anisogrid
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paper on anisogrid compositesTRANSCRIPT
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Cranfield University
Jamie Mountain
FE based design of A-6 inner upper wing
panels using Anisogrid Lattice concept
SCHOOL OF ENGINEERING
MSc THESIS
Cranfield University, 2007. All rights reserved. No part of this publication
maybe reproduced without the written permission of the copyright holder.
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Cranfield University
School of Engineering
College of Aeronautics
MSc Thesis
Academic Year 2006/2007
Jamie Mountain
FE based design of A-6 inner upper wing
panels using Anisogrid Lattice concept
SUPERVISOR: DR G. ALLEGRI
This thesis is submitted in partial fulfilment of the requirements for the degree
of Master of Science
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ABSTRACT
An anisogrid lattice is one potential method of improving the efficiency of CFRP
composite structures. The anisogrid lattice structure is comprised of a dense system of
unidirectional laminate ribs, manufactured by filament winding. Lattice structures can
be made with no skin, a single-sided skin, or a double-sided skin.
Composite anisogrid lattice structures have so far found application in only a select
number of specialist applications, usually in the space industry.
This paper aims to determine if the anisogrid lattice concept can find application in
aircraft structures; particularly, as a replacement for skin-stringer wing covers. A
traditional black aluminium composite wing cover is found, and an attempt to
redesign the cover to the same specifications, using the anisogrid lattice concept is
made.
Firstly, an analytical model is developed to determine the stiffness of the chosen
anisogrid lattice pattern. These methods are then used to perform the initial sizing of
the replacement anisogrid lattice upper wing cover, using an MS Excel spreadsheet
developed as a design tool, which incorporates the analytical model.
The replacement wing cover design is then modelled with an exact Finite Element
model, in order to verify the analytical methods developed. An excellent agreement is
found between the analytical and the exact FE model. An equivalent stiffness FE
model is then also developed, in order to avoid some of the shortcomings of the exact
FE model.
In conclusion, a composite anisogrid lattice upper wing cover is designed to replace a
black aluminium wing cover, and a weight saving of 23% is achieved. It is
therefore thought that anisogrid lattice structures represent an attractive step forward
for composite structures.
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Thesis Health Warning
This thesis has been assessed as of satisfactory standard for the award of a
Master of Science degree in Aerospace Vehicle Design. This thesis covers
the part of the assessment concerned with the Individual Research Project.
Readers must be aware that the work contained is not necessarily 100%
correct, and caution should be exercised if the thesis or the data it contains is
being used for future work. If in doubt, please refer to the supervisor named in
the thesis, or the Department of Aerospace Technology.
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ACKNOWLEDGMENTS
Love and thanks go to my parents for supporting me throughout my MSc, and life in
general.
Also, many thanks go to Dr. Allegri for his guidance and expertise throughout the
project.
And thanks to my friend Panos who has worked alongside me on a similar project.
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CONTENTS
1 Nomenclature..................................................................................................... 1
2 Introduction ....................................................................................................... 2
3 Project Objectives .............................................................................................. 3
4 Fundamentals of Lattice Structures .................................................................... 4
4.1 Lattice Structures ....................................................................................... 4
4.2 History and future of lattice structures ........................................................ 5
4.3 Manufacturing of composite lattice structures............................................. 8
4.4 Summary of Fundamentals ......................................................................... 9
5 Analytical Model ............................................................................................. 11
5.1 Analytical Model Selection ...................................................................... 11
5.2 Description of Analytical model ............................................................... 11
5.2.1 Unit cell Approach ........................................................................... 12
5.2.2 Stiffness of a Unit Cell ..................................................................... 13
5.3 Advantages/ Disadvantages of Analytical model ...................................... 16
6 Original A6 Composite Wing Cover Design .................................................... 17
6.1 A6 Airliner Description............................................................................ 17
6.2 Original Cover Design.............................................................................. 18
6.2.1 Design Criteria ................................................................................. 19
7 Design of Anisogrid Lattice Replacement for A6 Upper Wing Cover............... 21
7.1 Selection of Lattice Pattern....................................................................... 21
7.2 Carbon fibre grade and fibre volume fraction ........................................... 24
7.3 Design Variables ...................................................................................... 26
7.4 Design Constraints ................................................................................... 26
7.4.1 Minimum Skin Thickness ................................................................. 26
7.4.2 Euler buckling of Ribs ...................................................................... 28
7.4.3 Rib Beam constraint....................................................................... 29
7.5 Anisogrid Sizing and Optimisation Approach........................................... 29
8 Optimisation Results........................................................................................ 32
9 Finite Element modelling ................................................................................. 33
9.1 Description of models .............................................................................. 33
9.2 Exact Finite Element model...................................................................... 34
9.2.1 Results of Exact FE Model under Membrane Loads.......................... 35
9.2.2 Results of Exact FE Model under Shear Loads ................................. 39
9.2.3 Conclusions of Exact FE modeling ................................................... 40
9.3 Equivalent Finite Element model.............................................................. 41
9.3.1 Results of Equivalent FE Model under Membrane Loads.................. 42
9.3.2 Results of Equivalent FE Model under Shear Loads.......................... 45
9.3.3 Conclusions of equivalent Finite Element modelling ........................ 45
10 Equivalent FE Modelling of replacement A6 upper wing cover.................... 47
10.1 Aims of Modelling wing cover ................................................................. 47
10.2 Model Creation and boundary conditions ................................................. 47
10.3 Results of equivalent FE modelling of replacement A6 upper wing cover. 48
10.4 Conclusions of Equivalent FE Modelling of replacement A6 upper wing
cover 50
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11 Comparison between original A6 wing cover design and replacement
Anisogrid Lattice ..................................................................................................... 51
12 Additional Work .......................................................................................... 52
13 Conclusion................................................................................................... 53
14 References ................................................................................................... 54
15 Appendix A: Development of Analytical model ........................................... 56
15.1 Rib Stiffness matrices............................................................................... 56
15.1.1 Rotation matrix, [R].......................................................................... 56
15.1.2 Rib In-plane stiffness matrix, [A]...................................................... 57
15.1.3 Rib Flexural Stiffness Matrix, [D] .................................................... 61
15.1.4 Rib Out-of-plane Shear Stiffness Matrix, [H].................................... 64
15.2 Skin Stiffness Matrices............................................................................. 66
15.2.1 Skin In-plane stiffness matrix ........................................................... 66
15.2.2 Skin Flexural Stiffness Matrix .......................................................... 67
15.2.3 Skin Out-of-plane shear stiffness matrix ........................................... 68
15.3 Total Anisogrid Stiffness matrices............................................................ 68
16 Appendix B: Calculation of Materials Properties.......................................... 70
17 Appendix C: Mass of a Unit Cell.................................................................. 72
18 Appendix D: Wing Bending Loads Calculation............................................ 74
19 Appendix E: Validation of Analytical Model in Bending.............................. 79
19.1 Boundary conditions for pure bending...................................................... 79
19.2 Results of Exact FE model in pure bending .............................................. 80
20 Appendix F: Validation of Exact FE model in shear ..................................... 83
20.1 Results of Exact FE model under shear loads ........................................... 83
21 Appendix G: Derivation of Equivalent Stiffness FE model Properties .......... 84
22 Appendix H: Matlab File.............................................................................. 87
23 Appendix I: Engineering Drawings .............................................................. 94
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LIST OF FIGURES
Figure 1: Comparison of Isogrid and Anisogrid lattice structures ............................... 2
Figure 2: Anisogrid Lattice composite spacecraft attach-fitting adaptor [2] ................ 4
Figure 3: Vickers Viking aircraft [3] .......................................................................... 5
Figure 4: Isogrid fuselage barrel manufactured by filament winding [7]..................... 8
Figure 5: Geometry of a unit cell ............................................................................. 12
Figure 6: Cross section of Unit cell in 0 direction .................................................... 13
Figure 7: View of the interior of the A6 wing .......................................................... 18
Figure 8: Orthogrid and Anglegrid Lattice types ...................................................... 21
Figure 9: Orientation of anisogrid panel on upper wing............................................ 23
Figure 10: The effect of fibre selection on composite modulus, using epoxy matrix
and Vf=35% .................................................................................................... 24
Figure 11: Elastic Properties of Skin and Ribs ......................................................... 25
Figure 12: Skin thickness required versus relative velocity ...................................... 27
Figure 13: Flow chart showing the operation and functionality of Excel Spreadsheet
developed for sizing and optimising anisogrid lattice structure......................... 30
Figure 14: Selection of optimisation results ............................................................. 32
Figure 15: Exact FE model of 28 unit cells (shown with one skin hidden)................ 34
Figure 16: Comparison of strains and displacements predicted by FE and analytical
model............................................................................................................... 36
Figure 17: Minimum Principal strains in exact FE model for Case 10 (Compressive
Ny) .................................................................................................................. 37
Figure 18: Min Principal Strains in skin of middle unit cell (Case 10)...................... 37
Figure 19: Comparison between maximum and average strains predicted by exact FE
model............................................................................................................... 38
Figure 20: Boundary conditions for pure shear......................................................... 39
Figure 21: Middle Unit cell of Exact FE model under pure shear ............................. 39
Figure 22: Illustration of Equivalent stiffness concept.............................................. 41
Figure 23: Plot of displacements of Equivalent FE model under membrane loading
(Case 10) ......................................................................................................... 43
Figure 24: Close up of the circled area in Figure 23, showing constant displacements
(Case 10) ......................................................................................................... 43
Figure 25: Results of equivalent FE model under membrane loads........................... 44
Figure 26: Equivalent FE model under shear load .................................................... 45
Figure 27: Constraint forces in z axis on equivalent wing cover FE model ............... 48
Figure 28: Plot of displacements in Equivalent wing cover FE model ...................... 49
Figure 29: Minimum principal strains in Equivalent wing cover FE model .............. 49
Figure 30: Mass breakdown of original and replacement A6 wing cover.................. 51
Figure 31: Geometry for in-plane stiffness calculation ............................................. 58
Figure 32: Bending moment applied to unit cell....................................................... 62
Figure 33: Individual Elastic properties of HS Carbon fibre and Epoxy matrix ........ 71
Figure 34: Geometry for Moment of area calculation ............................................... 74
Figure 35: Stress distribution through thickness of upper unit cell............................ 76
Figure 36: A6 Wing Limit Bending Moment Diagram............................................. 78
Figure 37: Boundary conditions for pure bending ................................................... 79
Figure 38: Results of Exact FE model under bending............................................... 80
Figure 39: Maximum Principal in Exact FE model (Case 10)................................... 81
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Figure 40: Maximum Principal strains throughout Unit cell ..................................... 81
Figure 41: YY component of strains in ribs of unit cell ............................................ 82
Figure 42: Results of Exact FE model under Shear loads ......................................... 83
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1 Nomenclature
Symbol Description Units
A In-plane stiffness matrix N / mm
A Cross sectional area of ribs (with relevant subscript) mm2
B Coupling matrix N
b Rib width (with relevant subscript) mm
D Flexural stiffness matrix Nmm
d rib pitch (with relevant subscript) mm
E Youngs Modulus MPa
G Shear Modulus MPa
H Shear stiffness matrix N / mm
H Rib Height mm
I Second moment of area (with relevant subscript) mm4
J Torsional constant of rib cross-section mm4
k Curvature mm-1
l Length of ribs (with relevant subscript) mm
M Moment Nmm
M Mass kg
N Distributed load N / mm
P Concentrated force N
Q Stiffness matrix MPa
R Rotation matrix
T Torque Nmm
ts Skin thickness mm
V Shear force N
Poissons ratio
V Fibre volume fraction
0 (as subscript) Refers to zero degree ribs
90 (as subscript) Refers to 90 degree ribs
(as subscript) Refers to phi degree ribs
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2 Introduction
The aim of this thesis is to investigate the feasibility of using composite anisogrid
lattice construction in transport aircraft wing covers.
An anisogrid lattice is a method of reinforcing the skin using a geometrical pattern of
ribs. The anisogrid lattice is anisotropic with regard to its in-plane properties. The
anisogrid lattice is not to be confused with the isogrid lattice, which in contrast, is
isotropic in-plane. Many possible patterns for each concept exist, but examples of
these two concepts are shown below.
Figure 1: Comparison of Isogrid and Anisogrid lattice structures
Isogrid lattice construction was first used on the Vickers Wellesley bomber, (termed
geodetic construction). This form of construction was found to be extremely
resistant to damage due to the high redundancy of the lattice structure, and also saved
weight.
However, due to the difficulties of manufacturing such a structure using metallic
materials, this form of construction was abandoned in subsequent aircraft designs.
With the advent of fibre composite technology, and in particular a manufacturing
technique known as filament winding, isogrid and anisogrid lattice structures have
found use in a number of Russian and American spacecraft applications, and in a
small number of aircraft fuselages.
Isogrid Anisogrid
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The aerospace applications to date, however, have all been cylindrical or conical shell
structures, such as interstage structures in rockets.
Therefore, the aim of this thesis is to investigate whether the mass savings possible
using anisogrid construction in the manufacture of cylindrical or conical shells,
translates into mass savings in the skins of a transport aircraft wing.
3 Project Objectives
The objectives of this thesis are;
i. To develop an analytical method that allows the initial design and analysis of a
composite (CFRP) anisogrid lattice structure
ii. To design and optimise a composite anisogrid lattice structure to replace a
composite upper wing panel (comprising skin and stringers) of the A6 airliner
(2007 AVD GDP aircraft), using the analytical method derived
iii. To use the same design criteria as the original wing design in order to achieve
a fair comparison between the resulting designs
iv. To verify the analytical methods developed using Finite Element modelling
v. To produce a mass comparison between the original composite A6 cover
design, and the replacement anisogrid cover design
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4 Fundamentals of Lattice Structures
4.1 Lattice Structures
A lattice structure uses a geometrical arrangement of beams (termed ribs) as the
primary load bearing elements of the structure. Many different rib angles and lattice
patterns are available, but three or four different rib orientations (with respect to a
suitable reference axis) are commonly used in lattice structures.
The simplest lattice structures to design and analyse are isogrid lattices, which feature
ribs at zero, plus and minus sixty degrees and which therefore form equilateral
triangles. The resulting structure is therefore isotropic in-plane.
However, an investigation by Marco Regi [1] showed that a mass advantage can be
gained by using an anisogrid lattice, (which is anisotropic in-plane) so that the
structure is better tailored to resist the applied loads. The higher the applied loads, the
larger are the mass savings that can be expected from using an anisogrid lattice rather
than an isogrid lattice.
Figure 2: Anisogrid Lattice composite spacecraft attach-fitting adaptor [2]
Therefore, the mass savings offered by using an anisogrid lattice can usually outweigh
the added difficulty of designing and analysing an anisotropic structure.
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The most efficient lattice structure is one with no skin. However, certain applications
require a skin, such as the surface of an aircraft wing, and therefore it is possible to
manufacture lattice structures with either a single or double-sided skin. In the case of
a double-sided skin, the structure is akin to a sandwich panel, with the network of
reinforcing ribs replacing the typical honeycomb or foam core.
A typical lattice structure with no skin is shown in Figure 2 above, where the load
carrying ribs can be clearly seen. This particular structure has ribs arranged in three
different orientations; the orientation angles, dimensions, and pitches of the ribs
determine the characteristics of the lattice structure.
4.2 History and future of lattice structures
The first aeroplane to use a lattice structure was the British Vickers Wellesley light
bomber, designed in the mid 1930s by the pioneering Barnes Wallis, closely followed
by the Vickers Wellington medium bomber. These aircraft used a metallic geodetic
construction, which is a special case of isogrid construction where the ribs follow the
geodesic lines of the structure. The metallic lattice was covered with fabric.
Figure 3: Vickers Viking aircraft [3]
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The Vickers Viking aircraft was a civil development of the Wellington, which used
the same wings and engines, combined with a new fuselage and tail. The geodetic
lattice construction can be clearly seen in Figure 3 above.
Political decisions against Barnes Wallis then effectively ended the use of lattice
structures in subsequent British aircraft.
American lattice structures began in 1964, when a NASA project set out to establish
the optimum method of stiffening domes loaded in compression, concluded that the
best solution was to use metallic trusses forming equilateral triangles - isogrid
construction [4]. The study concluded that as well as saving mass, the structures were
also easily analysed due to their isotropy, and also resulted in less structural depth
than other stiffening concepts, thus maximising the internal volume of stiffened shells
such as fuel tanks.
In contrast to the British fabric skins, the early American lattice structures developed
used a relatively thick metallic skin, mechanically fastened to the metallic lattice. This
greatly increased the fail safety of the structure, as a crack beginning in the skin could
not propagate into the lattice, and vice-versa.
This kind of isogrid lattice construction was then used in the Delta family of rockets,
and also in the Skylab space station. In addition, McDonnell-Douglas used isogrid
construction in various aircraft secondary structures such as speed-brakes.
In the late 1960s, some American lattice structures were developed using composite
materials; however these were still based on the isogrid concept.
It appears that the first lattice structures to realise the added weight saving potential of
the anisogrid concept were developed in the former Soviet Union, in a research
institute which still exists today known as CRISM (Central Research Institute for
Special Machinery) [2].
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The first composite anisogrid lattice structure was developed in CRISM in 1981, and
after the structure failed in compressive load tests well after the predicted failure load,
a research program was begun to further the understanding of these kinds of
structures.
The research at CRISM found that by making the reinforcing ribs of unidirectional
CFRP, structures of very high specific strength and stiffness can be achieved - much
higher than so-called black aluminium structures - where CFRP is directly
substituted for aluminium in a structure, simply relying on the lower density of CFRP
to achieve a mass advantage over the original structure.
This research program resulted in the development of several CFRP anisogrid rocket
interstages, and also in 1986, the fabrication of a composite anisogrid lattice fuselage
barrel for the Ilyushin 114 regional airliner.
An investigation by Vasiliev [5] suggests that, in practice, weight savings of up to
40% can be achieved by substitution of anisogrid lattice structures for black
aluminium structures, which in turn usually offer weight savings of around 10-20%
over traditional aluminium structures.
The scope for the application of anisogrid lattices is high, with Vasiliev suggesting the
concept is suitable for application to fuselage frames and skins, rib and spar shear
webs in wings and tailplanes, and wing covers among others. However, at present, a
lack of understanding of the failure behaviour of lattice structures is holding back
their widespread introduction [6].
In conclusion, the utilisation of anisogrid lattice structures remains very low;
however, it is thought that as the use of composites in commercial aviation increases,
so too there will be a move away from black aluminium structures, towards
structural concepts that take better advantage of the properties of carbon-fibre
reinforced polymers. The anisogrid lattice is one such concept, which has proven
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advantages in cylinders loaded mainly in compression, which has led to its use in a
number of rocket applications.
This paper will aim to determine if similar advantages, in terms of mass saving, can
be gained from applying the anisogrid lattice concept to replace the black aluminium
skin-stringer panel design of the A6 wing covers.
4.3 Manufacturing of composite lattice structures
The anisogrid lattice structures which will be investigated in this paper are intended to
be manufactured from Carbon fibre reinforced polymer (CFRP), using a technique
known as filament winding. The manufacturing process will not be looked at in detail
within this thesis, and therefore the following paragraphs give a brief description of
the intended manufacturing technique.
Figure 4: Isogrid fuselage barrel manufactured by filament winding [7]
Filament winding is a process that is usually best suited to making hollow
components of circular or oval cross-section. Groups of pre-impregnated carbon fibres
(known as tows) are wound around a mandrel to produce a structure of the desired
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shape, laminate layup, and thickness. The laminate layup can be tailored by changing
the angle at which each layer of fibres is wound around the mandrel. The structure is
then cured, and lastly the mandrel is removed to leave the final structure. In order to
aid with the mandrel removal process, mandrels are sometimes inflatable, soluble, or
fusible. The main advantage of filament winding is that it can be a very fast method of
manufacturing components; however, its use is limited to convex components, and the
cost of mandrels can be high.
This manufacturing process can be adapted to produce an anisogrid lattice structure.
For example, in order to manufacture a single skinned anisogrid lattice structure, a
female rubber mould is placed around the mandrel. The rubber mould has the
anisogrid lattice pattern cut out of it, forming a number of channels that have the
dimensions of the ribs. The filament winding process is then used to wind into the
channels, producing ribs of unidirectional CFRP. Once the winding has reached the
height of the ribs and filled the channels, the process is then continued in order to
produce the skin to the correct thickness and layup. Figure 4 above shows an isogrid
fuselage barrel manufactured using this technique.
4.4 Summary of Fundamentals
In summary, composite anisogrid lattice structures are thought to offer the following
advantages;
Potentially offer up to 40% mass saving over black aluminium structures [2]
Ribs are unidirectional, and so they are very strong and stiff, and also do not
delaminate [8]
Anisogrid structures often require less structural depth than other structural
concepts (e.g. sandwich panels), therefore maximising internal volume [4]
The potential exists to completely automate the manufacturing process,
thereby reducing labour costs [8]
At present, composite anisogrid lattice structures also have the following
disadvantages;
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A Lack of understanding of the behaviour of lattice structures close to and
after failure [6]
This paper will aim to determine whether the potential 40% mass saving applies to the
covers of an upper wing surface, as it is not thought that a composite anisogrid lattice
has been used in such an application before. In order to do this, a suitable method of
designing an anisogrid lattice structure, to replace an existing black aluminium wing
cover design, had to be found. The following chapter will discuss this.
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5 Analytical Model
5.1 Analytical Model Selection
In order to design an anisogrid lattice structure, firstly it was necessary to find a
suitable analytical model of an anisogrid structure. Anisogrid lattices are able to
efficiently resist membrane (tensile or compressive), bending, torque, and shear loads,
and therefore it was important to find a model which could predict the behaviour of
the structure under each of these loading conditions.
Several alternatives were found in existing literature [2], [9]. However, the method
proposed by Vasiliev in reference [2] is valid only for one particular lattice pattern,
and the one discussed in reference [9] requires specialist software.
Therefore, the method found in reference [8] was deemed to be most suitable. The
reasons for selecting this model are that it does not require any specialist software,
and that this model can be easily applied to any lattice pattern, unlike the other
methods found which were derived for specific lattice types.
It is also possible to apply the model to un-skinned, single-skinned, or double-
skinned lattice structures.
5.2 Description of Analytical model
The analytical model found in Reference [8] is described as an Equivalent Stiffness
Model, and allows the stiffness of a chosen lattice structure to be determined when
subjected to membrane, bending or shear loads, or a combination of these.
Reference [8] gives the final stiffness relationships, in matrix form, for a particular
anisogrid lattice pattern. In order to further the authors understanding of the analytical
model, the same relationships were derived, and the full derivation of these can be
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found in Appendix A. However, the following sections will give a brief description of
the work carried out in determining the stiffness of the anisogrid lattice structure.
5.2.1 Unit cell Approach
The first step in determining the stiffness of an anisogrid lattice structure is to
determine the unit cell for the particular lattice pattern being used.
A unit cell consists of a number of ribs arranged in a particular manner, which when
periodically repeated, creates the geometry of the lattice structure. The particular
lattice pattern shown in Figure 5 below has ribs arranged in four different orientations;
these are termed 0, 90, and plus/minus degrees.
Figure 5: Geometry of a unit cell
In order that the unit cell correctly produces the structure when repeated, the ribs on
the outer edges of the unit cell are half the width of the corresponding ribs in the
centre. Therefore, for example, in Figure 5 above the centre 90 degree rib has a
90
0
l
0d9
0d
d Rib
Orientation
Ribs
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thickness of b90, and so the 90 degree ribs on the left and right of the unit cell are of
thickness b90/2.
This is illustrated in Figure 6 below, which shows the cross section of a unit cell,
considering a cut through in the 0 degree direction (see also Figure 5).
Figure 6: Cross section of Unit cell in 0 direction
5.2.2 Stiffness of a Unit Cell
Once the layout of the unit cell is known, it is then possible to consider each rib
individually, to determine the stiffness of a rib within its own reference axis when
subjected to membrane, bending or shear loads.
In order to account for the different orientation of ribs, the original stiffness matrices
are then rotated using a rotation matrix. The stiffness of the unit cell as a whole is then
found using the principle of superposition, summing the individual contributions of
each rib, in each direction.
The contribution of the laminate skins (if used) is also calculated separately using
laminate theory, and added to the ribs to determine the stiffness of the structure as a
whole.
2/90b 90b sin/b
H
2d90
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Final stiffness relationships are then found for membrane, bending, and shear loads
respectively, in the form1;
{ } [ ]{ } [ ]{ }kBAN += (1)
{ } [ ]{ } [ ]{ }BkDM += (2)
{ } [ ]{ }HV = (3)
Where the membrane stiffness matrix [ ]A , the bending stiffness matrix, [ ]D , and the
shear stiffness matrix, [ ]H , are the sums of the contributions of the skins and the ribs;
[ ] [ ] [ ]SKINRIB AAA += (4)
[ ] [ ] [ ]SKINRIB DDD += (5)
[ ] [ ] [ ]SKINRIB HHH += (6)
The coupling matrix, [ ]B , only exists for structures which are unsymmetrical; for example, a membrane load applied to a single-skinned lattice structure would induce
also some degree of curvature. The [ ]B matrix can be made zero by making the structure symmetrical; in practice this means either having no skins (the ribs are
unidirectional and therefore have a zero [ ]B matrix on their own), or having a double sided skin, where the top skin is symmetrical with regard to the bottom.
1 Where curly brackets represent a column vector, square brackets represent a square matrix
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The stiffness matrices (when inverted) then allow the calculation of the membrane
strains ([ ]A matrix);
x , y , xy
The bending curvatures ([ ]D matrix);
xk , yk , xyk
which can then be converted into strains using the relationship;
zkxx += 0
And the engineering shear strains ( [ ]H matrix);
xz , yz
The resulting strains in each direction can then be summed, using the principal of
superposition, and the maximum and minimum strains in the principal directions can
be found using the relationship;
2
2
2,122
xy
yxyx
+
+= (7)
The full stiffness matrices, and their derivation can be found in Appendix A.
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5.3 Advantages/ Disadvantages of Analytical model
The analytical model was developed with the goal of allowing the average stresses
and strains in an anisogrid structure to be quickly determined, in order to use them for
initial design purposes. The impact of dimensional changes to the structure, in terms
of strains, can be rapidly determined, and therefore the model can be used for
optimisation also. However, more detailed design and analysis is intended to be
performed using Finite Element Analysis.
The model has the following advantages;
The stiffness matrices can be determined for almost any lattice pattern
Suitable for un-skinned, single, or double-skinned lattice structures
Allows for skins to use different materials to ribs
Predicts strains under membrane, bending, or shear loads
Allows for ribs in different orientations to have different cross-sectional
dimensions (though this may be difficult from a manufacturing point of view)
Allows rapid analysis of the average strains in a lattice structure (using a
spreadsheet for example), therefore suitable for initial design and analysis
The model also has the following disadvantages;
The model does not predict any stress concentrations, where the local strains
may be higher than average, and where strains may therefore exceed limits
Buckling analysis has to be performed separately
The model cannot account for irregularities in the grid structure which may
cause stress raisers, such as where two lattices are joined together
With a suitable method of predicting the strains resulting from the applied loads on an
anisogrid structure now available, it is possible to begin the design of an anisogrid
lattice to replace a suitable black aluminium structure. The following Section will
discuss the structure selected to be redesigned using the anisogrid lattice concept.
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6 Original A6 Composite Wing Cover Design
This Section will briefly discuss the structure to be redesigned using the anisogrid
lattice concept, in order to determine if the weight benefits promised by the concept
are actually achievable.
It is important that the replacement anisogrid structure is designed to the same criteria
as the original structure, such that a fair mass comparison can be made between the
two.
6.1 A6 Airliner Description
The A6 airliner is the 2006/2007 AVD Group Design Project aircraft. The main intent
of the aircraft is to be environmentally friendly; this is mainly achieved through the
use of efficient engines and the extensive use of composite materials in the structure,
in a bid to lower the mass, and hence the fuel burn.
The aircraft also features very high aspect ratio wings, in order to reduce the induced
drag during the cruise. This means that the wings have a very large span, and
therefore develop very high bending moments close to the root, which generates very
high compression loads in the upper wing covers. The wing covers represent almost
56% of the total wing mass, and so any percentage reduction in the mass of the covers
is very desirable.
These factors make the wing covers of the A6 an excellent candidate to test the
benefits of the anisogrid lattice concept; the mass saved through using anisogrid
lattice construction is speculated to increase with load [1], and also the desire for
weight saving in the original design should mean the replacement anisogrid lattice can
be compared with the best available solution using conventional (i.e. black
aluminium) composite construction.
-
- 18 -
6.2 Original Cover Design
The original A6 composite wing cover design was performed by Raghuram
Dhanyamraju, and a full account of the design can be found in reference
[10]. A summary of the features important to this thesis will be made here.
The original A6 composite wing design has a two spar, distributed flange layout.
Tophat stringers are used, and close to the root there are 12 of these on each skin. The
rib pitch is 750mm. A cross section of the wing is shown below [10].
Figure 7: View of the interior of the A6 wing
It was decided to design an anisogrid panel to replace the skin and stringers between
the front and rear spars, and between two ribs. The section chosen is not the one right
at the root - in order to avoid the complex root fairing area and the landing gear
attachments, a section slightly further outboard has been taken. However, the
compressive loads due to bending are still very high in this area, and this led to a skin
thickness of 1.7cm, and stringer thickness of 1.2cm being required in the original
design.
The distance between the spars at this wing section is 3.8m, and with the
aforementioned rib pitch of 0.75m, this gives a panel area of 2.85m2. The total mass
Upper Wing Skin Tophat Stringers
Front Spar Rear Spar
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- 19 -
of this panel in the original design is 197kg, of which 62% is stringer mass and 38% is
skin mass.
6.2.1 Design Criteria
During the original wing design, the maximum loads the wing would ever be expected
to encounter in service were calculated, in accordance with airworthiness regulations.
These loads are termed Limit loads. The limit loads are then multiplied by a factor
of 1.5, to calculate the Ultimate Loads.
The design criteria for the wing, in terms of loading, were;
No material failure should occur under ultimate loads
Average strains must not exceed 0.4% at limit loads in order to account for
Barely Visible Impact Damage (BVID)
The structure must not buckle under limit loads
In order to simplify the original design, no account was made for degradation of
material properties due to thermal or moisture effects.
The bending moment diagram for the wing is shown in Appendix D. During the
original wing design it was estimated that the wing covers resist 86% of this bending
moment. Therefore, the replacement anisogrid panel will be designed to resist this
proportion of the loads also (i.e. it will be assumed that the upper and lower covers,
together, resist 86% of the bending moment).
The wing skins also resist the torque acting on the wing, and the shear due to lift in
the form of shear flow.
The loads acting on the upper skin and stringers at the section of the wing under
consideration (6m from fuselage centre-line) are;
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- 20 -
Compressive load due to bending of 5, 577 N/mm
Maximum Shear flow due to shear and torque of 800 N/mm
These same loads and design criteria will also be applied to the replacement anisogrid
lattice upper wing cover, in order to allow a fair mass comparison to be made.
The maximum and minimum principal strains, 1 and 2 , will be limited to 0.4%,
using equation (7) to calculate these from the average strain due to compression, y ,
and the average shear strain due to shear, xy .
For full calculation of the loads due to bending please see Appendix D.
-
- 21 -
7 Design of Anisogrid Lattice Replacement for
A6 Upper Wing Cover
7.1 Selection of Lattice Pattern
As aforementioned in Section 5, the methods used to develop the analytical model
allow the determination of the stiffness of almost any lattice pattern. However, due to
the time constraints of the project it is not possible to investigate the relative
advantages and disadvantages of various lattice patterns, and therefore a choice of one
has to be made.
It was decided to use the lattice pattern shown in Figure 5, as reference [8] describes
this as being sufficiently complex for most applications. Also, the stiffness matrices
have already been derived for this lattice pattern (these are given in appendix A).
This layout should be suitable, as it basically combines two lattice patterns known as
orthogrid and anglegrid [8] which are shown in Figure 8 below. The orthogrid has
high axial strength but low shear strength. Conversely, the anglegrid has high shear
strength, but low axial strength.
Figure 8: Orthogrid and Anglegrid Lattice types
Orthogrid Anglegrid
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- 22 -
The wing covers require high axial and high shear strength, and therefore combining
the orthogrid and anglegrid patterns into the pattern shown in Figure 5 should be a
good solution.
Also, it was decided to fix the angle phi at 60 degrees, again due to time constraints.
An angle of 60 degrees was chosen, as the wing is loaded mainly in uniaxial
compression and shear, and therefore a steep angle means the angular ribs contribute
more of their stiffness in resisting these loads (with the anisogrid oriented as shown in
Figure 9 below).
By fixing the angle phi at 60 degrees, setting one of the three rib pitches sets the other
rib pitches also, as it can be seen from Figure 5 that;
sin2 90dd = (8)
tan0
90
dd = (9)
Substituting equation 9 into equation 8, it is found that;
cos2sintan
2 00 d
dd == (10)
Furthermore, it can be seen from equation 10 that d is equal to 0d for an angle of 60
degrees (cos 60=0.5).
Therefore, the anisogrid lattice will be designed using the pattern shown in Figure 5,
and this will be oriented on the wing as shown in Figure 9 below.
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- 23 -
Figure 9: Orientation of anisogrid panel on upper wing
In order that no coupling between membrane loads and bending exists, it was decided
to use a double-skinned anisogrid lattice, with the top skin being symmetric with
respect to the bottom one. This makes all terms in the coupling matrix, [ ]B , equal to zero.
As the skin represents the external surface of the wing, it therefore needs to possess
good damage tolerance properties in order to resist damage from various impacts
ranging from hailstones to tool-drops during maintenance activities.
In order to achieve this, a quasi-isotropic laminate layup is used. The top skin
therefore takes the form n]45/45/90/0[ , and the lower skin is in the form
n]0/90/45/45[ .
The next section will discuss the materials and fibre volume fraction to be used.
Compression
due to
bending
Wing
Ribs Front Spar
Rear Spar o60= Wing spanwise direction
-
- 24 -
7.2 Carbon fibre grade and fibre volume fraction
The fibre volume fraction typically used in composite structures is between 60-65%.
In filament-wound lattice structures the fibre volume fraction in the ribs has to be
reduced somewhat, as where the lattice ribs meet, the fibre volume fraction is
effectively doubled.
A study by Vasiliev [2] shows that whilst specific stiffness of the ribs increases with
fibre volume fraction, the maximum specific strength of the ribs is achieved at 35%
fibre volume fraction, and therefore this is recommended for use. Changes to this will
not be investigated within this thesis, and a fibre volume fraction of 35% will be used
for the ribs.
The ribs will be manufactured from unidirectional laminates. In order to estimate the
stiffness and Poisson ratio of the resulting matrix/fibre composite, the Theory of
Elasticity is used. The formulae are given in Appendix B.
Figure 10: The effect of fibre selection on composite modulus, using epoxy matrix and Vf=35%
0
20
40
60
80
100
120
140
160
180
0 100 200 300 400 500
Fibre Modulus, E1 (GPa)
Composite Modulus, E1 (GPa)
Aluminium HS CF
IM CF
HM CF
UHM CF
-
- 25 -
The anisogrid lattice structure will be manufactured from carbon-fibre/ epoxy. Several
different grades of carbon fibres are available. An investigation into the effect of
using different grades on rib stiffness was performed, and the results are shown above
in Figure 10.
Using High Strength (HS) carbon fibres, it was found that the stiffness of the ribs is
approximately the same as if they were made of aluminium ( GPaE 80 ) as shown in
Figure 10. However, this is coupled with the fact that the resulting CFRP ribs have a
much lower density than aluminium, and so therefore still produce a weight
advantage.
The use of high modulus (HM) or ultra-high modulus (UHM) carbon fibres would
enable the modulus of the ribs to be greatly increased. However, the use of HM or
UHM fibres was rejected due to their extremely brittle nature, which leads to severe
processing difficulties. Therefore High Strength carbon epoxy will be used for both
skin and ribs.
The skins of the anisogrid lattice can be manufactured using a usual 65% fibre volume
fraction. The properties of both skin and rib materials were calculated using materials
data given in reference [11] for high strength carbon fibre and epoxy, and the elastic
properties of unidirectional laminates of skin and rib materials are given in Figure 11
below. In order to determine the stiffness of a quasi-isotropic skin laminate, normal
laminate theory is used, as shown in Appendix A.
Units Skin Material Rib Material
Longitudinal Youngs Modulus MPa 140, 000 80, 675
Transverse Youngs Modulus MPa 10, 000 5, 361
Shear Modulus MPa 5, 000 2, 264
Major Poissons ratio 0.3 0.285
Figure 11: Elastic Properties of Skin and Ribs
The following section will discuss the design variables used to design the structure.
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- 26 -
7.3 Design Variables
The anisogrid stiffness matrices derived (see Section 5.2) allow for the ribs in each
orientation to have different cross-sectional dimensions i.e. the ribs in the 0 degree
direction can have different thicknesses to those in the 90 degree direction.
However, it was decided that in order to ease the manufacturing difficulty of the
anisogrid structure, the ribs should all be of the same height, H, and width, b. Also, in
order to ease the optimisation process, the angle phi was set at 60 degrees. It is
possible that the use of a different angle could allow a lighter structure to be obtained,
but this is an area for further work.
Therefore, 4 variables are left which define the anisogrid structure. These are;
Zero degree rib pitch, d0
Total skin thickness, ts
Rib height, H
Rib width, b
Before beginning the sizing of the anisogrid structure, several constraints were
applied to these variables in order to ensure the structure designed is practical, as will
be discussed in the following chapter.
7.4 Design Constraints
7.4.1 Minimum Skin Thickness
Before commencing with the sizing of the anisogrid structure, it was necessary to
determine the minimum skin thickness that would be allowed (previous studies [2],[8]
have shown that the most efficient lattice structure is one with no skin at all). It was
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- 27 -
deemed that the most critical sizing criteria for the upper wing skin (disregarding the
loads) would be hailstone impact.
In order to determine the minimum skin thickness necessary to avoid damage from
hailstone impact, Reference [12] was found which details an investigation into the
relationship between hailstone kinetic energy and the onset of delaminations in the
impacted composite panel.
Using the data given in the paper, Figure 12 below was derived, which shows the
minimum skin thickness required to avoid damage when impacted by hailstones of
various diameters, at a range of relative velocities between the aircraft and the
hailstone.
Hail Impact
0
2
4
6
8
10
12
14
30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200
Relative Velocity (m/s)
Skin Thickness (mm)
0.5 inch
2.5 inches
Maximum horizontal relative speed
Max Vertical Relative Speed Limit
4.5 inches
Figure 12: Skin thickness required versus relative velocity
The A6 airliner has a maximum climb rate of around 30m/s, and from reference [12]
the terminal velocity achieved by a hailstone of 2.5inches diameter is 32m/s.The
maximum relative vertical velocity between a hailstone and the upper wing skin is
-
- 28 -
therefore taken to be around 62m/s (though hailstone terminal velocity increases with
diameter).
The three hail diameters shown in the graph, of 0.5 inches, 2.5 inches, and 4.5 inches,
correspond to small, medium, and large hail respectively.
It was decided to design the structure to withstand 2.5inches diameter hail impact at
the maximum relative vertical velocity, as 4.5inch hail is very uncommon and would
result in an overly conservative design. The structure will therefore be designed with a
skin thickness of at least 1 millimetre for each skin (top and bottom).
7.4.2 Euler buckling of Ribs
As mentioned in Section 6.2.1, the anisogrid lattice structure must not be allowed to
buckle when subjected to limit loads. The ribs will be checked for buckling using the
Euler buckling theory, assuming the ribs are beams, subjected to a compressive load,
and simply supported at each end;
2
2
L
IEP RIBcrit
=
(11)
This is a conservative assumption, as the ribs are supported top and bottom by the
skins and therefore the support conditions of the ribs lies somewhere between simply
supported and fully clamped.
Due to the time constraints of the project, it was not possible to formulate criteria for
the buckling of the skins between the ribs; it will therefore be assumed that if the ribs
do not buckle, nor will the skins. This is thought to be a reasonable assumption, as the
dense system of reinforcing ribs provides very good support to the skins. Also,
reference [2] suggests that often material failure will occur prior to buckling in lattice
structures.
-
- 29 -
7.4.3 Rib Beam constraint
In order that the Euler-Bernoulli beam theory used in the derivation of the Euler
buckling formula is applicable to the ribs, it is important that they can be considered
as beams. Therefore, the following criteria for ribs of rectangular cross-section and of
Height, H, width, b, and length, L, was decided, stating that the length of the ribs
(distance between supports) must be at least 5 times greater than the characteristic
length;
522
+ bH
L (12)
With reference to Figure 5 and Figure 9, it can be seen that the length of the ribs is
effectively controlled by the rib pitch.
7.5 Anisogrid Sizing and Optimisation Approach
In order to size and optimise the anisogrid lattice structure, an MS Excel spreadsheet
was developed which, when given an input of the four independent design variables,
(see Section 7.3), and the membrane, bending, and shear loads acting upon a unit cell,
calculates all the data required to predict the average strains present in the structure.
Figure 13 below is a flow chart showing the main functionality of the spreadsheet.
Due to the large number of formulae that had to be input into Excel in order to
achieve the desired outputs, it was possible that mistakes would be made when
building the spreadsheet.
In order to verify that the equations had been input into the Excel spreadsheet
correctly, they were also input into Matlab, and a copy of the script is provided in
-
- 30 -
Appendix H. It was checked that Excel and Matlab predicted the same strains for a
given anisogrid size, and a given set of loads, and a complete agreement was found.
Figure 13: Flow chart showing the operation and functionality of Excel Spreadsheet developed
for sizing and optimising anisogrid lattice structure
With the knowledge that the analytical model had been correctly input into the Excel
spreadsheet, it was possible to begin the sizing of the structure. The loads acting upon
the upper wing cover, from Section 6.2.1, were therefore input. The design constraints
Input 4 independent design
variables: d0, b, H,ts
Excel Calculates other constants dependant on input
variables, such as; Rib areas, Moments of areas, laminate
layup, Mass of unit cell
Calculates Skin A, D matrices, Rib A, D, H matrices, sums
to obtain full anisogrid stiffness matrices. Then inverts
total stiffness matrices
Output Strains;
Direct strains: xyyx ,,
Curvatures: xyyx kkk ,,
Shear strains: yzxz ,
Input Loads;
Membrane: xyyx NNN ,,
Bending: xyyx MMM ,,
Shear: yx VV ,
Are all
constraints
satisfied? No
Change input
variables
-
- 31 -
(strain limits, minimum skin thickness etc) could then be satisfied by varying the four
input variables.
Clearly, there are a huge number of different combinations of rib pitches, rib
dimensions, and skin thicknesses that would enable all the design constraints to be
satisfied. Therefore, in order to try and determine the solution that would produce a
unit cell of minimum mass and yet still satisfy all the design constraints, the solver
tool in Excel was used.
The solver tool is designed to determine the minimum value of a function, whilst
satisfying given constraints, by varying up to 500 variables that are included in the
function.
Therefore, a function which determines the mass of a unit cell from the 4 independent
variables (all other dimensions are dependant on these) was derived (the formula is
given in Appendix C), and the constraints given in Sections 6.2.1 and 7.4 were
applied. The solver tool was then set up to determine the minimum of the mass
function.
In order to ensure that the solver determined the solution of the lowest possible mass,
and did not just find local minima, a large number of runs were conducted, varying
slightly the initial constraint conditions (such as minimum rib pitch).
It was noted that for a given set of constraints, solver does converge to the same
solution repeatedly. However, it was also found that solver will not run if a feasible
solution that satisfies all the constraints is not first input into the spreadsheet.
Therefore, a solution which satisfies all constraints has to be found and input
manually; then solver can be run and will vary the 4 inputs in order to minimise the
mass.
A selection of the different solutions obtained using the solver tool is given in the
following section.
-
- 32 -
8 Optimisation Results
A large number of solver runs were performed, each time varying slightly the initial
constraints in order to vary the optimisation process performed by Excel, and give a
good possibility of the lightest available solution being found.
No. Total skin
thickness, ts,
mm
Rib Height,
H, mm
Rib width, b,
mm
Zero degree
rib pitch, d0,
mm
Mass of unit
cell, kg
1 7.00 30 24 134 2.19
2 16.50 32 14 149 2.54
3 14.00 30 18 153 2.68
4 14.50 41 17 192 4.31
5 21.25 30 30 184 5.54
Figure 14: Selection of optimisation results
A selection of results generated using the solver tool is given in Figure 14 above. It
can be seen that case number 1 represents the lightest solution. This also satisfies all
the constraints laid out in Sections 6.2.1 and 7.4, and therefore this structure was
selected.
A total of 68 unit cells have to be used to create an anisogrid lattice panel of the size
of the wing panel being replaced. This results in a panel mass of 151kg, which
represents a 23% mass saving over the original wing cover design (197kg).
In order to validate the analytical model used to perform the design of the anisogrid
lattice, Finite Element modelling was used. The next section will discuss the
modelling carried out, and the results obtained.
-
- 33 -
9 Finite Element modelling
Linear-static Finite Element (FE) modelling of the anisogrid lattice was performed, in
order to validate the analytical model, and also to apply the lattice structure to the
curved upper wing surface. In order to do this, two different models were created; an
exact model, and an equivalent model.
The MSC Patran pre- and post-processor was used, together with the Nastran solver.
9.1 Description of models
Firstly, an exact model of a large, flat plate of unit cells was created using shell
(Quad4) elements to represent ribs and skins. The exact model gives more accurate
results, and also shows the stresses in each individual component (ribs and skins) and
therefore allows stress concentrations to be seen. However, the model is time
consuming to create and run, due to the large number of elements required to simulate
all the individual ribs and skins.
Also, the model cannot be altered easily - for example, changing the rib height would
require the redrawing of each rib, therefore making the model almost useless for
optimisation purposes. The exact model would also be very difficult to apply to the
curved wing surface. Therefore, once the analytical model was validated using the
exact FE model, an equivalent FE model was then created.
The equivalent model simply uses a flat surface meshed by shell elements, but which
has an equivalent stiffness (in membrane and shear) to the exact model. The
equivalent model, therefore, no longer shows the stresses present in ribs and skins
individually, but rather shows average stresses throughout the whole anisogrid
structure.
-
- 34 -
However, the equivalent model takes much less computational time, and is also easy
to apply to the curved wing surface in order to determine the effect (if any) of the
curvature of the wing surface on the stresses and strains induced.
The equivalent model is also much more appropriate for optimisation, as in order to
change the height of the ribs, for example, a change in the material properties to
represent the change in stiffness is all that is required.
9.2 Exact Finite Element model
The exact FE model was created using Quad4 shell elements to represent both ribs
and skins. Firstly, a single unit cell was created. This was then copied a number of
times in order to create a large, flat plate of 28 unit cells, as shown in Figure 15
below. An engineering drawing of the structure is provided in Appendix I.
Figure 15: Exact FE model of 28 unit cells (shown with one skin hidden)
The reason for creating a large (roughly 1m2) plate of unit cells was that large local
stresses and strains are developed where forces are applied to a node. In order to
simulate a unidirectional, uniformly distributed membrane load, the plate is clamped
at one edge, and each node along the opposite edge is loaded with a force. The edge of
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- 35 -
the plate therefore incurs local areas of stresses and strains which are incomparable
with the analytical model, as this assumes uniformly distributed loads are applied.
It is possible to avoid these local effects by considering a unit cell in the middle of the
plate. In this area, the stress distribution is more uniform, in accordance with Saint
Venant theory. The unit cell in the middle of the plate can therefore be analysed to
verify the analytical model.
The replacement wing cover is subjected only to a compressive membrane load due to
bending, Ny, and a shear flow, Nxy. However, in order to fully validate the analytical
model under membrane loads, compression in the other direction, Nx, was also
investigated, and the results are given in the following section.
Bending moment is not directly applied to the wing covers, but it was decided to
attempt to validate the analytical model in bending for future use, and the results of
this can be seen in Appendix E.
9.2.1 Results of Exact FE Model under Membrane Loads
In order to verify the analytical model under membrane loads, 5 different load cases in
both the x and y directions were run on the Exact FE model.
The strains and displacements calculated by the FE model were then compared with
the strains and displacements predicted by the analytical model in the Excel
spreadsheet.
When reading displacements, average values were taken along the edge of the plate in
order to avoid effects of stress concentrations at points of load application. Average
strains were read by considering one of the unit cells close to the middle of the plate.
More accurate reading of the results was achieved by using the plot markers
function in Patran, which displays the strains present in each element.
-
- 36 -
A table of the results obtained and comparison with the analytical model is given in
Figure 16 below. (Please note that all loads are compressive, and therefore strains are
compressive but are quoted positive for ease of reading).
Average Strains Average Displacements
Case Nx,
N/mm x Excel, *10
-3
x Patran, 10
-3
%
difference xs Excel,
mm
xs Patran,
mm
1 1000 1.14 1.12 2 1.24 1.22
2 2000 2.29 2.26 1 2.48 2.47
3 3000 3.43 3.39 1 3.72 3.68
4 4000 4.58 4.50 2 4.97 4.88
5 5000 5.72 5.65 2 6.21 6.13
Ny,
N/mm y Excel, *10
-3
y Patran, *10
-3
%
difference ys Excel,
mm
ys Patran,
mm
6 2000 1.24 1.21 2 1.33 1.20
7 3000 1.86 1.80 3 2.00 1.93
8 4000 2.48 2.24 3 2.67 2.58
9 5000 3.11 2.99 3 3.33 3.22
10 55772 3.46 3.36 3 3.72 3.61
Figure 16: Comparison of strains and displacements predicted by FE and analytical model
The variation in percentage difference is thought to be mainly due to the difficulty
of consistently interpreting the finite element output in exactly the same way.
It can be seen that a very good agreement was found between the analytical and the
exact FE model, under membrane loading in both directions.
Figure 17 below shows the minimum principal strains throughout the exact FE model,
for load case 10 (see Figure 16). The effect of the loads on the edge of the plate can be
clearly seen, causing very high local strains. Also, the fully clamped boundary
conditions on the opposite edge of the plate cause similar local effects.
It can be seen however, that away from the edges of the plate the strain distribution is
much more uniform, and therefore more comparable with the analytical model.
2 Corresponding to actual applied load resulting from bending of wing
-
- 37 -
Figure 17: Minimum Principal strains in exact FE model for Case 10 (Compressive Ny)
In order to avoid the local effects around the edges of the plate, and view the strain
distribution throughout a unit cell in more detail, it was possible to plot only the
middle unit cell in Patran, and this is shown in Figure 18 below.
Figure 18: Min Principal Strains in skin of middle unit cell (Case 10)
-
- 38 -
Using the plot markers function, it was found that the strains in the skin (the white
region in Figure 18) agree very well with the analytical model.
One of the design criteria is that the strains in the anisogrid lattice should be limited to
a maximum value of 0.4%. However, the analytical model predicts only average
strains, and therefore, limits the average strains to 0.4%. There may exist stress
concentrations in the lattice structure, which cause the local strains to increase above
this.
In order to determine whether this is the case, and to what extent stress concentrations
exist, the average strains found for Cases 1 to 10 and listed in Figure 16, were
compared with the maximum strains present in the middle unit cell. A ratio was then
taken between the maximum strain and the average strain for each case, and the
results are listed in Figure 19 below.
Case Average Strain
(Patran)*10-3
Maximum Strain
(Patran) *10-3
Maximum strain
/Average strain
Nx x xMAX
1 1000 1.12 1.14 1.018
2 2000 2.26 2.27 1.004
3 3000 3.39 3.41 1.006
4 4000 4.50 4.55 1.011
5 5000 5.65 5.69 1.007
Ny y yMAX
6 2000 1.21 1.23 1.017
7 3000 1.80 1.82 1.011
8 4000 2.24 2.28 1.018
9 5000 2.99 3.00 1.003
10 5577 3.36 3.37 1.003
Figure 19: Comparison between maximum and average strains predicted by exact FE model
The average ratio of maximum to minimum strain is 1.01 (i.e. maximum strains are
1% higher than average). Therefore, it can be concluded that under membrane
loading, limiting the average strains to 0.4% is probably adequate as no major
stress/strain concentrations exist.
-
- 39 -
9.2.2 Results of Exact FE Model under Shear Loads
In order to simulate pure shear on the exact FE model, different boundary conditions
to the membrane cases were used. Only three nodes, on three corners of the plate,
were assigned boundary conditions, and each edge was subjected to an equal shear
flow of 800 N/mm. The boundary conditions used are shown in Figure 20.
Figure 20: Boundary conditions for pure shear
As with membrane, a number of different runs with different shear loads were
performed, and the results can be seen in Appendix F.
Figure 21: Middle Unit cell of Exact FE model under pure shear
800 N/mm
0=== zyx
Rotations allowed
0=z Rotations allowed
0== zy
Rotations allowed
x
z
y
-
- 40 -
A very good agreement was found with the analytical model, as with membrane
loading. Figure 21 above shows the results for the load case corresponding to the
loads applied on the wing covers; 800N/mm. An average strain in the xy
direction, xy , of around 2.4 milli-strain is found, which agrees reasonably well with
the Excel prediction of 2.55milli-strain.
However, the analytical model and the exact FE model do not agree quite as well in
shear as they do in membrane - with an average of around 5% difference for the cases
investigated. However, this is still an acceptable difference, and due to the time
constraints of the project it is not possible to investigate the reasons for this small
discrepancy.
9.2.3 Conclusions of Exact FE modeling
An exact FE model of a flat plate of 28 unit cells was created, in order to validate the
analytical model.
Results were taken from the middle of the plate, in order to avoid local edge effects,
and obtain a fair comparison with the predictions of the analytical model.
The exact FE model has been compared with the analytical model under membrane,
bending, and in-plane shear loads. (For bending results see Appendix E). An excellent
agreement has been found in all cases investigated.
Therefore, the analytical model, and Excel spreadsheet in which the analytical model
is implemented, have both been validated. It can be concluded therefore that the Excel
spreadsheet is a useful initial design tool for anisogrid lattice structures, accurately
predicting the average strains in the structure due to applied loading.
The exact FE model shows that no apparent problem areas exist in terms of stress and
strain concentrations within the anisogrid lattice structure.
-
- 41 -
9.3 Equivalent Finite Element model
After validation of the analytical model using the exact FE model, an equivalent
stiffness FE model was also created. The aim of the equivalent FE model was to
reduce the properties of the anisogrid lattice into a single surface of shell elements.
The advantages of doing this are that the equivalent FE model can then be easily
applied on the upper wing surface, thus avoiding the difficulties of creating the
geometry of an anisogrid lattice structure with the curvature of the wing surface,
which an exact model would require. Also, the equivalent stiffness model requires
much less processor time to run, and is much more suitable to use for optimisation.
Only an accurate representation of membrane and shear stiffness was required, as
these are the only loads applied to the wing cover being redesigned. The method
derived therefore only provides correct properties under membrane and shear loading.
The way in which this was achieved was to effectively smear the ribs of the lattice,
such that the gaps between the ribs are filled, and the ribs can be effectively
considered as another continuous layer of skin, whilst the whole structure maintains a
constant volume. Therefore, an equivalent thickness of the anisogrid lattice structure
is created. Figure 22 below illustrates this concept.
Figure 22: Illustration of Equivalent stiffness concept
Unit cell
Ribs
Skin Smeared Unit cell
Equivalent
thickness
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- 42 -
It was then necessary to provide this smeared plate of equivalent thickness with
materials properties that would then accurately represent the behaviour of the
anisogrid lattice structure, in terms of stiffness.
As the anisogrid lattice structure is anisotropic, it was therefore necessary to define a
2 dimensional orthotropic material in order to provide the correct behaviour in both x
and y directions. The equivalent stiffness was determined by taking a ratio of the
Youngs modulus of skin and rib materials in each direction, based on their areas. The
equations derived for calculating the equivalent thickness, and stiffnesses are given in
Appendix G.
The materials properties found were:
E1 = 45,664MPa; E2 = 24,115MPa; G12 = 17,727MPa
Shell elements can then be assigned the equivalent thickness, and the orthotropic
material properties in Patran, and then used to replace the global FE model. In order to
validate the process, an Equivalent FE model was created to replace the exact FE
model, and the same 10 membrane load cases as applied to the exact FE model (given
previously in Figure 16) were applied also to the equivalent model. The model was
also tested in shear, and the results are summarised over the following sections.
9.3.1 Results of Equivalent FE Model under Membrane Loads
As with the exact FE model, the equivalent model was fully clamped along one edge,
and membrane loads applied along the opposite edge. It was found that the edge
effects were not as severe on the equivalent FE model as on the exact FE model, with
only the corners of the plate experiencing excessive deformation.
Therefore the area in the middle of the loaded edge of the plate undergoes a constant
displacement, and therefore the displacements were read from this region. This area is
circled in Figure 23.
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- 43 -
Figure 23: Plot of displacements of Equivalent FE model under membrane loading (Case 10)
Figure 24 below shows a close up of the area circled in Figure 23, using the plot
markers function in Patran to show the displacements at each node. It can be seen
that constant displacements occur in this area, allowing accurate results to be taken.
Figure 24: Close up of the circled area in Figure 23, showing constant displacements (Case 10)
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- 44 -
Strains were read from the centre of the plate, as with the exact FE model, in order to
completely avoid any edge effects. The same 10 cases as with the exact FE model
were run, and the results are given below, in Figure 25.
Figure 25: Results of equivalent FE model under membrane loads
It was quickly noticed that the strains and displacements vary perfectly linearly with
applied load - this is of course expected to be the case, but this linear relationship was
not observed quite as perfectly in the exact FE model due to the more pronounced
edge effects, and the difficulties of therefore reading strains and displacements in the
same way each time.
As this linear behaviour is observed, it was therefore easy to correct the difference
between the analytical model and the equivalent FE model, simply by using algebra to
correct slightly the Youngs modulus values in the equivalent stiffness model. This
was done, and the equivalent model now agrees exactly with the analytical model
under membrane loading (this can be seen in Figure 24 which actually shows the
corrected equivalent FE model for Case 10).
Average Strains Average Displacements
Case Nx x Excel, *10
-3
x Patran, *10
-3
% xs Excel, mm xs Patran,
mm
1 1000 1.14 1.25 2 1.24 1.22
2 2000 2.29 2.25 2 2.48 2.44
3 3000 3.43 3.37 2 3.72 3.66
4 4000 4.58 4.50 2 4.97 4.88
5 5000 5.72 5.62 2 6.21 6.10
Case Ny y Excel, *10
-3
y Patran, *10
-3
% ys Excel, mm ys Patran,
mm
6 2000 1.24 1.23 1 1.33 1.32
7 3000 1.86 1.84 1 2.00 1.98
8 4000 2.48 2.46 1 2.68 2.64
9 5000 3.11 3.07 1 3.33 3.30
10 5577 3.46 3.43 1 3.72 3.68
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9.3.2 Results of Equivalent FE Model under Shear Loads
The same boundary conditions for shear were applied as in the exact FE model, and
equal shear loads were applied to each edge. Figure 26 below shows the equivalent FE
model under shear load; it is apparent that there are again some local effects due to the
effects of the constraints in three of the corners. However, a good agreement is found
in the rest of the plate (dark green area).
Figure 26: Equivalent FE model under shear load
As in the earlier case of membrane loading, a slight discrepancy was found between
the Excel predictions and the equivalent FE model in shear; this difference was
reduced by correcting the equivalent shear modulus value in the equivalent FE model.
9.3.3 Conclusions of equivalent Finite Element modelling
A simple method has been derived for providing a surface meshed with shell elements
the equivalent stiffness properties of the analytical and exact FE models of the
anisogrid structure, under membrane and shear loading.
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- 46 -
The equivalent model has been validated against the analytical model, and a very
good agreement was found. Small corrections have then been made to the equivalent
properties of the shell elements, in order to increase the degree of agreement with the
analytical model even further, under both membrane and shear loading.
As intended, the equivalent FE model requires much less processor time than the
exact FE model, due to the fewer number of elements, whilst still correctly predicting
the average strains undergone by the anisogrid structure.
The equivalent FE model is also much more suitable for optimisation purposes, as to
incorporate changes to the dimensions of the anisogrid structure into the equivalent
FE model, a change in materials properties and equivalent thickness of the shell
elements is all that is required.
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- 47 -
10 Equivalent FE Modelling of replacement A6 upper wing cover
10.1 Aims of Modelling wing cover
With the analytical model validated using an exact FE model, it was desirable to
model also the replacement upper wing cover in FE, in order to determine if the
curvature of the upper wing surface has any effect on the strains developed.
Unfortunately, due the time constraints of the project, it was only possible to apply the
compressive membrane load due to bending to the model.
10.2 Model Creation and boundary conditions
Due to the difficulties of creating an exact model of the anisogrid lattice structure with
the complex curvature of the upper wing surface, the equivalent FE approach
developed earlier was used.
The CATIA model of the replacement upper wing cover was imported to Patran from
the original A6 aircraft model, as a surface.
The surface was then meshed with Quad4 shell elements, and the properties of the
equivalent stiffness model were applied to these elements.
It was not possible to apply the membrane load whilst simply clamping the wing
surface at one end, as was done in the previous cases. The reason for this is that the
wing surface is not perfectly aligned with the horizontal (y) axis, and therefore
applying a load along this axis would impart a bending moment on the surface.
Therefore, the surface would bend instead of being compressed as intended.
This problem was avoided by clamping the wing surface along one edge, and by then
constraining the movement of all remaining nodes in the z axis. As the angular offset
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of the wing from the y axis is very small, this has a negligible effect on the strains
developed.
This was verified by plotting the constraint forces, in the z axis, in Patran. The results
are shown in Figure 27 below, and it can be seen that the forces required to prevent
movement in the z axis are very small (a maximum of only 5.92 Newtons).
Figure 27: Constraint forces in z axis on equivalent wing cover FE model
10.3 Results of equivalent FE modelling of replacement A6
upper wing cover
The compressive load due to bending, Ny, of 5577 N/mm was applied to the
equivalent wing model, and an excellent agreement between the displacements and
strains was achieved, with respect to the analytical model.
Figure 28 below shows a plot of the displacements undergone by the surface. The
average displacement along the middle of the wing edge (again avoiding the effects at
the corners) is 2.57mm. The prediction from the analytical model is 2.60mm.
-
- 49 -
Figure 28: Plot of displacements in Equivalent wing cover FE model
The minimum principal strains due to compression were also plotted, and these are
shown in Figure 29 below. Again a very close agreement is found, with the analytical
model predicting 3.46milli-strain and the FE model showing 3.34milli-strain.
Figure 29: Minimum principal strains in Equivalent wing cover FE model
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- 50 -
10.4 Conclusions of Equivalent FE Modelling of replacement A6 upper wing cover
The replacement A6 anisogrid lattice upper wing cover was modelled in Patran, using
the equivalent stiffness FE method.
An excellent agreement between the analytical and equivalent FE models was found,
when applying a compressive membrane load due to bending. This therefore suggests
that the curvature of the wing has no significant effect on the strains induced due to
compression.
The anisogrid lattice design obtained by using the analytical model implemented in
the Excel spreadsheet is therefore adequate to withstand the applied loads, whilst
satisfying all constraints.
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- 51 -
11 Comparison between original A6 wing cover
design and replacement Anisogrid Lattice
The FE analysis has led to the conclusion that the replacement A6 anisogrid lattice
cover designed using the analytical model adequately meets all design criteria, and
therefore is an adequate replacement of the original cover design.
A comparison of the mass breakdown of the original and replacement cover designs is
made in Figure 30 below.
Original Design Replacement Design
Skin thickness, mm 17 7
Stringer / Rib thickness, mm 12 24
Skin Mass, kg 122 33
Stringer / Rib mass, kg 75 118
Total Mass 197 151
Figure 30: Mass breakdown of original and replacement A6 wing cover
Overall, the replacement anisogrid lattice cover design achieves a 23% weight saving
over the original black aluminium cover design.
Therefore, the 40% weight saving speculated by Vasiliev [2], has not been achieved.
A 23% weight saving is still very significant however; assuming that the same
percentage weight saving can be achieved by redesigning the whole A6 wing cover
using the anisogrid lattice method, a saving of around 3,100kg can be made on the
original cover ma