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A study of locomotiveundercarriage equipment

case design using compositematerials and finite element

analysis

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Additional Information:

• A Master's Thesis. Submitted in partial fulfilment of the requirements forthe award of Master of Philosophy of Loughborough University.

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Publisher: c© Carolyn Hardaker

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LOUGHBOROUGH UNIVERSITY OF TECHNOLOGY

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I

A STUDY OF LOCOMOTIVE UNDERCARRIAGE EQUIPMENT CASE DESIGN

USING COMPOSITE MATERIALS AND FINITE ELEMENT ANALYSIS.

by

Carolyn Hardaker B.Sc (Hons)

A Thesis submitted in partial fulfilment of the requirements for the award of

Master of Philosophy of the Lough borough University of Technology.

September 1990

(c) by Carolyn Hardaker 1990

Loughborough University of Technol0qy Library

Date Mo-...,, Class

~~ o3.6ooo76~

ACKNOWLEDGEMENTS

I wish to express my thanks to Dr. P.W. ShaJlllan of the Depanment of Transport Technology for the advice, direction-and suppon he has given throughout the research. f \._)

Thanks are also due to Brush Electrical Machines Ltd. for providing sponsorship and Computing Services at Leicester Polytechnic for their help and co-operation.

Finally, I wish to thank my parents, Jen and John and friends Andy and Manin for helping in the production of this thesis.

SUMMARY

The research, commissioned by Brush Electrical Machines Ltd. of Loughborough, centres around

the design of locomotive undercarriage equipment cases. These cases are generally of mild steel

construction and are used to carry a wide variety of equipment from electromechanical switchgear

to electronic monitoring equipment. Reviewing their design showed that they tend to be

overdesigned, complex structures, with their manufacture ahd assembly being labour intensive and

costly. In the competitive traction engineering market, with severe weight penalties featuring in all

present day contracts it is important minimise weight and 'costs. Hence, it was proposed to

investigate the possibility of redesigning a cas·e in composite material in order to produce a light

weight, less complex structure to satisfy the standard railway service loads at a reduced cost.

Finite Element Analysis was used extensively in the research, initially to evaluate the structural

integrity of a typical steel case indicating the weak points of the design and providing an accurate

value for the mass of the case, it was then used in the development of an equivalent composite

model. However, as the Finite Element Analysis of composite structures is a relatively new field, it

was necessary to perform extensive software testing as a precursor to composite case development

in order to assure accuracy of results in terms of stress and displacements. Once confidence in the

software had been established an experimental model was developed from uni-directional and

woven cloth Glass/Epoxy composites, this was analysed and compared with the earlier analysis of

the typical case.

The case developed was found to be equivalent in structural integrity to the isotropic case analysed,

both were shown to have acceptable stress levels, minimal deflections and satisfactory fatigue life

when subjected to the standard railway service loads. The mass of the composite model, was

calculated to be approximately 72% of the equivalent isotropic case, which would save a ballpark

IV

figure of IOOkg per case, with funher investigation in alternative composite materials leading to

additional weight reduction. This advantage combined with a reduction in internal complexity

through integral moulded equipment attachment points and a high resistance to corrosion make the

proposition of a composite undercarriage equipment case viable. Hence it can be concluded that in

the long term, the use of composite materials in an equipment case application would result in a

reduction of both manufacturing and operational costs, although it should be noted that initial

invesnnent would be required for further development work and tooling.

The research concludes also that Finite Element Analysis is an effective tool to use in the design and

development of both isotropic and composite structures, providing an accurate stress and

displacement analyses, without the need for expensive prototypes.

v

NOTATION

M Bending Moment

m Moment Intensity

EF Young's Modulus of the Fibre

Et.t Young's Modulus of Matrix

VF Fibre Volume fraction

VM Matrix Volume fraction

E1 Young's Modulus of the material in the fibre direction.

E2 Young's Modulus of the material transverse to the fibre direction.

G12 In - plane Shear Modulus

~~ Strain in the fibre direction.

E2 Strain of the material transverse to the fibre direction.

¢1 Shear Strain .

01 Direct Stress in the fibre direction.

02 Direct Stress in the material transverse to the fibre direction.

"TI2 In plane Shear Stress.

vi

LIST OF CONTENTS

Page

Summary IV

Notation vi

Chapter 1. Introduction to the Research. 1-1

Chapter 2. An Overview of Equipment Case Design. 2.1 Introduction 2-1 2.2 Main Features of Design 2-1 2.3 Review of Possible Design Improvements 2-4 2.4 Conclusions 2-7

Figures 2-8

Chapter 3. Analysis of an Isotropic Case. 3.1 Introduction 3-1 3.2 The Equipment Case 3-1 3.3 Finite Element Modelling 3-2 3.4 Load Cases 3-3 3.5 Load Case Idealisation 3-4 3.6 Results 3-5 3.7 Discussion 3-10 3.8 Conclusions 3-11 Figures 3-12

Chapter 4. Orthotropic Element Testing. 4.1 Introduction 4-1 4.2 Orthotropic Element Selection 4-1 4.3 The Element Test Model 4-2 4.4 Element Tests 4-4 4.5 Theoretical Effects 4-6 4.6 Results 4-7 4.7 Discussion 4-12 4.8 Conclusions 4-13 Figures 4-14

Chapter S. Composite Case Design. 5.1 Introduction 5-1 5.2 Case Rationalisation 5-1 5.3 Preliminary Design Calculations 5-2 5.4 Side Structure Idealisation 5-4 5.5 Skeletal Model Idealisation 5-10

. 5.6 Results 5-12 5.7 Discussion 5-17

vu

Chapter 6.

Chapter 7.

References

Appendices

5.8 Conclusions Figures

Design Proposals and Recommendations. 6.1 Introduction 6.2 Design Features 6.3 Further Refinements to Design

Conclusions.

VlI

5-17 5-18

6-1 6-1 6-5

7-1

8-1

9-1

LIST OF FIGURES

Figure Number Page

2.1 A General View of a Typical Equipment Case 2-8 ( H6 Tube Stock)

2.2 Three Views of a Typical Structure (ALRV ) 2-9 2.3 Internal Detail of the ALRV Case 2-10 2.4 Side View of a Typical Pannier showing Internal Mounting 2-11

Arrangements ( Class 318 ) 2.5 Composite Contactor Housing (ALRV ) 2-12 2.6 Pannier under assembly (ALRV) 2-12 2.7 View of the Internal Strucmre of a Case ( ALRV ) 2-13

3.1 Quarter Case Idealisation 3-12 3.2 Full Model Idealisation 3-13 3.3 Stress Contour Plot for the Full Case 3-14

( Longitudinal Load Case) 3.4 Displacement Plot for the Vertical 1.5g down Load Case 3-15 3.5 Displacement Plot for the Lateral Load Case ( Idealisation I) 3-16 3.6 Displacement Plot for the Lateral Load Case ( Idealisation 2) 3-17 3.7 Displacement Plot for the Longitudinal Load Case 3-18 3.8 Graph of UX and UY Displacements along the Upper Beam 3-19

Vertical 1.5g down Load Case 3.9 Graph of UX and UY Displacements along the Central Web 3-20

Vertical 1.5g down Load Case 3.10 Graph of UX and UY Displacements along the Lower Beam 3-21

Vertical 1.5g down Load Case 3.11 Graph of Von Mises Stress along the Upper Beam 3-22

Vertical 1.5g down Load Case 3.12 Graph of Von Mises Stress along the Central Web 3-23

Vertical 1.5g down Load Case 3.13 Graph of Von Mises Stress along the Lower Beam 3-24

Vertical1.5g down Load Case 3.14 Graph of UX and UY Displacements along the RH Upper Beam 3-25

Lateral Load Case 3.15 Graph of UX and UY Displacements along the LH Upper Beam 3-26

Lateral Load Case 3.16 Graph of UX and UY Displacements along the Central Web 3-27

Lateral Load Case 3.17 Graph of UX and UY Displacements along the RH Lower Beam 3-28

Lateral Load Case 3.18 Graph of UX and UY Displacements along the LH Lower Beam 3-29

Lateral Load Case 3.19 Von Mises Stresses at a Support Location 3-8 3.20 Graph of Von Mises Stress along the RH Upper Beam 3-30

Lateral Load Case

ix

Figure Number Page

3.21 Graph of Von Mises Stress along the RH Lower Beam 3-31 Lateral Load Case

3.22 Graph of Von Mises Stress along the Central Web 3-32 Lateral Load Case

3.23 Graph of Von Mises Stress along the LH Upper Beam 3-33 Lateral Load Case

3.24 Graph of Von Mises Stress along the LH Lower Beam 3-34 Lateral Load Case

3.25 Graph of UX, UY and UZ Displacements along the 3-35 RH Upper Beam, Longitudinal Load Case

3.26 Graph of UX, UY and UZ Displacements along the 3-36 RH Central Web, Longitudinal Load Case

3.27 Graph of UX, UY and UZ Displacements along the 3-37 RH Lower Beam, Longitudinal Load Case

3.28 Graph of Von Mises Stress along the RH Upper Beam 3-38 Longitudinal Load Case

3.29 Graph of Von Mises Stress along the RH Central Web 3-39 Longitudinal Load Case

3.30 Graph of Von Mises Stress along the RH Lower Beam 3-40 Longitudinal Load Case

3.31 Fatigue Analysis Data 3-10

4.1 Element Test Model 4-2 4.2 Material Constants for a Graphite / Epoxy Composite 4-3

(VF = 60%) 4.3 Lamina / Laminate Lay - Ups Used 4-3 4.4 Loading and Restraints for Tension and Bending Tests 4-4 4.5 Tests Performed 4-5 4.6 Mid Ply Stress Results for Tension Tests 1,3 and 5 4-14 4.7 Stress Results for the Symmetric Laminate in Bending, Test 2 4-15 4.8 Stress Results for the Generally Orthotropic Lamina in Bending, 4-16

Test 4 4.9 Displacement Results for the Tension Tests I, 3 and 5 4-17 4.10 Displacement Results for the Bending Tests 2 and 4 4-18 4.11 Symmetric Laminate in Tension, Test I, 36215 Element 4-19 4.12 Symmetric Laminate in Tension, Test 1,44215 Element 4-20 4.13 Symmetric Laminate in Tension, Test 1,43215 Element 4-21 4.14 Generally Orthotropic Lamina in Tension, Test 3, 36215 Element 4-22 4.15 Generally Orthotropic Lamina in Tension, Test 3,44215 Element 4-23 4.16 Generally Orthotropic Lamina in Tension, Test 3, 43215 Element 4-24 4.17 Un symmetric Laminate in Tension, Test 5, 36215 Element 4-25 4.18 Unsymmetric Laminate in Tension, Test 5, 44215 Element 4-26 4.19 Unsymmetric Laminate in Tension, Test 5, 43215 Element 4-27 4.20 Symmetric Laminate in Bending Test 2, 44215 Element 4-28 4.21 Symmetric Laminate in Bending Test 2, 43215 Element 4-29 4.22 Generally Orthotropic Lamina in Bending Test 4, 44215 Element 4-30

x

Figure Number Page

4.23 Generally Orthotropic Lamina in Bending Test 4,43215 Element 4-31

5.1 Preliminary Design Calculation Model 5-2 5.2 Longitudinal Beam Idealisation - First Model 5-19 5.3 Support Structure Idealisation - First Model 5-20 5.4 Longitudinal Beam Idealisation - Refined Model 5-21 5.5 Support Structure Idealisation - Refined Model 5-22 5.6 Skeletal Model Idealisation 5-23 5.7 Displacement Plot Vertical 1.5g Load Case - First Model 5-24 5.8 Displacement Plot Vertical1.5g Load Case - Refined Model 5-25 5.9 Displacement Plot Longitudinal Load Case - Refined Model 5-26 5.10 Displacement Plot Lateral Load Case - Skeletal Model 2 5-27 5-11 Direct Stress Distribution at the Support and along the Upper Flange 5-28

A Longitudinal Beam, Vertical Load Case - First Model 5.12 Direct Stress Distribution at the Support and along the Upper Flange 5-29

A Longitudinal Beam, Vertical Load Case - Refined Model 5.13 Direct Stress Distribution at the Support and along the Upper Flange 5-30

A Longitudinal Beam, Longitudinal Load Case - Refined Model 5.14 Direct Stress Distribution at the Support and along the Upper Flange 5-31

Skeletal Model, Lateral Load Case - Model 1 5.15 Direct Stress Distribution at the Support and along the Upper Flange 5-32

Skeletal Model, Lateral Load Case - Model 2 5.16 Graph of Vertical ( UY ) Displacements along the Upper Flange 5-33

First Model, Vertical 1.5g down Load Case 5.17 Graph of Vertical ( UY ) Displacements across the Support 5-34

First Model, Vertical1.5g down Load Case 5.18 Graph of Lateral ( UZ ) Displacements through the Choke Load 5-35

Path, First Model, Vertical 1.5g down Load Case 5.19 Graph of Vertical (UY ) Displacements along the Upper Flange 5-36

Refined Model, Vertical 1.5g down Load Case 5.20 Graph of Vertical (UY ) Displacements across the Support 5-37

Refined Model, Vertical 1.5g down Load Case 5.21 Graph of Vertical ( UY ) Displacements along the Upper Flange 5-38

Refined Model, Longitudinal Load Case 5.22 Graph of Vertical ( UY ) Displacements across the Support 5-39

Refined Model, Longitudinal Load Case 5.23 Graph of Lateral ( UZ ) Displacements along the RHS Upper Flange 5-40

Skeletal Model I, Lateral Load Case 5.24 Graph of Lateral ( UZ ) Displacements along the RHS Choke 5-41

Location, Skeletal Model 1, Lateral Load Case 5.25 Graph of Vetical ( UY ) Displacements along the RHS Upper 5-42

Flange, Skeletal Model I, Lateral Load Case 5.26 Graph of Lateral ( UZ ) Displacements along the RHS Upper 5-43

Flange, Skeletal Model 2, Lateral Load Case 5.27 Graph of Lateral ( UZ ) Displacements along the RHS Lower 5-44

Flange, Skeletal Model 2, Lateral Load Case

xi

Figure Number Page

5.28 Graph of Lateral ( UZ ) Displacements along the LHS Upper Flange 5-45 Skeletal Model 2, Lateral Load Case

5.29 Graph of Lateral ( UZ ) Displacements along the LHS Lower Flange 5-46 Skeletal Model 2, Lateral Load Case

5.30 Graph of Lateral ( UZ ) Displacements along the RHS Choke 5-47 Location, Skeletal Model 2, Lateral Load Case

5.31 Graph of Lateral ( UZ ) Displacements along the LHS Lower Flange 5-48 Skeletal Model 2, Lateral Load Case

5.32 Graph of Longitudinal ( UX ) Displacements along the Second 5-49 Lateral Cross Member, Skeletal Model 2, Lateral Load Case

5.33 Graph of Longitudinal ( UX ) Displacements along the First Lateral 5-50 Cross Member, Skeletal Model 2, Lateral Load Case

5.34 Graph of Vertical ( UY ) Displacements along the RHS Upper 5-51 Flange, Skeletal Model 2, Lateral Load Case

5.35 Graph of Vertical ( UY ) Displacements along the RHS Lower 5-52 Flange, Skeletal Model 2, Lateral Load Case

5.36 Graph of Venical ( UY ) Displacements along the RHS Lower 5-53 Flange, Skeletal Model 2, Lateral Load Case

5.37 Graph of Venical (UY ) Displacements along the LHS Upper 5-54 Flange, Skeletal Model 2, Lateral Load Case

6.1 Side View of the Skeletal Model 6-2 6.2 Actual Flange Section 6-3 6.3 Actual Web Venical Stiffener 6-4 6.4 Mounting Point Details 6-5

xii

LIST OF APPENDICES

Page

Appendix Al Typical Loading Conditions for Undercarriage 9-1 Equipment Cases.

Appendix BI Engineering Drawing of a Typical 9-2 Undercarriage Equipment Case.

Appendix B2 Fatigue Analysis of Welded Joints. 9-3

Appendix Cl Sample Output from CoALA and Pafec for 9-4 Test 3.

Appendix C2 Stress Analysis of a Generally Orthotropic 9-14 Lamina in Tension.

Appendix C3 Calculation of Displacements for a Symmetric 9-16 Laminate in Tension.

Appendix C4 Stress Analysis of a Generally Orthotropic 9-18 Laminate in Bending.

Appendix CS Calculation of Displacements and Stresses for 9-20 a Symmetric Laminate in Bending.

xiii

CHAPTER 1

Introduction to the Research

This research was commissioned by the Traction Division of Brush Elecnical Machines Ltd.,

Loughborough, one of the leading Traction Engineering companies in the U.K, supplying

locomotives and electromechanical equipment worldwide.

The research centres around the design of locomotive undercarriage equipment cases with the major

objectives being as follows:-

i. To investigate and improve current equipment case design, by the use of Composite

Materials and Finite Element Analysis.

ll. To develop expenise in the use of Finite Element Analysis for the stress analysis

of isotropic and composite structures.

Undercarriage equipment cases are generally mild steel structures, carrying a large variety of

electromechanical equipment suspended either directly from the case or from internal frameworks.

At present cases are designed using basic stressing methods and thus tend to be overdesigned, with

their manufacture and assembly being very labour intensive. This suggests the possibilities of design

improvements to save weight, reduction of internal complexity and hence manufacturing costs.

To remain competitive in the world market all these considerations must be investigated in order to

produce well designed lightweight structures at low cost, this philosophy applying not only to

equipment cases but the total product as a whole. Weight savings are panicularly imponant in

present day locomotive engineering, with severe weight penalties clauses written into all contracts.

The research involved the extensive use of Finite Element Analysis, a powerful computer aided

engineering tool usi here to perfonn stress analyses and evaluate displacements for idealisations of

" structures subjected to a variety of loading conditions. The technique enables new features and

1-1

design changes to be simulated without the need for expensive prototypes. The Finite Element

software used throughout the research was the Pafec 75 integrated suite of software, pre and

post-processing being done via PIGS, version 4.2 and the analyses performed with Pafec,

version 6.2.

The initial research into equipment case design involved a survey of recent designs and the Finite

Element Analysis of a typical case to evaluate structural integrity and possible design improvements.

The survey established that equipment case design, although adequate could be improved and led to

a feasibility study of a composite case, with the aim to develop a model which would satisfy the

standard service loads and to achieve a series of design proposals for a lightweight, less complex

structure.

Finite Element Analysis of composite structures is a relatively new field, hence before any case

idealisation could take place, the accuracy of Pafec Orthotropic Finite Elements was investigated in

terms of displacements and stress analysis, via extensive element testing. On assuring satisfactory

accuracy, but limited with Pafec Orthotropic Finite Elements, work commenced on composite case

design. It should be noted that this research did not investigate the intricacies of composite material

design or material selection, this area to be the objective of further work.

The research introduced above involved several investigations which have been consolidated as a

whole in the following report. Each part of the research has been detailed in a specific chapter

which if necessary can be read independently.

1-2

CHAPTER 2

A Review of Equipment Case Design

2.1 Introduction

Undercarriage equipment cases support the control gear required for the operation of locomotive

traction motors. The equipment contained within the cases varies from large masses such as chokes

to smaller components of electromechanical control gear with also a vast array of electronics control

and monitoring equipment. They are a self contained assembly and as their name suggests affixed to

the locomotive undercarriage. The number of motors each case controls is locomotive dependent,

varying from two to a maximum of eight.

An overview of case design reveals their design to be complex and manufacture highly labour

intensive. In order to be competitive and adhere to the strict weight restrictions imposed in

locomotive specifications, it is necessary to produce designs which are :-

i. effective in their purpose of supporting and enclosing the equipment.

ii. capable of supporting all the load cases imposed by the locomotive requirements.

iii. economical both in initial production cost and in subsequent durability and maintenance

requirements.

iv. accessible for maintenance or replacement of equipment.

v. of minimum weight consistent with achieving the above four objectives.

The following chapter discusses current equipment case design, highlighting the main features and

possible improvements, concluding with a series of objectives for the ensuing research.

2.2 Main features of design

2.2.1 Equipment Case Review

Equipment cases are currently individually designed to suit each new type of locomotive and as such

2-1

their design varies considerably. There are however many fundamental features common to all

designs. Typical examples reviewed are the equipment cases designed by Brush Electrical

Machines Ltd. of Loughborough for the following vehicles:-

i. Aniculated Light Rail Vehicle (ALRV) - An articulated street car.

ii. H6 Tube Stock - A London Underground locomotive.

iii. Taiwan Locomotive - A general purpose locomotive.

iv. Class 318 - A British Rail locomotive.

Sketches. photographs and fabrication drawings of these cases can be seen in figures 2.1-2.8. which

on examination. the following points were noted.

2.2.2 General Layout and Construction

Equipment cases are large compartments mounted under the chassis of an electrically powered

locomotive which carry all the electrical equipment associated with the traction motors. An

equipment case usually weighs around 2.5 tonnes total with approximately two thirds its weight

being equipment. The severe weight penalty clauses in present day contracts mean that it is

imperative to keep this figure to a minimum. a typical penalty clause can charge the supplier $42/kg

overweight (1990 figure) for every equipment case ordered in the contract. However. weight is not

the only design consideration. there are other important factors such as to provide adequate

equipment protection from the elements and to ensure ease of access for maintenance.

A typical current design of equipment case is constructed from mild steel sheet of 3-5 mm thickness

and rectangular section mild steel tube. although 2-3mm Stainless Steel (Austenitic Type 304L) was

used for the Taiwan locomotive.

The basic structure comprises of two end frames connected by two longitudinal beams which

support both the structure and the contents of the equipment case. The structure is completed by

2-2

lateral frames at appropriate positions and by sheet metal panels to increase stiffness and provide

enclosure. Removable skin sections are required for access to the equipment.

In some designs there has been a limited use of composite materials in lightly loaded parts of the

structure but as far as is known composite material has never been used for the main load carrying

parts of the structure. A typical example of the use of composites in existing designs is a contactor

housing used on the ALRV locomotive but here the composite material did not support the weight of

the contactor, see figure 2.5.

The heavy bulk items of equipment carried in the equipment case such as chokes and some control

gear are usually located in the space between the longitudinal beams. The heaviest item, normally

the choke (l200kg) is supported by its own short cross beams which attach to the upper part of the

main longitudinal beams with the choke being suspended below the beams. Lighter components,

with a typical mass of 20kg are in general mounted on brackets attached to whatever part of the

structure is dictated by convenience of layout. Figures 2.3 and 2.7 show the internal detail of the

ALRV case, it should be noted that figure 2.7 shows the case in the process of assembly and as such

is upside down.

All the electronic control and monitoring equipment is contained within the side panniers along with

lighter items of control gear, figure 2.6 shows a view of a typical of pannier in the process of

assembly. Again the equipment is attached either directly to the casing or to frameworks within the

panniers. Detachable GRP covers are usually used to protect pannier contents from the environment.

As the cases hold a vast array of equipment, the associated wiring is complex, with cable runs and

supports contributing a great deal to complexity and overall cost. The electrical power connection

from the case to the motors is also complex, comprising of numerous heavy cable connections.

The whole equipment case is attached to the locomotive frame at four suspension points located at

2-3

each end of the longitudinal support beams, either by vertical bolts or horizontally via a yoke fitting

around the beam section.

The whole structure including mountings is designed to meet, as a minimum, vertical, longitudinal

and transverse limit loading conditions specified in each individual contract. Typical values are

shown in Appendix A 1. Both proof load cases, in which no significant permanent distortion should

result from the application of the limit loads and the fatigue case, in which no fatigue failure should

occur in a specified number of fatigue cycles are considered. The typical contract load cases are are

specified as 'g' loads, the actual loading thus depending on the mass and mass distribution of the

structure.

The type of equipment case described above has been in service with many railway operators for a

considerable time and has proved to be generally satisfactory. Thus scope for improvement by minor

changes to the basic design is limited and the only possibilities for a large step improvement are

either new electrical design, which could result in the use of lighter equipment or a total rethink of

the structural design.

The first option is unlikely in the short term although future developments in electrical equipment

will no doubt bring benefits in this area. The second option of improving the support structure is

considered in the following research which together with possible improvements in equipment

layout would achieve a more economic case.

2.3 Review of Possible Design Improvements

Areas where design improvements are possible can be divide into several categories such as weight

reduction, reduced complexity, cost reduction, improved maintainability and customer appeal. The

following sections indicate where structural improvements and better layout can contribute to a

better prOduct.

2-4

2.3.1 Weight reduction

In the early days of the railway locomotive, weight was considered to be desirable, within the limits

of the supporting strength of the track, as it improved traction. Recent designs with J111 wheel drive

and traction control make excessive weight undesirable, which combined with the strict weight

penalty clauses result in a requirement to reduce structural weight.

Considering the existing component weights and layouts the typical equipment case structure is

reasonably efficient within the limitations of steel construction and avoidance of expensive

manufacturing techniques such as chemical etching to suit local loading conditions used widely in

the aircraft industry. Thus the potential for large weight reductions in the structure is limited unless

new materials are considered.

The possibilities for new construction material are in essence a light alloy option (such as aluminium

and magnesium) or composite materials. This research investigates the use of composite material as

these materials were considered more suitable for the application than light alloys for the following

reasons.

i. High stiffness per unit weight.

ii. Good fatigue life and vibration damping qualities.

iii. Good electrical insulation properties.

iv. Non magnetic properties.

v. Chemical inertness and biocompatability ie complete resistance to corrosion.

Additional possibilities in weight reduction could come from optimum layout of equipment to

reduce cable runs etc.

2.3.2 Reduced Complexity

If the electrical circuit is considered to be fixed, reductions in complexity can only be achieved by

2-5

layout improvements, grouping inter connected components as closely as possible.

There are also potential improvements in the structure particularly in a composite design to make

some of the many mounting brackets integral with the structural members and thus reducing the

number of bolted joints. Figure 2.4 illustrates the complexity of typical mounting arrangements,

with each of the mounting bolts shown is an attachment for a piece of equipment or a cable clamp.

2.3.3 Cost Reduction

Structural material costs for the typical current design are low, particularly if mild steel is used and

are unlikely to be undercut by the use of an alternative material. There is however scope for reduced

labour costs by reducing hand fitting work by means of more integral brackets on structural

members, reduced layout complexity and shorter cable runs.

It is unlikely that any improved equipment case on the lines considered in this report would be

substantially cheaper to produce, in the limited quantities required, than the current item. However,

if overall costs can be kept substantially the same for a better performing product the change would

be worthwhile.

2.3.4 Maintainability

Given equipment reliability, maintainability can be divided into ease of access for routine attention

and component changes and the effectiveness and life of the corrosion protection system. Attention

to component layout could improve access and the use of a material which did not need corrosion

protection would be a considerable advantage.

2.3.5 Customer Appeal

The sale of railway locomotives is a competitive business, where th~ number of potential customers

is limited. Thus advanced design features are a marketing advantage when comparing similar

designs from competing companies. This is not an engineering reason for change but is another

2-6

factor to be considered in the design strategy for a new locomotive.

2.4. Conclusions

The review of existing designs of locomotive equipment cases and the considerations of possible

routes to improvement carried out in the previous sections has indicated various areas where

improvements are possible.

It can be concluded that it is a worthwhile proposition to investigate the possible structural

improvements which may come from a change of material from steel to glass reinforced composite

material for the main case structure and produce a series of design proposals for such a case. It is

also noted that further benefits could be achieved through improvements in equipment layout and by

reducing the wiring complexity to save weight and cost.

2-7

Figure 2.1 A General View of a Typical Equipment Case (86 Tube Stock)

,.l.

2-8

Figure 2.2 Three Views of a Typical Structure (ALRV)

~ ..... ---,

: . ,

-" .. -.

2-9

"

q "

, r·-· 2 •

, ,

o I

Figure 2.3 Internal Detail of the ALRV Case.

. -.

. ,

11 ..

fl)'

o ... _~ ::2~~T ..... ___ --=--_.:.-:===0,= .

1-.o.L1.o.;==;';~. --- --~ "-::;'1

2-10

'·8 r. u ~. , , ,

...

.~

i

lid: " , I

'"

i I

II~ I J I' • I .r

Figure 2.4 Side View of a Typical Pannier showing Internal Mounting Arrangements (Class 318)

o Sl

01.S.

2.2.~

2-11

.70

Figure 2.5 Composite Contactor Housing ( ALRV )

Figure 2.6 Pannier under assembly (ALRV)

•

2-12

Figure 2.7 View of the internal structure of a Case (ALRV)

,.

2-13

3.1lntroduction

CHAPTER 3

Analysis of an Isotropic Case

A detailed analysis of a typical isotropic equipment case was performed, in order to achieve a set of

results for stresses and displacements under the defined standard service load cases. The mass of the

unloaded structure was also established. These results were used to assess the efficiency of the

structure with a view to improving its design , reducing weight and for comparison with future

models. The investigation involved a static and elastic stress analysis of the structure using the

Pafec Finite Element software. The typical Finite Element approach was adopted throughout the

investigation, initially analysing a fairly coarse mesh model of the case and progressing to further

enhanced models and analyses.

The chapter discusses the Finite Element idealisation of the model, the analysis technique and finally

the results and conclusions.

3.2 The Equipment Case

The case to be modelled can be seen in Appendix B I. It is a hybrid case incorporating many features

typical in equipment case design. The case is constructed from Austenitic Stainless Steel, Type 340L

with two lateral mild steel sections for additional strength and has overall dimensions of

3.4 x 2.3 x 0.8 m. The case has of two sets of longitudinal double box section beams situated directly

above each other and separated by a vertical panel. The ends of the upper beams serve as attachment

points to the undercarriage of the locomotive. Equipment is supported on either side of these beams,

the choke is situated between them and on either side in panniers is the switchgear and electronics.

The equipment is protected from the environment by steel floor, roof, and side panels with

removable covers over the panniers. These covers being generally formed in composite materials.

3-1

3.3 Finite Element Modelling

The equipment case was idealised using the PIGS pre·processor, all significant dimensions being

read from the drawing in Appendix Bl. The case displays two planes of symmetry, so for the

symmetrical venicalload case a quarter of the case was modelled. The lateral and longitudinal load

cases being anti symmetric involved using a full model which was achieved simply from the quarter

model by mirror and copy commands. Initially a coarse mesh model was generated and analysed,

which resulted in a series of model developments until a satisfactory analysis was achieved whose

results could be viewed with confidence.

Cenain simplifying assumptions were made in the modelling process:-

I. The less structurally significant stiffeners were modelled initially as tension rods and

later as beams.

ii. Any filleted corners were modelled as sharp.

iii. Any welded joints between plates were represented by planes meeting at sharp edges.

The initial coarse mesh model was developed using the 'in-plane', first order quadrilateral elements,

type 36200 and tension bars, type 34000, to model the less significant stiffeners, see Reference I for

detailed information on element types. However, on mesh refinement and a change to the second

order version of the quadrilateral element (36210), the results from the analysis could not be viewed

with confidence in cenain areas of the structure where 'out of plane' effects take place. These areas

being the venical shear panel, lower beam panel and suppon locations. The 'out of plane' effects

caused by loads applied in these areas had not been catered for in the element type selection. A

funher change of element type was made to the 44210 element, a second order facet shell element

and the beam element 34400, with both element types incorporating rotational degrees of freedom.

The final model for a quarter case can be seen in figure 3.1 and the full model case can be seen in

3-2

figure 3.2. The quarter case has 557 nodes, 233 elements, 3114 degrees offreedom, the full case has

1975 nodes, 932 elements, 6590 degrees of freedom.

3.4 Load Cases

Four proof load cases and a body fatigue load case were considered in the investigation, these load

cases being taken from the standard cases featured in Appendix A 1. The details of each are as

follows:-

3.4.1 Proof Load Cases

The equipment mountings must be able to withstand the mass of the equipment, choke and pannier

loads when subjected to the following accelerations, all loads being reacted at the support.

i. Vertical LOg (up)

ii. Vertical l.5g (down)

iii. Lateral l.1g ± l.Og vertical (down)

iv. Longitudinal 3.0g ± l.Og vertical (down)

3.4.2 Fatigue Load Case

All equipment mountings must be designed to have a fatigue life of not less than 107 cycles for the

loads produced by the following accelerations acting on the mass of the equipment, ie choke and

pannier loads.

i. Vertical LOg (down) ± 0.3g

ii. Lateral ±O.3g

iii. Longitudinal ±O.2g

3.5 Load Case Idealisation

The case must be self-supporting and carry the following loads:-

i. Choke, mass 1.2 tonnes applied as point load at four fixing locations.

3-3

ii. Pannier loads of 700 kg each side, distributed along the length of the case. The pannier

loads were applied on both upper and lower beams to simulate equipment loadings more

realistically.

3.5.1 Vertical Load Case

Since this load case has two axes of symmetry a quarter model of the equipment case was used

subjected to a quarter of the total load carried The pannier loads were distributed along the upper

and lower beam sections, half the load being carried by each section. The loading across these

structures was applied to the nodes along the overhanging edge of the bracket section and was

adjusted in the ratio of 1:4:1 across each element to obtain the second order 44210 elements. The

choke load was carried by the two nodes that provided the connection points for the channel section

and main case.

As only a quarter model was loaded, the boundaries along the axes of symmetry of the model had to

be restrained to ensure that symmetry with the other three quadrants was preserved. Hence along the

longitudinal axis of symmetry, displacements in the lateral direction and rotations around the

longitudinal and vertical axes were restrained. Along the other axis of symmetry, the lateral axis,

displacements in the longitudinal direction and rotations about the vertical and lateral axes were

restrained.

The reaction to the applied loads was achieved by restraining aII displacements at the support nodes.

3.5.2 Lateral and Longitudinal Load Cases

Both of these load cases are anti symmetric, the lateral load case is anti symmetric about the

longitudinal plane and visa versa. A fuII case idealisation was used to model both these load cases as

combinations of loadcases were required, with the loading being applied as described above for the

vertical load case. Restraints were applied to the four support loactions to prevent any

3-4

displacements at these points.

3.6 Results

3.6.1 Proof Load Cases

The displacement and stress contour plots obtained from the proof load analyses can be seen in

figures 3.3·3.7. The stress contour plots are uninformative having few contours and so only one has

been included as an example of the output obtained. On identification of the areas of high

displacement and stress, namely the supports, beams and separating vertical shear panel, several

graphs have been plotted out via the PIGS post-processor to clarify the results for each load case.

The graphs of stress distribution use Von Mises equivalent stress criterion evaluated along the mid

surface of the shell elements. The Von Mises stress was evaluated as the stresses are two

dimensional and this criterion combines the two direct and one shear stress into a single equivalent < ~ stress based on a shear distortion energy limit at yield, Reference 2. The resulting graphs all have

sharp, jagged contours which have been smoothed out to illustrate the stress distribution more

realistically. Considering each proof load case in turn:-

3.6.1.1 Vertical Load Cases

i. l.Og Vertical up

ii. l.5g Vertical down

The results for load case (i), as expected were found to be proportionally smaller and in the opposite ? \., ~

direction to load case (i). The results described here are for the more severe load case (ii).

The displacement graphs, figures 3.8 - 3.10, show the vertical and lateral components of

displacement experienced by the upper and lower beams and their separating shear panel, the central

web. It should be noted that the graphs for this load case tertninate at the centre of the case, due to

the symmetrical nature of the load case. It can be seen that the vertical displacements are negative

3-5

throughout, they have a maximum value at the ends of the beams and a minimum value at the centre,

the actual value being very small, in the order of 10- 1 mm . The maximum vertical displacement

occurs at the centre of the lower beam, magnitude 2.Dmm.

Lateral displacements are again very small, in the order of 10- 1 mm and are positive throughout

showing that the right hand side of the case tends to bow outwards, in the positive x direction. Along

the upper beam the lateral displacements are virtually constant and negJigble at O.OO3mm, however

along the central web and lower beam they are at a minimum at the ends rising sharply to a

maximum at the centre, the maximum displacement being 2.2mm at the centre of the central web.

Longitudinal displacements throughout were in the order of 10- 2 mm and therefore considered

negligble.

The examination of the stress output files reveals modest stress levels, in the order of 101 Nmm- 2

well below the yield stress of 204 Nmm- 2 for Austenitic Stainless Steel.The highest stress levels

were found at the suppons, with a typical value of 55 Nmm- 2 ie 27% of yield. The graphs, figures

3.11 - 3.13, show the stress distribution along the upper and lower beam and also the central web.

All the curves tend to maximum towards the ends of the structure and a minimum centrally. The

maximum Von Mises stress in these areas is 15.8 Nmm- 2 ie 7.7% of yield.

3.6.1.2 Lateral Load Case

±l.lg lateral + l.Og venically down

Results were only obtained for the positive lateral load case. The negative load case would produce

the mirror image of results discussed here, therefore was not necessary for the assessment of

. displacement and stress variations. The lateral load case is anti symmetrical and so a full model of the

equipment case was used in the analysis. When submitted initially, the displacement output as shown

in figure 3.5 was produced. The quilting effect that can be seen was assumed to be the result of

3-6

unrealistic loading in this load case. The pannier loads were evenly distributed along both the upper

and lower beams, which under lateral loading resulted in these lateral loads impinging along the

vertical shear panel wall, the central web. In an actual case the equipment these loads are

representing are attached via frames and side walls to the shear panel rather than as overhanging

weights. These frames and the supporting structures would provide additional vertical and lateral

stiffness. The revised lateral load case therefore was modelled with the lateral loads displaced to the

upper beam, where they could be distributed through the upper shear panel, the displacement output

as seen in figure 3.6, shows the results of the second set of loading conditions.

Several graphs have been plotted to illustrate areas of interest. Graphs are included from both the left

and right hand sides of the structure to show the effects of anti symmetric loading. Displacements

throughout the whole structure can be seen to be negligble, in the order of 10- 1 mm. Vertical and

lateral displacements have been considered, the longitudinal component being negligable, in the

order of 10- 2 mm. The displacement graphs can be seen in figures 3.14- 3.18. The displacements

along the upper right and left beams show both vertical and lateral components to be similar curves,

with the right hand curves being slightly more exaggerated, a maximum of 2% larger. Vertical

deflections are negative throughout and are of a maximum value at the ends and a minimum

centrally, -0.35mm for the right hand side. Lateral deflections are positive showing a tendency for

both sides of the case to bow in the positive x direction. Lateral deflections are at a minimum at the

ends and rise to a 0.14mm maximum at the centre of the right hand beam. Similarly the vertical

component of deflection along the left and right hand lower beams show a similar distribution, a

maximum value at both ends with a central plateau minimum of -1.2mm. The lateral components

differ from the left to right hand side. The left hand side being at a minimum in the centre, bowing

in the negative x direction by 0.1 8mm, the right hand deflection shows a bow in the positive x

3-7

direction, maximum of 0.23mm. Along the right hand side central web, the venical component is

almost constant at -0.33mm, the lateral component shows a large rise towards the centre, with a

maximum of 1.2mm positive bow.

The deflections discussed above are of a negligble size, however they do indicate the weaknesses in

the structure, ie lack of stiffness in the venical shear panel under lateral loading. The stress output

file again shows modest stress levels of the order of 101 Nmm- 2, these being well below yield for

the material. The table, figure 3.19 details stresses at the suppons and reveals that these are highly

stressed in comparison with the rest of the structure.

Von Mises Stresses at a Support Location

Loadcase Support Node (488) Support Node (491)

Vertical (ii) 54.16 57.98

Lateral 104.00 52.40

Longitudinal 56.10 47.52

All stresses ore in Nmm2

Figure 3.19

The graphs, figures 3.20- 3.24 show the Von Mises stress distribution along the regions of 'high'

stress in the structure. Again graphs along both left and right hand sides are included to illustrate the

effects of anti symmetry .

The upper beams show high stress at the ends and a minimum in the centre. The left hand side is

slightly higher stressed than the right hand side, by 6%. Similarly the lower beam exhibits the same

stress features along its length, again the left hand side being slightly more exaggerated, by 6%.

3-8

Along the right hand'central web, high stresses appear at the ends 14 Nmm- 2, dropping steeply to a

minimum - 3 Nmm- 2 in the centre.

3.6.1.3 Longitudinal Load Case

± 3.0 g Longitudinal + 1.0 g Vertically down

As in the lateral load case only the positive case is considered, the negative load case giving the

'mirror image' of the results described here, The nature of the load case involves both sides of the

model being equally loaded and hence only the right hand side of the case is considered.

Displacements through the structure can be seen to be negligble, in the order of 10- 1 mm. Several

graphs along the regions of high displacement can be seen in figs 3,25 - 3.27 and show longitudinal,

lateral and vertical components of deflection. The upper beam sho.ws a positive longitudinal

displacement, the displacement being constant along the beam at 0.6mm. The vertical displacements

show a negative displacement and varies unsymmetrically along the length ~th maximum

displacement being -0.34mm. Lateral displacements are negligble here. The lower beam exhibits a

similar constant value of 0.7- 0.9 mm of longitudinal displacement. The vertical deflection here is at

a 'maximum at the ends and a minimum of -1.28":.1m centrally, with lateral displacements again

negligble. The central web has a high positive lateral displacement varying from a minimum at the

ends to a maximum of 1.1mm centrally, this hig~ lateral displacement again is due to the lack of

stiffness in the vertical shear panel. The longitudinal component shows a similar distribution to the

upper beam with deflections between 0.4 - 0.6 mm. The vertical component again is negative and

varies un symmetrically along the length of having a minimum value of -0.344 mm. The stress

output file again shows no stress that exceeds yield for the material. The support locations are the

most highly stressed with stresses of the order of 102 Nmm- 2. The Von Mises stress distributions

along the upper and lower beams and central webs again exhibit stresses in the order of 101 Nmm- 2

3-9

and can be seen in figures 3.28 -3.30.

3.6.2 Fatigue Load Case

The fatigue analysis involved using three sample nodes from the highly stressed areas of the

structure ie the suppons and the upper and lower beams. The applied stress range at these locations

was determined and compared with standard fatigue data, Reference 3. The results are tabulated in

figure 3.31 and details of the calculations can be found in Appendix B2.

Fatigue Analysis Data

Location (Node) -2 Stress Range Nmm Weld Class Reserve Factor

Support (488) 83.8 B 1.2

Upper Beam (236) 0.4 F 100

Lower Beam (394) 1.9 F 21

Figure 3.31

The results show that the fatigue reserve factors are high for the majority of the structure and that a

grade F weld is acceptable. However the suppon locations experience a high stress range and would

require a very high quality grade B weld to withstand the stated fatigue conditions. This would be

impractical and hence it is likely that cracks would initiate and grow at the suppon locations.

3.6.3 Case Mass

The finite element analysis determined the mass of the unloaded structure to be 360kg.

3.7 Discussion

The analyses revealed that the structure was relatively unstressed when subjected to the service proof

and fatigue loads and was 360 kg when unloaded. The results of the investigation, show that there

are areas for concern in the' case design:-

3-10

i. High stresses are experienced at the suppons compared with the rest of the structure.

The fatigue analysis revealed that high quality grade B welds would be required at the

suppon locations in order to withstand the fatigue loadings. Lower quality welds would

cause the case to fail by cracking the suppons.

ii. There is a tendency for some lateral deflection when the case is subjected to

antisymmetric loading caused by the lack of lateral stiffness in the vertical shear

panel separating the supponing upper and lower beams.

Hence, design improvements for the case would be to introduce some venical stiffeners into the

venical shear panel to prevent the tendency for lateral deflection and to reinforce the suppon area.

3.8 Conclusions

The analysis of the isotropic case has revealed that the case is over designed, although there are

several areas that are exhibit high stresses and displacements and require design improvements. It

can be concluded that a redesign in composite materials to produce a more efficient and lighter case

is a feasible proposition.

3-11

Figure 3.1 Quarter Case Idealisation

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3-12

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3-13

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3-16

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3-17

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TT TI C" IIV AMn 117 m~PI ArrMrMT~ AI n~lI: Tf.t~ Qf.t I nU~Q QnM 3.0G LONG.

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• & W • W U

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Figure 4.15

~ ~~~ ... • • IL1\.-.., • • •

~ .~"

I !; •

Generally Orthotropic Lamina in Tension, Test 3, Element 44215

" i : • 6

P I ~! . ... .;

. " • a, lifa Ii .~ w I!I~ .w M _

l! • l! :/:i wl!! Ui • = be

~ i == ;;;§

o 0

4-23

Figure 4.16 Generally Orthotropic Lamina in Tension, Test 3, Element 43215

i : N

~ ~~~ . ~ I :!~ --- !5 ' , - 11 .. fl.....1\-..... ••• ~! ,;

~ .~N

- " • t

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4-24

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4.1 Introduction

CHAPTER 4

Orthotropic Element Testing

Single element testing has been the subject of several papers in the past, References 4 and 5,

however Iinle work has been performed in evaluation of orthotropic elements. Hence, in order to

justify the use of Pafec Finite Element software in the analysis of a composite equipment case, the

performance in terms of accuracy of displacements and stress analysis of three onhotropic elements

was investigated, via a series of element tests.

The results of the tests were assessed using a proven laminate design analysis package, CoALA and

'first principle' calculations, where applicabk Finally the most appropriate element for the problem

in hand was selected.

4.2 Orthotropic Element Selection

The composite equipment case will be in essence a thin plate structure, hence the elements tested

were all thin plate elements capable of modelling orthorropic properties. Three element families

were selected, with their second order quadrilateral element being used. The elements tested are

described below, see Reference 1 for more detail.

i. The 36215 iso-parametric membrane element used only for in plane stress situations.

ii. The 44215 facet shell element used where both in-plane and out-of-plane effects are

important.

iii. The 43215 semi-Loof curved shell element, again used where both in-plane and

out-of-plane effects are imponant. The semi-Loofis a more sophisticated element than

the 44215 and can be used in curved and folded shell problems, however it can be

degenerated to a flat plate as in the following tests.

4-1

4.3 The Element Test Model

The evaluation of the Pafec orthotropic elements was carried out with a series of tests involving

idealised loading and suppon conditions appropriate in the stress analysis of composite materials. As

the composite case will be subjected to both bending and tension, each element was tested to see if it

accurately modelled a variety of composite material properties in tension and, where applicable, in

bending.

The model used for the element tests was a cantilever bar 50mm long and IOmm wide, idealised

into five elements each with an aspect ratio of unity, see figure 4.1.

Element Test Model

y I '--~~11'0~

Lx 50mm

Figure 4.1

The bar was modelled in a unidirectional Carbon/Epoxy composite, using typical material constants

for a fibre volume fraction of 60% as tabulated in figure 4.2.

4-2

Material constants for 0 Carbon /Epoxy composite ( Vr = 60% )

Material Constants Value

Young's Modulus:-

Longitudinal E, 140.0

Transverse E 2 10.0

Shear Modulus 5.0

Poisson's Ratio v'2 0.3

Figure 4.2

GN/m2

GN/m2

GN/m2

Three common lay-ups were used in the tests as described below and illustrated in figure 4.3.

4-3

Lamina/Laminate Lay-ups Used.

Specially orthotropic laminate 2[0/90] s

Generally orthotropic lamina [45]

Unsymmetric lominate [45/-45]

Figure 4.3

1. Symmetric laminate using specially onhotropic plies ie those with a fibre orientation of

OOand 90~ The laminate lay-up is 2[0/90] •.

ii. Generally orthotropic lamina, fibre orientation of 45:

iii. Un symmetric laminate, lay-up [45/-45].

The theoretical propenies of these materials are described in section 4.5.

4.4 Element Tests

Each element type was tested with the three different lay-ups in tension, tests I, 3 and 5. Bending

tests were performed on the 44215 and 43215 elements with the symmetric laminate and lamina,

propenies, tests 2 and 4 respectively. The un symmetric laminate was not tested in bending, as there

was no satisfactory basis for comparison with a proven system. The test models were loaded and

restrained as shown in figure 4.4. The loads applied conformed with the formulation for elements

4-4

with mid side nodes.

Tension Test Loading and Restraints

~ E166.6 N

666.6 N

iH:l-_ ....... _.....L __ L-_-'-_--I_166.6 N

Total Force = 1000N

Bending Test Loading and Restraints

::; It ::::::: "Al..· __ .L-_....L __ ..L._....JI.-_...I. t- 0.166Nm

t Total Moment = 1 Nm =

Moment Intensity = 102 N

3 10 Nmm

( ie 102

Nmm/mm )

Figure 4.4

A summary of the tests performed can be seen in figure 4.5.

4·5

Tests Performed

No. Elements Lay-I.-P Test Ply mm

36215 2[0/9Ols Tension 1 4421g 1

4321

2 4421g 4321 2[0/9Ols Bending 1

3 3621g 4421 [+45] Tension 2 43215

4 44215 43215 [+45] Bending 2

36215 [+45/-45] Tension 1 5 44215 43215

Figure 4.S

The laminate design analysis package, CoALA was used as a check on the resultant stresses and

displacements obtained from the element tests, together with hand calculations in the simpler cases

for funher verification, detailed in Appendix C2 - CS. CoALA software is a commercially available

package from Cranfield Institute of Technology and provides a detailed analysis of a lamina or

laminate under a specific loadcase. It computes laminate stiffness, elastic and physical engineering

constants, strain, strength and stress analyses on a ply to ply basis, see Reference 6 . Sample output

from Pafec and CoALA for test 3 can be found in Appendix Cl.

4.5 Theoretical Effects

Theoretical laminate analysis, detailed in Reference 7 predicts various effects from these test

situations as described below:-

4-6

4.5.1 Test 1

Symmetric laminate in tension, theory predicts that there is 'in-plane orthotropy' ie no shear coupling

effects, ie straight extension.

4.5.2 Test 2

Symmetric laminate in bending, similarly theory predicts 'bending orthotropy', ie no bend-twist

coupling effect.

4.5.3 Test 3

Generally orthotropic lamina in tension, here the shear coupling terms are present in the stiffness

matrix. Hence shear is predicted along with extensional effects.

4.5.4 Test 4

Generally orthotropic lamina in bending, the presence of fibres at orientations of other than 0 and 90

involve the bend-twist coupling terms in the bending matrix.

4.5.5 Test 5

Unsymmetric laminate in tension, theory predicts that membrane-bending coupling effects are

introduced in the stiffness matrix. These terms mean that in-plane loads cause both in-plane and

out-of-plane deformations and vice versa.

4.6 Results

The stress results from the element tests are tabulated in the tables 4.6 - 4.8. Table 4.6 details the

results of the tension tests 1,3 and 5. Table 4.7 details the results from test 2 and table 4.8 the results

from test 4. Displacement results are tabulated in tables 4.9 - 4.10, table 4.9 detailing the results

from the tension tests and table 4.10 the results from the bending tests. Pafec displacement plots can

be seen, figures 4.11 - 4.23.

4-7

4.6.1 Tension Tests

4.6.1.1 Stress Results

The results from the tension tests I, 3 and 5 can be seen in tables 4.6. The resulting output from the

tests gave a variety of stress data, with both packages tested generating a ply by ply stress solution.

The Pafec 36215 element details both in plane principal stresses and stresses along the material axes.

The 44215 element provides stresses on the 'positive', 'neutral' and 'negative' surfaces in the material

directions ie along and transverse to the fibre direction. The 43215 semi·Loof element provides

'upper' and 'lower' surface stresses again in the material directions. CoALA provides mid ply stresses

only. The stresses tabulated from both packages are those at the mid ply in the principal material

direction.

On examination of table 4.6 it can be seen that the three element types tested produced similar stress

results in tests 1 and 3. There were slight discrepancies in the value of shear, however as the order of

magnitude for this result is 10- 7 Nmm- 2 , they can be ignored. Comparison with the CoALA results

show an accurate correlation in magnitude and direction for direct stress. The Pafec results for shear

are of the same order of magnitude as the CoALA results, which being minimal can be approximated

to zero, the theoretical value for shear in such a test. A hand calculation can be found in Appendix

C2 which correlates exactly with both Pafec and CoALA providing funher verification.

Comparison of the Pafec results for test 5, figure 4.6 the un symmetric laminate in tension show that

they correlate accurately in shear, however significant differences occur in the values of direct stress

between the plane stress element, 36215 and the bending elements, 44215 and 43215. On

comparison with the CoALA results for this test, it can be seen that direct stress values obtained by

the bending elements are very accurate, typically with a 0.02% variation from the CoALA result.

The plane stress element proved less accurate, in direct stress there was a maximum deviation of

4-8

26.3% from the CoALA results. The value of shear obtained from all elements compared exactly

with the CoALA result, however directions were reversed in Pafec.

4.6.1.2 Displacement Results

The displacement plots, figures 4.11 - 4.19 display the various coupling effects predicted in section

4.5, for the tension tests. Each element exhibits straight extension when tested in tension with

symmetric laminate properties, test 1, figures 4.11- 4.13 verifying that no shear coupling took place.

This was due to there being only specially oriliotropic plies present in the laminate. The test 3

displacement plots figures 4.14 - 4.16 shows the effect of shear coupling terms in the stiffness matrix

by the presence of both shear and extensional displacements. The displacement plots for test 5 for

elements 44215 and 43215, figures 4.18 and 4.19 show the effect of both extension and curvature

for the unsymmetric laminate case, predicted by the presence of membrane-twist terms in the

stiffness matrix. The plot for the 36215 element, figure 4.17 in this case does not give a fair

representation of the effects, due to it not being able to achieve the out-of-plane effects.

Comparison of actual displacement results between elements can be seen in table 4.9, with

displacements at the three nodes along the loaded end of the cantilever being tabulated. Theoretical

calculations for test I, using both CoALA derived constants and composite analogies can be found in

Appendix C3 to verify the Pafec data. It can be seen that the results obtained for the three elements

correlate exactly in tests I and 3 and with the theoretical values. Test 5, the unsymmetric laminate

showed that there was accurate correlation between the two bending elements for in plane

displacements, with out of plane displacements showing some deviation. The displacements here for

the in plane element along the bar were found to be 87% of the equivalent value recorded for the out

of plane elements with transverse displacements being 19% greater than those derived by the

bending elements. Obviously, the in plane element produced no out of plane displacements.

4-9

4.6.2 Bending Tests

4.6.2.1 Stress Results

The results from the bending tests 2 and 4 can be seen in tables 4.7 and 4.8 respectively. As in the

tension tests the stresses tabulated from both packages are at the mid ply in the principal material

direction (ie along and transverse to the fibre direction).

Comparisons were made between the neutral surface stresses obtained at the centroid of the mid bar

element for the Pafec test model, element 3, these being the equivalent stresses to those obtained at

the centroid of the CoALA single element. On comparison of the results for the symmetric laminate

in bending, test 2, between the Pafec 44215 element and CoALA it can be seen that the Pafec

results are greater for direct stress. Direct stress in the fibre direction in the outer plies were 78%

greater and for the inner plies was 2.5 times greater. Direct stress perpendicular to the material

direction was 1.5 times greater in the outer plies and 82% greater in the inner plies. Shear values

obtained for the 44215 element were of an order of magnitude smaller than those derived by

CoALA. On considering stress directions, the directions obtained by the 44215 element for shear

correspond with CoALA, however direct stress directions are opposite throughout the depth of the

laminate.

The Pafec 43215 element performs more consistently in test 2 to the 44215 element. Comparing

mid ply stresses, interpolated across the depth of the ply with the CoALA mid ply stresses it can be

seen that the values obtained for direct stress are greater than the CoALA results. Direct stress values

in the outer plies along the material direction were found to be 33% greater, with inner ply stresses

being 28% greater. Direct stresses transverse to the material direction were consistently greater, by

33% in both inner and outer plies. The results for shear were similar to the 44215 element, with the

shear values obtained being of an order of magnitude smaller than the CoALA derived results. Stress

4-10

directions obtained by the 43215 element were the same as those from the 44215 element, ie they

corresponded with CoALA in shear and were opposite in direct stress.

The results for the generally orthotropic lamina bending test, test 4 are shown in figure 4.8. Here, the

CoALA results are taken to a variation to test 4. This variation involved using two 45 plies to

idealise a single lamina of 45. This was necessary due to CoALA only calculating centroidal

stresses, which meant on analysis of a single lamina, CoALA correctly predicts zero stress and hence

provides no basis for comparison with the Pafec results.

The uni-directional lamina test predicted stresses at the centroid of each lamina which by

extrapolation through the depth of the lamina gives the results detailed in table 4.8. The results

from the 44215 element shows that the positive and negative surface stresses are greater than

CoALA, 2.6 times in the fibre direction, 1.9 times transverse to the fibre and 1.9 times greater in

shear. The neutral surface stresses tend to zero in Pafec and are exactly zero in CoALA, which

corresponds to bending theory. The results from the 43215 element in this case show that the stress

results are of the same order of magnitude as CoALA, being in general about 66% of the equivalent

CoALA result. Extrapolating over the depth of the ply shows that mid ply stresses are zero, which

again corresponds with CoALA and bending theory. Both elements predict the same stress directions

as CoALA. The results of a first principles calculation can be seen in Appendix C4, which verifies

that the direct and shear results generated by CoALA are correct.

4.6.2.2 Displacement Results

The displacement plots for these tests can be seen in figures 4.20 - 4.23 feature the various coupling

effects predicted in section 4.5. The plots for the symmetric laminate in bending show that both

elements types exhibit pure bending with no bend-twist effects, figures 4.20 and 4.21. The generally

orthotropic lamina in bending does feature these effects, as exhibited by both element types in

4-11

figures 4.22 - 4.23.

On comparison of the actual values obtained for displacements, tables 4.9 and 4.10, it can be seen

that the 44215 element produces greater displacements. It should be noted that in order to obtain

realistic values for displacements the applied test moment was reduced from 107 Nmm to ID"

Nmm. Theoretical values derived for the symmetric laminate in bending, see Appendix CS, show

that the 44215 elements are more accurate than the 43215 results, being 13.2% too large and the

43215 results being 29.7% too small. The generally onhotropic lamina in bending showed again

that the 44215 element produced larger displacements than the 43215 element, in both cases

displacements were not constant across the width of the bar, due to the presence of the bend twist

coupling effect.

4.7 Discussion

The justification of using CoALA to verify the Pafec results obtained in the element tests was

assured by a series of 'first principles' calculations, which can be found in Appendices C2 - CS. To

summarise the results from the tests, it can be said that the use of the Pafec onhotropic mid side

noded quadrilateral elements, 36215,44215,43215 for the analysis of plane stress situations using

single laminae or symmetric laminates is justified in terms of both displacement and stress analysis.

Bending applications, involving the 44215 and 43215 elements showed that for the symmetric

laminate application direct stress results were higher than the CoALA generated results. The

semi-Loof element produced stresses which were 33.3% greater through the depth of the laminate.

The 44215 element again produced higher stresses which although were not a consistant amount

greater through the depth of the laminate can be assumed to be twice the equivalent CoALA result.

Lamina stresses obtained again were greater for the 44215 element by a factor of 2 and less for the

semi-Loof, 66% of the CoALA values. Displacements values obtained in both the symmetric

4-12

laminate and lamina test showed that the 44215 element produced a more accurate value than the

semi-Loof element.

The results for the symmetric laminate tension test establish that the direct stresses are significantly

different between the plane stress element and CoALA, although the shear stresses correlated. The

stresses generated by the bending elements corresponded accurately with CoALA.

The actual application of Finite Element Analysis in this research involves a composite case

subjected to tension and local bending, using generally and specially onhotropic laminae. The above

results demonstrate that the element to use is the 44215 plate bending element, providing accurate

results for stress and displacement in tension which covers the majority of the structure and

reasonable results for displacement in bending, with stress results here providing a safety factor of

more than 2 for lamina applications.

4.8 Conclusions

It can be concluded that the most appropriate element to use for the analysis of the composite

structure in terms of accuracy of stress analysis and displacernents for the are( under plane stress is , the 44215, facet shell element. Where, local bending is imponant the 44215 will give conservative

stress results.

The poor stress results for the laminated elements under bending is a cause for concern for the

vendors and users of Pafec and these conclusions have been transmitted to that company.

4-13

Fig 4.6 Stress Results for Tension Tests 1, 3 and 5

Test No. Ply

0 1 90

9~

3 45

5 45 -45

0 1 90

9~

3 45

5 45 -45

0 1 90

90 0

3 45

5 45 -45

0 1 90

90 0

3 45

5 45 -45

-2 All Stresses in Nmm

Mid Ply Stresses

(11 (12

Pafec Element 36215

46.700 0.867 -0.867 3.299

:~.~~~ ;.~~~

25.000 25.000

45.83 4.166 45.83 4.166

Pafec Element 44215

46.700 0.867 -0.867 3.300 -0.867 ;.;~~ 46.700

25.000 25.000

36.850 13.030 36.850 13.030

Pafec Element 43215

46.700 0.867 -0.867 3.300

:~.~~~ ~.;~~ 25.000 25.000

36.850 13.030 36.850 13.030

CoALA Results

46.701 0.868 -0.868 3.298

~~:~~~ ~~~~ .B 8

25.000 25.000

36.952 13.048 36.952 13.048

4-14

T12

0.397E-7 0.169E-6 0.169E-6 n'l07<"·_7

-25.000

25.000 -25.000

0.140E-6 1.200E-6

~.~~~~:~ -25.000

25.000 25.000

0.350E-7 0.350E-7

~.~~~~-:~ -25.000

25.000 -25.000

0.157E-7 0.166E-6

0.16~f-6 0.157 7

-25.000

25.000 -25.000

Fig 4.7 Stress Results for the Symmetric Lominote In Bending. Test 2

Mid Ply Stresses

Element Ply Ifl 1f2 T12

Pofec Element 44215

0 -0.531E2 -0. 183E1 -0.103E2 1 90 0.364El -o.133El 0.345El

90 -0.364El 0.133El -0.345El 0 0.531E2 O.I83El 0.103E2

0 -0.554E2 -0.515 -0.148El 2 90 0.758El -o.116El 0.495

90 -0.758El 0.116El -0.495 0 0.554E2 0.515 0.148El

0 -0.567E2 -o.105El -0.151 3 90 0.340 -O.I34El 0.504E-l

90 -0.340 O.I34El -0.504E-l 0 0.561£2 0.105El 0.151

0 -0.566E2 -0.905 -0.105E-l 4 90 0.103El -0.132El 0.350E-2

90 -0.103E 1 0.132El -0.350E-2 0 0.566E2 0.905 0.105E-l

0 -0.565E2 -0.984 0.826E-3 5 90 0.649 -0.132El -0.276E-3

90 -0.649 0.132El 0.276E-3 0 0.565E2 0.984 -0.826E-3

Pofec Element 43215

0 -0.423E2 -0.665 -0.670E-2 1 90 0.825 -0.985 0.224E-2

90 -0.825 0.985 -0.224E-2 0 0.423E2 0.665 0.670E-2

0 -0.425E2 -0.572 -0.593E-l 2 90 0.127El -0.980 0.197E-l

90 -0.127El 0.980 -0.197E-l 0 0.425E2 0.572 0.593E-l

0 -0.424E2 -0.564 -0.316 3 90 0.131El -0.980 0.645E-l

90 -0.131El 0.980 -0.645E-l 0 0.424E2 0.564 0.316

0 -0.424E2 -0.561 -0.923 4 90 0.131El -0.980 0.307

90 -0.131El 0.980 0.307 0 0.424E2 0.564 0.923

0 -0.424E2 -0.564 -0.440El 5 90 0.132El -0.980 0.146El

90 -0.132El 0.980 -0.146El 0 0.424E2 0.564 0.44OEl

CoALA Results

0 0.318E2 0.426 -O.I44El 90 -0.987 0.734 0.416El 90 0.987 -0.734 -0.416El

0 -0.318E2 -0.426 O.I44El

All stresses in Nmm-2

4-15

t--00

Fig 4.10 Displacements Results for the Bending Tests 2 and 4

Test 2 - Symmetric Laminate in Bending

Node (i) Node (ii) Element Type

ux uy uz ux uy uz

44215 0 0 -1.17 0 0 -1.35 43215 0 0 -2.02 0 0 -2.02 CoAlA 0 0 -1.89 0 0 -1.89

Test 4 - Generally Orthotropic Lamina in Bending

Node (i) Element Type

ux uy uz

44215 0 0 -0.94 43215 0 0 -1.39

Results shown for on applied moment of 104

Nmm All displacements in mm

ux

0 0

ux displacements ore along the bar uy displacements ore perpendicular to the length of the bar uz out of plane displacements

Node (ii)

uy uz

0 -0.89 0 -1.31

ux

0 0 0

ux

0 0

Node (iii)

uy uz

0 -1.26 0 -2.02 0 -1.89

I

Node (iii)

uy uz

0 -0.84 0 -1.22

Figure 4.11 Symmetric Laminate in Tension, Test 1,36215 Element

i : ng . ~ ~

N

I DO':

~ i p. .1 ·a" IL..J'-...-. • • • .. ,;

~ .~N

I - .. ! a. h ~ I N

~ N IS§ . - !! • !! i:: i: lal a: a

~ L ~

~

:.

=-r:.

...... _-- .............. - SI

:. Ii f!'~

:, IS ; i • · ..... L... -• W

& W ~ w --....... ---........ c .. ~ ~ ~

&

~ ~

~ L ...... _ .. _----- ..... ':! 0 r .. • • r .. • !:! .. .... _._ .. _------ .... • w

· ~

• • • :: L 0 -..... __ ...... __ ...... -• - · 0

:! I • 0 - • · • ~

• ~

~: :: y

"'a w L Ut ..

" w

L: ~ --

4-19

: ~ · • " "

o o r

o r .. • o M • W -! • c • ~ .. o • -o z -• o

~ I\....J\-..,

~

Figure 4.12 Symmetric Laminate in Tension, Test 1,44215 Element

H

~~~ • 15 ... !~ , ,

••• _~H

- ~

I o· w

11! I!Ig

• ';ID 11

4-20

li ::

~! .!I~

.:

~ h i . - !! !! i: ;~; hg

! E ~ ;;;!

o 0

• :;

0 "~ i

c w • C C

o o r

o r .. • o

• • w -• ~ • ~ .. o • -o c -• o

• J J ~

U W .. • W J -:

~ Il..-Jl...-.-.

~

Figure 4.13 Symmetric Laminate in Tension, Test 1, Element 43215

N

~~~ il ---•• ••• •• N

-" t

O· w Bg e ~o III

4·21

i = I I!

_~ft

" • h fi ! . - 11 • 11

i~ _ w_ . ~ . hg

! i~ ;;!

o 0

Figure 4.14 Generally Orthotropic Lamina in Tension, Test 3, Element 36215

. i :: 0

~ Ill! • . ~ I od':

~ ~ P. I\....J\.-.-, • • • • 51

-aft ~

. ~. I " - " i I a •

~! ~ .. .8 -• • 11 • 11 ~ ad i~ ;; Hi

I i IS

2

:;..:

:.

=-r:.

I! r----; .........• ~ ::. ft' i-!'~ : ~ · :, i · • · • ...JL... • : · i r--+ ........... ; .. , .. : E .; .. ~ .. · : : • · • • ~ : E • E t--t ... __ ....... ~ • -~ • L · : • •• ft · i · · · · · : · • · · • ~ •

-1 ft r-f ...•.. __ ... _-• ::: ': ! :

& : c

• : ::: : L

~--.... ---------.. a &

:; • -• • : - : I & a · - ~ p ~ · ~ : c

% & • :0::-.. %c • .. UI! ..

" .. L: ~ ---

4-22

~

• w

Figure 4.15 Generally Orthotropic Lamina in Tension, Test 3, Element 44215

N i ~ ~~~ . ~ I ... P • • II......J\..... ....

II ~ "~N

~ " I

... w ~w ~

Ill! --;:i wl!i f :I III

:

-~~ ,;

• i .~

If • If ~w_ i ~ .

le i!

~ i

lj;: :.! ....

C w - . w

~

• •

:: -• w ~

!

"' c • ~ L .. "' ~ .. • ~ "' .. ~ --c • w • w .. .. -~ ~

4-23

Figure 4.16 Generally Orthotropic Lamina in Tension, Test 3, Element 43215

i : N

~ ~~~ • li J noO i -~- !i . , ~!

- ~ " Il...J'\..--, •••

~ ... ~ ~

- " • o , ~ I w ~o !i

1!1 ~ - II! M _

II • II

~ a;i wl!i _ w _

-= a ~ a hi

~ i tF-

:;;!

o 0

~ 0 • 0 ~

;; :; W M

E E ~

~ ~ .. " • • • 0

;;; • = ! • c • ::: .. 0 • ~ 0 • ~ ; • 0

~ ~ ;; ~ .. c • '" w ,,~

• "'c w u~ u

w ~

:1 L~ : '~

4-24

Figure 4.17 Unsymmetric Laminate in Tension, Test 5, Element 36215

i = 118 • li

~ M ... I .d':

~ ! p 1I

- R ~ ;;: " fLJ\-.-., • • • ..

~ .~N

I L • D • h i ~ I w

"'i . - !I • !I ~

w a.; N~ - ~ -~

• :2 i be i

I E ~ 2

::..: : ~, ::

SI r. •••••••••••••• , :: "~ r~

: i If -'-

He·············· •

-• Ir w

" w • w

: [··'··T"·" " • " " • !! · · · · · ~ · · · · • · ~ · · · · · ft · · · ~ .............. • ft · • · • ! • • = ! · .............. !

• · ~ · • i i w i • : ~ w : r

" :r • i r-~

:r~ • • · · u~ ,. W ,:;

D D

= L; :

4-25

Figure 4.18 UnsymmetricLaminate in Tension, Test 5, Element 44215

• i ::

~ In . ~ I ... P • •

,I • iI .. ILft-... •••

~ ••• .: ..

~ I ...

h i w

I!sE ~ . !I • !I ~ ~:i i~ » ~ » ;:

III hi

i ti IF-

:o!

D D

• ID

: ~ · · c ~

• ~

~

~ · • ~ · • • ! ~

• w · ! • C • I ~

.. • · . w

11 c c :z: • ",. ~ :z:C • " ui! w " ~

L~ · :

4-26

Figure 4.19 Unsymmetric Laminate in Tension, Test S, Element 43215

H i ::

~ ~~~ . ~ I ... i: ' , n....J\...-. ••• I!

• !I ft

~ .~H H

• t

o· 11 0 Ii ! w

I!o • ~ it

;,! ~ - If • If ~ i: iH III

~ i!i o 0

•

• ;; 0 0 J

;; W -C

~ ~

J L

ft

• • ;. • • • ~ -• w ~

~

• c • ; u

• " ~

w

= c c % • :o:~ M %c • u~ " w ;: ~

0 0

: Li

4·27

c c

~ o • a • .; y

• a • .. • ! • c • ~ L a • -a • -• a

• --c w W L M

W

= :

~ f\......J'\--.,

~

Figure 4.20 Symmetric Laminate in Bending, Test 2, Element 44215

.. ~~~ • Zi M_ --- p • • • •• M."

I ii§

~ 11';

a •

4-28

i :

A

5! -I"

,;

h i ~ M _ l .. l ''is

f~ iH

~ f li o 0

• " " -c W M

C C

" " r

D r .; u c

" c w • ! • C

• ::: .. " .. · D Z · .. D

~ I\.......J'\-..

~

Figure 4.21 Symmetric: Laminate in Bending, Test 2, Element 43215

N

~~~ • P,! ... P , , • • •

• ~N - ~ " . I " ~.: I:

= ;;

4-29

J :

I J!

.tjft .:

• r h i ~ ft _

!I • !I

i~ ; ~ ; hi

~ i ~ to!

" "

~-:rC '-'I'

~

L:

~

• w

" W J W

c

" "

" · · .. • " • w • : • c • ~ .. " • -" .. -• " ~

J J C • W • W .. W

~ :

Figure 4.22 Generally Orthotropic Lamina in Bending, Test 4, Element 44215

i H

~ ~~ i! I ... ••

!! ~ •••

~ "~H

;;11 I I!I' h

~ -~ a wlii ;:

• I!

4-30

:

• § " ,;

• !i ! \I • \I ;:~;: hi

~ .~

f M!

" "

R r

-.. W K W ... W

.. W M .. .. .. ~ -· .. .. o .. w • !

!:! L o .. -o .. -• o ~ ... ... c • w .. w .. w ... -:

Figure 4.23 Generally Orthotropic Lamina in Bending, Test 4, Element 43215

i N

~ ~~~ il I .. - A ••

~! Il.....1\....-, •••

~ .~N

I a;:

~! ~ , -~ § i~

4-31

:

• iI ~ ,;

• Ii ~ If • If ;;; hA

~ E :F-

t;!

o 0

S.llntroduction

CHAPTERS

Composite Case Development

This chapter investigates the design and development of a composite undercarriage equipment case

using Finite Element Analysis. It discusses case rationalisation, initial design calculations, the finite

element analysis process and finally details the results and recommendations.

The investigation so far into equipment case design has revealed that they tend to be overdesigned

and have several inherent problems; weak support design leading to high local stresses and a

tendency to exhibit excessive flexiblity due to the 'open frame' design. These points have been taken

into consideration in the following case development.

The confidence of using the Pafec orthotropic finite elements in such an analysis has also been

assured through extensive element testing, see chapter 4.

S.2 Case Rationalisation

The investigation of previous cases has shown that they are of a skeletal nature. The main structure

consists of two parallel, longitudinal side beams of deep section with appropriately positioned load

bearing cross members, the outer skin and panniers provide no significant contribution to the

strength of the case.

The design of the two longitudinal beams was considered initially as these form the basis of the case

and are used to provide the support mechanisms for all the equipment carried and also the case itself.

In previous designs, these structures have been found to be lacking in lateral stiffness and have been

highly stressed in the area of the case supports.

On establishing an efficient beam design, under vertical and longitudinal loading conditions, the

structure was extended with the addition of cross members, to form the basic skeletal case. This was

5-1

subjected to lateral load conditions.

5.3 Preliminary Design Calculations

Design calculations were carried out to establish some size guidelines for the longitudinal beams

structure. The structure was modelled as a uniform beam. simply supported at both ends. The beam

was considered to carry two loads:-

i. A uniformly distributed load of 2.1 kN/m. representing the equipment carried by the case

and its self weight.

ii. Two point loads of 3 kN to represent half the choke load. the heaviest single item of

equipment.

Figure 5.1 illustrates the model.

0.85

L = 3.40

Figure 5.1

Considering vertical equilibrium

RA+Ro = wL+2P= 7.14+6.0= 13.14kN

Where w=2.1 kN/m. L=3.4 m and P=3 kN.

The loading is symmetrical. hence.

5-2

w

/

~ = Ro = 13.14/2 = 6.57 kN

Maximum bending moment occurs centrally.

Considering section BC

Bending Moment M = ~ x - wx2 - P(x - 0.85)

Centrally x = 1.7m, hence

M = 5.14kNm

The depth of the case is 0.8m. hence the approximate compressive and tensile loads in the top and

bottom flanges respectively can be calculated from.

End Load = Bending Moment 1 Depth

ie. End Load = 5.14/0.8 = 6.5 kN

The end load was adjusted to 7.0 kN to provide a small reserve factor against changes in the loading

pattern which must occur due to alterations of the equipment layout at a later date. Considering a

ballpark figure for allowable direct stress of 100 MN/m2 for a Glass Fibrel Epoxy Resin composite

structure.

Hence from.

Beam flange cross section = End Load 1 Direct Stress

Flange cross section = 7 x lOS 1100 x 106 = 70 x 10- 6 m2

ie. Flange cross section = 70 mm2

The above calculations show that the cross sectional area of a beam flanges to carry the specified

loads is very small. so it can be deduced that the above model is not strength critical and the material

cross section required are dictated by stiffness considerations. The initial size guidelines were taken

from similar features in the standard metal isotropic case modified to take account of the composite

material specification.

5-3

5.4 Longitudinal Beam Idealisation

These are symmetrical about the vertical axis, hence half the structure was modelled. The

dimensions for the structure, 3.4 x 0.8 m were taken from the hybrid case analysed in chapter 3. The

model consists of a thin vertical web with vertical stiffeners, positioned at either end and at a point

centrally where the choke load is located. Upper and lower horizontal flanges complete the structure

providing further stiffness. The upper flange is extended to provide the case support, with a

triangular gusset panel to aid load transfer.

The structure was modelled using two composite materials. The web, vertical stiffeners, outer

portions of the upper and lower horizontal flanges and gusset panel at the support were modelled in

a woven glass lamina of varying thickness dependent on the location. The initial material sizings

being, 4mm thick for the web and outer portions of the flanges and 8 mm thick material for the

vertical stiffeners The inner portion of the upper and lower flanges were modelled in a unidirectional

lamina consisting of glass rovings in an epoxy resin to provide longitudinal stiffness. This portion of

the flange was initially designed 50mm wide using 8mm thick unidirectional material. The

torsional stiffness of the flanges was increased by the use of woven cloth to form the outer edges of

the upper and lower flanges, the total flange width being 150 mm. These dimensions give an initial

flange cross section of 1000 mm2 , well above the preliminary design calculations of 70 mm2 • A

volume fraction analysis of both materials can be found in section 5.4.2 of this chapter.

5.4.1 Finite Element Modelling

5.4.1.1 Model 1

The initial coarse mesh model, can be seen in figures 5.2 and 5.3 shows the small degree of mesh

refinement at the support locations. The mid side noded onhotropic facet shell elements, 44215 and

44115 (a triangular element) were used. These elements were the most reliable of the onhotropic

5-4

elements tested in tension and bending situations. It was shown that stress and displacement results

achieved by the 44215 element in the tension tests correlated accurately with the proven analysis

package, CoALA and theoretical calculations. In bending the element did not function as well, with

the stress results achieved being a factor of 2 too great and the displacements being 7 % too great.

The use of the triangular 44115 elements was restricted, as these elements by the nature are not as

accurate in terms of stress and displacement analysis as their quadrilateral counterparts.

The initial model had 166 nodes, 599 elements and 3174 degrees of freedom.

5.4.1.2 Model 2

The results of the above analysis could not be viewed with confidence in the suppon region. The

model was refined around the suppon location using the same element types. The original gusset

panel was replaced by two similar panels offset by 25mm from the central axis. A small rib of

woven cloth was added across the edge of the upper flange to reduce transverse flexibility. Extra

longitudinal gusset panels were included into the web to aid load transfer. The stiffness of the

suppons was increased by reinforcing the triangular gusset plates to double thickness, 8 mm and the

top surface elements surrounding the exact suppon location to five times the original thickness, 40

mm. This later development simulating a washer to distribute the load. The final design scheme is

shown in figure 5.4 and the refined suppon can be seen in figure 5.5.

The final model had 688 nodes, 186 elements and 3510 degrees of freedom.

5.4.2 Material Selection

The two materials selected for the composite case were:-

i. A unidirectional lamina, consisting of 8 mm diameter Glass Rovings in an epoxy resin.

ii. A woven glass cloth lamina, again in an epoxy resin.

Laminas were selected as it was considered too complex and expensive to specify multi-layer

5-5

composites at this stage, especially as the structure being considered was subjected to direct stresses

with no torsional effects, an ideal situation for the use of unidirectional materials.

A volume fraction analysis was carried out on each of these materials to determine their Young's'

and Shear Moduli.

5.4.2.1 The Uni-directionallamina

Propenies:-

E for fibre q, = 38 GPa

E for matrix EM = 3 GPa

Considering a fibre volume fraction of 0.6, hence

VF = 0.6

VM = 0.4

Rule of Mixtures Theory, Reference IO details:-

Modulus in fibre direction El = q, VF + EM Viol

Modulus in transverse direction 1/E2 = VF IEF + VM IEM

An accepted approximation for the in plane Shear Modulus gives,

GI 2 = 0.4 X El ... (3)

Hence on substitution,

El = 24.0GPa

E2 = 6.7 GPa

Gl2 = 9.6 GPa

... (\)

... (2)

The Pafec onhotropic material module within the data file requires these properties to be

transformed into the appropriate compliances, as detailed in the following matrix.

5-6

£1 Sl1 SI2 0 0"1

£2 = S21 S22 0 0"2

$12 0 0 S66 T!2

From composite theory, Reference 11.

SII = llEl · .. (4)

S22 = llE2 · .. (5)

SI2 =-~12IE2 ., . (6)

S66 = 1/012 · .. (7)

Hence on substitution in (4), (5), (6) and (7) the compliances for the unidirectional laminate are,

Sl1 = 4.167 x 10- 11 Pa- I

S22 = 1.496 X 10- 1 0 Pa- I

SI2 = -1.083 X 10- 11 Pa- I

S66 = 1.042 X 10- 10 Pa- 1

5.4.2.2 The Woven Glass Cloth lamina

A similar analysis for the woven material gives,

Propenies:-

E for fibre q = 38 OPa

E for matrix Et.! = 3 GPa

Considering a fibre volume fraction of 0.6, hence

VF = 0.6

VM = 0.4

An accepted approximation for the in plane Shear Modulus gives,

5-7

G12 = 0.4 X El

On substitution into equations (1) and (3) from the previous analysis gives,

El = 9.6GPa

E2 = 9.6GPa

G12 = 3.8 GPa

Note E2 is assumed to be the same as El due to the nature of the woven material.

The compliances are, on substitution into (4), (5), (6) and (7):-

Sll = 0.104 x 10- 9 Pa- 1

S22 = 0.104 X 10- 9 Pa- 1

S12 = -2.708 X 10- 11 Pa- 1

S66 = 2.630 X 10- 10 Pa- 1

5.4.3 Load Cases

The standard load cases can be found in Appendix AI, the cases considered in this analysis as

follows:-

5.4.3.1 Proof Load Cases

The equipment mountings must be able to withstand the mass of the equipment when subjected to

the following accelerations, all loads being reacted at the supports.

Two proof load cases were applied to the structure:-

i. Vertical1.5g down

ii. Longitudinal ±3.0g + 1.0g vertical down

It should be noted that the vertical 1.0g up case, load case (i) in Appendix A 1 has not been applied

here, the vertical 1.5g down being a 'worst case'. Only the positive longitudinal case was analysed, it

was assumed that on obtaining satisfactory results in this case, the negative case would produce

5-8

satisfactory results. A lateral load case was not applied here, a true indication of lateral stiffness can

only be established on testing a model of the whole case structure.

5.4.3.2 Fatigue Load Case

All equipment mountings must be designed to have a fatigue life of not less than 107 cycles for the

loads produced by the following accelerations acting on the mass of the equipment, ie choke and

pannier loads.

i. Vertical l.Og down ± O.3g

ii. Longitudinal ±O.2g

Lateral effects were not considered here (±O.3g) standard case, Appendix AI. The fatigue life

determined from the summation of the effects of venical and longitudinal loads is anticipated to be

much greater than the stated requirements.

5.4.5 Load Case Idealisation

The structure is assumed to carry half the mass of the equipment in the case and half the mass of the

case itself. The following loads were applied:-

1. A uniformly disoibuted load of 2.1 kN/m, along the length of the longitudinal beam

to represent equipment loading and self weight.

ii. Choke load, total mass l.2 tonnes applied as a point load at two locations on each side

structure.

The structure was loaded under venical and longitudinal load cases with loads applied along the

central axis of the upper flange, in the required ratio 1:4: 1 across each of the second order elements

used.

The structure was restrained in the venical, lateral and longitudinal directions along the axis of

symmetry and at the suppon node in the venical and lateral directions. The longitudinal direction

5-9

was not restrained at the support to allow a slotted hole to be modelled. In the analysis of the first

model, the support node was taken to be the central node at the edge of the upper flange extension,

however in the refined model the restraint was applied 7 mm from the edge at the next suitable node

in order to model the mounting position more realistically.

5.5 Skeletal Model Idealisation

5.5.1 Finite Element Modelling

The side structure was extended to form a skeletal case. This was achieved by the addition of four

lateral cross members, placed at either end of the case and internally at the choke mounting

locations. The cross members were modelled as vertical webs the full height of the case, with upper

and lower horizontal flanges to providing further stiffness. The cross members were modeJled in the

same materials as the longitudinal beams with the web, stiffeners and initially the outer portions of

the upper and lower flanges being modelled in the woven cloth composite, the inner portion of the

flanges being modelled in the unidirectional laminate to provide lateral stiffness.

5.5.1.1 Model 1

Even though the case displays two axes of symmetry, it was necessary to model half the case as the

model is to be subjected to an anti symmetric load case. The side structure used previously was

extended by the addition of two cross members, with a certain amount of remodelling being done

around the cross member locations. The half model of the case was achieved by mirroring about the

longitudinal axis. The model was idealised again using the facet shell elements, 44215 and 44115,

with again the use of triangular elements being kept to a minimum. The model can be seen in

figure 5.6.

5.5.1.2 Model 2

The analysis of model 1 revealed that there was high lateral deflections. To overcome this the

5-10

stiffness of the upper and lower flanges on the beams and cross members was increased, by using

unidirectional rovings across the full width of the flanges instead of having a central core of rovings

and outer portion of woven cloth. Woven cloth would be wrapped around the flanges to provide an

outer skin and increased torsional stiffness, this was not modelled. The depth of the flanges was also

increased to 32mm, again to increase stiffness. To effect the above changes, no remodelling was

required, material properties of the appropriate groups of elements were changed in the Pafec data

flIe.

5.5.2 Material Selection

See section 5.4.2

5.5.3 Load Cases

The skeletal case was tested under lateral load conditions and as in previous specifications, the

equipment mountings must be able to withstand the mass of the equipment when subjected to the

following loading, all loads being reacted at the supports. The actual loading tested was;-

± l.lg + l.Og vertical down

It should be noted that only the positive lateral case was analysed, it was assumed that on obtaining

satisfactory results in this case, the negative case would produce satisfactory results. As the skeletal

model was analysed under the lateral load case only, no fatigue analysis was performed.

5.5.3 Load Case Idealisation

The skeletal case was tested under lateral loading conditions, subjected to the same loads as the

longitudinal beams, ie the uniformly distributed load to represent equipment and self weight and the

point loads to represent the choke. In the previous model all loads were applied along the upper

surface of the side structure as their load paths passed straight through the structure, when loaded

vertically and laterally. To simulate realistic loading in the lateral load case, loads were applied

5-11

along the upper and lower flanges to achieve the effect of equipment being distributed throughout

the case.

Restraints were applied at the axis of symmetry to prevent longitudinal displacements An

additional restraint was also applied in this plane to prevent rotation around the vertical axis. At the

supports vertical and lateral displacements were restrained. longitudinal displacements were

allowed. to simulate the effect of a slotted hole mounting.

5.6 Results

The displacement plots obtained from the proof load analyses for side structure and skeletal model

can be seen in figures 5.7 - 5.10. There are no stress contour plots included. these plots can only be

obtained for composite material analyses by using PIG S version 4.3 and later releases to

post-process. Unfortunately this revision of software was not available at the time of the research.

On identification of the areas of high displacement. several graphs have been plotted via PIGS post­

processor to clarify the results for each load case. The stress output files were examined and areas of

high stress have been plotted out manuaIly. Figures 5.11 - 5.13 details the direct stress at the

element centroid in the material direction along the positive surface for elements along the upper

flange and the support locations in both for the two side structure models analysed. Figures

5.14- 5.15 provides the same information for the skeletal model.

5.6.1 Longitudinal Beams

5.6.1.1 Vertical Load Case

The analysis of the first model. revealed the structure to be relatively unstressed. with stresses of the

order 106 Nm- 2 throughout the structure. These stresses being well within the maximum working

stress level of loB Nm- 2• which is approximately half the Ultimate Tensile Strength of the glass

rovings used. these giving a safety factor of 2. The support area however was highly stressed with

5-12

stresses of the order of IOS Nm- 2 being found, with actual values higher than the working level,

see figure 5.11. Elements in the gusset panels were also more highly stressed With stresses of the

order of 107 Nm- 2. Local elements to the support area in the main web, however experienced low

stresses of the order of 106 N m - 2 •

The displacement curves for the fust model, figures 5.16 - 5.18 show that there is a maximum

vertical displacement of 4.72 mm centrally along the upper surface under the verticall.5g down load

case. with approximately 3.00 mm deflection across the fust element of the support area.

Transversely across the top flange at the support position there was a symmetrical deflection of

3.12 mm as shown in figure 5.17. Lateral displacements were minimal throughout, of the order of

10-3 mm, as can be seen in figure 5.18.

The structure was then modified several times to improve the design in the support location, as

described in section 5.4.1.2. The mesh was refined in the support area, to enable more accurate

results to be obtained. Modelling refinements included the addition of a small transverse rib of

woven cloth across the termination of the upper flange to prevent transverse flexibility. The gusset

plate elements and the upper flange support elements were increased in thickness, the gusset

elements by a factor of two and the support elements by a factor of 5. This model was then analysed

under vertical and longitudinal load cases.

The refined model subjected to the vertical load case, again showed that the majority of the case

was relatively unstressed, with stresses of the order of 106 Nm- 2. The support location experienced

higher stresses of the order of 107 Nm- 2, however these are well within the maximum working

stress level of IOS Nm- 2, see figure 5.12. Displacements for the vertical load case showed a

reduction in vertical displacements, with a maximum of 2.7Omm occurring centrally and a

proportionate deflection over the support region. Across the support displacements were minimal of

5-13

the order of 10- 2 mm. Graphs of these displacements can be seen in figures 5.19 . 5.20. On checking

lateral and longitudinal displacements throughout the structure, both were found to be negligible.

Displacement plots are also included for both models analysed here and can be seen in figures 5.7 -

5.8.

5.6.1.2 Longitudinal Load Case

When subjected to the longitudinal load case, again the majority of the case was relatively

unstressed, with stresses of the order of 106 Nm- 2 throughout the structure. The support location

experienced higher stresses of the order of 107 Nm- 2 , however these are still well within the

maximum working stress level of loB Nm- 2 , see figure 5.13. Displacements in the vertical direction

along the upper flange increased for this load case and were at a maximum of 3.07 mm centrally

with a proportionate deflection over the support region. Across the support displacements were

minimal of the order of 10- 5 mm. Graphs of these displacements can be seen in figures 5.21 - 5.22.

Lateral displacements throughout the structure were minimal throughout the structure, of the order of

10- 3 mm. Longitudinal displacements at the mounting point were minimal, maximum of 1.24mm

and in the real case would be absorbed by a slotted hole or similar mechanism at one end mounting.

A displacement plot can be seen in figure 5.9.

5.6.1.3 Fatigue Load Case

The support area of the structure which experienced the highest stresses was considered in the

fatigue analysis. A typical stress range experienced by a node at the support was calculated to be

1.46 x 107 Nm- 2. From standard data sheets the fatigue strength of a Glass/Epoxy composite

( VF = 60%) at 107 cycles is approximately 20% of its Ulimate Tensile Strength. The U. T.S of the

material used in this analysis is 3.24 x loB Nm- 2 ,hence the maximum stress range allowable is 6.4

x 107 Nm- 2. Hence the stress range experienced is well within the allowable range, with a reserve

5-14

o

factor of 4.4 and the fatigue life of the case is assured for 107 cycles.

5.6.1.4 Mass of the structure

The mass of the longitudinal beams were 29.2kg for the first model and 31.2kg for the refined

model. These masses being quoted for a full single complete beam.

5.6.2 Skeletal Case

5.6.2.1 Lateral Load Case

The analysis of the first model revealed the majority of the structure to be relatively unstressed, with

stresses of the order of 106 Nm- 2 or lower throughout. However there were several areas of high

stress identified as shown in figure 5.14, where stresses of the order of 107 Nm- 2 were found.

These occurred at the intersection points between the side structures and lateral cross members.

These high stresses are to be expected as there is no fillet modelled at the corners to smooth the

stress distribution.

High lateral displacements were experienced see figure 5.23 with a maximum of 22.89 mm at the

centre of the right hand side upper flange, at the supports the lateral displacement is negligible.

Figure 5.23 also shows that the model does not bend in a smooth manner, there is a sharp gradient to

the graph before the location of the internal cross member. The gradient at this position is

substantially reduced as would be expected, with the cross member providing lateral bending

stiffness. Towards the centre of the case the gradient of the curve increases and is a maximum at the

centre. An example of lateral displacements through the depth of the case can also be seen in figure

5.24, and shows that the deflection is relatively uniform through the depth of the case, the lower

flange being displaced an extra 5.4% at this position. Vertical displacements are low, negligible in

the support locations with a typical maximum displacement of 1.04 mm occurring at the centrally,

figure 5.25.

5·15

o

These results showed a lack of lateral stiffness and a lack of strength at the points where the cross

members merge, which led to the development of the second model.

The second model with redesigned upper and lower flanges showed on analysis that the regions of

high stress were substantially lower, of the order of 106 Nm- 2. Stresses throughout the structure

were again of the order of 106 Nm- 2 or lower, see figure 5.1.

Lateral displacements, figures 5.26 - 5.29, were greatly reduced to what was thought as an

acceptable 3.05 mm occurring centrally along the right hand upper flange. Displacements along the

left hand flanges again were acceptable and 2.3% greater than the right hand side. A deflection of

3.12 mm occurred centrally along the left hand side upper flange. The case defonned laterally in the

same manner as before, as can be seen from the similar shaped curves in figures 5.26 - 5.29 to figure

5.23. The graphs show that there is small deflections at the ends of the case rising to a maximum

centrally. Lateral displacements through the depth of the case were found to increase

proportionately, figures 5.30 - 5.31. The graphs of displacement in the longitudinal direction along

the two cross members, figures 5.32 - 5.33 detail very small displacements, and deform in an's'

shape. The displacements are minimal of the order 10-3 mm and hence there is no cause for

concern due to non-linear effects. Vertical displacements along the flanges were at a maximum

centrally, 0.47mm is the maximum displacement along the right hand side upper flange is a typical

value. The lower flanges exhibited smaller vertical deflections, 4.8% smaller. The vertical

displacements along the left hand flanges were substantially greater, with a maximum vertical

displacement occurring centrally along the upper flange of 0.65 mm, 18% greater. Again the lower

flange exhibited smaller vertical displacements than the upper flange , 1.2% smaller centrally.

Unlike the lateral displacement curves, figures 5.34 -5.37, the vertical displacernents graphs do not

have similar curves showing that both sides of the case do not deform symmetrically in the vertical

5-16

sense. However, as these displacements are negligible any un symmetrical deformation is very slight

and can be ignored.

5.6.2.2 Mass of the structure

The mass of the resulting unloaded skeletal case was found to be 222kg, for a full model.

5.7 Discussion

The resulting design for a skeletal case on analysis reveals that it compares very well with the

similar analysis of the isotropic case, chapter 3. The model is relatively unstressed throughout, with

higher stresses occurring around the suppon locations and the junctions of the lateral cross members

and the main body of the case. However, these can be alleviated by the addition of a fillet at the

intersection points. Lateral stiffness again proved to be a cause for concern in the initial model of the

case, however with the change of flange design to one of completely unidirectional rovings, a

satisfactory level of stiffness was achieved. The displacements obtained were of a similar order to

those experienced by the isotropic case and well within the working limits of normal traction

engineering. A fatigue analysis using stress results from the 'highest' stressed area of the refined

model established that the structure would not suffer from fatigue damage, up to 107 cycles.

The mass of the skeletal case compares extremely favourably, the isotropic case having a mass of

360kg and the composite skeletal model a mass of 222kg, with funher refinements additional

weight savings could be achieved. Of course the mass of the skeletal model will increase with the

addition of outer panels, covers and internal frameworks, a reasonable estimate of total weight would

be 260 kg, giving a total saving of lOOkg ie a weight advantage of 0.72: 1.

Changing the flange material to a carbon fibre composite would theoretically enable funher weight

reduction, its greater value of Young's Modulus would acheive a case of equivalent structural

integrity using less material. However, practically this would be of no benefit as a minimum

5·17

thickness design could not be used due to manufacturing difficulties.

5.8 Conclusions

The skeletal model analysed was relatively unstressed, with acceptable stresses at the support

locations. Deflections throughout the structure were minimal and acceptable. On further refinement

the final mass of 222kg could be reduced, however it still compares favourably with the mass of the

isotropic case analysed.

5-\8

Figure 5.2 Longitudinal Beam Idealisation· First Model

000 Z NM ..... 0 -.... < .... " 11 11 0 a: >< >- N

\ \

w

~ U ..... o

5-19

er: w Z w ..... ..... t­V'I

~ w 1:

~ ..... o V'I

x «

~ -'"

-t-

'" I

'" o

SUPPORT POINT

VERTICAL STIFFENER--

TITLE SUPPORT STRUCTURE - FIRST MODEL

ROTATION

X = 20 y = 30 z = 10

.., ~.

til !" tH

m c "Cl

'g ::l m -Cl ~ til ;;: et ~. -g' .., 3: :: 8-!!.

Figure 5.4 Longitudinal Beam Idealisation· Refined Model

\

000 z NM.­o ..... < to- 11 11 11 o a::><>-N

rJi=~ \~~ ~~ ~~

5-21

..J .... co co :0::

co .... z -"­.... a:

.... § ... u => a: ... '" w co

'" W ...J I-

I-

'f '" '"

.-r::: -­LONGITUDINAL STIFFENERS

TITLE SUPPORT STRUCTURE - REFINED MOOn

ROTATION

X = 20 Y = 30 z = 10

.., ~.

nl Ut In Cl! C '0

3. Cl! :; C n C nl ... Co

~ [ g-

" to ::I ::I

It

== &. !!.

V1 V1

S U ....J < & W f-2:

Figure 5.6 Skeletal Model Idealisation

~uu ~DD

DD

5-23

000 z C'lM ...... 0 -~ < ~ 11 11 11 0 a: )( >- N

....J W Cl o 1:

Jt

V1 V1

S U Cl Z w

z Cl --< '" ...J -< .... Cl

...J .... Cl Cl >: ...J -< ~ .... ...J .... x '" W -l I--I-

Figure 5.7 Displacement Plot Verticall.5g Load Case· First Model

I i :

~~~ gJi . ~ ~

I . II = I -... i

1I . I ·

~ ••• ... ~ d .115 .. • I • I; If ! w

11" " Cl 11 • 11 S i: iH ~.J 0

J i li :; :; r: :::. ~T

11

:::. r!" ; _i

r- • • ...; .... :. , ' ,',' 1"-,',' .... · ~~ ,',I . , .... ; :::: ....

. ... jiff! " . . ' .' · .: , -:"t.': .' · ..... .' · : ::;: .' · ,0' • . ' · ,.", .' -.--.,~, .: ,

t :::: .' : ..... . ' " .. , 0" • ',I" ", . . .... ,11 • · · ...... i;;; ." . ," I · · .... ", . • · "." . " , • · .....

,00 • ..... ". , : · :::: : ," . ," . ! : .. ...... ~.;;; . " · !.:: : : :,':: : : :;:: y ',,' : !!:: : .. : .. ,

• : .'. '" , ! "', D .", , "7;: - ,",

D .... • =:::: .. ," , : ..., ........ : • ...... .;,.;;;.; .. : .. : .. • - • , "' ... ' "', : :::: .. ,", ! · - I" : • · ...... ~i - '. • • • I:: : • w • • ::F. : • IJlLL : -: I • • • • w1l ~ ... • • . :: .. : I • • · .... ~, • · .. • • "' .... .. /J/JL ~r i • : I':: D • . ,:: .. • D • .... -i;;i .. ~., • : c: : • !:::: - , % • ",,' "'-=

... ,' %-.. ~: up .. L!

.. .. : -

,

5-24

Figure 5.8 Displacement Plot Verticall.5g Load Case· Refined Model

f§ ~~~ gJi _R_

~ ••• _.-I i •

!:: i r;

, c..... ".j

mri!'t=+-~I :; ! '

i ! i ••• (

w

'U. M

.J ,

«

i u a

! « D ~

! !, ~

'. « ~ ii ~ ". • T w i • • ~ r w r ~ a D • r···i • .. ! .. .. • .. ~ ~

~

,

5~2S

:

! .~

I ;2

::.,!'

::..: ::. :T

I!

~::

1':~ -'-

I . j . , : , I: . , = '; ~

~ -: i :re ... ~ . !

L~ ; I 's i

Figure 5.9 Displacement Plot Longitudinal Load Case • Refined Model

9Ji e i ;

~ ~~~ i.~ I .... .• p I /LA-, ••• .1 -,-~

•• N - .; 1-- i I • D':'

~; i

IB! --- 11 • 11 j; i~ - ; -.. i.ei

I ~li ~,!'

':-.:

~ ,!" 11

:. ,!'; • :, ,.:~ , I -''-.

,

, ~; ; : i :; , i!!

, ,

' .. t; f···.

.., I··· .

r i . , ... . .. i --w

, 0 ; - ! 0

f ~ : ~ jl) i 'J -.. ! - .1. 't 0 D

UJ/. .. - tiff :: .. T • ! ~ ! UfJ ,

'F-~ I .. a f 0

~'" .. . ... ~ .. a ; ",11 .. !

:;~: zo-o. ",-.. """,,,' ",i! • ..

L! ~ --5-26

Figure S.10 Displacement Plot Lateral Load Case • Skeletal Model 2

I! i :

~ ~~~

~ i . ti I --- -I;~ I I1 .1 ---0

1L...J\,...., • • • w . ' . L ~

•• N

.U r I !i iD! !I·~ I! • !I !; ~;;ao i~ aB •

! i!i ..

w • • .. .. • .. J

J • • w -• J

J W

I • D .. J 11 c D -W Z J r-w z" • • u~ w -J

L: ---5-27

Figure 5.11 Direct Stress Distribution at the Support and along the Upper Flange.

Support Position

A Longitudinal Beam, Vertical Load Case - First Model

Upper Flange Plan View y

4.9E7 3.4E6 3.0E6 2.7E6 2.3E6

4.8E8 4.8E8 8.9E7 6.0E7 1.0E7 4.7E8 4.7E8 8.6E7 5.9E7 9.0E6

3.0E6 3.2E6 2.9E6 2.5E6 2.3E6

y

Support Location - Side View

1.lE7 2.5E6 1.5E6

Local Web Elements

Mid point of Side Structure

1.8E6

9.0E6 9.0E6

1.8E6

A Gusset Panel

9.4E6

Vertical Stiffener - Section YY

8.1E6 3.1 E6

1.8E6 6.6E6

4.4E6 3.4E6

Upper Elements only

Stress values taken are from the positive surface, in the material direction at the centroid of the element. All Stresses in Nm-~

5-28

Figure 5.12 Direct Stress Distribution ot the Support and along the Upper Flange. A Longitudinal Beam. Vertical Load Case - Refined Model

Upper Flange Plan View

1.4E6

3.3E6

1.2E6

1.4E6

1.9E6 7.1E6

3.4E7 4.6E7

8.3E7 3.7E7

1.9E6 2.7E6

y

E 1.5E7 1.7E7

E 1.9E7 1.9E7

E 2.0E7 2.1E7

El 1.6E7 1.9E7

y Transverse Rib End View Support Position

Suppart Location - Side View

4.9E6 3.3E6

1.3E6

A Gusset Panel

Longitudinal Gusset Panels - Plan View

E6

Vertical Stiffener

3.6E6 8.2E5

1.2E7

9.5E6

Section YY

~

Local Web Elements

Mid point of Si de Structure

2.8E6

3.6E6

9.oE6

2.8E6

9.4E6 1.9E7 Upper Elements anly

1.9E7 2. lE7

2.6E7 2.6E7

Stress values taken are from the positive surface. in the material direction at the centroid of the element. All Stresses in Nm-~

5-29

Figure 5.13 Direct Stress Distribution ot the Support and along the Upper Flange. A Longitudinal Beam, Longitudinal Load Case - Refined Model

Upper Flange Plan View

g.6E5

2.2E6

7.8E5

g.4£5

I.5E6

3.8E6

1.7E7

1.6E6

6.oE6

4.0£6

7.5E6

5.1E6

y

E 1.3£7 I.5E7

E 1.7£7 1.9E7

E 1.7£7 1.8£7

E 1.4£7 I.5E7

y Transverse Rib End View Support Position

Support Location - Side View

2.2E6

6.1E5

A Gusset Panel E5

Longitudinal Gusset Panels - Plan View

Vertical Stiffener

2.3E6

8.4E6

6.1E8

Section YY

6.1E5

Local Web Elements

Mid point of Side Structure

3.4£6

8.6E6

8.6E6

3.4£6

7.1E6 7.3E6 Upper Elements only

1.4E7 1.4E7

2.0E7 1.9E7

Stress values token ore from the positive surface, in the material direction at the centroid of the element. All Stresses in Nm-~

5-30

VI • w -

Support Po

Transverse Rib End View

Figure 5.14 Direct Stress Distribution at the Support and along the Upper Flange Skeletal Model. Lateral Load Case - Model #1

Upper Flange - Plan View y

~E E4 E~ E4 6.OE4 E4 E4 E4

sition End Lateral Internal Lateral

~.5E 3.3E EE 3.9E6 Cross Member 1.6E6 E6 E6 8.9E6 Cross Member

3.4E5 ~.8E 6.9E EE 2.3E7 5.4[6 4.2[6 3.4E6 [6 [6 4.9E 5.7[7

7.3[6 ~7E 12.9[ E 9.4E6 4.5[6 4.2E6 7.7E6 E6 [6 9.3[6 5.5[6

5.3E6 .IE 2.3[ EE 3.4E6 3.1[6 4.9[6 5.2E6 E6 E6 6.5E6 5.6[6

4.8E5 .9EE 2.6E EE 2.7E6 2.7E6 4.2E6 2.6E6 E6 [6 2.4EE 2.3E6

~

y

Support Location - Side View Vertical Stiffener - Section YY

2.2E5 4.1[5

1.4[6 2.3E6

Mid poir Side Str

E6

E6

[6

E6 ~

Local Web Elements Upper Elements only 2.3E6 2.3E6

t of ucture

Longitudinal Gusset Panels - Plan View

1[612.1[611.3[6 .

IEs!2.3[611.4E6 Stress values taken ore from the positive surface. in the material direction at the centroid of the element. All Stresses in Nm-~

v. , W to..>

Support Po

Transverse Rib End View

Figure 5.15 Direct Stress Distribution at the Support and along the Upper Flange Skeletal Model, Lateral Load Case - Model #2

Upper Flange - Plan View y

.4£:5 E4 E~ 2.2E6 2.4E5 E4 E4 E4

:ition End Lateral Internal Lateral .2E 2.8E E6 503E6 Cross Member 4.5E6 E6 E5 4.3EI Cross Member

1.lE5 .7E1 .OEI E6 1.9E6 6.0E6 1.5E6 2.2E6 E5 E6 1.0E6 1.2E6

1.7E6 ~.5E~ .7EE E5 1.lE6 3.0E6 1.8E8 7.0E5 E5 E5 8.2E~ 3.1E5

1.lE6 ~o3E~ ,,"4E E5 6.8E5 1.2E6 3.4E6 2.0E8 E5 E5 3.2E5 5.3E5

1.7E6 ~.3E ~.OE E5 1.7E6 3.2E6 7.8E4 3.7E6 E5 E5 3.2E5 6.7E5

y

Support Location Side View Vertical Stiffener - Section YY

6.6E5 4.:1E5

1.5E5 1.0E4

Mid pair Side Str

2.3E6

2.3E6

2.3E6

2.2E6

Local Web Elements Upper Elements only 4.0E5 2.2£5

t of ucture

Longitudinal Gusset Panels - Plan View

IE5I E4 E4

IE:4I (4 E4 Stress values token ore from the positive surface, in the material direction at the centroid of the element. All Stresses in Nm-~

NODES 283 509 14 356 18 15 19 82 85 4 142 145 148 150 6

0.0000 I I I I I I I I I I I I I J

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DISPLACEMENTS (UYI MM LOAOCASE 1.5G VERTICAL (DO\lNI

TT TI F: VERTICAL IUYI DISPLACEMENTS ALONG THE UPPER FLANGE - FIRST MODEL

.." I

w '"

281 282 0.0000 I

-3. 1323..j / /

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t-----------------------------lIWO\lNI T T T I I=" !UYl 01 SPLACEMENTS ACROSS THE SUppnRT - F TRST MnnFL

'" I ..., '"

NODES UPPER FLANGE

1

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r-------------------------------------------------------------41100WNI TT TI ~ I ATFRAI 11171 nrsPI ArFMFNTS THROllhH THF rHOKE LOAn PATH - FIRST MODEL

3.2118_~ NODES

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1.5G VERT ICAt (DOIIN)

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T T T I ~ (UY) D I SPLACEMENTS ACROSS THE SUPPORT - REF I NED MODEL

I

. 3.1131_ NODES 28 '~09 11 356 18 15 19 82 85 1 . 112 115 118 150 6

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DISPLACEMENTS CUY) MM E-1 LOADCASE 1.0G VERTICAL (DOIIN)

TT TI I=" VERTICAL C UY) DTSPLACfMfNTS ALONG THE UPPER FLANGE- REFINED MODEL 3.DG LONG.

NODES O. 12~_1 290 509 292 562

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'" I .,. 0

01 SPLACEMENTS (uZ) MM

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LOADCASE 1.0G VERTICAL

~------------------------------------------------------------~I(DOWN)

T f T I ~ LATERAL (uZ) 0 I SPLACEMENTS ALONG THE RHS UPPER FLANGE - SKELET At MODEL 4 4(' I • T

, \

'" I .,. ~

NODES

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15.118

T r T I F LA TERAL I UZ I 0 ISPLACEMENTS ALONG THE RHS CHOKE LOCA T!-Ott SKELETAL MODEL 1 it: I AT

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LOADCASE DISPLACEHENTS (UY) MM E-1 1.0G VERTICAL (DOWN)

TITLF VERTICAL (UY) DrSPLACEMENTS ALONG THE RHS UPPER FLANGE - SKELETAL MODEL 4""'.1'"

V1 I ...

w

DISPlACEMENTS (UZI MM [-1 30.511

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20.919

11.122

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11.328

8.1302

4.9329

1.1356

"11 ~.

;;:

~

KEY _. UZ DISP. - - UY DISP.

1

LOADCASE . ~ 1.0G VERTI CAl ';

r-----------------------------------------------------------~I(DOWNI I

TIT L E LATERAL (UZ I D I SPLACEMENTS ALONG THE RHS UPPER FLANGE - SKELETAL MODEL "2 1. 11: I AT

'" I .,.. '"

DISPLACEMENTS IUZI MM 3.3349

3.0163

2.8111

2.5591

2.3005

2.0420

1.1831

1.5218

1.2662

1.0016

0.1191 1~ 11S~ 22 32 94~ 91

KEY

'"I ~.

;;l tit \..> ...,

- UZ DISP. - - UY DISP.

LOADCASE 1. OG VERTI CAL i ,

r-------------------------------------------------------------~IIDOWN) TT TI ~ LATFRAI 11171 m5PI ArFMFNT5 AI nNr, THF RH5 I nUFR FI ANr,F - SI<F:I.FTAL MnnF!. If? •• ,.. I aT

u. 1-u.

DISPLACEMENTS IUZI MM E-1 31. 161

21.881

24.601

21.321

18.041 .., ~.

ftl en

14.163 lie:

11.183

8.2032

KEY 4.Q230 ---:. UZ DISP.

1.612Q

-1.6312 1801 LOADCASE

1.0G VERTICAL t----------------------------~IIDO\lNI

TTTI F LATERAL IUZI DISPLACEMENTS ALONG THE LHS UPPER FLANGE - SKELETAL MODEL n? 1. 1G LA T.

'" I .... Cl'

DISPLACEMENTS IUZI 3. 181l.

2.931

2.6934.

2.449

2.2051

1.960

1.1161.

1.412

1.2281

0.984

0.1400 159

MM

KEY

"lj

~.

;;! Ut N \0

- UZ DISP.

LOADCASE 1.0G VERTICAL

~--------__ ------------------------------------------------~IIDOWNI TT TI C I ATFRAI 11171 nrsPI ArFMt='NTS AI nNr. nIt=' II-IS I mlFR FI ANr.1=- - C:I<FI FlAl MnnFI tI? , •• ~ 0 ....

'" I ~ ....,

NODES

.., ~.

ri

~

3 KEY _. UZ DISP.

LOADCASE 22.245 22.113 23.~81 23.S48

DISPLACEMENTS IUZI MM E-1 1.0G VERTICAL r-------------------------------------------------------------~IIDOWNI

21.310 21.118

TT TI I=" LATERAL IUZI OISPLACEMENTS ALONG THE RHS CHOKE LOCATHlt+ SKELETAL MODEL tl2 I •• ,. I IT

~" I ....

00

NODES 150

150

151iJ ""- '"1 iQ' c

151~ ~ ~ Ut f... ...

152 KEY -- UZ DISP.

153

LOADCASE 15311~ >

21. 11 ~ ~ ~ I 1. OG \lERTI CAL r-------------------------------------------------------------~I(DOWNI

21.1"13 22.1)86 23,'151 23.1)28 22.21"1 DISPLACEMENTS CUZI MM E-1

T I TLELATERAL IUZI DISPlACEMENTS ALONG THE lHS CHOKE LOCATHltt SKELETAL MODEL 112 I 1.1r. I AT,

VI I ..,. '"

10. 130

8.3655

6.0006

3.6351

NODES 8 110 10A3 119 116 112 6~3 1911 1911 1920

-1.0941

-3.4590

-5.8239

-8.1888

-10.554

-12.919

KEY

:!l ~ ~ Ut l.­N

- UX DISP. - - UY DISP.

LOADCASE 1.0G VERTI CAL

r-------------------------------------------~~~~=---~I(DOWN)

DISPLACEMENTS lUX) MM E-2 SKELET AL MODEL

TT TI F LONGITUDINAL DISPLACEMENTS ALONG THF SFcnNn LATERAL CRnss MFMRFR 1. it: I AT.

8.5598

5.5991

2.6396 NODES 16 1942 1952 1514

-0.3206

-3.2801 '" I

'"

"PI ~.

~ 0 Ut

-6.2409 I:

-9.2010

-12.161

KEY -15.121 -- UX OISP.

- - Uy DISP. -18.081

-21.042 DISPLACEMENTS lUX) MM [-2

LOADCASE

I

SKELETAL MODEL 1 1•OG VERTICAL r------------------------------------------------------------~(DOWN)

TTTI F LONGITUDINAL lUX) DISPLAC[MENTS ALONG THE FIRST LATERAL CROSS MEMBER 1.1G LAT.

NODES 010rf13.3· 1338 356 15 82 801 1 902 115 150 . ""F~. I " ~- ~-

-0. 3816~ ~ ......

~

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\ \

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-4.294Q -~ ....... , "- ,

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DISPLACEMENTS (UY) MM E-1 1.0G UERTI CAL (DOWN)

TITLE VERTICAL (UYI DISPLACEMENTS ALONG THE RHS UPPER FLANGE - SKELETAL MODEL ~2 I .. ~ I IT

NODES o. 18191t ~ 1150 22 32 'l41 'l1'l 'lO 100 5

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(DOWN)

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NODES 1599 2301 1600 1391 2110 2118 1131 1141 1151

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-1.9151_

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'" .,

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------ - -... - - UY DISP. , ... , -5.8363--1

, ..... ... ... ... .... ... -6.4818J "- -- -- I LOADCASE DISPLACEMENTS (UYI MM E-1

1.0G VERTI CAL

TT TI F VERTICAL !UYI DISPLACEHENTS ALONG THE LHS LOWER FLANGE - SKELETAL MODEL 1t2 . (DOWN I 1 • • ..... _

O. Sl12 NOOES 1801, 216q 18SS 1'126 1'130 2000 1502 2011 1181 11QS ,

L _I -o.1281~ " I

\ \

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\ -.I

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....... .... -6. 181SJ .... ---.

DISPLACEMENTS (UY) MM E-1 I LOADCASE 1.0G VERTICAL (DOWN)

TT TI ~ . VERTICAL CUY) DISPLACEMENTS ALONG THE LHS UPPER FLANGE - SKELETAL MODEL 112 111(~IAT

6.1 Introduction

CHAPTER 6

Composite Case Design Proposals

The case design investigated in chapter 5 was for the pmpose of analysis reduced to a skeletal

structure consisting of longitudinal and lateral beams of idealised cross section. This idealisation

would require some change to make it a practical manufacturing proposition. The following chapter

details some possible construction features which would niaintain similar stress levels to those

derived in the analysis and suit the properties and manufacturing processes of glass fibre composite

materials. Not all features are shown, but the principles illustrated could be extended to all parts of

the design.

6.2 Design Features

The principal design features discussed below are illustrated in figures 6.2 - 6.4, with figure 6.1

detailing their exact locations in the model.

6.2.1 Main Flanges

The flange design adopted in the skeletal model is shown in figure 6.2, and consists of a rectangular

section of unidirectional material. An outer layer of woven cloth would be wrapped around the

section to provide protection and consolidate the junction between the flange and vertical web as

described in section 6.1.2.

6-1

>.. ... ..., Cl

E E CX1 E

>.. 0 III ... 0

'" ·x cC « «

Qj "0 0

:::;;

"0 ..., Cl ... Qj Cl

E .:.< t:

III Cl ~ ... ... ~

Cl :;; .r- III ..., ... 0 0 u

~ ..., ...

.~ Cl

> > Cl

E "0 Vi Lll

<'! ~ 0 ID ..., Cl

... ... 0 :l a. c> a.

:l LL.. III

u CD

6·2

Actual Flange Section

Detail A - Vertical Cross Section

150mm

32ml • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

Figure 6.2

6.2.2 The Webs

• • • • • • • • • ••• •• • • Unl-directional glass ravings

Web bonded into machined slot In flange.

Outer cloth wrap

Vertical web

The webs are modelled as flat sheets of woven material of constant thickness.' In practice these

would be moulded separately from the flanges, with integral thickened sections for the venical

stiffeners as shown in figure 6.3. Assembly would be by bonding the web panels into moulded slots

in the flanges, figure 6.2. In addition an external layer of woven material applied by wet lay up

methods would be added after assembly around the flanges extending onto the webs, figure 6.2.

This layer would provide protection for the load carrying members and joints from external damage.

6.2.3 Lateral Cross Members

The lateral cross members are proposed to be of identical construction to the side members. The

method of attachment to the side structure is again via a moulded slot in the appropriate venical

6-3

stiffener, figure 6.3. At the merger point for the longitudinal and lateral flanges, some interleaving

would be required to avoid a plane of pure matrix material, with continuity being maintained

longitudinally.

Actual Web Vertical Stiffener

Detail B - Horizontal Cross Section

Moulded random fibre fillet

All woven cloth laminas

Figure 6.3

6.2.4 Main Mounting Points

L ngitudinal beam

ateral cross member

The main method of mounting must be in the form of metal bushed holes. Composite material is

unsuitable for the high bolt clamping loads, due to the potential failure of the matrix material in

compression resulting in local delamination. The bushes may be bonded in or arranged to provide a

controlled 'nip' as shown in figure 6.4.

6-4

Mounting Point Details

Detail C - Vertical Cross Section through Upper Flange

Uni-d gloss ro~'incls

Figure 6.4

provides !=or)tr~oll!;1p nip when

htened.

In order to achieve the modelled conditions the main mountings must have minimal longitudinal

restraint. This could be achieved by a slotted hole in the metal locomotive chassis or preferably by a

short swinging link or rubber bushed mounting.

6.2.5 Attachment Points

It is proposed to use threaded metal inserts bonded in at the various equipment and cable clamping

positions. Similarly integral brackets could be fonned as pan of the structure, to reduce complexity

and assist rapid assembly.

6.2.6 Protection from Service Damage

The size guidelines proposed for the flanges, although adequate for the load cases, may not be

sufficient to withstand the general handling abuse likely to be experienced in railway service.

Additional protective laminas could be added in areas subjected to damage without excessively

increasing the weight.

6·5

6.3 Further Refinements of Design

Alternative materials such as Carbon fibre or Kevlar could be introduced into the structure to

improve local areas. Carbon fibre has a high Youngs's Modulus, approximately 3 times that of glass

and would be useful in reducing deflections. Carbon is also 50% stronger and 33% lighter than glass.

Kevlar has high strength but a relatively low Young's modulus making it very flexible and ideal for

areas subjected to high impact damage.

These specialised fibres are much more costly than glass and thus likely only to be used in local

areas, rather than in the main members of the equipment case.

6-6

The research concludes that:-

CHAPTER 7

Conclusions

7.1 The Finite Element Analysis of Composite Structures has proven to be an effective method of

analysing practical applications of these structures.

7.2 Extensive element testing has shown that the Pafec 75 Suite of Finite Element software is a

reasonably effective package to use for this purpose.

7.3 The proposed composite undercarriage equipment case has been shown to have acceptable stress

levels, minimal deflections and satisfactory fatigue life under the standard railway service load

conditions.

7.4 The main advantages associated with composite construction compared with a conventional

steel structure are; reduced weight, reduced complexity and high resistance to corrosion. On

application to an equipment case these should result in reduced manufacturing and operating costs

in the long term, however initial investment would be required to develop the manufacturing

processes and tooling.

7.5 The analyses have shown that a weight advantage ratio for a composite structure of 0.72: 1 is

easily achieved with low risk stress levels and a relatively simple design, with the additional

advantage of integral equipment mounting positions to reduce complexity.

7.6 In addition to the engineering advantages, there is the marketing advantage of incorporating

advanced technology in engineering products, which may justify the additional development costs.

7-1

REFERENCES

1. Pafec Data Prep Manual

2. Warnock F. and Benham P.P.,"Mechanics of Solids and Strength of Materials," Pitman Publications; 1969, pp 364-369.

3. Gurney T. R., ''Fatigue Design Rules for Welded Steel Joints," The Welded Institue Research Bulletin; Vo117, May 1976.

4. Sharrnan P.W.,"Stress Quality Testing of Iso-parametric Membrane Elements in Uniform Thickness and Stiffened Benchmarks," NAFEMS Conference; May 1987.

5 .. Somashekar B.R. and Prathap G.,"A field consistency error model for the single element test for the eight noded brick element," Quality Assurance in FEM Technology, Edited by J.Robinson; 1987, pp 171 -180.

6. Dahto D., "CoALA Primer," Cranfield Institute of Technology Publication; 1987.

7. Hoskins B.C. and Baker A.A., "Composite Materials for Aircraft Structures," AlAA Education Series; 1985, pp 108 - 111.

8. Hoskins B.C. and Baker A.A., "Composite Materials for Aircraft Structures," AIAA Education Series; 1985, pp 95 - 98.

9. Todd J.D., "Structural Theory and Analysis," The Macmillan Press Ltd.; 1979, pp 138 - 152.

10. Vinson J.R. and Sierakoski R.L., "The Behaviour of Structures Composed of Composite Materials," Martinus NijhoffPublishers; 1986, pp 29 - 49.

11. Hoskins B.C. and Baker A.A., "Composite Materials for Aircraft Structures," AIAA Education Series; 1985, pp 64 -67.

8-1

APPENDIX Al

Typical Loading Conditions for Undercarriage Equipment Cases

i. Proof Load Cases

The equipment mountings must be able to withstand the mass of the equipment and the case

self-weight when subjected to the following accelerations. all loads being reacted at the supports.

1. Vertical l.Og (up).

11. Vertical

111. Lateral

IV. Longitudinal

ii. Fatigue Load Cases

l.5g (down).

± l.1g + l.Og vertical (down).

± 3.0g + l.Og vertical (down).

All equipment mountings shall be designed to have a fatigue life of not less than 107 cycles for the

loads produced by the following accelerations acting on the mass of the equipment.

i. Vertical

11. Lateral

iii. Longitudinal

l.Og (down) + O.3g.

±O.3g.

±O.2g.

9-1

APPENDIXBl

Engineering Drawing of a Typical Undercarriage Equipment Case,

( Courtesy of Brush Electrical Machines Ltd. )

Enclosed at the back of the thesis.

9-2

APPENDIXB2

Fatigue Analysis of Welded Joints

The fatigue analysis requires the applied stress range to be detennined at the welded joints in the

structure. Three sample locations were taken at the most highly stressed areas ie the upper and lower

beams and the support locations.

Stress ranges were calculated by subtracting stresses obtained from the following superimposed

loadcases:-

Stress I = Vertical 1.3g down + 0.3g lateral + 0.2g longitudinal

Stress 2 = Vertical 0.7g down - 0.3g lateral - 0.2g longitudinal

A typical calculation for node 488, located at a support is as follows:-

Maximum principal stresses, taken along the middle surface at node 488 are:-

Vertical 1.3 g down 106 Nmm- 2

Vertical 0.7g down 57 Nmm- 2

Lateral +O.3g 19 Nmm- 2

Longitudinal +0.2g 1.8 Nmm- 2

Hence substituting in (1) and (2) gives,

Stress 1 = 124 Nmm- 2

Stress 2 = 40 Nmm- 2

Stress range = 84 Nmm- 2 ~

The resulting stress range is thus 84 Nmm- 2 • (

... (1)

... (2)

On refering to Gurney, Reference 3 reveals that a class B is required for this stress range over 107

cycles. A maximum allowable stress range here would be 100 Nmm- 2 ,hence a class B weld would

r> have a reserve factor of@

9-3

APPENDIX Cl

Sample Output from CoALA and Pafec for Test 3

9-4

L..l.L..L..l.CLL CCCCCCCC

CC CC CC CC CC CC

LCCCCCCC CCCCCCCC

COLLEGE

t .. t S ~l" I­_ .... I-

h"," , l

(;'-'lL.F'HH 1. \FE"i~:' J

tl ... AtlAA LL t'lAAAAA AAAAHA LL AAAAAA

AA AA LL AA HA 000000 AA AA LL AA Af..t 000000 HA I.A LL AA AA

00 "~O AAAAkAfli-lAk LL .. AIiA,",,~o,\i~/"-l

00 OU AAAHAtl",AAtI LL AAAAAAAAAA 00 00 AA AA LL AA AA

000000 AA AA LLLLLLLLLL AA AA 000000 AA AA LLLLLLLLLL AA AA

01' AER01~AUTJCS LA1'lINATE ANALYSIS

Craniield Institute of Technolo9v

j'j ;~ r E Fo: 1 A L

j

t:: ..:.:: I'~U 12 ,: .. c '1.e

e;: .. c t: '.' .. c:

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9-5

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.. li.=)IiI~IE.,..i:15

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.. E;5i.:lil,'E-c:.·2

.. 25~::(jE--~11

.. 280"H:-12J4 .. ~':::~~1C:)(LIE -t-[qZi

... _ ...... ------

L U H iJ

i.. ... r1 I N H T E CON F I 6 U R H T ION

FLy PLy I-IATERIAL PLY TEI-IP. MOIST. NUl-lE-ER THrc~;I~ESS NUN8ER AI~GLE DIFF. ,~

------ --------- -------- ----- ----- ------

1 • 20G(;)Eo"'01 1 45.~ib .o1J\l .121",

9-6

.:',

L U H ii C H S E 1

NON-I·IECHHIH C,..L i fHERI·IAL AN" Hu I::; T Uk;:: I LliHU::'

Ui.kI::SIRAINED RESIDUAL STRESSES AND STRAINS

F L Y 5 T R E S S E S iat centr-oidi

NO. .. 1 .. ;, ------ ------

1 • "'lUil~"tE +~12} .liil1lf1lI1JI1lE+f1l0

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9-7

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t: I.! U i \.: H L ~ N I LDHL:f:

9·8

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kE.SUU AN'I .~IECHANICAL + RESIDUAL> STRESSES AND STRAINS ------------------------------------------------------

P LiS T RES S E 5 ~at centr-oidj

F-- L r 5 T R H i I'J S

c .'2

9·9

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Faiiur-e lnde::

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2

LHI·W"HIt:. CCJN51l"fUIIVE EQUic,nON FuR THE ':'1~r~L'r5ED CONFIGURATION

.I3Es50E+05

.6B50E+05

.6542E+05

.oB5IdE+11I5

.BB50E+05

.6542E+1iJ5

"':'.8.0" STIFFNESS ~IHTRIi.

• 6542E+11I5 • 6542E+05 .7247E+05

.11I01110E+00 • 0000E+00

• 000111E+00 • 0000E+fU0 • 0000E+00 • 0000E+00

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. 0('10i11E +GG .. IZl01b0E: +1')G . G0Q'10E +[1,0 .. 0GGGE -t-;Zlrz; ------------------------------------+------------------------------------

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--------------------------_._-----------._-------------

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• 11 ~

9-10

..... .... : : ... : .. :; ..

Parec Data File, Test 3

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, III CONTIICL I , 12. STRESS ,. 13' CLEAR. FILES

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I 39. 16 O. 50000£-02 O. 10000£-01 , 40. 17 0.00000 o.~-02

( 411 18 o. l:KK)OE-ol 0.00000 42. 19 O. 20000£-01 0.~OE-02

43' 20 O. l~OOE-OJ o. tOOOOE-Ol 44. 21 0.25000£-01 0.00000 451 22 0.30000£-01 0.~E-02

46' 23 0.25000£-01 O. l0000E-Ol 471 24 0.35000£-01 0.00000

48' 25 O. 40000£-01 O. 50000E-02 49. 26 0.3,000£-01 O. 10000£-01

50' 27 0.45000£-01 0.00000

I 511 28 0.50000E-01 O. ~0E-02 I 52> 2'1 0.45000E-ol O. 10000£-0.

( 54) C ._-- ._--.. _--I 55. El.IEJ£NTS , 56 • .... .,.,.1 I n. ELEJO- 36215 I N. ~ROP·l I 59. -- TOPO I 60. I I 2 4 3· 14 17 " I. I 611 2 2 S 3 8 18 U 19 20 I 62. 3 S 12 S 13 21 19 22 23 I 63. 4 12 ID 13 9 24 22 25 26 I 64. 5 10 7 9 11 27 25 28 29 I 65. e

I

I .... LNU .... TES I .7. ...... ORTH. llATEAIAL. IUlBER LOWER "PER IlADI IIAD2 AUS. SET _I - -=-I 68. I I -iI.OOI 0,001 0004500 I 691 e I 701 ORTHDTRDP le. ""TEA J "'-I 711 MIICIER 811 syv 8U .IV SYZ all BHIV BHYZ 1IH1l

I I 72. I 7.69£-121.43£-10 0 ""'. I SE-12 0 0 I. 66E-10 0 0 I 73. e ( 74. UIADS ( 7," CASE - DIRE - YALU

) ( 76' I 7 I 0 166.666 ~ .~ Icnrv ('-l.

I 77. I 28 I 0 666.666 ( 781 I U I 0 166.666 ( 791 e ( 80. IIESTRAINTS I all MlDE.NUIISER PLIIIIE DIRECTION I 821 I I I ( 83, 17 0 2 ( 84, C ( S5I OUT. DRAW ( 86. OR .... "LOT ( .7. I 5 ( 88. 2 30 I 89. END. CIF. DATA

_ CIF DATA o ERRORS

9-11

I

I I I •

•

IIY8TEII L£YE1. 6. 2 .AA.V 1986

_NO. 7 "-MT8 >£RE

Parec Displacement File, Test 3

PO'P.... """""""",,, EEEEEI :CCCC .. .. A A FF E IC CC .... A AFF E C PO'P.... """"" "" EEE ic pp No A" EE -c: pp AA A" EE ~ pp AA AV EE C pp AA A F EEEE£ ccecc

LaYEL TMEN - 07128/86 AT 1:'5 PIt

TITLE OENERALLV IIRTHDTRDPIC .M IH 1DIBIIIN .. 5 PLV THI_

........................................ • • • STATICS BDLUTIDN ay lLOCJ(, FRONT • • IN DOJILE PRECISION • • • • 81RUCT\RE CONTAINS S2 FREEDCI'IS .• • It ELEI'ENTS • • NCD 11£ FRONT SIZE IS 16 • • • ........................................

o ... A BLOCKED FRONT SDLUTJON ....a .lIEN RECM..E:STED ... o THE .uJCK BIlE IB 272 I OD J 8 ~ LAC E " ENT S FOR LOA D C A 8 E I --------------------OSNtPLE OF LAROEST DISPLACEP£HTS

0 __________ _

NODE UX

11 0.1952426£-03 29 O. 19~2424£-03

7 O. 1952418£-03 29 O. 1757184£-03 27 0.1757177£-03

9 O. 1'61940E-03

NODE UY

11 -0. 1703635£-03 28 -0. 1691380£-03

7 -0. 1679124£-03 ~ -0. 1534495£-03 27 -0.1509983£-03

9 -0. 136'3~E-03

•

NJDE RESUL. TANT

11 0.2'91204£-03 2B 0.2583.6.E-03

7 '0.2'75149£-03 29 O. 2332889£-03 27 0.2316834E-03

9 0.2074573E-03

NOTE - (2' THE HlSTOQAM INDICATES THE _ITUDE OF THE RES\A.TIWT TRANSLATION AT EACH NODE. EACH STAR • ~EPAESENT8 0.2591E-04 UNITS

(3, " STAR • IN It. DISPLACDtENT ~ INDICATES ntIIT A CONSTRAINT HAS IIEEN APPLIED.

(4' -.V STRUCT'\MAL NOIIES ARE OIIIEN IH THE TAIlLE .£LOW

I

CASE I TA.....sLATlDNS REaL TNfT TRAHSI..ATltlt SCALED COORDINATES NODE ~TIPLIED BY lE 6 ftULTIPLIED 8y lE 6 flJL.TlPLIED IY lE 0

...... EA Ul UV U HISTOORNt I V

I • 1.226 1.226 0.00 0.00 2 39.048 -32.602 110.869 •• 0.010 0.00 3 39.048 -35.053 112.474 - 0.010 0.010 4 • -1.226 1.226 0.00 0.010 5 78.097 -66.429 102.53 -- 0.020 0.00 7 ,.-..'95.24 -167.91 257. :u ._- 0.050 0.00 8 78.097 -68.880 104.13 -- 0.020 0.010

• 156, 19 -136.54 1107.46 ........ 0.040 0.010 10 1S6. 19 -134.08 1I05.B5 ._. 0.040 0.00 11 .......--1195. :24 -170.36 259.12 ._- 0.050 O. OlD 12 117.15 -100.26 IM.19 ...... 0.030 0.00 13 117. III -102. 71 155.79

_ .. 0.030 0.010

14 19. 524 -15.688 25.046 • 0.005 0.00 15 39. 04B -33.827 111.663 •• 0.010 0.005 16 19.1124 -lB. 139 26.650 .. 0.005 0.010 17 • • • 0.00 0.005 IB lIB. lI73 -49. :ns 76.698 ... 0.015 0.00 19 7B.097 -67 .• " 103.33 .... 0.020 0.005 110 58. 573 -5 •. 967 78.303 ••• 0.015 0.010 2. 97. 621 -83.343 128.36 ..... 0.025 0.00 22 117.15 -101. 48 154.99 •••••• 0.030 0.005 =>3 97.621 -a5, 794 129.96 ..... 0.0;5 0.010 24 136.67 -117.17 180.02 ....... 0.035 0.00 25 156. 19 -13',31 1106.65 ........ 0.040 0.005 26 136.67 -119.62 181.63 ....... 0.035 0.010 27 1'5.72 -1:51.00 231.68 ......... 0.045 0.00 28 "'"' 195. 24 -169. 14 2118.32 .......... 0.050 0.005 29 175.72 -153 .• ' 233.29 ......... 0.045 0.0'0 -

9-12

•

I

PIWIE NO. 9 IITMTB .... E

6.2

Parec Stress File, Test 3

",PP. -. FFFFF EEEEE CCCCC ~ P A A FF E C CC .... " "FF E C _p -. I'FF EEE C

"" AA AF EE C .... No "F EE C pp AA AF EE c

&.EW:L TAIi£N - 07 nBl86 AT 1: 15 PIt I

.... No "F EEEEE CCCCC:

TItLE CIDER .. LV IIIT1CJ111OPIC _ IN tDelDN ,

~ 196 _KSPACE IGDIA.E IIELETED IN IICI'98OB IlEFAUL T BTIIE&S. ELDENTS ItIIDIA..E CREATED.

a.DENT TYPE :16215 II-tIOIIE ISlPMOI£TRIC -.rLAlERALI I'LNE BTIIEBS I IIITNOTIIOPrc

""INCI'AL BTIIESSES - 81_-1 IS T>£ ""U ..... ""'-~ OF BTItEIIS IN 1>1£ po..- IF nE ELE>IENT 81at1A-2 IS nE ftlN.1"U'I YALUE CF B11E8S , .. nE PLANE CF 'nE EI.EPENT

~DF.I_' - THE ",AST ANOLE • ""'"8un , IS THE NCL.E 8ETWEEN THE MTERIAL DIRECTION AND SIOl'lA-l . THE SECOND ANCIL£ • EI~ • IS THE ANOLE BETWEEN THE EL.£PENT X-AXIS AND 'hE MTERIAL DIRECTION: aOTH ANOLES ME ~ED IN A POSITJ\IE SENSE ABOUT 'nE ELEJ£NT Z-AXIS

MTERIAL. STRESSES &TRESS CD'PCN!NT8 REFERRED TO MTEIUAL AlES

1 1 1 1 1 1 1 1 1

2 2 2 2 2 2 2 2 2

3 3 3 3 3 3 3 3 3

4 4 4 4 4 4 4 4 4

11 11 11 11 , 11 11 , 11

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 I 1

1 1 1 1 ., 1 1 1 1

I 1 1 1 I I 1 1 1

I I I 1 I 1 1 1 1

1 5. 000£ 07 -3. 320E 02 2. 500£ 07 .4 5.000£ 07 -9.2OOE 01 2.500£ 07

2 ~. 000£ 07 6. BOO£ 01 2. 500£ 07 17 ~.OOO£ 07 -6.000E 01 2.500£ 07

• ~. 000£ 07 -1.2OOE 02 2.500£ 07 111 5. oooE 07 -2. 600£ 02 2. 500£ 07 4 5.000£ 07 -1.000E 02 2.5OOE 07

16 5.000£ 07 -4.6OOE 02 a.5OOE 07 3 5.000£ 07 -B. 960£ 02 2.5OOE 07

a 5.000£ 07 -5.640£ 02 2.~ 07 19 ~.OOOE 07 4.0&0£ 02 2. ~ 07

:5 ~. 000£ 07 6. 960£ 02 a. 500£ 07 15 5.000£ 07 -1.8OOE 02 a.5OOE 07

• 5.000£ 01 -6.000£ 01 2.5OOE 07 19 5.000E 07 -6.290£ 02 2.500£ 07

3 5.oooE 07 -1. 490£ 02 2.50OE 07 20 5.000£ 07 -9.760£ 02 2. ~OE 07

9 ~.OOOE 07 -2.300£ 03 2.500£ 07

11 5.000E 07 -3. OBOE 02 2.5OOE 07 21 11. oooE 07 8. 1160£ 02 2. 500E 07 12 11. oooE 07 7. 840£ 02 2. IOOE 07 19 II.OOOE 07 -2.760£ 02 2.~ 07

• 5.000£ 07 -1.72OE 02 2.1OOE 07 Z2 5.oooE 07 -1.3OBE 03 2.5OOE 07 a 5.oooE 07 -4. l20E 02 2. 500E 07

1I3 5. oooE 07 -1. 34SE 03 2. 500E 07 13 1I.000E 07 -3.1I6OE 03 2.5OOE 07

12 5.OOOE 07 -1.0lIBE 03 2.1OOE 07 24 5.OOOE 07 1.2BOE 03 2. lODE 07 10 5. OOOE 07 1.972E 03 2. _ 07 Z2 1I.000E 07 -4. ,_ 02 2. 500E 07

• 5.OOOE 07 2.4OOE 01 2.5OOE 07 211 1I.000E 07 -1.104£ 03 2._ 07 13 1I.000E 07 -4. OBOE 02 2. _ 07 1I6 11. OOOE 07 -1. _ 03 2. IOOE 07

9 S. 000£ 07 -4. 780E 03 2. DOGE 07

10 5.OOOE 07 -1I.040E 02 2._ 07 lI7 1I.000E 07 1. _ 03 2.5OOE 07

7 II.OOOE 07 2.6116£ 03 2.5OOE 07 211 5.oooE 07 -3.000E 02 2.1OOE·07

• 5.000E 07 1.240£ 02 2.5OOE 07 lIB 5. oooE 07 -So 840£ 02 2. 500E 07

9 5.oooE 07 -&.II2OE 02 2.5OOE 07 1I9 11. oooE 07 -2. 304E 03 2. 500E 07 11 4.999E 07 -II.IBSE 03 2.5OOE 07

9-13

•

-45.0 -45.0 -45.0 -45.0 -45.0 -45.0 -4'.0 -45.0 -45.0

-45.0 -4'.0 -45.0 -4'.0 -4'.0 -"5.0 -45.0 -45.0 -45.0

-411.0 -411.0 -45.0 -45.0 -45.0 -4'.0 -411.0 -4'.0 -45.0

-411.0 -411.0 -411.0 -411.0· -45.0 -411.0 -411.0 -45.0 -45.0

-45.0 -411.0 -45.0 -4'.0 -411.0 -411.0 -45.0 -411.0 -4'.0

.,. 0 2. ~ 07 2. 500E 07 -2. 500£ 07 4',0 2.5OOE 07 2.:tOOE 07 -2. SOOE 07 45.0 2.5OOE 07 2. 500E 07 -2. 500E 07 45.0 ;.500£ 07 a. SOOE 07 -2.5OOE 07 .,. 0 2. ~ 07 2. 500E 07 -2. 500£ 07 .,. 0 2. ~ 07 2. 500E 07 -2. 500£ 07 "5, 0 a. 500E 07 2. &ooE 07 -2. 500£ 07 .,. 0 2. ~ 07 2. 500E 07 -2. 500£ 07 .,. 0 2. ~ 07 2. 500E 07 -2. 500£ 07

.,. 0 2. ~ 07 2. 500E 07 -2. 500£ 07 "'.0 2. 500£ 07 a. 500E 07 -2. M)OE 07 .,. 0 a. 500E 07 2. 500£ 07 -2. 500E 07 .5. 0 2. 500E 07 2. 500E 07 -2.:tOOE 07 .'.0 2. ~ 07 2.5OOE 07 -2.500£ 07 "'00 2. $OOE 07 2.5OOE 07 -2.500£ 07 .5, 0 2. SOOE 07 2. ~ 07 -2.500£ 07 .5.0 2.:tOOE 07 2.5OOE 07 -2. 500£ 07 4', 0 a. 500£ 07 2. 500£ 07 -a. 500£ 07

411.0 lI. SOOE 07 2. 500E 07 -2. 50DE 07 45.0 2. ~ 07 2. 500E 07 -2. 500£ 07 411.0 2.~ 07 2.5OOE 07 -2.5OOE 07 .5.0 2.~ 07 2.5OOE 07 -2.5OOE 07 45. 0 2. SOOE 07 2. 500E 07 -2. 50DE 07 45.0 2. ~ 07 2. 500E 07 -2. 500E 07 411.0 2.~ 07 2.1100£ 07 -2.1IOOE 07 411.0 2.SOOE 07 lI.lIOOE 07 -2.5OOE 07 411.0 lI. 499£ 07 a. IIOOE 07 -2. 500E 07

411.0 2.SOOE 07 a. lODE 07 -2.5OOE 07 411. 0 2. IIOOE 07 a. lODE 07 -2. 1100£ 07 45.0 2. ~ 07 lI. 500E 07 -2. 500£ 07 45.0 2. ~ 07 a. ~ 07 -2. 50DE 07 411. 0 a. ~ 07 a. IIOOE 07 -2. 500£ 07 411.0 2. IIOOE 07 2.1IOOE 07 -2. IIOOE 07 45. 0 lI. ~ 07 lI. ~ 07 -2. 500£ 07 .11.0 2. IIOOE 07 2. IIOOE 07 -2. 500£ 07 45.0 2.499E 07 lI.lODE 07 -2.5OOE 07

411.0 2.~ 07 2.~ 07 -2.1IOOE 07 45.0 2. ~ 07 a. ~ 07 -2. lODE 07 45.0 a. ~ 07 2. IIOOE 07 -2. IIOOE 07 45.0 R.~ 07 2.~ 07 -2.1IOOE 07 411.0 2.~ 07 2.5OOE07 -2.1100£ 07 .11.0 2. IIOOE 07 2.!IODE 07 -2. 500£ 07 .5. 0 R. 500E 07 2. 500E 07 -2. 500£ 07 .11.0 2.~ 07 2.~ 07 -2.5OOE 07 45. 0 2 .• 99E 07 2. ~ 07 -2. 500£ 07

APPENDIXC2

Stress Analysis of a Generally Orthotropic Lamina in Tension

Fibre direction 45° Ply thickness 2mm

c Considering the above 45 single lamina in tension used in Test 3.

_1000 N

From standard text, Reference 8, using a transformation matrix to obtain stresses in the material

directions.

[

ax 1 ay I Txy,

..J r

cos2a

= sin2 a sinacosa ~

For a 45 ply, a=45, hence

[

0.5 0.5 0.5

sin2a cos2a sin2 a

0.5 0.5

-0.5

-2 sina cosa l -2 sina cosa cos2 a - sin2 a

-~] [:] o T12

... (I)

In test 3 er; = 1000/20 = 50 Nmm- 2 ,O""y = 0 Nmm- 2 and Txy = 0 Nmm- 2 ,substituting these

values in in matrix (1) to give the following simultaneous equations.

50 = 0.5 01 + 0.5 (Y,? - Ti 2

0= 0.5 01 + 0.5(Y,? + T12

o = 0.5 0""1 - 0.5 ~

Solving equations (2), (3) and (4) gives,

01 =25 Nmm- 2

(Y,? = 25 Nmm- 2

9-14

(2)

(3)

... (4)

T! 2 = -25 Nmm- 2

These values agree with the results shown in figure 4.6.

• I

9-15

APPENDIXC3

Calculation of Displacements for a Symmetric Laminate in Tension

i. Theoretical Verification

Considering the cross section of the symmetric laminate bar ,2[0/90]. used in Test I.

10 mm

This is equivalent to,

. ,,'"'' """'1::::::::"""'"'''''''' "" :::::::::::: ::: ::::: ::: :::::::: :::,': :::::::::::: ::::: :::::: :::: I

x ::::::::::::::,' --x 4 mm E --- .......................... --

1 ':::::::::::::: ............................................................. ............................................................. .............................................................

1>,= 10 mm • Not to scale

Where by proportions,

" , (1)

AsEI=140xl<J3 Nmm- 2,E2=lOxl<J3 Nmm-2andbl=lOmm

b2 ='0,714 mm

9·16

The cross section area of the bar analogy is,

Bar Analogy Cross Section = (2 x I x 10) + (2 x 0.714) = 21.43 mm2

There are no shear coupling effects when the bar is subjected to tension, due to there being only

specially onhotropic plies present in the laminate. Displacement occurs along the length of the beam

only.

Displacement can be calculated from,

Displacement = Load x Length of Bar Cross Sectional Area x E

..• (2)

As Load = 1000 N, Length of Bar = 50 mm, Cross Sectional Area = 21.43 mm2

and E = 140 x 161 Nmm- 2

Displacement = 1.67 x 10- 4 mm

ii. CoALA Verification

CoALA provides several equivalent engineering elastic constants for the laminate analysed, see

Appendix Cl. / ""'--1 According to CoALA the overall E for the laminate in tension is 0.7536 x/IO 5 Nmm- 2

~ Which on substitution into (2), with Cross Sectional Area = 4 x 10 = 40 mm2 gives,

Displacement = 1.67 x 10- 4 mm

Hence CoALA agrees with the theoretical result derived above. 1,1 FIi c. (

~\~~

9-17

APPENDIXC4

Stress Analysis of a Generally Orthotropic Lamina in Bending

ibre direction

Considering a triangular element of the above bar, subjected to the following moments intensities

(moments per unit width).

Resolving moments (m, direction)

m, = mn (b / cos a ) cos a + mt ( b / cos a ) sin a

m, = mn + mt tan a ... (1)

Resolving moments (my direction)

my = mn ( b / cos a ) sin a + mt (b / cos a ) cos a

As there is only m, present in test 4, my = 0, hence

0= mn tan a -mt

mt = mn tan a ... (2)

9-18

Hence substituting for (2) in (I),

mx = mn + fin tan2 a

Hence,

fin = mx ( I + tan2 a) . .. (3)

Considering a cross section of material in the fibre direction of unit width,

From the standard formula,

.!!!.=j[ I y

Where 0-= 01, m = mn, I = il /12 and y = t / 2

Hence,

In Test 4 mx = 102 N (moment intensity), t = 2mm and a = 45°,

Hence, \

01 = 0.75 X 102 Nmm- 2

This result agrees with the CoALA result as shown in figure 4.8.

9-19

APPENDIX CS

Calculation of Displacements and Stresses for a Symmetric Laminate in Bending

i. Theoretical Verification of Displacements

Considering the cross section of the symmetric laminate bar ,2[0/90]. used in Test 2.

~ ::.:.:.:.:.: ..•. : ... : ... : ... :.:.:.: .. :.: ....... :.:.:.: ..•••... : •..... :.:.:.:.:.:.:.:.:.:.:.:.: ... : ..•••... :. 4 mm

I

··················:·················· .. ···-y-·············1 1 E 1 _ ....................................•.......•..............•

10 mm

This is equivalent to,

.""':" ..... '1.,::::::.,:::::::,:",,: , " :::::::: ...... : .. : ...... ::::: :::: ...... : .. : .. ::::: ...... : .. : .. ::::: .. ::::: .. : ::: :: I

b,= 10 mm

Not to scale

Where by proportions,

... (1)

b2 = 0.714 mm

9-20

Hence Ixx can be calculated from standard Second Moment of Area fonnula

Ixx =47.14mm4

There are no bend twist effects when the bar is subjected to a constant moment, due to there being

only specially orthotropic plies present. Hence from the standard bending fonnula, Reference 8, for

venical displacements at the end of a cantilevered bar,subjected to constant moment.

Displacement = Moment x ( Length of Bar )2

2 El !xx ~

... (2) , ' .'

For, Moment = I<rNmm, Length of Bar = 50 mm, El = 140 x IOS Nmm- 2 , Ixx = 47.14 mm4,

Displacement = 1.89 mm

CoALA Verification

CoALA provides several equivalent engineering elastic constants for the laminate analysed, see

Appendix Cl. , .

According to CoALA the overall E for the laminate in bending mode is 0.12421 x 10 6 Nmm- 2

Which on substitution into (2), with Ixx = 53,13 mm4 (simply from bd3/12)

Displacement = 1.89 mm

Hence CoALA agrees with the theoretical result derived above.

ii. Theoretical Verification of Stresses

Considering the upper ply, from

OJ =~ ... (3)

Ix x "

Where M =,"107 Nmm, y = 1.5 mm, Ix x = 47.14 mm 4,

Hence, on substitution in (3)

OJ = 0.318 X 106 Nmm- 2

This result agrees with the CoALA value shown in figure 4.7.

9-21

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...•. .. . . Authori~edby: Simon Cockbill Issue 2 Page 1 of 1

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