AMTS STANDARD WORKSHOP PRACTICE - AMT … · Composite Design Section 1 of 3 AMTS-SWP-0042-F-2011 AMTS STANDARD WORKSHOP PRACTICE _____ Composite design Section 1 of 3 ... 10.1 Beam

Composite Design Section 1 of 3 AMTS-SWP-0042-F-2011

AMTS STANDARD WORKSHOP PRACTICE _________________________________________

Composite design Section 1 of 3: Composite design theory

Reference Number:

AMTS-SWP-0042-F-2011

Date:

October 2010

Version:

Final


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Contents 1 Scope ............................................................................................................................ 2 2 Technical terms.............................................................................................................. 2 3 Primary references ......................................................................................................... 4 4 Definition ........................................................................................................................ 4

4.1 Definition explanation .............................................................................................. 5 5 Classification of the composite ....................................................................................... 5

5.1 The reinforcement ................................................................................................... 5 5.2 The matrix ............................................................................................................... 6 5.3 The manufacturing .................................................................................................. 6

6 Basic Terminology used in Composite design ................................................................ 7 6.1 Important terminology ............................................................................................. 7

7 Theoretical Lamina equations ...................................................................................... 10 7.1 Orthotropic lamina properties ................................................................................ 10 7.2 Orthotropic properties for in plane stresses ........................................................... 12

7.2.1 Transformation of Coordinates ............................................................................ 12 7.2.2 Thermal Moisture effect ....................................................................................... 14

7.3 Methods for obtaining lamina properties ............................................................... 14 7.3.1 Experimentally .................................................................................................... 14 7.3.2 Theoretically ........................................................................................................ 16

8 Laminate Analysis ........................................................................................................ 16 8.1 Deformation due to extending and bending ........................................................... 16 8.2 Force and moment resultants ............................................................................... 17 8.3 ADB matrices ........................................................................................................ 18

8.3.1 Laminate code .................................................................................................... 20 8.3.2 Special laminates ................................................................................................ 20 8.3.3 Symmetric laminates ........................................................................................... 20 8.3.4 Balanced laminates ............................................................................................. 21 8.3.5 Quasi-isotropic laminates .................................................................................... 21

8.4 Stresses in the plies .............................................................................................. 21 9 Static strength and life of composites ........................................................................... 21

9.1 Fibre properties ..................................................................................................... 22 9.2 Tensile failure and compressive failure ................................................................. 22 9.3 Failure criteria ....................................................................................................... 23

9.3.1 Maximum stress .................................................................................................. 23 9.3.2 Maximum strain ................................................................................................... 23 9.3.3 Tsai-Wu .............................................................................................................. 23

9.4 Fatigue life ............................................................................................................ 23 10 Beam analysis ............................................................................................................. 24

10.1 Beam fundamentals .............................................................................................. 24 10.2 Symmetric rectangular beams............................................................................... 25 10.3 Laminated I-beam ................................................................................................. 27

10.3.1 Symmetric Narrow flanges ................................................................................ 28 10.3.2 Stress distribution in the web ............................................................................. 28

11 Conclusion ................................................................................................................... 30


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1 Scope A sophisticated analysis plays an important role in the development of aerospace structures.

It is required that many specialities are combined on an overall problem in any complex design structure. For each of these specialities the specific knowledge must be known by the designer. [1]

This SWP is part 1 of three SWP’s which covers the following information: SWP 42: Composite design Section 1 of 3 - Composite definition

- Composite classification - Basic terminology - Lamina Theory - Laminate Analysis Theory - Static strength life of composites

Theory - Beam Analysis theory

SWP 48: Composite design Section 2 of 3 - Fundamentals of Composite design

decisions - Advantages / Disadvantages - General guidelines for composite

designs

SWP 49: Composite design Section 3 of 3 - Interacting of software - Comparison of various software tools - Considerations in software - Links to various software

-

2 Technical terms

Aramid Aromatic polyamide fibres in which at least 85 percent of the amide linkages are directly attached to two aromatic rings, giving high tensile strength characteristics (Often referred to as Kevlar, DuPont’s trademark)

Buckling Failure mode usually characterized by unstable lateral deflection, rather than breakage, under compressive force.

Carbon fibre Produced by pyrolysis of an organic precursor fibre, such as PAN (polyacrylonitrile), rayon or pitch, in an inert atmosphere at temperatures above 982ºC/1800ºF. “Carbon” is often used interchangeably with “graphite” but carbon fibres are typically carbonized at about 1315ºC/2400ºF and contain 93% to 95% carbon while graphite fibres are carbon fibres submitted to graphitization at 1900º to 2480ºC (3450ºF to 4500ºF) after which they contain more than 99% elemental carbon

Composite Three-dimensional combination of at least two materials differing in form or composition, with a distinct interface separating the components. Composite materials are usually manmade and created to obtain properties that cannot be achieved by any of the components acting alone.

Fibre One or more filaments in an ordered assemblage, such as roving or tow


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Fibreglass Reinforcing fibre made by drawing molten glass through bushings. The predominant reinforcement used in polymer matrix composites, it is known for its good strength, processability and low cost.

Fibre-reinforcement polymer (FRP)

General term for a polymer-matrix composite that is reinforced with cloth, mat, or other fibre form. However, in practise, the term is most often used in reference to glass fibre-reinforced plastics

Filler A constituent, usually inert, added to a matrix to modify a composite’s properties (e.g. increase viscosity, improve appearance or decrease density) or to decrease part material cost.

Hand layup A fabrication method in which reinforcement layers, pre-impregnated or wet-laid are placed and arranged in a mould manually. (In contrast to spray up or automated methods, such as fibre placement)

Impregnate To saturate the voids and interstices of a reinforcement with resin Laminate To unite or bond two or more layers or laminate (often with the aid of

pressure and/or heat) Any fibre- or fabric-reinforced composite consisting of laminate with one or more orientations with respect to a particular reference direction

Laminate coordinate axes

Set of coordinate axes, usually right-handed Cartesian, used as a reference in describing the directional properties and geometrical structure of the laminate. Usually the x-axis and the y-axis lie in the plane of the laminate and the x-axis is the reference axis from which ply angle is measured. The x-axis is often in the principal load direction of the laminate and/or in the direction of the laminate principal axis (see principal axis, off-axis laminate and x-axis)

Matrix Material in which reinforcing fibre of a composite is embedded. Matrix materials include thermosetting and thermoplastic polymers, metals and ceramic compounds.

Ply A single layer (or lamina) used to fabricate a laminate. Also, the number of single yarns twisted together to form a plied yarn

Reinforcement The key element that, when combined with a matrix to make a composite, provides the required properties (primarily strength). Reinforcement forms range from individual short fibres to complex braided, woven or stitched textile using continuous fibres

Resin A solid or pseudo-solid polymeric material, often of high molecular weight, which exhibits a tendency to flow when subjected to stress, usually has a softening or melting range, and usually fractures conchoidally. As composite matrices, resins bind together reinforcement fibres and work with them to produce specified performance properties.

Weave To interlace fibres in a pattern, often based on a 0°/90°grid, the fabric pattern formed by interlacing yarns. Interlacing patterns vary. In plain weave, for instance, warp and fill fibres alternate to make both fabric faces identical. A satin weave pattern is produced by a warp tow over several fills tow and under one fill tow (e.g. eight-harness satin features one warp tow over seven fill tows and under the eighth)

Wet layup Application of a resin to dry reinforcements in the mould


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3 Primary references The main sources used for this document are indicated below. At the time of publication, the editions indicated were valid. All standards are subject to revision, and parties to agreements based on this document are encouraged to investigate the possibility of applying the most recent editions of the standards indicated below:

[1] MARSHALL,A.C. 1994. Composite basics fourth edition. Marshall consulting: USA.

Chapter 9,10.

[2] BAILIE, J.A., LEY, R.P. & PASRICHA, A. 1997. A summery and review of composite

laminate design guidelines, Task 22 NASA Contract NAS1-19347. Military aircraft

Systems Division.

[3] JONKER, A.S. 2003. Laminate analysis of composite materials. NWU

Potchefstroom. (Original coursework at NWU Potchefstroom campus.)

[4] WIKIPEDIA, 2011. Poison Ratio. http://en.wikipedia.org/wiki/Poisson's_ratio . Last

access: 20 October 2011

[5] ESACOMP, 1999. Theoretical background of Esacomp Analysis v1.5, Part III

Laminates. Esacomp

[6] CHAPTER 7, 2010. Experimental Characterization of Composite Materials.

http://www.ncat.edu/~ccmradm/ccmr/meen613/Ch7ExptMethds.pdf . Last date of

access: 20 October 2011.

4 Definition A composite can be defined as two or more a distinct constituents of a mixture. In order for a material to be considered as a matrix, the following criteria must be met [3]:

1. Both constituents must be present in reasonable proportions of more than 5%

2. The composites properties must differ from the constituent’s properties.

3. Various means of intimate mixing and combinations of constituents are used to produce the composite material.

http://en.wikipedia.org/wiki/Poisson's_ratio

http://www.ncat.edu/~ccmradm/ccmr/meen613/Ch7ExptMethds.pdf


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4.1 Definition explanation From the first part of the definition, it would be thought that materials like plastic and two-phase microstructure metal can be considered as composites, but when the other criteria’s are considered it is not the case. Although plastic contains various quantities of fillers and lubricants, criteria 1 and 2 is not met. Metal, on the other hand, due met criteria 1 and 2, but not criteria 3. [3]

This, and the proceeding SWP’s on composite design, will consider composites as a mixture consisting of a polymer matrix with some sort of reinforcing fibre. [3]

5 Classification of the composite As previously mentioned, the composite in this SWP’s will be consisting of polymer matrixes with some sort of reinforcing fibres. These components will be discussed with a brief overlook on the manufacturing methods available.

5.1 The reinforcement Composites can consist of numerous different fibres. In general three fibre types are considered with high performance polymer matrixes. These are glass fibre, carbon fibre and aramid fibres. The following table will give a brief over look on these fibres, but SWP 2 Raw Materials can be consulted for further detailed information, like fibre weaves sizing and compatibility.

Table 1: mostly used fibre properties.

Fibre Properties

Glass Fibre

Colour: White

Most widely used due to: Low cost Corrosion resistance Availability Manufacturing use Low stiffness Moderate strength to weight Chemical and weathering resistant

Carbon Fibre

Colour: Dark Grey/Black

Second most widely used: High strength to weight High stiffness to weight Properties vary according to manufacturing method. Different grades: high strength to high modulus.

Aramid Fibre

Colour: Yellow

Higher strength and stiffness light weight Lower compressive strength Good toughness in impact Ductile behaviour compared to other fibres that fails brittle.


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5.2 The matrix The purpose of the matrix is to protect and stabilise the reinforcement. Most applications use polymeric matrices, which consist of two classes, namely thermosets and thermoplastics. These matrices are discussed in detailed in SWP 2 Raw materials and SWP 4 Mixing of resins.

Thermosets (SWP 2 , 4)

Thermosets are one-time irreversible chemical reaction processes. Thermosets can be dived into three main categories, namely polyesters, vinyl-esters and epoxies:

Polyesters

Polyesters are a general purpose resin with the lowest cost and performance. The matrix cures with a catalyst reaction, usually MEKP, which is highly corrosive. Working time is low and reactions can be violently exothermic.

Vinyl-esters

Have similar physical properties to polyester. It does however have increased mechanical properties with a better controlled pot life (working time), but is more expensive than polyesters.

Epoxies

Epoxies have the most superior performance to the other thermosets, but are also the most expensive. The pot life can be designed for up to 4 hours and it have increased mechanical properties. Epoxies cure with stoichiometric reactions and therefore the mixing ratios are very critical.

Thermoplastics (SWP 32)

Thermoplastics are reversible reaction processes and can be formed by repeated application of heat.

5.3 The manufacturing There are various composite manufacturing methods, all of which are discussed in other SWP’s, of which the most widely used is hand lay up to the flexibility of the method. Manufacturing methods include:

Hand layup or wet layup (SWP 14)

Vacuum bagging (SWP 16)

Resin transfer methods, like vacuum infusion processing (SWP 34)

Filament winding and ply wrapping (SWP 33)


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6 Basic Terminology used in Composite design

Design functions cannot logically proceed unless the designer has an understanding of the loads imposed on a structure. An understanding on how the structure carries and transfers the loads is needed before a composite can be fully designed. There is an assumption that this mathematics are to complex and are only applicable for composite engineers, but with a little background of these calculations most of the loads and stresses can calculated on simple structures.

6.1 Important terminology Important terms in the calculation of stresses and loads are explained as follows:

LOAD (P)

The gross or total amount of force applied to a structure, usually expressed in Newton or Pounds. A load has a size and a direction.

Figure 1: Load illustration

WING LOADING (F)

The amount of air load applied per unit of area of the wing or vertical/horizontal tail areas. F = P/A

STRESS ( )

Stress (Ơ) is the amount of load (P) applied per unit of cross-sectional area (A) of

the structural member.

Figure 2: Strain and tension illustration

ALLOWABLE STRESS ( )

Allowable stress is the exact number of loading that a piece of material, of 1 square meter cross section, can carry without breaking.

LOAD DIRECTION

As important as the load itself, as most structural composite materials are stronger in one direction than the other to comply with the load in that direction.


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STRENGTH DIRECTION

Is the exact direction within a structural material which a quoted amount of allowable stress value may be carried.

STRENGTH

Strength is the total load that a given structural member can carry in the given direction.

STRAIN ( )

The amount of deflection that a loaded structural member experience due to the load it is carrying.

i) ii)

Figure 3: i) Illustration of strain and deflection ii) Section A enlarged to show reaction, shear, tension and compression.

MODULUS

The precise measurement of exactly how much a material will stretch under a given load. A high modulus means that a structure does not move much. The shear modulus for isentropic materials is given by the following relationship from Hook’s law, where G is the shear modulus, E the elasticity modulus and ν the Poison’s ratio:

TENSION

Tension is a load that acts along a single line, which tries to stretch the member that is being loaded, like in a cable.

COMPRESSION

Compression is a load that acts along a under load.

SHEAR ( )

Shear is a load that acts in opposite direction, but still parallel, to an equal resisting force of to one side.

POISON RATIO ( )

The poison effect occurs when a material is compressed in one direction and then tends to expand in the other two perpendicular directions of compression. The poison ratio is the fraction (percentage) of that expansion divided by the percentage compression for small values of these changes.


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Figure 4: Poison effect [4]

BEARING

A small member such as a load imposes a special case of compression upon the hole in which it is located, when the load is at the right angle to the axis of the bolt

BENDING ( )

Bending is a complex load, which usually entails compression, shear and bearing loads. The bending load on a structure can be calculated as the resultant internal moment (M) on the structure multiplied by the perpendicular distance from the neutral axis where bending load is applied, divided by its moment of inertia (I).

LOAD COMPONENT

Loads sometimes have odd directions, that is not in the simple coordinates and then the loads are divided into components. Each of which is in a direction that is easier to analyse. The sum of the components is equal (geometrically, not arithmetically) to the original load and is given by the equation:

√

LOAD RESULTANT

Loads acting upon a part at the same time can be geometrically added to get a single load, the resultant load. This single load can then be separated into components that are easier to analyse or that fit the structure better.

Figure 5: Load components and Load resultants


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MATRIX

In mathematics, a matrix (plural matrices, or less commonly matrixes) is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. Figure 7 shows an m by n matrix. [4]

Figure 6: m by n matrix [4]

7 Theoretical Lamina equations Equations are required to understand the coupling of membrane and bending responses. These equations are derived and discussed by various sources, but will be briefly discussed for the theoretical background required for most computer composite analysis tools. Properties of Fibre Composites differ from normal engineering materials in that it is highly directional. This directionality affects the usage of the material. The fibres in the component can be arranged to accommodate the highest load with the highest strength and stiffness. An individual lamina (Ply) will be examined and will form the basic building block where the relationships will be applied to layers of lamina. (All of the following information are the exact material of: [3] JONKER, A.S. 2003. Laminate analysis of composite materials. NWU Potchefstroom. (Original coursework at NWU Potchefstroom campus.)

NOTE: In this SWP the terms Lamina and Ply are used interchangeably throughout the document.

7.1 Orthotropic lamina properties In Figure 7 an orthotropic material is shown in a unidirectional layer with the fibre direction coordinate system 1, 2 and 3. Next to that is the global coordinate system X, Y in relationship to the fibre direction coordinate system.


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Figure 7, Lamina (Ply) axis [2]

The material properties are different in the 1 and two directions as this is an orthotropic material. The lamina 1 and 2 axes are, respectively, along the fibre direction and at right angles to it within the plane of the ply. The stress ( ), strain ( ) and stiffness ( ) of these directions can then respectively be given by [3]:

An orthotropic material’s Poison ratio can be defines as a stress in direction 1 that will result in a strain in direction 2. This relationship is shown by [3]:

The strain in the thickness of the lamina is given by [3]:

The following equations can be derived if the strain in direction 1 due to the loadings in all three the directions are considered [3]:

Similar results can be obtained for all the directions and this can all be arranged in a m by n matrix [3]:

{ } [ ]{ }

12

31

23

3

2

1

12

13

23

3322

23

11

13

33

32

2211

12

33

31

22

21

11

12

31

23

3

2

1

1

100000

01

0000

001

000

0001

0001

0001

G

G

G

EEE

EEE

EEE


12

The S matrix is symmetric for elastic materials so that [3]:

7.2 Orthotropic properties for in plane stresses From these equations a set of equations can be derived for an engineering structure of a laminate with the assumption that the laminates are thin in direction 3 and thus that [3]:

Thus obtaining the matrix for strains:

12

2

1

12

2211

12

22

21

11

12

2

1

1

100

01

01

G

EE

EE

And the stress matrix:

12

2

1

66

2221

1211

12

2

1

00

0

0

Q

QG

QQ

With [Q] as:

12

2112

22

2112

1121

2112

1121

2112

11

00

011

011

G

EE

EE

Q

7.2.1 Transformation of Coordinates

The local stresses of direction 1 and 2 can be written in terms of global x –y stresses. These equations can be derived using the same method as for the Mohr circle:

xy

y

x

T

12

2

1

With

22

22

22

sincoscossincossin

cossin2cossin

cossin2sincos

T

( 7: 8)

( 7: 9)

( 7: 10)

( 7: 11)

( 7: 12)


13

The inverse relation is given by

12

1

Txy

With

22

22

22

1

sincoscossincossin

cossin2cossin

cossin2sincos

T

The same transformation can be used with strain components if one bears in mind that the tensor definition of strain is the engineering strain divided by two. It is thus possible to show that

xy

y

x

xy

y

x

Q

with

44

66

22

6612221166

3

662212

3

66121126

3

662212

3

66121116

4

22

22

6612

4

1122

44

12

22

66221112

4

22

22

6612

4

1111

cossincossin22

cossin2cossin2

cossin2cossin2

coscossin)2(2sin

cossincossin4

sincossin)2(2cos

QQQQQQ

QQQQQQQ

QQQQQQQ

QQQQQ

QQQQQ

QQQQQ

The significance of the different Q terms lay in that they determine the coupling between

the different axes. 11Q Signifies that there is coupling between strain in the x direction and

stress in that direction. 12Q Show that if there is strain in the y direction there will be a

stress component in the x direction. This is clearer if the previous equation is written as:

xyyxxy

xyyxy

xyyxx

QQQ

QQQ

QQQ

662616

262221

161211

( 7: 13)

( 7: 14)

( 7: 15)

( 7: 16)

( 7: 17)


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7.2.2 Thermal Moisture effect

The moisture and temperature changes of a composite material can also induce stresses and strains on the material. The coefficients for directions 1 and 2 of moisture are given by (α1, α2) and the thermal by (β1, β2). Deriving the following equations:

0

22

11

12

2

1

12

2

1

mT

mT

S

This can be written only for the temperature effects, as:

T

T

T

Q

xyxy

yy

xx

xy

y

x

7.3 Methods for obtaining lamina properties To perform the analysis on the lamina, the properties of the lamina need to be obtained. These properties can be obtained by two methods, first experimentally or theoretically.

The stiffness properties that need to be obtained are the following

Young’s modulus in the 1 direction E11,

Young’s modulus in the 2 direction, E22,

the in plane shear modulus G12,

One of the in plane Poisson ratios v12 or v21.

7.3.1 Experimentally

All the values can be obtained by normal tension tests with unidirectional test samples in the 0°, 90° and 45° which are equipped with strain gauges (Black blocks) as illustrated in the following figures:

( 7: 18)

( 7: 19)


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Figure 8: 45 angle ply laminate showing the stresses and strains. [6]

E11 and v12

As the stress in uniaxial in the fibre direction the modulus is just the ratio of the stress to the strain in the 1 direction. Poisson ratio v12 is simply the ratio of the strain in the 2 direction to the strain in the 1 direction -ε2/ ε1.

E22

Again the modulus is just the ratio of the stress to the strain in the 2 direction.

G12

The response to this test can then be used to calculate the value of G12 as follows:

1

12

21

12 2114

4

EEEE

G

x

Ex

1

2

1

2

1 2

( 7: 20)


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7.3.2 Theoretically

The simple rule of mixture models can be used to obtain the stiffness properties of a composite, provided that the mixing ratios are known. Say Vf and Vm are the volume fractions of the fibre and resin respectively, and then the properties can be obtained as follows:

m

m

f

f

m

m

f

f

mmff

mmff

G

V

G

V

G

E

V

E

V

E

vVvVv

EVEVE

121212

2222

121212

11

1

1

This method, however, is not as easy to use as the experimental method, as the volume fractions needs to be theoretically correct. Obtaining of this volume fractions can be obtained by measuring the amount of resin before and after the lay-up was done and subtracting the amount left. The fibre volume is a bit more difficult, but can be obtained by weighting the fibres before and after lay-up. The weight ratios will be known after this and if divided by the density given on the datasheets, the volume fraction can be determined.

8 Laminate Analysis In the previous chapter relationships were developed for the stress and strain of a single lamina made from an orthotropic material. In this chapter these relationships will be extended to a laminate made from a number of lamina. Each of the lamina can be orientated in a different direction and thus contributes differently to the stiffness of the laminate.

8.1 Deformation due to extending and bending Considers a flat plate made up of a number of thin individual layers. These layers are bonded perfectly together so that when it is deformed the layers do no slip over each other. When this plate deforms, the Kirchhoff –Love hypothesis assumes that there is no through the thickness deformation and normal to the centre line remains normal after deformation. The deflection of the plate can then be considered to be a combination of two modes. This first is axial deformation or extension and the second is bending.

( 7: 21)


17

Axial deformation is handled in the normal way. When the plate bend, the deflection of the centre line of the plate form an elastic curve with radius ρ as shown in Figure 3.1 The inner surface of the plate in bending experience compression while the outer skin experience compression. The strain in laminate is a function of the curvature and the position form the neutral axis as shown in the Figure 3-2.

The total strain is thus the extension strain plus the strain due to bending. This can be calculated for both the x and y directions as well as for the shear deformation. It can be written in matrix notation as follows:

xy

y

x

xy

y

x

z

z

xy

y

x

0

0

0

0

8.2 Force and moment resultants The next step in the development of a method for laminate analysis is to relate the external forces and moment to the stresses experienced by the laminate. In this context the stresses will refer to the average stress integrated over the thickness of the laminate due to an applied force per unit width. A similar interpretation will be given to the moment resultant. The laminate axis is illustrated in Figure 10 along with the three membrane stress resultants Nx , Ny , Nxy.

Figure 9: Laminate axis and stress resultants [2]

Consider only the x direction. It follows from equilibrium that the applied force result in an internal moment in the laminate which is simply the force divided by the area as shown by the following equation:

( 8: 1)


18

2/

2/

h

h

xx dzN

The laminate thickness is given by (h). This result can again be extended to all the directions and also developed for the moments as given in the following equations:

The integrals can simply be obtained by summing the integrals over each ply. Remember that z is defined from the centre of the total laminate in the positive y sense. The definition of the layers are given in the following diagram

Figure 10: Notion for laminate thicknesses and ply location

8.3 ADB matrices The last step in the development is to relate the force and moment resultants to the strain in the laminate. This relationship is defined by the matrices called the ABD matrices and is given by the following equation:

0

DB

BA

M

N

A B and D are each a 3x3 matrix defined as follows:

( 8: 2)

( 8: 3)

( 8: 4)


19

3

2

3

1

3

1

2

1

2

1

1

1

kkN

kkijij

kkN

kkijij

kk

N

kkijij

hhQC

hhQB

hhQA

It follows that each of the ABD matrix components are dependent on the Q matrix for the specific layer. That is, the contribution of the stiffness of each layer is added to the stiffness of the complete laminate. The ABD matrices describe the coupling between the applied forces and moments and the deformation behaviour of the laminate. This is more clearly visible if the matrixes are written out better as in the following equation

xy

y

x

xy

y

x

xy

y

x

xy

y

x

dddbbb

dddbbb

dddbbb

bbbaaa

bbbaaa

bbbaaa

M

M

M

N

N

N

666261666261

262221262221

161211161211

366261666261

262221262221

161211161211

It follows from the equation that the [B] matrix gives coupling between plane deformation {ε} and bending deformation {κ}. If {B} =0 then there will be no coupling. The elements of the [A] matrix give the coupling between the different extensional deformation element, εx, εy and γxy.

The elements of the [D] matrix give the coupling between the elements of the bending deformation.

( 8: 5)

( 8: 6)


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Figure 11: ABD matrix description to deformation

8.3.1 Laminate code

A system was devised to allow the quick and clear representation of the composition of a laminate. In this system the direction of each ply is specified in degrees. A slash divides the different layers. A numerical subscript indicates the number of layers with the same orientation. The letter s indicates that the laminate is symmetric about this point.

[02/90/45/90] Two layers at 90° followed by a layer at 90°, one at 45° and a

final layer at 90°. [90/45/0]s 90°, 45 °, 0° 0°,45°,90°

8.3.2 Special laminates

Laminates can be designed so that they have certain desirable properties. Two special laminates are the so-called symmetric and balanced laminates.

8.3.3 Symmetric laminates

A symmetric laminate is one which is symmetric to the mid plane of the laminate. This has the effect of letting the [B] matrix be zero. As was shown in section 3.3 this is a very desirable condition as it prevents coupling between the extension strains and the bending strains.


21

8.3.4 Balanced laminates

In a balanced laminate there are an equal number of plies in any give +θ direction as there is in the -θ direction. This results in the A16 and A26 terms being zero with the result that the in-plane extension and shear responses are uncoupled.

8.3.5 Quasi-isotropic laminates

This layup is used to simulate a normal anisotropic material where there is no change of properties with direction. This can be achieved with both the following laminates: [0/45/-45/90] s and [0/60/-60] s These laminates are used where there is not a clear knowledge of the direction of the loading. These laminates are also often called “black aluminium” for they can substitute normal materials without redesign.

8.4 Stresses in the plies The stresses in the individual plies can easily be obtained with the previous equations. For a normal analysis the laminate layup is normally known and the stresses due to external forces must be computed. This can be done as follows: Firstly calculate the ABD matrices and obtain the inverse of the ABD matrix. This can then be used to get the centreline strains and curvatures for the laminate.

M

N

DB

BA10

The strain distribution can then be obtained for each layer from Equation 3-1 as follows:

z 0

These strains are the mid plane strains if the value of z is for the mid plane. Strain values at the top and bottom of each ply can also be calculated if the stress variation over the ply is of interest. The strains can then be transformed from the xy system to the 1-2 system with the previous coordinate transformations after which the stress in each layer can be calculated from the stress by multiplying it with the [Q] matrix.

9 Static strength and life of composites Advanced fibre composites have very good strength to weight ratios and are often used in strength critical applications. They also display very good fatigue properties. This chapter will first look at the static strength and failure criteria after which some fatigue properties will be discussed.

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22

9.1 Fibre properties The strength of a fibre composite is largely determined by the properties of the fibre. Fibres alone have extremely high tensile strengths with values between 1.8 and 4 GPa. If the fibre and matrix combination is however considered, the strength values are much lower, with values between 200 MPa en 1200 Mpa. This is due to the matrix which is very much weaker than the matrix. The strength is therefore directly related to the volume fraction of the fibres in the composite. A maximum fibre volume fraction of around 50% is attainable with hand lay-up and about 65% with machine impregnation. Inclusions of air bubbles and impurities, which are difficult to avoid, also serve to reduce the strength of the composite.

9.2 Tensile failure and compressive failure The maximum tensile strength of the composite is largely determined by the tensile strength of the fibres. The inclusion of air bubbles etc. during the manufacturing period lowers the strength of the composite, but the matrix also serves to strengthen it. The maximum tensile strength of the fibres depends on surface defects on a micro scale which create stress concentrations in the fibres. The statically distribution of these flaws, in a fibre bundle, results in the levelling of theses stress concentrations. The matrix therefore serves to bridge adjacent flaws from fibre to fibre and thus strengthening the composite.

Figure 12: Fibre pre-curve in woven fabrics

During compression the matrix supports the fibre bundles against buckling. With straight fibre bundles such as roving’s in a spar cap the compressive strength approaches the tensile strength, but it is much lower with woven fabrics. The reason therefore is that the weave pre-curve the fibre bundles which lowers the fibre buckling stress. The twill weaves, where each bundles passes under every second transverse fibre bundle has a higher compressive strength than the conventional weave patterns. It was found that the fibre is sometimes the limiting factor in compressive strength. The relative low compressive strength of Aramid fibres are due to the fact that the fibres fail laterally, without micro buckling being the main mechanism. The adhesion between the fibre and the matrix is also an important consideration in compressive strength. If the adhesion is low the matrix cannot fully support the fibres.


23

9.3 Failure criteria There exist a number of failure criteria that is applicable to composite materials. They are more complex than for isotropic materials due to the fact that the material properties of composites vary in the different direction.

9.3.1 Maximum stress

The maximum stress criterion states that the stresses in each of the main fibre directions must be lower than the maximum allowable stress in that direction.

121222112211 SSSSS cccctttt

9.3.2 Maximum strain

The maximum strain criterion states that the strain in each of the main fibre directions must be lower than the maximum allowable strain in that direction.

121222112211 SrSrSrSrSr cccctttt

9.3.3 Tsai-Wu

The Tsai-Wu is a multi-axial stress criterion and is similar to Von Misses for isotropic materials. It was however specially developed for composite materials. Tsai-Wu states:

ctctctct SSF

SSF

SF

SSF

SSF

FFFFFF

22

22

22

22

12

66

11

11

11

1

12662112

2

22222

2

11111

1111111

12

9.4 Fatigue life As fibre composites are a relative new material there was very little known about the fatigue life. The development of high performance glider and wind turbine blade from composites has led to the development of fatigue data for glass fibre. It was found that the 45° plies in the web of a beam is more sensitive to fatigue failure than the flange material. Fatigue data was therefore developed for 45° glass fibre plies. The following diagram gives the SN curve for this situation.

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24

Figure 13: S-N Curve for 45° glass fibre

The two curves represents stress ratios of R=0.1 and R = -1. That is the ratio of the maximum strain over the minimum strain. The y-axis of the curve shows the strain in a percentage.

10 Beam analysis Fibre composites are often used in beam element, due to the common nature of beams in structures. Due to the orthotropic nature of composites, classical beam theory alone cannot be used in the design and analysis of beams from composite materials. This chapter will extend basic beam theory to be applicable to composite materials. Beams are structural units that are designed to support a bending moment. Bridges and aircraft wings are typical beam structures.

10.1 Beam fundamentals The basic fundamentals for isotropic beams serve are the same for composite beams. These following assumptions are mad in the analysis of beams:

Moment loading is one dimensional

Axial force is zero

Plane sections remain plane during deformation The following figure give the basic sign conventions used for beam analysis.

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6 7 8 9 10Log N

Str

ain

in

45

° fi

be

r (%

)

R=-1

R =0.1


25

Figure 14: Sign convention used for beam analysis

The deflection equation for a beam with these sign conventions are thus:

beamMdx

wdEI

2

2

The curvature of the beam is given by

2

2

x

wx

so that the moment-curve relation becomes

The axial stress distribution in the beam is given by the classical beam equation

I

My

This formula does not hold for composite beams due to fact that it is based on a uniformly varying stress distribution. The stresses in composite beams can be very non-uniform due to the changes in material properties from layer to layer. The following sections will now develop expressions for the stiffness and stress distributions in composite beams.

10.2 Symmetric rectangular beams Consider a beam with rectangular cross-section as shown in the figure. The beam is constructed with layer upon layer of composite materials.

beamx MEI

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( 10: 4)


26

Figure 5-2: Rectangular laminated beam

Assume that the lay-up is balanced and symmetric. This will prevent distortion and temperature induced deformations. For a balanced layup, A16, A26 and D16 and D26 varnish. It is also assume with composite beams, as with isotropic beams that the strain distribution is linear. Due to the different properties of the different layer, the stress distribution is not linear. The element on lamination theory and beam theory must therefore be combined to be able to predict the stress distribution in a composite beam. The first step is to consider Eq. 3.6 without the zero terms.

xy

y

x

xy

y

x

D

M

M

M

][

Note that the moment results in this equation are unit based and must be multiplied by the beam width to get the total moment as used in beam theory.

bMM xbeam

We can now differentiate between cases, which are narrow and wide beams. The difference lies in the transverse distortion of the beam due to Poison effect. A wide beam shows this effect only at the outer edges while a narrow beam shows in more pronounced. Mathematically the effects can be described by setting the transverse moment My = 0 for the narrow case and the transverse curvature κy = 0 for the wide case. This gives the following results. Narrow beams My = Mxy = 0

0

0][ 1

x

xy

y

x M

D

z

Mid plane

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27

1

11

1

11

D

bEI

EID

bbMM

eff

xxxbeam

Wide beams κy = κxy =0

xy

y

x

y

x

DM

M

][

0

11

11

bDEI

EIbDbMM

eff

xxxbeam

The value of EIeff can now be used to solve for the deflection in fibre composite beams. The curvature and strain in the x-direction are calculated from

xx

beamx

z

EI

M

The transverse strains can be calculated from the preceding equations. Narrow beams

yy

xxy

z

D

DMD

1

11

1

211

21

Wide beams

yz 0

The strains can then again be transformed into the fibre directions for the plies of interest.

10.3 Laminated I-beam Structural shapes like I-beams can easily be made from composite materials. It is also very easy to tailor the lay-up of the different components be well suited to the stresses experienced there. It will therefore be sound to use a largely unidirectional lay-up in the

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28

flanges where axial stresses are experienced, with a 45° Bi-directional lay-up in the web to counter the shear stresses. It is thus necessary to develop a beam model for I-beams. The following figure shows the layout of a composite I-beam.

Figure 5-3: I-beam cross section

10.3.1 Symmetric Narrow flanges

For symmetric narrow flanges, the stress resultant Ny =0. In addition the My =0. The expression for the effective EI is given by:

z

EI

M

DA

bA

hEI

xx

eff

beam

flange

flange

f

web

eff

0

,111

11

2

1

1

11

3

212

10.3.2 Stress distribution in the web

The web experience three different loads. These are the axial load due to flange deformation, shear load and a compressive load due to the resultant of the axial loads in the flange.

Figure 15: Wings being load tested with bags of cement to apply the load.

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29

COMPRESSIVE LOAD

The bending of the spar results in a compressive load on the web. This can be calculated from (VDI 2013) as:

2

2

m

flangebeam

Ah

MP

The unit of PA is N/m and can be used directly in LAP (Laminate Analysis Program; See SWP 49 on Composite design software).

SHEAR LOAD

The maximum shear load can be calculated with three different methods. These are normal beam theory, laminate analysis combined with beam theory and the height approximation. Beam theory

It

yVA

Nxy = tweb web

Laminate analysis

a. Calculate the shear stress distribution in the web due to the shear load can be

calculated with the following equation as shown by Swanson (1997,221)

1444

22

*

1124

1FF

bh

AEIt

V f

webeffw

-The position in the flange where the shear stress is to be calculated

b. The shear stress can now be used to calculate the stress with respect to the

fibre directions. The stress resultant Nxy = tweb web. This value can now be used

in LAP to calculate the ply stresses.

Height approximation Nxy = Vweb/hm The first and last method was used in this calculation. The largest value for Nxy was then use for design purposes. The following figure shows all the loadings on the web just below the flange.

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30

11 Conclusion This SWP explained the entire basic theory involving composite design and how to retrieve specific information experimentally and theoretically. This SWP is followed by SWP 48, which give case specific guidelines which should be enforced when designing any part.

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