
spatial directions (ABAQUS), Nx
on top, Ny in the middle, Nz at the
bottom
55
E.2. SINGLE STIFFENED PLATE (CENTER)
APPENDIX E. VERIFICATION PLOTS
Fig. E.11: The absolute error of the
center stiffened plate section forces
Fig. E.12: The relative error of the
center stiffened plate section forces
56
APPENDIX E. VERIFICATION PLOTS
E.3
E.3. SINGLE STIFFENED PLATE (OFF-CENTER)
Single stiffened plate (Off-center)
Fig. E.13: The displacements of the
off-center stiffened plate in all three
spatial directions (Optimiser), U1
on top, U2 in the middle, and U3
at the bottom
Fig. E.14: The displacements of the
off-center stiffened plate in all three
spatial directions (ABAQUS), U1
on top, U2 in the middle, and U3
at the bottom
57
E.3. SINGLE STIFFENED PLATE (OFF-CENTER)
Fig. E.15: The section forces of the
off-center stiffened plate in all three
spatial directions (Optimiser), Nx
on top, Ny in the middle, Nz at the
bottom
APPENDIX E. VERIFICATION PLOTS
Fig. E.16: The section forces of the
off-center stiffened plate in all three
spatial directions (ABAQUS), Nx
on top, Ny in the middle, Nz at the
bottom
58
APPENDIX E. VERIFICATION PLOTS
E.3. SINGLE STIFFENED PLATE (OFF-CENTER)
Fig. E.17: The absolute error of
the off-center stiffened plate section
forces
Fig. E.18: The relative error of
the off-center stiffened plate section
forces
59
E.4. DOUBLE STIFFENED PLATE (SYMMETRIC)
E.4
APPENDIX E. VERIFICATION PLOTS
Double stiffened plate (Symmetric)
Fig. E.19: The displacements of the
unstiffened plate in all three spatial
directions (Optimiser), U1 on top,
U2 in the middle, and U3 at the
bottom
Fig. E.20: The displacements of the
unstiffened plate in all three spatial
directions (ABAQUS), U1 on top,
U2 in the middle, and U3 at the
bottom
60
APPENDIX E. VERIFICATION PLOTS
E.4. DOUBLE STIFFENED PLATE (SYMMETRIC)
Fig. E.21: The section forces of the
unstiffened plate in all three spatial
directions (Optimiser), Nx on top,
Ny in the middle, Nz at the bottom
Fig. E.22: The section forces of the
unstiffened plate in all three spatial
directions (ABAQUS), Nx on top,
Ny in the middle, Nz at the bottom
61
E.4. DOUBLE STIFFENED PLATE (SYMMETRIC)
APPENDIX E. VERIFICATION PLOTS
Fig. E.24: The relative arror of the
unstiffened plate section forces
Fig. E.23: The absolute arror of the
unstiffened plate section forces
62
APPENDIX E. VERIFICATION PLOTS E.5. DOUBLE STIFFENED PLATE (ASYMMETRIC)
E.5
Double stiffened plate (Asymmetric)
Fig. E.25: The displacements of the
unstiffened plate in all three spatial
directions (Optimiser), U1 on top,
U2 in the middle, and U3 at the
bottom
Fig. E.26: The displacements of
the double asymmetric stiffened
plate in all three spatial directions
(ABAQUS), U1 on top, U2 in the
middle, and U3 at the bottom
63
E.5. DOUBLE STIFFENED PLATE (ASYMMETRIC) APPENDIX E. VERIFICATION PLOTS
Fig. E.27: The section forces of the
unstiffened plate in all three spatial
directions (Optimiser), Nx on top,
Ny in the middle, Nz at the bottom
Fig. E.28: The section forces of the
unstiffened plate in all three spatial
directions (ABAQUS), Nx on top,
Ny in the middle, Nz at the bottom
64
APPENDIX E. VERIFICATION PLOTS E.5. DOUBLE STIFFENED PLATE (ASYMMETRIC)
Fig. E.29: The absolute arror of the
unstiffened plate section forces
Fig. E.30: The relative arror of the
unstiffened plate section forces
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