Marius Müller

Reducing weight of an energy efficient vehicle

using Altair OptistructTM for composite analysis

Bachelorthesis

submitted in fullfilment of the requirements for the degree of

Bachelor of Science (BSc.)

at

Graz University of Technology

Supervisors

Ass.Prof. DDDipl.-Ing. Dr.mont. Josef Domitner

Institute of Materials Science, Joining and Forming

Faculty of Mechanical Engineering

Dipl.-Ing. Vladimir Boskovic

Institute of Materials Science, Joining and Forming

Faculty of Mechanical Engineering

Graz, December 2018

In cooperation with

TERA TU Graz

Institution for Energy Efficient Vehicle Technologies

Inffeldgasse 25 D

Austria – 8010 Graz

Acknowledgment

At this point I want to thank everybody who was involved in creating this thesis:

Sincere thanks goes to all employees of the Institute of Materials Science, Joining and Forming,

especially to the employees of the research group Tools & Forming. Many thanks go to my

supervisors Ass.Prof. DDDipl.-Ing. Dr.mont. Josef Domitner and Dipl.-Ing. Vladimir Boskovic

who supported me day after day and who made my wish of working on this topic come true.

I also want to thank Arash Shafiee Sabet for proofreading my thesis and for giving lots of

feedback and input.

Furthermore I want to thank all my colleagues from TERA TU Graz who supported and

motivated me since I joined the team three years ago.

I want to thank all my work colleagues of the company Prisma Engineering Maschinen- und

Motorentechnik GmbH for introducing me to the exciting world of the finite element simulation.

Without you this thesis would never have been realised.

Additional I want to thank all my friends who supported me in good and bad times and who never

stopped believing in me. Special thanks go to Adnan Duranović, Emir Jagodić, Manuel Regouz

and Jennifer Stippich who spend part of their time with proofreading and giving invaluable

feedback.

Further I want to thank my family for their active support and for enabling me this education.

Last but not least I want to thank the company Y’plus for proofreading my thesis.

AFFIDAVIT

I declare that I have authored this thesis independently, that I have not used other than the

declared sources / resources, and that I have explicitly marked all material, which has been quoted

either literally or by content from the used sources.

………………………………….. ………………………………………………….

(Place, date) (Signature)

Abstract

Electric vehicles have been discussed so often in recent days, maybe even more in times of bad

air quality and rising prices for gasoline and diesel. Electric cars might not bring the solution for

all problems, but they can take us another step down the road towards a bright and green future.

Apart from the energy source and the drive technology it is obvious, that the need for energy will

rise in coming years and decades. Developing nations desire to reach the same standard of living

as the industrialised countries. The number of vehicles will rise and so the energy consumption,

not only in the mobility sector.

There are a few ways to lower the energy consumption of cars and one efficient way is the weight

reduction of the vehicles. By using light-weight materials like carbon fibre reinforced plastics, the

overall vehicle mass can be reduced drastically. In addition to a smart choice of the used

materials, the application of computer-aided simulation tools allows a fast construction and

design process and the need for time consuming test series of prototypes can be reduced

significantly.

This thesis is intended as a useful and practical guide for students participating in the Shell Eco-

Marathon and involved in building light-weight and energy efficient vehicles. The process of

creating and analysing a finite element model of a monocoque made of carbon fibre reinforced

plastics is discussed within this paper.

Keywords: automotive engineering, lightweight construction, composites, carbon fibres, Finite

Element Method (FEM), Shell Eco-Marathon

Content 1. Introduction ....................................................................................................................................... 1

1.1. About TERA TU Graz ................................................................................................................ 1

1.2. Shell Eco-Marathon ..................................................................................................................... 3

2. Theoretical background .................................................................................................................... 4

2.1. Composite materials .................................................................................................................... 4

2.2. Fibres ........................................................................................................................................... 4

2.3. Matrix of carbon fibre reinforced plastics ................................................................................... 7

2.4. Prepregs ....................................................................................................................................... 8

2.5. Sandwich constructions ............................................................................................................. 11

2.6. Introduction to the Finite Element Method ............................................................................... 13

2.7. FE simulation with Altair HyperWorks .................................................................................... 16

2.8. Modelling of composites ........................................................................................................... 16

3. Motivation and goals ...................................................................................................................... 23

4. Finite element analysis of composite monocoque ......................................................................... 24

4.1. Design constraints of an energy efficient vehicle ...................................................................... 24

4.2. Geometry of the monocoque ..................................................................................................... 25

4.3. Material data of the monocoque ................................................................................................ 26

4.4. FE-model of Fennek .................................................................................................................. 28

4.5. Load cases ................................................................................................................................. 29

4.6. Results ....................................................................................................................................... 39

4.7. Optimization .............................................................................................................................. 43

5. Summary .......................................................................................................................................... 46

6. Outlook ............................................................................................................................................. 46

7. References ........................................................................................................................................ 47

8. List of figures ................................................................................................................................... 49

9. List of tables ..................................................................................................................................... 50

10. List of abbreviations ...................................................................................................................... 50

11. Symbol directory ........................................................................................................................... 51

Page 1

1. Introduction

1.1. About TERA TU Graz TERA TU Graz is a student association which has the passion and the core competence of

developing high energy efficient vehicles. Graz students from different universities and

institutions of higher education are working together closely within the structure of this group to

gather knowledge and shape the future of mobility.

1.1.1. TERA’s history in the Shell Eco-Marathon

In 2010 the team took part in the Shell Eco-Marathon (see chapter 1.2) for the first time. The

team participated with the first version of the Fennek on the Lausitzring in Germany and was

honoured with the Newcomer-Award. In 2011 the Fennek was again designed from zero-base and

was fully battery-driven. TERA TU Graz won the competition with astonishing 843 km/kWh. In

2012 the team participated in the Shell Eco-Marathon with two vehicles. The Fennek reached the

second place in the prototype class and the Panther participated in the UrbanConcept class. In

2013 the team again started with the Fennek coming in fourth. The energy consumption had been

significantly reduced by 30% over the previous year. In 2014 TERA TU Graz was once again able

to win the Shell Eco-Marathon and moreover set a new world record for the most energy efficient

vehicle (1092 km/kWh ~ 9707 km/litre1) with the photovoltaic-supported, battery-driven vehicle.

__________________________________________________________________________________ 1) calculated with a caloric value of 8.89 kWh/l

d) c)

b) a)

Figure 1.1:

a) TERA Fennek 2010 b) TERA Fennek 2011/2012 c) TERA Panther d) TERA Fennek 2013/2014

Page 2

1.1.2. TERA Ibex

To continue the success stories of the student research and development group the team decided to

develop something special: the TERA Ibex – with the goal of achieving legal public road use.

The prototype was driven for the first time at the 5th anniversary of TERA TU Graz in early 2015.

Since then a carbon fibre reinforced plastic monocoque with an aluminium frame was developed

and built. In summer 2017 the team began to assemble all components from the prototype to the

new chassis.

1.1.3. Fennek 2017

TERA TU Graz took part in the Shell Eco-Marathon again after two years of absence. The

competition took place in London and the team came in third with 596 km/kWh.

1.1.4. Goals for 2018

TERA TU Graz aimed to participate in the Shell Eco-Marathon 2018 in London. New regulations

meant the team had to introduce further optimizations to the old chassis.

The largest planned changes were the improvement of the aerodynamics and weight reduction of

the vehicle. The goal for the 2018 season was once again a top position.

Figure 1.3: TERA Fennek 2017

a) b) Figure 1.2:

a) TERA Ibex - The prototype b) TERA Ibex - Rolling chassis

Page 3

1.2. Shell Eco-Marathon Shell Eco-marathon is organised by Royal Dutch Shell and is a unique competition that challenges

students from around the world to design, build and drive the most energy-efficient car. After a

year of preparation, student teams compete on a track in their self-built cars to see who can drive

further using the least energy. Over 500 teams from more than 50 different countries have only a

few days to battle it out on urban circuits during three annual events in Asia, America and Europe.

[1]

In 2017, the main Shell Eco-marathon events took place in three locations around the world: [1]

Shell Eco-marathon Asia: March 16-19 in Singapore, SG.

Shell Eco-marathon Americas: April 27-29 in Detroit, Michigan, USA.

Shell Eco-marathon Europe: May 25-28, London, United Kingdom.

1.2.1. The competition

The competition is split into two classes or categories. The Prototype class focuses on maximum

efficiency, while passenger comfort is not considered. The UrbanConcept class encourages more

practical designs for the road. [1]

Cars are divided by energy type: [1]

Internal combustion engine fuels include petrol, diesel, liquid fuel made from natural gas

and ethanol.

In the electric mobility category, vehicles are powered by hydrogen fuel cells and lithium-

based batteries.

Over several days, the teams have four attempts to travel furthest on the equivalent of one litre of

fuel. Cars drive a fixed number of laps around the circuit within a given time limit. The organisers

calculate their energy efficiencies and name a winner in each class and for each energy source. Off-

track awards recognise other achievements including safety, teamwork and design.

The competition inspires the engineers of the future to turn their visions of sustainable mobility

into realities. It also sparks and fuels passionate debate about what could one day be possible for

cars on the road. [1]

Figure 1.4: TERA TU Graz team makes the 3rd place with 596 km/kWh at Shell Eco-

Marathon 2017

Persons in the picture from left to right:

Standing: Michael Wild, Jakob Möderl, Matthias Greiner, Thomas Hofer, Lukas Scheucher

Kneeing: Dominik Novak, Sabrina Hammerschmid, Konrad Fellner, Julian Naderer

Page 4

2. Theoretical background

2.1. Composite materials A composite material is a material that is formed by combining two or more materials on a

macroscopic scale. [2]

2.1.1. Particulate composites

Particulate composites are materials that are manufactured by spreading pieces of chopped fibre

material onto a film of matrix material. [2]

A schematic sketch of a particulate composite is shown in Figure 2.1.

2.1.2. Laminated composites

Laminated composite materials are materials that are made up of any number of layered

materials, of the same or different orientation, bonded together with a matrix material. The layers

of a laminated composite are usually called plies. Plies can be made from a wide range of

materials, including adhesive plies, metallic-foil plies, fibre-matrix plies of various fibre and

matrix material combinations, and core plies of various core materials. [2]

2.1.2.1. Fibre-matrix laminated composites

Fibre-matrix laminated composite materials are materials made up of any number of fibre-matrix

plies, of the same or different fibre orientation, layered and bonded together with a matrix

material. [2]

Fibre-matrix ply materials can be classified into unidirectional plies and weave plies (see chapter

2.4.1.1 and chapter 2.4.1.2).

2.2. Fibres In order to be designated as lightweight construction materials, fibres must have high stiffness

and strengths as well as low densities. All the requirements are fulfilled by elements of the first

two rows in the periodic system: Boron, Carbon and Silicon. These elements have theoretically

high atomic binding energies. The attempt is made to achieve the resultant high stiffness and

strengths by manufacturing fibres. [3]

Four main mechanisms are responsible for the good properties of these fibres: [3]

1. Influence of the magnitude effect

By using a model concept of a chain, it can be said that the weakest link of the chain determines

the strength of the chain (weakest link theory). With an increasing number of links the chance that

Figure 2.1: Schematic sketch of a particulate composite ply material [2]

Page 5

a weaker link will occur rises. This means the number of defects in a material volume increases

with enlargement of the volume. This effect is shown in Figure 2.2 for the example of a fibre. [3]

Furthermore, small fibre diameters are really interesting for the manufacturing process. These

fibres can be formed easily, because of their low bending stiffness. This good drapability is very

important for the lay-up process of the laminate because the layers fit in small radii. [3]

2. Influence of orientation

The orientation of the highest atomic bindings in fibre longitudinal direction is the best way to

achieve high tensile strengths. For crystalline materials highest atomic bindings are the crystal

planes, for polymers the highest atomic bindings are the molecular chains. The high strengths and

stiffness in longitudinal direction are causing reduced strengths and stiffness in cross direction of

the fibre. [3]

3. Avoidance of defects and notches

Tiny notches with high form factors reduce the strength significantly. Fibres are usually made of

brittle materials. Brittle materials do not have the ability of plastic flow, so the notch effect plays

a very important role. Defects perpendicular to the loading axis in particular are critical. By

stretching the fibres in longitudinal direction the defects are adjusted, stretched and flattened.

Very often crack formation starts at the surface of a body. Surface defects of fibres can be

removed by etching or by soothing the notch root. [3]

4. Effect of the fibre form on the failure progress

Cracks can grow unhindered in a compact material and can cause abrupt material failure for

brittle materials. However, in a fibre composite material cracks are stopped at every fibre

boundary. The load of one fibre is distributed to the neighbour fibres after failure. The crack also

goes to the neighbour fibre and is stopped there in the first instance. At this point the crack causes

a notch effect. The intension of this notch effect is the smaller the wider the gap between the

fibres and the ductile the matrix. After several load changes the neighbour fibre also fails and the

crack is passed on to the next fibre. [3]

The properties of fibre and matrix which slow down the crack growth are: [3]

large gaps between the fibres which are equal to a low fibre content

a fracture toughness matrix which dulls crack tips and lowers the notch effect

ductile and impact toughness fibres

Figure 2.2: Influence of fibre volume on the strength (recreated after [3])

Fibre diameter in mm

Str

ength

in

N/m

m2

Page 6

2.2.1. Classification of fibres

Fibres can be divided into the following categories: [3]

Natural fibres:

hairs, wools, silk, cotton, flax, sisal, hemp, jute, ramie, banana fibres

Organic fibres:

polyethylene, polypropylene, polyamide, polyester, polyacrylonitrile, aramid, carbon

Inorganic fibres:

glass, basalt, quartz, SiC, Al2O3, boron, asbestos

Metal fibres:

steel, aluminium, copper, nickel, beryllium, tungsten

Natural fibres are only available in limited lengths. Fibres of limited lengths are called staple

fibres. Synthetic fibres are producible with endless length. An endless fibre is known as a

filament. Car interiors are being made ever more frequently from plant fibres. Aluminium fibre

and copper fibre reinforced components are used for electrical shielding. [3]

2.2.1.1. Carbon fibres

Carbon fibres have the best properties of all reinforced fibres. The technical breakthrough of the

production of fibrous carbon started with the process of degradation of preformed organic fibres.

In 1961 the research result was published that the widely used textile fibre polyacrylonitrile

(PAN) is a good base material for carbon fibres. Today, 90 percent of the market is based on this

precursor [3]. A carbon yield of 55 percent by weight can be achieved. Furthermore, young’s modulus of 450,000 N/mm

2, tensile strengths of 7,000 N/mm

2 and breaking elongations of three

percent can be reached [3]. Another industrially used precursor for carbon fibres is petroleum- or

black coal pitch. Through thermal treatment the pitch is converted into a mesophase (liquid

crystal phase). In a melt spinning process the mesophase is subsequently manufactured as a fibre

with very high orientation parallel to the fibre axis. [3]

Graphite has a structure built up of several layered sheets (see Figure 2.3). High strength and

high young’s moduli are based on a strong bonding of the graphite crystals within the atomic

layers. Theoretically a yield modulus of 1,050,000 N/mm2 and a tensile strength of 100,000

N/mm2 can be reached [3]. Defects like a low amount of fibrils, defects in the precursor, damages

during manufacturing and processing of the layers cause a deterioration in these properties.

Surface defects can be reduced by etching the surface. Levelling of surface notches has a very

positive effect of the strength of the fibre. [3]

Figure 2.3: Unit cell of graphite crystal (recreated after [3])

strong covalent bond

weak Van der Waals

bond

fibre cross direction

fibre longitudinal

direction

6.7 Å

Page 7

The use of carbon fibres is welcomed, because of the following properties: [3] [4]

Very light; density of around 1.8 g/cm3

Extremely high stiffness and very high yield’s moduli which are adjustable in the

manufacturing process

Excellent fatigue stiffness

Strong anisotropy: carbon fibres accept the loads and therefore lighten the loads for the

matrix and the adhesion

Thermal expansion coefficient positive in fibre cross direction, negative in fibre

longitudinal direction

Good heat- and electrical conduction

Suitable for electromagnetic shielding

Compatibility with most of the acids and alkalis

Compatibility with all synthetic polymers, human tissues and bones

Permeability for X-rays

Some disadvantages are: [3]

The fracture behaviour is brittle; elongation at fracture sometimes too low

Low compressive strength parallel to the fibre direction

High costs

In the presence of electrolytes an electrochemical voltage difference can occur between

carbon fibres and metals. Some metals become anodic and corrode very strong. Special

isolation measures should be included in the design.

2.3. Matrix of carbon fibre reinforced plastics The matrix has to support the fibres and bond them together. It transfers any applied loads to the

fibres and keeps the fibres in the chosen position and orientation. It also makes a main

contribution to the service temperature of a component. Furthermore, the matrix determines

environmental resistance and maximum service temperature of the ply. [5]

There are three main matrix types: Epoxy, phenolic, Bismaleimide and polyimide [5]

2.3.1. Epoxy

Epoxy matrixes show excellent mechanical performance as well as good environmental resistance

and great toughness. Furthermore these matrixes have good adhesion for honeycomb bonding and

are quite easy to process. Curing temperatures are range from 120 °C to 180 °C and the maximum

service temperature lies up to 100 °C. [5]

2.3.2. Phenolic

Phenolic matrixes are used for fire resistance products as they have low smoke and low toxic

emissions. These matrixes are curing very fast in an economic processing. The maximum service

temperature lies up to 100 °C. [5]

2.3.3. Bismaleimide and polyimide

Bismaleimide and polyimide matrixes have an excellent resistance to high temperatures, chemical

agents, fire and radiation. Service temperature goes up to 260 °C. [5]

Page 8

Figure 2.5: Unidirectional ply material [2]

2.4. Prepregs Prepregs are fibre-reinforced resins that cure under heat and pressure to form exceptionally strong

yet lightweight components. Prepregs consist of a combination of a matrix and fibre

reinforcement. The position of prepreg technology in terms of performance and production

volumes is compared in Figure 2.4 with other fabrication processes. [5]

2.4.1. Types of reinforcements

There are many reinforcement forms and each type offers different advantages. The two main

types are listed below:

2.4.1.1. Unidirectional reinforcements

Unidirectional ply materials are built up of straight fibres embedded in a matrix material. Typical

UD plies have fibre volumes which range from 45 to 70 percent. [2]

Unidirectional reinforcements are used for components which are requiring predominant strength

and stiffness in one direction. Pressure vessels, drive shafts and tubes are examples of parts which

are produced using UD reinforcements [5]. Figure 2.5 shows a schematic of a UD ply material.

Figure 2.4: Performance and production volume of different fabrication processes [5]

Page 9

2.4.1.2. Weave plies

Fabrics consist of at least two threads which are woven together - the warp and the weft. Weave

plies have reduced longitudinal stiffness compared to their equivalent UD ply product forms. The

reduced stiffness is due to the undulation of the fibres. The weave style can be varied according to

crimp and drapability. Low crimp gives better mechanical performance because straighter fibres

carry greater loads. A drapable fabric is easier to lay up over complex geometries. A mixture of

different fibre and matrix material combinations is possible. [2] [5]

The most important weave styles are shown in Figure 2.6.

2.4.2. Prepreg processing techniques

The most common prepreg processing techniques are the vacuum bag oven process, the autoclave

process, the match moulding process, the tube rolling process and the pressure bag process. The

processing method is determined by the quality, cost and type of component being manufactured.

[5]

The main prepreg processing techniques which are used at TERA are discussed below:

2.4.2.1. Vacuum bag oven process

A flexible bag, the vacuum bag, is placed over the composite lay-up and is subsequently sealed

(see Figure 2.7a). The remaining air in the bag is removed by a vacuum pump. The removal of

air forces the bag down onto the lay-up with a pressure of up to 1 atmosphere (Figure 2.7b and

Figure 2.7d). The complete assembly, with vacuum still applied, is placed inside an oven or a

heated mould with good air circulation. The course of the temperature can be seen in Figure 2.7c.

Vacuum bag processing is used to manufacture components of varying thickness and large

sandwich structures. Examples are parts used in aerospace engineering, in the marine industry, for

railway interiors, wind energy and automotive purposes. [5]

Figure 2.6: Three main weave styles of fabrics (recreated after [5])

Plain weave Satin weave

(4, 5, 8, 11)

Twill weave

(2/1, 3/1, 2/2)

Low drapeability / high crimp Good drapeability / low crimp Average drapeability / average crimp

Page 10

a) b)

c)

c) d)

a) b)

2.4.2.2. Autoclave processing

The autoclave processing is used to produce superior quality structural components containing

high fibre volume and low void contents. For this process a vacuum bag is needed. The autoclave

is a pressure vessel which controls the application of the heat up rate and the cure temperature

(Figure 2.8a), the pressure (Figure 2.8b) and the vacuum (Figure 2.8c), The manufacturability

of high standard honeycomb sandwich structures is possible as well, typically at lower pressures.

Long cure cycles are required because the large autoclave mass takes a long time to heat up and

cool down and a temperature distribution on the tooling and composite components is required.

[5]

Figure 2.7: Vacuum bag processing [5]

a) Assembly b) Atmospheric pressure is acting c) Cure temperature d) Vacuum trend

Figure 2.8: Autoclave process [5]

a) Cure temperature and heat up rate b) Pressure trend c) Vacuum trend

Page 11

2.5. Sandwich constructions Honeycomb sandwich construction is one of the most valued structural engineering innovations

developed by the composites industry [6]. Used extensively in aerospace and other industries [6],

the honeycomb sandwich provides the following key benefits over conventional materials: [6] [7]

Very low weight and durability

High stiffness compared to its own weight

High mechanical strength at low densities

Production cost saving

Absorption capacity relating to deformation energy (crash) and vibrations (structure-borne

sound)

The facing skins of a sandwich panel can be compared to the flanges of an I-beam, as they carry

the bending stresses to which the beam is subjected, with one skin in compression and the other

one in tension (Figure 2.9). The core resists the shear-load, increases the second moment of area

and so the stiffness of the structure by holding the facing skin apart (Figure 2.10). The core-to-

skin adhesive rigidly joins the sandwich components and allows them to act as one unit with a

high torsional and bending rigidity. [6]

2.5.1. Materials

The composite structure provides great versatility as a wide range of core and facing material

combinations can be selected. [6]

2.5.1.1. Skin materials

Typical skin materials are epoxy carbon, epoxy glass and epoxy aramid. [6]

2.5.1.2. Core materials

Core materials are available in a wide range of materials including foam through to honeycombs.

Honeycomb materials are made of aluminium, aramid, Korex, Kevlar, fibreglass and carbon [6].

Some characteristics of aramid honeycombs are: [6]

Very low weight

High stiffness

Durability

Production cost savings

Figure 2.9: Construction of a sandwich panel compared to an

I beam [6]

Figure 2.10: Relative stiffness and weight of

sandwich panels compared to solid panels [6]

Page 12

Figure 2.11: Manufacture of sandwich constructions [5]

2.5.1.3. Adhesive materials

The adhesives should create adequate bonding between core and skin materials. Following

criteria are important to ensure good performance of the adhesives: [6]

Fillet forming:

The adhesive should flow sufficiently to form a fillet without running away from the skin

to core joint.

Bond line control:

The adhesive needs to fill any gaps between the bonding surfaces.

2.5.2. Sandwich construction processing methods

Sandwich constructions with prepreg skins can be manufactured by autoclave process, vacuum

bag oven process or by a pressing process. For autoclave or press processing sandwich

constructions can usually be laid up and cured as a single shot process. For the vacuum bag

curing of large components, however, it may be necessary to lay-up and cure in two or more

steps. This will improve the quality of the component, ensuring against voids and telegraphing

(honeycomb cells are visible through the composite skins). Excessive pressures which can lead to

movement of the core or eventually core crush should be avoided. [5]

The principal setup for the manufacturing of a sandwich structure with prepreg is shown in

Figure 2.11. [5]

Page 13

2.6. Introduction to the Finite Element Method The descriptions of the laws of physics for space- and time-dependent problems are usually

expressed in terms of partial differential equations. Often the partial differential equations cannot

be solved analytically. In these cases an approximation of the equations can be constructed,

typically based on different types of discretization. These discretization methods approximate the

PDEs with numerical model equations, which can be solved using numerical methods. The finite

element method (FEM) is used to compute such approximations. [8]

In FEM, elements are used to discretize a given problem. The elements are assigned to the part in

a so called preprocessor. The calculations are solved in a solver and can be reviewed

subsequently in a postprocessor. [8]

2.6.1. Linear static analysis

Linear means straight line in the stress-strain-diagram. In real life, after crossing the yield point,

the material follows a nonlinear curve. Solvers follow this straight line from base to deformed

state, even after crossing the yield point, so solvers never show failure when passing the yield

strength. The analyst has to conclude whether the component is safe by comparing the maximum

stress values with yield or ultimate stress values. The solver solves the basic finite element

equation: K*u=P (global stiffness matrix times displacement vector is equal to external force

vector applied to the structure). The equilibrium equation is solved by a direct or iterative solver,

the unknown displacements are solved simultaneously by using a Gauss elimination method that

exploits symmetry of the stiffness matrix K (see chapter 2.5.5), for computational efficiency.

Once the nodal points of the elements are calculated, the stresses can be computed by using the

constitutive relation for the material, e.g. Hooke’s law: σ=E*ε. Non term of the equations is dependent on time or displacement. Thus, for a linear static analysis only two independent

material constants are required: Young’s modulus and the Poisson's ratio. [9]

Generally many complex problems which can be approximated to a static system can be

simplified and solved as a static analysis. Care must be taken principally while evaluating results

to ensure that the linear assumptions are still valid. [9]

2.6.2. Nonlinear analysis

Generally there are three cases when nonlinear analyses have to be used: [9]

Nonlinear geometry:

When large deformations occur, the stiffness matrix is a function of displacement and is

recalculated after a certain predefined displacement.

Contacts:

The stiffness matrix also changes as a function of displacement, when two parts get in

contact or separate.

Material:

Some materials do not show a linear elastic behaviour. In these cases nonlinear material

properties have to be applied. The stress-strain diagram, along with the hardening rule for

the material, is required as input data.

2.6.3. Buckling analysis

Buckling analysis is only applicable for compressive load. The bending stiffness of the part is

much smaller than the axial stiffness, as a result the lateral deformation is quite large.

This type of load is usual for slender beams and thin-walled sheet metal parts. [9]

Page 14

2.6.4. Nodes and elements

All real life objects are continuous and made up of atoms which are bonded together by the force

of attraction. The basis of all numerical methods is to simplify the problem by discretizing.

In a Finite Element Analysis nodes thus work like atoms and the gap between the nodes is filled

by an entity termed an element. Calculations are made at the nodes and the results are interpolated

for the elements. Generally, a greater number of calculation points (=nodes) improves the

accuracy, but the solution time is directly proportional to (DOF)n, where n is the number of

calculation points. The total DOF for a given mesh model is equal to the number of nodes

multiplied by the number of DOF per node. [9]

2.6.4.1. 1D Elements

1D elements are used to connect nodes together, attach dissimilar meshes and distribute loads.

These elements support six DOF. [9]

In Altair OptistructTM

(see chapter 2.6) simple beams with constant properties may be modelled

with CROD or CBAR elements. CBEAM elements are used to model more complicated

geometries with varying cross sections. [9]

CROD elements support tension and compression only. In other words, the nodes of a CROD

element only have translation degrees of freedom (still this element has a torsional stiffness).

CBAR and CBEAM elements allow bending as well. A CBAR element is a kind of simplified

CBEAM element and should be used whenever the cross-section of the structure and its

properties is constant and symmetrical. This element requires that its shear centre and the neutral

axis coincide. Due to this requirement CBAR elements are not useful for modelling structures that

may warp, such as open channel-sections. This is because the cross-sections of CBAR elements

remain plane. This limitation is not present in CBEAM elements. [9]

Connectors are geometric entities that define connections between other entities. In Altair

OptistructTM

there are two types of connection elements: RBE2 and RBE3. RBE2 elements are

rigid links to transfer motion from the independent node to the dependent node(s). RBE3

elements are rigid links to distribute loads. They will not induce stiffness. [9]

More 1D elements available in Altair OptistructTM

of course. One example is that of the spring

elements which directly couple two degrees of freedom with a stiffness value.

2.6.4.2. 2D Elements

2D elements are used when two of the dimensions are very large in comparison to the third

dimension. Mathematically the element thickness specified by the user is assigned half on the

element top side and half on the element bottom side. In order to present the geometry

appropriately it is necessary to create the middle surface (midsurface) and mesh on this surface.

2D elements support six DOF. [9]

2.6.4.3. 3D Elements

3D solid elements have been formulated with three translational DOFs and no rotational DOF due

to high bending and torsion stiffness. [9]

2.6.5. Stiffness and stiffness matrix

For a structural analysis the stiffness is a really important property. In FEM the stiffness is a

characteristic property of an element. Formulations of the stiffness property for any geometry can

be performed by meshing the part and then solving the equation K*u=f (global stiffness matrix

Page 15

times displacement vector is equal to external force vector applied to the structure) for a given

shape [9]. Methods for formulating the stiffness matrix are shown below: [9]

1. Direct method

2. Variational method

3. Weighted residual method

All software codes use either the Variational or weighted residual method formulation because

these methods are much easier to formulate than the direct method. [9]

2.6.5.1. Basic principle of a discrete mechanical system

For a better understanding of how a stiffness matrix is formulated, a small example is shown in

Figure 2.12.

The equilibriums of forces at a node n due to stiffness k of adjacent elements are calculated. 𝑘1 ∗ 𝑢1 + 𝑘12 ∗ (𝑢1 − 𝑢2) = 𝑓1 (2.1)

𝑘2 ∗ 𝑢2 + 𝑘12 ∗ (𝑢2 − 𝑢1) = 𝑓2 (2.2)

The calculated equilibriums are then written in a compact matrix form:

[𝑘1 + 𝑘12 −𝑘12−𝑘12 𝑘2 + 𝑘12] ∗ (𝑢1𝑢2) = (𝑓1𝑓2) (2.3)

The abbreviated formulation reads as followed:

K*u=f (2.4)

By inverting the stiffness matrix, the formulation for the displacement vector can be found:

u=K-1

*f (2.5)

Figure 2.12: Example of a simple FE-structure [10]

Page 16

2.6.5.2. Properties of a stiffness matrix: [9] [10]

Diagonal terms have the largest contributions.

Off-diagonal terms show the coupling stiffness between nodes.

The matrix is symmetric for linear structures.

A symmetric stiffness matrix shows the force is directly proportional to displacement.

Systematic formalism for assembly of K-matrix.

Order of the stiffness matrix corresponds to the total DOFs

Boundary conditions eliminate DOF’s. A singular stiffness matrix means the structure is unconstrained and has rigid body

motion.

Each column of the stiffness matrix is an equilibrium set of nodal forces required to

produce the unit respective DOF.

Diagonal terms of the matrix are always positive meaning a force directed in say the left

direction cannot produce a displacement in the right direction. Diagonal terms will be zero

or negative only if the structure is unstable.

Linear equation system, so a superposition of load cases is possible.

The solver first solves displacements u for the system, and then computes stresses for each

element.

2.7. FE simulation with Altair HyperWorks

Altair HyperMeshTM

Altair HyperMeshTM

is a pre-processor for high fidelity modelling. With automatic and semi-

automatic shell, tetra, and hexa meshing capabilities, HyperMeshTM

simplifies the modelling

process of complex geometries. Furthermore, it supports a wide variety of CAD and solver

interfaces. The ability to generate high quality mesh quickly is one of HyperMeshTM’s core

competencies. Modelling of laminate composites is supported by advanced creation, editing

and visualization tools. For review purposes composites structures can be visualized in 3D,

individual layers isolated and ply orientations visualized graphically. [11]

Altair OptistructTM

Altair OptistructTM

is an analysis solver for linear and nonlinear problems under static and

dynamic loadings. Inter alia, OptistructTM offers solutions for the design and optimisation

of advanced materials such as laminate composites. [12]

Altair HyperViewTM

Altair HyperViewTM

is a complete post-processing and visualization environment for finite

element analysis, CFD and multi-body system data. [13]

2.8. Modelling of composites In general composites can be modelled by using single layer shells, multi-layer shells and/or

solids. In case of using solids, each ply has to be modelled with at least one layer of solid

elements. This requires a huge number of solid elements. Using this method it should be kept in

mind that the aspect ratio should not be greater than five to achieve a good element quality. In

Altair OptistructTM

solid elements are assigned with the property card PSOLID. [14]

Page 17

Building a laminate from shell elements requires a creation of property cards to define the ply and

sequence information. In Altair OptistructTM

there are three main types of property cards for

creating shell laminate properties: [14]

PCOMP:

PCOMP defines all the laminate properties such as ply material, thickness and orientation

and also the stacking sequence in one property card. These have no information about

plies that are also part of other regions. During post-processing, this requires a great deal

of book keeping work to track ply and stacking information for each PCOMP. [15]

PCOMPG:

PCOMPG is similar to PCOMP and additionally stores global ply identification number,

so plies that are part of many regions can be tracked across the regions, reducing the effort

of keeping track of the ply properties and stacking information. [15]

PCOMPP:

The PCOMPP card has an external information relationship. This implies that the ply

definition, the stacking order, the assignment and the thickness are defined separately in

other cards: PLY, STACK and LAMINATE card. The PLY card defines fibre orientation

and layout as well as ply thickness. The STACK card sets the sequence of PLYs into a

laminate. [14]

PCOMP and PCOMPG are zone based properties used for modelling zone based laminates

whereas PCOMPP is a ply based modelling property. [14]

2.8.1. Composite zone-based shell modelling

Composite shell zone-based modelling is the traditional approach for building composite models.

It requires a property definition for each laminate zone within a composite structure. Thus another

laminate zone property definition must be defined at each ply drop or addition location (Figure

2.13). Plies that extend through multiple laminate zones must be redefined in each laminate zone

property definition. [14]

Figure 2.13: Zone based modelling [15]

Page 18

2.8.2. Composite ply-based shell modelling

Composite shell ply-based modelling is a relatively new technique for building composite models

that attempts to mimic the composite manufacturing process of cutting and stacking plies on top

of each other. In this modelling approach each ply is meshed with elements which are assigned to

a ply in a following step. The principal advantage of ply-based modelling is the ability to easily

make design updates to composite models via addition and subtraction of plies, modification of

ply shapes and fibre orientation. [14]

The list below gives a short overview of how to setup a composite in Altair HypermeshTM

by

using the ply-base modelling approach: [14] [15]

1. Importation/Creation of the 2D geometry

2. Defeaturing the model geometry

3. Generation of the mesh: usage of QUAD4 elements

4. Alignment of the elements in line with the main laminate direction and definition of the

element normal:

The element normal defines the stacking sequence. The plies are listed from the bottom

surface upwards, with respect to the element’s normal direction (Figure 2.14)

The material orientation (Figure 2.15) establishes a reference for ply angles. Ply angles

can be specified relative to a

i. Element coordinate system

ii. Vector projected onto elements

iii. Coordinate system (global or local coordinate system)

The element coordinate system is strongly dependent upon the node numbering in the

individual elements. It is advisable to prescribe a coordinate system for composite

elements and define ply angles relative to this system. Furthermore, the material

orientation is very important for the definition of E1 and E2. E1 corresponds to the x-axis

of the elements material coordinate system and direction E2 corresponds to its y-axis.

Figure 2.14: Ply stacking sequence with respect to the element normal [14]

Page 19

5. Definition of the composite material

Typical material model used for composites is MAT8, which is a planar orthotropic

material. The use of isotropic MAT1 or general anisotropic MAT2 ply properties is

also supported.

Usually used parameters: E1, E2, ρ, NU12, G12, G1Z, G2Z, Xt, Xc, Yt, Yc, S

6. Definition of the property

PCOMP, PCOMPG, PCOMPP

7. Assignment of the properties to the components

8. Application of boundary conditions and loads

9. Creation of a load step with the constraints and loads

10. Running of the analysis

11. Post processing

Contour plot of element strain and element stress in the global system or analysis

system.

The most important results for composite models are the ply-level mechanical strains

and stresses in the material coordinate system.

2.8.3. Composite ply-by-ply solid modelling

Composite ply-by-ply solid modelling is the most accurate modelling technique and is the only

method which can accurately capture through-thickness effects. However, ply-by-ply solid

modelling requires a solid layer of elements for each ply and thus a huge number of solid

elements. [14]

2.8.4. Material cards for composite modelling

Usually composite materials are modelled with an orthotropic material. MAT8 is the commonly

used orthotropic material model for shells and MAT9 or MAT9ORT for solids. [14]

2.8.4.1. MAT8

MAT8 is used to define the material properties for linear temperature-independent orthotropic

material for two-dimensional elements. The MAT8 card is most commonly used to represent

planar orthotropic material for composites. [14]

Figure 2.15: Alignment of elements according to the laminate main direction [15]

Page 20

2.8.5. Ply conventions

In the case of fibre-matrix unidirectional composite plies, the material coordinate system 1-axis

defines the fibre direction, the material coordinate system 2-axis defines the transverse matrix

direction, and the material coordinate system 3-axis defines the through-thickness direction of a

ply. The fibre orientation of a ply, theta, is defined relative to the global system x-axis using

right-hand rule to define positive theta as shown in Figure 2.16. [2]

2.8.6. Symmetric laminates

A symmetric laminate is defined as a laminate that is composed of plies such that the thickness,

angle (theta), and material of the plies are symmetric about the middle surface of the laminate.

For symmetric laminates, there are no extensional–bending or shear–twisting coupling

behaviours. [2]

Examples of symmetric laminates using engineering notation, assuming all plies have the same

thickness and material, are given below: [2]

[0/45/90/45/0], which can also be written as [0/45/90]s.

[45/-45/90/0/0/90/−45/45], which can also be written as [45/−45/90/0]s.

Figure 2.16: Definition of fibre direction, transverse matrix direction and the fibre orientation

(recreated after [2])

1-fibre

z, 3-thickness

1-matrix

Page 21

Figure 2.17: Different modelling styles of sandwich structures [15]

2.8.7. Modelling honeycomb sandwich panels for finite element analyses

In general the shear forces normal to the panel will be carried by the honeycomb core. Bending

moments and in-plane forces on the panel will be carried as membrane forces in the facing skins.

For many practical cases, where the span of the panel is large compared to its thickness, the shear

deflection will be negligible. In these cases, it may be possible to obtain reasonable results by

modelling the structure using composite shell elements. It should be noted that the in-plane

stiffness of the honeycomb is negligible compared to that of the facing skins. When a more

detailed model is needed the honeycomb can be modelled with solid 3D elements. Attempts to

model the individual cells of the honeycomb should be avoided for normal engineering analyses.

[6]

Figure 2.18: Plot of displacement of different sandwich structure modelling methods [15]

Page 22

As shown in Figure 2.17 and Figure 2.18 there is a small difference in the result of using just

shell elements or cores modelled with solids. On the one hand modelling the sandwich structure

just with shell elements makes the structure a bit too stiff, on the other hand it gives fast and

relative good results. The modelling method with solid elements gives more accurate results.

2.8.8. Maximum strain theory This theory is similar to St. Venant’s maximum normal strain and Tresca’s maximum shear stress theory applied for isotropic materials. The strains in the lamina are resolved to the local axes and

the failure modes of a composite lamina are predicted by comparing the individual strains with

respect to their ultimate strains. [14]

Page 23

3. Motivation and goals The main goal of Shell Eco-Marathon is as mentioned, to travel further with the lowest energy

consumption.

In order to keep the energy consumption low, it is important to reduce the overall vehicle drag.

Five different types of driving drags can occur when considering an accelerating three-wheel-

vehicle (e.g. Fennek) going uphill on a road without cornering: [16] [17]

1. Tire drag: 𝐹𝑅 ≔ 𝜇𝑅 ∗ 𝐹𝑁 = 𝜇𝑅 ∗ 𝑚𝑣 ∗ 𝑔 (3.1)

2. Wheel bearing drag: 𝐹𝐵 ≔ 𝑃𝐵𝑙𝑜𝑠𝑠𝑣 (3.2)

𝑃𝐵𝑙𝑜𝑠𝑠 ≔ 1.05 ∗ 10−4 ∗ 𝑀𝐵 ∗ 𝑛 (3.3) 𝑀𝐵 ≔ 𝑀𝑟𝑟 ∗ 𝑀𝑠𝑙 ∗ 𝑀𝑠𝑒𝑎𝑙 ∗ 𝑀𝑑𝑟𝑎𝑔 (3.4)

3. Aerodynamic drag: 𝐹𝐴 ≔ 12 ∗ 𝜌𝑎𝑖𝑟 ∗ 𝐴 ∗ 𝑐𝑥 ∗ 𝑣2 (3.5)

4. Uphill drag: 𝐹𝑈 ≔ 𝑚𝑣 ∗ 𝑔 ∗ sin(𝛿) (3.6)

5. Acceleration drag: 𝐹𝐴𝑐𝑐 ≔ 𝑚𝑣 ∗ 𝑎 (3.7)

All equations beside the equation for the aerodynamic drag contain the vehicle mass.

A reduction in drag is proportional to a reduction in energy consumption. A reduction of the

vehicle’s mass increases the influence of the aerodynamic drag. An aerodynamic optimization

should thus also achieved, but that is not part of this thesis.

Project goals and approach

This thesis should give the members of TERA TU Graz a detailed overview about the usage of

finite element analysis for composite materials and in particular for simulating different load cases

of vehicles built for Shell Eco-Marathons. The main goal of this scientific work is the weight

reduction for the monocoque of an energy efficient vehicle. The overall mass of the vehicle should

be reduced by defining a smart layup of carbon fibre reinforced plastics for the monocoque. Next

to the weight, the stiffness of the vehicle must not be neglected. In addition the use of honeycomb

sandwich structures is wished for. The main tasks of the thesis are divided into five subsections:

Literature research

Defining load cases

Pre- and Post-Processing of the monocoque

Carbon fibre reinforced plastic model: layup defined manually

Documentation of the results

Using the gathered know-how to build the Fennek 2018

Page 24

4. Finite element analysis of composite monocoque

A monocoque is like an insect skeleton, it carries the loads without any additional chassis or

frame. Goal of a FEA of a monocoque is to optimize diverse parameters like material properties,

wall thickness, laminate lay-up angles and additional usage of sandwich structures in order to

reduce the overall vehicle mass to lower the energy consumption during the competition.

4.1. Design constraints of an energy efficient vehicle The development of an energy efficient vehicle is a very complex procedure. Many work steps

are necessary to achieve good results. The position of the FEM calculations in the product

development process is shown in Figure 4.1:

The most important category are the SEM regulations. Not following these would result in

exclusion from the competition. Ergonomics are considering the pilot’s physical dimensions. The pilot needs to feel save in the vehicle as well as comfortable enough to drive the vehicle without

needing to maintain excessive concentration throughout the run [16]. Furthermore, the

manufacturing process must not be neglected, since all rigidity calculations are worthless if the

parts are not producible. Last but not least the aerodynamics are very important. The factor

A * cx in the Formula (3.5) has to be minimized to reduce the aerodynamic drag of the vehicle.

Keeping all of these constraints in mind and after a fully defined vehicle body was designed, the

rigidity calculation can be started.

Figure 4.1: Design constraints of an energy efficient vehicle

Page 25

4.2. Geometry of the monocoque

Geometry scale 1:20

Figure 4.2: Geometric properties of Fennek

2600

634

1010.5

Points of wheel

suspension

Page 26

4.3. Material data of the monocoque

13 12

18

1 2

17 16

4

5

10

14

15

19 11

20 3 8 6

7

9

Figure 4.3: Material data of Fennek

Geometry scale 1:20

Page 27

CFK Fennek and CFK wing

Material: MTM49-3

E1 = 65544 MPa [18]

E2 = 63124 MPa [18]

NU12 = 0.3 (experience)

G12 = 3200 MPa [19]

Rho = 1.23e-009 t/mm3 [19]

Figure 4.4: Alignment of elements according to the laminate main direction; example part: wing

Table 4.1: Material data of Fennek 2017

Number Material name definition Material Stacking Wall thickness [mm]

1 Laminate layer 1 CFK Fennek [45°/0°/45°] 0.66

2 Laminate layer 2 CFK Fennek [45°/0°/45°] 0.66

3 Laminate layer 3 CFK Fennek [45°/0°/45°] 0.66

4 Laminate layer 4 CFK Fennek [45°/0°/45°] 0.66

5 Laminate layer 5 CFK Fennek [45°/0°/45°] 0.66

6 Laminate layer 6 CFK Fennek [45°/0°/45°] 0.66

7 Laminate layer 7 CFK Fennek [45°/0°/45°] 0.66

8 Laminate layer 8 CFK Fennek [45°/0°/45°] 0.66

9 Laminate wing CFK wing and Rohacell IG/IG-F [45°/0°/0°/0°/0°/0°/0°/45°] 1.86

10 Laminate core side CFK Fennek and Rohacell IG/IG-F [45°/0°/45°/core/45°/0°/45°] 6.32

11 Laminate core side back CFK Fennek and Rohacell IG/IG-F [45°/0°/45°/core/45°/0°/45°] 6.32

12 Laminate core bottom CFK Fennek and Rohacell IG/IG-F [45°/0°/45°/core/45°/0°/45°] 6.32

13 Laminate core bottom back CFK Fennek and Rohacell IG/IG-F [45°/0°/45°/core/45°/0°/45°] 6.32

14 Laminate core ingress thick CFK Fennek and Rohacell IG/IG-F [45°/0°/45°/core/45°/0°/45°] 6.32

15 Laminate core ingress thin CFK Fennek and Rohacell IG/IG-F [45°/0°/45°/core/45°/0°/45°] 6.32

16 Laminate partition wall CFK Fennek [45°/0°/45°] 0.66

17 Laminate core partition wall CFK Fennek and Rohacell IG/IG-F [45°/0°/45°/core/45°/0°/45°] 6.32

18 Laminate overlap CFK Fennek [45°/0°/45°/45°/0°/45°] 1.32

19 Window front Polycarbonate / 2.00

20 Window side Polycarbonate / 2.00

Page 28

4.4. FE-model of Fennek

Figure 4.5: FE-mesh of Fennek

Page 29

Monocoque

QUAD 4 elements (first order shell elements) and MAT8 material card

Windows

QUAD 4 elements (first order shell elements) and MAT1 material card

Wheel suspension

CTETRA elements (second order tetrahedron elements) and MAT1 material card

Spokes

CBAR elements and MAT1 material card

Rim

QUAD 4 elements (first order shell elements) and MAT1 material card

The wheel suspension, the spokes and the rim were only modelled in order to obtain the correct

stiffness for the calculation. No closer look was taken into the stresses there.

First order shell elements were used for the monocoque because applying the orientation of the

materials is easier on first order elements.

4.5. Load cases Finding the right load cases (LC) for a vehicle built for Shell Eco-Marathon is relatively difficult

process, since there are no norms and little literature for this issue. Some load cases are obvious;

others are very challenging to find, understand and handle.

4.5.1. Pilot ingress

Every time the driver enters the car or gets out of it, the vehicle is loaded in a specific area

through feet or hands acting on the monocoque. The areas of present loads were dedicated by the

process of having the 2017 driver get off and on the Fennek several times.

For the first step, only the load caused by one foot in the monocoque was taken into account. A

shoe of size 37 (European Union) has a length of 240 millimetres and a width of 90.5 millimetres

[20]. A driver mass of 50 kilograms multiplied with the gravity constant and divided by the area

spanned by one foot leads to a pressure of 0.23 MPa. For the second step it was assumed that the

driver is leaning on the frame and only the hands have contact to the monocoque. Here the hand

force only is acting on the vehicle’s frame. For both situations the vehicle was fixed in 2- and 3-

direction at the contact patches of the front wheels to the ground. At the rear wheel the vehicle

Figure 4.6: Feet and hands acting on the monocoque

Person in the picture: Sabrina Hammerschmid,

driver of Fennek 2017

Figure 4.7: Hands acting on the monocoque

Person in the picture: Sabrina Hammerschmid,

driver of Fennek 2017

Page 30

was fixed in 1- and 3-direction at the contact patches to the ground. This boundary condition

simulates the brakes are acting on the rear wheels during the ingress.

4.5.2. Pilot is lying in the vehicle

Most of the time the vehicle is in motion, but from time to time the driver has to lie in the

stationary vehicle. These situations can occur during technical inspection for example. To get the

right mass distribution in the vehicle, a dummy driver was designed and meshed:

Hand force = 2 x 250 N

240 9

0.5

Foot Pressure = 0.23 MPa equal 500 N

1185

Figure 4.9: FE-model of the dummy driver

Head

Torso

Upper arm

Forearm

Femoral

Lower leg

BC 2, 3

fixed

BC 2, 3

fixed

BC 1,3

fixed

Geometry scale 1:20

Figure 4.8: Boundary conditions and acting loads of load case pilot ingress

Page 31

4.5.2.1. Dimensions of body parts

The density of an adult weighing around 50 kg is equal to 1076 kg/m3, the volume is equal to

0.046 m3 (linear interpolation of values in Table 1 of [21]). The mass distribution of the different

body parts is shown in Table 4.2. Basis for the calculations are a drivermass of 50 kilograms.

Average driversize is 1.6 meters.

The lengths of the different body parts can be calculated by applying the values of Figure 4.10 to

the average driversize. The resulting values are summarised in Table 4.3.

Considering these values, simplified geometries for each body part can be found.

4.5.2.1.1. Dimensions of torso

The torso, for example, can be simplified by a geometrical body with a cross section of an ellipse,

a length of 600 millimetres and a mass of 21.5 kilograms. Due to the dimensions of the

monocoque the large half-axis of the ellipse is equal to 135 millimetres. Taking these values into

account, the size of the minor half-axis can be calculated:

Part of body Percentage [%] Number in human body Total percentage [%] Mass of part [kg] Total mass [kg]

Head 7.00 1.00 7.00 3.50 3.50

Torso 43.00 1.00 43.00 21.50 21.50

Upper arm 3.00 2.00 6.00 1.50 3.00

Forearm 2.00 2.00 4.00 1.00 2.00

Hand 1.00 2.00 2.00 0.50 1.00

Femoral 12.00 2.00 24.00 6.00 12.00

Lower leg 5.00 2.00 10.00 2.50 5.00

Foot 2.00 2.00 4.00 1.00 2.00

Sum 100.00 50.00

Table 4.2: Mass distribution of body parts (calculated from [22])

Table 4.3: Length distribution of body parts (calculated from [23])

Part of body Percentage [%] Part length [mm]

Head 0.125 200

Torso 0.375 600

Arm 0.375 600

Leg 0.500 800

Figure 4.10: Lengths of body parts

(recreated after [23])

Head

1/8

Legs

1/2

Knees

1/4

Page 32

Cross section of an ellipse: 𝐴𝑒𝑙𝑙 = 𝜋 ∗ 𝑎𝑒𝑙𝑙 ∗ 𝑏𝑒𝑙𝑙 (4.1)

Volume of the simplified torso with length LT, cross section AT mass mT and

density ρH: 𝑉𝑇 = 𝐴𝑒𝑙𝑙 ∗ 𝐿𝑇 (4.2) 𝑉𝑇 = 𝑚𝑇/𝜌𝑇 (4.3)

Length of the minor half-axis:

𝑏𝑒𝑙𝑙 = 𝑚𝑇𝜌𝑇∗𝐿𝑇∗𝜋∗𝑎𝑒𝑙𝑙 (4.4)

There the length of the minor half axis is thus equal to 79 millimetres.

4.5.2.1.2. Dimensions of head The total body surface is defined as the outer body surface which is covered with skin. The

average body surface of a woman is equal to 1.6 square metres [24]. The head surface comes to 9

percent of the total body surface [25]. The head can be simplified as cylinder with a mass of 3.5

kilograms and a length of 200 millimetres.

Shell surface of a cylinder with circumference U and length h: 𝑀 = 𝑈 ∗ ℎ (4.5)

Deck area of a cylinder with radius rC: 𝐷𝐴 = 𝑟𝐶2 ∗ 𝜋 (4.6)

Total body surface of a cylinder: 𝐻𝑆 = 𝑀 + 2 ∗ 𝐷𝐴 (4.7)

Resulting radius of the wanted cylinder:

𝑟𝐶 = −2∗𝜋∗ℎ+√(2∗𝜋∗ℎ)2+8∗𝜋∗𝐻𝑆8∗𝜋 (4.8)

Figure 4.11: Dimensions of an ellipse

Page 33

This results in a cylinder radius of 80 millimetres. A radius of 80 millimetres and a head length of

200 millimetres leads to a head mass of 4.3 kilograms. This is a little too high and is due to the

fact that the head geometry is not a cylinder in real life. By considering a head mass of 3.5

kilograms and keeping the cylinder radius of 80 millimetres, a new head length can be found by

solving following equation: ℎ = 𝑀𝑟𝐶2∗𝜋∗𝜌 (4.9)

This leads to a new head length of 162 millimetres.

4.5.2.1.3. Dimensions of arms and legs Arms and legs can be seen as truncated cones. The locations of the centres of gravity of each

body part are shown in Figure 4.12.

By solving the equation 𝑉 = ∫ 𝑟(𝑥)2 ∗ 𝜋 ∗ 𝑑𝑥𝑥=𝐿𝑇𝑐𝑥=0

(4.10)

with 𝑟(𝑥) = 𝑑2 + (𝐿 − 𝑥) ∗ tan[arctan (𝐷−𝑑2∗𝐿 )] (4.11)

the formula for the volume can be found: 𝑉 = 𝐿 ∗ 𝜋 ∗ (𝐷2+𝐷∗𝑑+𝑑212 ) (4.12)

Figure 4.12: Location of centre of

gravity of body parts [22]

Figure 4.13: Calculation of volume and centre of gravity of a truncated cone

Page 34

The centre of gravity can be calculated by solving the following equation: 𝑙𝑆 = 1𝑉 ∫ 𝑥 ∗ 𝑟(𝑥)2 ∗ 𝜋 ∗ 𝑑𝑥𝑥=𝐿𝑇𝑐𝑥=0 (4.13)

which leads to the following formula: 𝑙𝑠 = 14 ∗ 𝐷2+2∗𝐷∗𝑑+3∗𝑑2𝐷2+𝐷∗𝑑+𝑑2 (4.14)

By putting (4.14) in an Excel sheet, ls can be calculated as a function of d and D, where D is set to

one specific value and d was set to different numbers till a match between centre of gravity and

mass was found:

This Excel sheet was created for all four truncated cones, the upper arm, the forearm, the femoral

and the lower leg. The result table is shown in Table 4.5.

After creating all body parts, they were fixed to each other by rigid elements. These elements

have no mass properties and so they are not influencing the mass distribution of the other parts.

The whole body was fixed via spring elements to the monocoque. The springs were subjected to a

very high stiffness in the 3-direction and to no stiffness in the other two directions.

4.5.2.2. Acting loads The vehicle was again fixed in 2- and 3-direction at the contact patches of the front wheels and in

1 and 3directions at the contact patches of the rear wheel to the ground. This boundary condition

simulates that the brakes are acting on all three wheels during the pilot is lying in the vehicle.

Furthermore, an acceleration of the size of the gravity constant was applied on the model in

global 3-direction.

Table 4.4: Calculation of the smaller diameter of a truncated cone, shown example: calculation of upper arm

Wanted ls [mm] L [mm] D [mm] d [mm] Calculated ls [mm] Density [kg/m3] Wanted mass [kg] Calculated mass [kg]

141 300 85 74 143.09 1076 1.5 1.60

73 142.42 1.59

72 141.74 1.57

71 141.05 1.55

70 140.35 1.53

69 139.65 1.51

68 138.93 1.49

67 138.21 1.47

Compromise between mass and density

Table 4.5: Calculation of the smaller diameter of a truncated cone, all results

Body part L [mm] D [mm] d [mm] Wanted ls [mm] Calculated ls [mm] Density [kg/m3] Wanted mass [kg] Calculated mass [kg]

Upper arm 300 85 71 141 141 1076 1.5 1.55

Forearm 300 70 43 126 127 1076 1 0.82

Femoral 400 160 110 176 176 1076 6 6.23

Lower leg 400 110 65 168 166 1076 2.5 2.65

Page 35

4.5.3. Vertical impact

In this load case the vehicle is travelling in a straight line and is loaded by an acceleration of 3g in

the global 3-direction [26].The vehicle was fixed in the 1 and 3 directions at the contact patches

to the ground at the rear wheel and in direction 2 and 3 on the contact patches to the ground at the

two front wheels.

4.5.4. Pushing the vehicle

The roll bar of the vehicle must withstand a static load of 700 N applied in a vertical, a horizontal

and/or a perpendicular direction without deforming in any direction [27]. The brakes are acting on

each wheel and so the vehicle was fixed in the 1, 2 and 3 directions at the contact patches to the

ground at each wheel.

Gravity constant g=9.81 m/s2

Geometry scale 1:20

Figure 4.14: Boundary conditions and acting loads of load case pilot is lying in the vehicle

BC 2, 3

fixed

BC 2, 3

fixed

BC 1, 3

fixed

3 times gravity constant 3*g=3*9.81 m/s2

Geometry scale 1:20

Figure 4.15: Boundary conditions and acting loads of load case vertical impact

BC 2, 3

fixed

BC 2, 3

fixed

BC 1, 3

fixed

Page 36

4.5.5. Pulling the vehicle

Before the start it can happen that the vehicle is pulled to the start line by a track marshal.

Assuming that the track marshal is handling the vehicle with care, a load of 750 N can be found

(driver mass + vehicle mass equals 75 kg and with an acceleration of around 10 m/s2). The

vehicle was fixed in the 1, 2 and 3 directions at the contact patches to the ground at the rear wheel

and in directions 2 and 3 on the contact patches to the ground at the two front wheels.

Gravity constant g=9.81 m/s2

700 N

Gravity constant g=9.81 m/s2 700 N

(700 N)

BC 1, 2, 3

fixed

BC 1, 2 ,3

fixed

BC 1 ,2, 3

fixed

700 N

(700 N)

BC 2, 3

fixed

BC 2, 3

fixed

BC 1, 3

fixed

700 N

Geometry scale 1:20

Figure 4.16: Boundary conditions and acting loads of load case pushing the vehicle

Geometry scale 1:20

Figure 4.17: Boundary conditions and acting loads of load case pulling the vehicle

Page 37

4.5.6. Vehicle is cornering

The vehicle is moving with a constant velocity v and then is cornering with a constant turn radius

R. The worst case is a velocity of 30 km/h and a turn radius of 8 meters (8 meters turning radius

is the minimum radius at SEM). Taking an overall mass (vehicle plus driver) of 75 kg into

account, the centrifugal force, which is acting on the centre of gravity, can be calculated:

𝐹𝐶 = 𝑚∗𝑣2𝑅 (4.15)

Figure 4.18 shows the cornering situation of a three-wheeled vehicle.

Figure 4.18: Vehicle cornering situation

Shortcut Quantity Unit

L 1379.5 mm

a 1010.5 mm

R 5400 mm

β 11.3 degree

βi 16 degree

βo 13.4 degree

μ0 0.9 1

Table 4.6: Needed numbers for calculation of the cornering acceleration

(μ0 after [28])

Page 38

The forces acting in longitudinal direction of the vehicle are very small and the minimum turning

radius is relatively large. For these reasons, the Ackermann steering can be neglected and the

centrifugal force can be applied in global y-direction. By applying Newton (F=m*a), the

cornering acceleration can be calculated: 𝑎𝐶 = 𝐹𝐶𝑚 (4.16)

This cornering acceleration is then applied on the model as well as the gravity acting on the

vehicle. The vehicle was fixed in the 1, 2 and 3 directions at the contact patches to the ground at

the rear wheel and in direction 2 and 3 on the contact patches to the ground at the two front

wheels. This means that the contact patches to the ground cannot move in direction 2, which is a

good approximation to a cornering situation.

Furthermore, the body is not only connected via springs to the monocoque in the global 3

direction, but also in the global 2 direction on the y+ side. The action of the springs in the global

2 direction has a very high stiffness in y-direction but no stiffness in the other two directions. This

way the reaction forces from the cornering acceleration can directly directed into the monocoque.

Geometry scale 1:20

Figure 4.19: Boundary conditions and acting loads of load case vehicle is cornering

Gravity constant g=9.81 m/s2

BC 2, 3

fixed

BC 2, 3

fixed

BC 1, 2, 3

fixed

Cornering acceleration aC=8.61 m/s2

Springs connecting the upper arm with the monocoque

Page 39

4.6. Results After calculating the six load cases above, it is clear that the load case “vehicle is cornering” is the most harming load case of all. For this reason, only the results for this case are shown in this

thesis. The results of the other load cases can be purchased by contacting the author.

4.7.1. Deformation

Blue lines display undeformed body;

deformation scale: 1 [mm] in the

drawing displays 1000 [µm] in reality

Geometry scale 1:20

Figure 4.20: Deformations due to cornering acceleration

Page 40

4.7.2. Stresses

In Figure 4.21 and Figure 4.22 the principal normal stresses are displayed just on the

monocoque. The stresses in the suspension are not displayed here.

56 [MPa]

Figure 4.21: Principal stresses due to cornering acceleration; detailed view

Page 41

As shown in the illustrations above, the stresses in the monocoque are not higher than 56 MPa.

This means that no critical area was dedicated.

Figure 4.22: Principal stresses due to cornering acceleration; overview

Page 42

4.7.3. Strains

In order to state that the monocoque is able to carry the loads, the maximum strain theory

(Chapter 2.9.8) was applied, whereby a maximum strain of 0.5% (experience) is allowed. The

result is displayed in Figure 4.23.

The maximum strain of the monocoque is equal to 0.1 %. This means the monocoque is

sufficiently stiff to carry all the loads applied.

No critical area was dedicated on the wing.

0.1 [%] Figure 4.23: Strains due to cornering acceleration

Page 43

4.7. Optimization There are two ways of optimizing the body: [16]

Pragmatic method:

Setting up the FEM-model and vary parameters like wall thickness and/or material

properties manually.

Parametric optimization on evolutionary algorithms

A symmetric layup should always be used in order to avoid unwished coupling behaviours

between the individual plies (see Chapter 2.9.6). Since the TERA team has only one prepreg

type available, the layup of the monocoque cannot be changed because the minimum number of

plies was already used for it. The team therefore decided to concentrate on the wing.

The wing of Fennek 2017 has the following layup: [45°/0°/0°/0°/0°/0°/0°/45°]

After simulating a series of different optimizations, the following was determined to be the best

lay-up: [45°/0°/0°/0°/45°].

4.7.1. Comparison of stresses of the wing

Wing Fennek 2017 Optimisation of wing

Figure 4.24: Comparison of stresses of the wing

Page 44

4.7.2. Comparison of strains of the wing

The maximum strain at the wing of the optimization is equal to 0.1 %. This means the wing is still

sufficiently stiff to carry all the loads. The number of wing plies was not further reduced because the

team decided that a mass reduction of 510 grams compared to Fennek 2017 is adequate. This mass

reduction corresponds to a total mass reduction of the monocoque of 7.5 %.

Wing Fennek 2017 Optimisation of wing

Figure 4.25: Comparison of stresses of the wing

0.1 %

Mass: 764 g Mass: 1274 g

Page 45

Figure 4.26: TERA TU Graz team and Fennek 2018 at Shell Eco-Marathon 2018

Persons in the picture from left to right:

Standing: Rene Pöschl, Konrad Fellner, David Fink, Hans-Georg Hirt, Marius Müller, Juan Campos Alonso

Kneeing: Jonas Naderer, Julian Naderer, Nicole Kelz

Page 46

5. Summary The goal of the Shell Eco-Marathon is to challenge students from around the world to build the

most efficient vehicles. One of these student teams comes from Graz, Austria, and this is TERA

TU Graz. The Graz team has participated in SEM since 2010 and has made consistent,

determined and successful efforts to improve the Fennek every time. In order to make the vehicle

even more efficient, one of the goals for the season 2017/2018 was the weight reduction of the

monocoque. The design of the monocoque was already defined, so changing the layup of the plies

(amount of plies, orientation and stacking order) were the only changes which could be made.

The team decided to use the FEM in order to achieve this goal. Since there are no norms and little

literature available, one of the most difficult parts of this thesis was the definition of the most

critical load cases. After the calculation of six different LC, the most harming LC (Vehicle is

Cornering, Chapter 4.5.6) was documented here. The simulations clearly indicated that the

monocoque was built to a very high standard of excellence and only very few changes could be

introduced on the wing. By changing the amount of plies of the wing, its mass could be reduced

by 510 grams (Optimization), which corresponds to a total mass reduction of the monocoque of

7.5 %.

6. Outlook This thesis is intended as a useful and practical guide for students participating in the Shell Eco-

Marathon and involved in building light-weight and energy efficient vehicles.

One of the main issues of simulating carbon reinforced parts is the availability of material data.

The best way to receive material data is to make material tests (tensile test or three point bending

tests).

In order to improve the Fennek even further, another prepreg with a thinner wall thickness should

be used. Furthermore, buckling and fatigue analyses can also be carried out. The results of the

Finite element analysis (FEA) should be proved by test driving with strain gauges fixed on the

monocoque.

Furthermore, a very close review and examination of the entire Fennek concept is proposed. It is

possible that the wing can be replaced with wishbone wheel suspension to reduce the number of

aluminium parts.

Page 47

7. References

[1] http://www.shell.com/energy-and-innovation/shell-ecomarathon/about.html

last access: 2018/12/09

[2] WOLLSCHLAGER J. A.: Introduction to the Design and Analysis of Composite Structures;

An Engineers Practical Guide Using Optistruct®,

2014, Wollschlager, Mill Creek, Wash

[3] SCHÜRMANN H.: Konstruieren mit Faserverbundwerkstoffen,

2004, Springer, Berlin

[4] RYBANDT S., LIES C., HIPKE T. “Triebkopfkabine aus Aluminiumschaumverbund“ Lightweight.design [Wiesbaden, Germany] July 2017: A16, A17. Print

[5] HexPly® Prepreg Technology,

January 2013, Publication No. FGU 017c

[6] HexWebTM

HONEYCOMB SANDWICH DESIGN TECHNOLOGY

December 2000, Publication No. AGU 075b

[7] WEISE J., LEHMHUS D., BAUMEISTER J. “Leichtbaustrukturen auf Basis von Aluminium-

schaumgranulat“Lightweight.design [Wiesbaden, Germany] July 2017: A10. Print

[8] https://www.comsol.com/multiphysics/finite-element-method last access: 2018/12/09

[9] Altair University: Practical Aspects of Finite Element Simulation;

A Study Guide, 3rd

edition released 05/2015

[10] Modelling and FE-Simulation in Automotive Structural Mechanics,

Graz University of Technology, Lecture number: 331.095, summer term 2017

[11] https://www.altairhyperworks.com/product/HyperMesh last access: 2018/12/09

[12] https://www.altairhyperworks.com/product/OptiStruct last access: 2018/12/09

[13] https://www.altairhyperworks.com/product/HyperView last access: 2018/12/09

[14] Altair University: Introduction to Practical Aspects of Composites,

1st edition released 05/2018

[15] https://altairuniversity.com/wp-content/uploads/2017/10/Composite_Pre_Postprocessing.pdf

last access: 2018/12/09

[16] SANTIN J.J, ONDER C.H, BERNARD J., ISLER D., KOBLER P., KOLB F.,

WEIDMANN N., GUZZELLA L.:

The world’s most fuel efficient vehicle – Design and development of PAC-Car II,

2007, vdf Hochschulverlag AG an der ETH Zürich, Zürich/Singen

Page 48

[17] http://www.skf.com/de/products/bearings-units-housings/principles/bearing-selection-

process/operating-temperature-and-speed/friction-powerloss-startingtorque/index.html

last access: 2018/12/09

[18] INSAM G.: Auslegung, Konstruktion und Validierung des Kohlefasermonocoques

eines Formula Student Rennfahrzeuges,

2008, Inst. für Verbrennungskraftmaschinen und Thermodynamik,

Graz, University of Technology

[19] http://www.lavender-ce.com/wp-content/uploads/pds1059-mtm49series-7.pdf

last access: 2018/12/09

[20] https://ribbelmonster.de/groessentabelle-fuer-schuhe-schuhgroessen/ last access: 2018/12/09

[21] https://apps.dtic.mil/dtic/tr/fulltext/u2/639241.pdf last access: 2018/12/09

[22] http://www.arsmartialis.com/index.html?name=http://www.arsmartialis.com/faq/m

anteil.html last access: 2018/12/09

[23] http://www.kunst-malerei.info/menschen-malen.html#WoMi70jOWUk

last access: 2018/12/09

[24] http://flexikon.doccheck.com/de/Körperoberfläche last access: 2018/12/09

[25] http://www.brandverletzte-leben.de/brandverletzten-ratgeber/verbrennungsgrade/

last access: 2018/12/09

[26] HEIßING B., ERSOY M., GIES S., Fahrwerkshandbuch – Grundlagen, Fahrdynamik,

Komponenten, Systeme, Mechatronik, Perspektiven,

3., überarbeitete und erweiterte Auflage, Praxis, ATZ/MTZ-Fachbuch

[27] Shell Eco-Marathon, 2018 Official Rules, Chapter I

[28] https://www.schweizer-fn.de/stoff/reibwerte/reibwerte.php last access: 2018/10/01

Page 49

8. List of figures

1.1a TERA Fennek 2010 1

1.1b TERA Fennek 2011/2012 1

1.1c TERA Panther 1

1.1d TERA Fennek 2013/2014 1

1.2a TERA Ibex - The prototype 2

1.2b TERA Ibex - Rolling chassis 2

1.3 TERA Fennek 2017 2

1.4 TERA TU Graz team makes the 3rd place with 596 km/kWh at

Schell Eco-Marathon 2017 3

2.1 Schematic sketch of a particulate composite ply material [2] 4

2.2 Influence of fibre volume on the strength (recreated after [3]) 5

2.3 Unit cell of a graphite crystal (recreated after [2]) 6

2.4 Performance and production volume of different fabrication processes [5] 8

2.5 Unidirectional ply material [2] 8

2.6 Three main weave styles of fabrics (recreated after [5]) 9

2.7a Vacuum bag processing - Assembly [5] 10

2.7b Atmospheric pressure is acting [5] 10

2.7c Vacuum bag process - Cure temperature [5] 10

2.7d Vacuum bag process - Vacuum trend [5] 10

2.8a Autoclave process - Cure temperature and heat up rate [5] 10

2.8b Autoclave process - Pressure trend [5] 10

2.8c Autoclave process - Vacuum trend [5] 10

2.9 Construction of a sandwich panel compared to an I beam [6] 11

2.10 Relative stiffness and weight of sandwich panels compared to solid panels [6] 11

2.11 Manufacture of sandwich constructions [5] 13

2.12 Example of a simple FE-structure [10] 15

2.13 Zone based modelling [15] 17

2.14 Element normal [14] 18

2.15 Alignment of elements according to the laminate main direction [15] 19

2.16 Definition of fibre direction, transverse matrix direction and the fibre orientation

(recreated after [2]) 20

2.17 Different modelling styles of sandwich structures [15] 21

2.18 Plot of displacement of different sandwich structure modelling methods [15] 21

4.1 Design constraints of energy efficient vehicle 24

4.2 Geometry properties of Fennek 25

4.3 Material data of Fennek 26

4.4 Alignment of elements according to the laminate main direction; example part: wing 27

4.5 FE-mesh of Fennek 28

4.6 Feet and hands acting on the monocoque 29

4.7 Hands acting on the monocoque 29

4.8 Boundary conditions and acting loads of load case pilot ingress 30

4.9 FE-model of the dummy driver 30

4.10 Lengths of body parts (recreated after [23]) 31

Page 50

4.11 Dimensions of an ellipse 32

4.12 Location of centre of gravity of body parts [22] 33

4.13 Calculation of volume and centre of gravity of a truncated cone 33

4.14 Boundary conditions and acting loads of load case pilot is lying in the vehicle 35

4.15 Boundary conditions and acting loads of load case vertical impact 35

4.16 Boundary conditions and acting loads of load case pushing the vehicle 36

4.17 Boundary conditions and acting loads of load case pulling the vehicle 36

4.18 Vehicle cornering situation 37

4.19 Boundary conditions and acting loads of load case vehicle is cornering 38

4.20 Deformations due to cornering acceleration 39

4.21 Principal stresses due to cornering acceleration; detailed view 40

4.22 Principal stresses due to cornering acceleration; overview 41

4.23 Strains due to cornering acceleration 42

4.24 Comparison of stresses of the wing 43

4.25 Comparison of strains of the wing 44

4.26 TERA TU Graz team and Fennek 2018 at Shell Eco-Marathon 2018 45

9. List of tables

4.1 Material data of Fennek 2017 27

4.2 Mass distribution of body parts (calculated from [22]) 31

4.3 Length distribution of body parts (calculated from [23]) 31

4.4 Calculation of the smaller diameter of a truncated cone, shown example:

Calculation of upper arm 34

4.5 Calculation of the smaller diameter of a truncated cone, all results 34

4.6 Needed numbers for calculation of the cornering acceleration (μ0 after [28]) 37

10. List of abbreviations

SEM Shell Eco-Marathon

UD Unidirectional Reinforcement

FEM Finite Element Method

FEA Finite Element Analysis

1D One dimensional

2D Two dimensional

3D Three dimensional

DOF Degree of freedom

LC Load case

Page 51

11. Symbol directory

Symbol Unit SI unit Description

FR N N Rolling resistance force

μR - - Rolling resistance coefficient

FN N N Normal force

mv t kg Vehicle mass

g mms-2

ms-2

Gravitational constant

FB N N Wheel bearing drag

PBloss W W Bearing frictional power loss

v mm/s m/s Vehicle speed

MB Nmm Nm Total frictional moment of bearing

N rpm 1/s Rotational speed of bearing

Mrr Nmm Nm Rolling frictional moment, includes

effects of lubricant starvation and inlet

shear heating

Msl Nmm Nm Sliding frictional moment, includes the

effects of the quality of lubrication

conditions

Mseal Nmm Nm Frictional moment from integral seals

Mdrag Nmm Nm Frictional moment from drag losses,

churning, splashing, etc., in an oil bath

FA N N Aerodynamic drag

ρair kg/m3

kg/m3

Density of air

A mm2 m

2 Reference area for aerodynamic drag

cx - - Drag coefficient

FU N N Uphill drag

δ Degree Degree Angle of gradient

Facc N N Acceleration drag force

a ms-2 ms-

2 Acceleration of vehicle

Page 52

Ef║ N/mm2 N/m2 Yield modulus of the fibre in fibre direction

Ef┴ N/mm2 N/m2 Yield modulus of the fibre perpendicular to

fibre direction

E1 N/mm2 N/m2 Young's modulus in longitudinal direction of

the defined coordinate system; also defined

as fibre direction or 1-direction

E2 N/mm2 N/m2 Young's modulus in lateral direction of the

defined coordinate system; also defined as

matrix direction or 2-direction

ρ t/mm3 kg/m

3 Mass density of the ply material

NU12 - - Poisson's ratio (ε2/ε1 for uniaxial loading in 1-direction)

G12 N/mm2 N/m

2 Inplane shear modulus

G1Z N/mm2 N/m

2 Inplane shear modulus for shear in 1-Z plane

G2Z N/mm2 N/m

2 Inplane shear modulus for shear in 2-Z plane

Xt N/mm2 N/m

2 Tensile strength in x-direction

Xc N/mm2 N/m

2 Compression strength in x-direction

Yt N/mm2

N/m2 Tensile strength in y-direction

Yc N/mm2 N/m

2 Compression strength in y-direction

S N/mm2 N/m

2 Shear strength

Aell mm2 m

2 Cross section of an ellipse

aell mm m Length of the large half-axis of an ellipse

bell mm m Length of the minor half-axis of an ellipse

VT mm3

m3

Volume of the simplified torso

LT mm m Length of simplified torso

mt t kg Mass of torso

ρT t/mm3 kg/m

3 Density of torso

M mm2

m2

Shell surface of the cylinder

U mm m Circumference of the cylinder

h mm m Height of the cylinder

Page 53

DA mm2 m

2 Deck area of the cylinder

rC mm m Radius of the cylinder

HS mm2

m2

Total body surface of a cylinder

V mm3 m

3 Volume of truncated cone

r(x) mm m Radius of truncated cone

LTc mm m Length of truncated cone

D mm m Major diameter of truncated cone

d mm m Minor diameter of truncated cone

ls mm m Position of centre of gravity of trunc. Cone

FC N N Centrifugal force during cornering

m t kg Mass of driver plus vehicle

R mm m Turning Radius

FYi N N Force acting on the inner wheel of the

vehicle

FYo N N Force acting on the outer wheel of the

vehicle

FYr N N Force acting on the rear wheel of the vehicle

XG mm m x-position of the centre of gravity

YG mm m y-position of the centre of gravity

ZG mm m z-position of the centre of gravity

L mm m Distance between the contact patches to the

ground of the front wheels to the rear wheel

a mm m Distance between the contact patches to the

ground of the front wheels

μ0 1 1 Friction coefficient between asphalt and

wheels

β Degree Degree Steering angle

βi Degree Degree Inner steering angle

βo Degree Degree Outer steering angle

aC mms-2

ms-2

Cornering acceleration