anisogrid

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.

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

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.

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.

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.

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

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

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

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

- 1 -

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

- 2 -

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

- 3 -

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

- 4 -

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.

- 5 -

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]

- 6 -

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].

- 7 -

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

- 8 -

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

- 9 -

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;

- 10 -

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.

- 11 -

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

- 12 -

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

- 13 -

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

- 14 -

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

- 15 -

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.

- 16 -

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.

- 17 -

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

- 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;

- 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

- 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.

- 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.

- 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

- 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

- 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

- 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.

- 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)

- 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

- 45 -

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.

- 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.

- 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

- 48 -

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

- 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.

- 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