FUNDAMENTALS OF COMPOSITES MANUFACTURING: Materials, Methods and Applications, 2nd Edition

Chapter 10

Design BY RIKARD BENTON HESLEHURST What you will learn/overview (bullets)

x That design of composite structures and components is complex, but with a very good understanding of the principle characteristics and issues good structures and components will result.

x That a good understanding of the influences and issues of the individual constituent materials

(fibre and resin) is needed to produce good designs.

x That attention to detail in design and manufacture is needed to achieve the desired outcomes in behaviour and performance.

x The design and manufacture (fabrication) are intimately involved with good composite structure

design.

x There are many more variable in the design of composite structures that need to be understood and addressed.

Nomenclature 1,2,3 Laminate Orthogonal Directions A Cross Sectional Area A Extensorial Stiffness a Panel Length B Extensorial/Flexural Coupling Stiffness b Panel Width C Generalized Stiffness CTE Coefficient of Thermal Expansion D Flexural Stiffness E Young’s Modulus E’ Storage Modulus E” Loss Modulus f Fibre G Shear Modulus J Polar Moment of Area k Element stiffness k Radius of Curvature L Element Length M Edge Moment per unit width m Matrix m,n Integers N Load per unit Width P Applied Load Q In-plane Ply Stiffness

QI Quasi-isotropic R Panel Aspect Ratio R Tube Radius t Thickness T Torque T Transverse Direction V Volume fraction W Weight x,y,z Ply Orthogonal Directions 'T Temperature Difference G� Deformation H Strain I Angle of Twist K Damping Loss Factor P Poison’s Ratio U Density V Axial Stress W Shear Stress Z Natural Vibration Frequency

Contents

A. Methodology of Composite Structure Design,

B. Basic Stress Types,

C. Laminate Theory,

D. Rule of Mixtures,

E. Modeling and Finite Element Analysis,

F. Lay-up Notation,

G. Symmetric and Balanced Laminates,

H. Cracking in Composites,

I. Vibration and Damping,

J. Smart Structures,

K. Fatigue,

L. Comparison To Metals,

M. Residual Stresses

N. Case Study

A. Methodology of Composite Structure Design The basis of the composite structure design methodology can be undertaken by a series of development procedures. These development procedures are outline in the following flowchart, Figure A-1. A brief description of each step follows the flowchart.

Figure A-1: Composite Design Methodology Flowchart

Step 1 – Design Specification is the most important of the steps in design. The development of a thorough and detailed Functional Specification will always save time and money. The end result is a produce that works the first time and meets the customer requirements. This is a very crucial step in composite structures as there are many more ways of doing the design the wrong way when compared with traditional materials, such as metals. A very good approach to developing the design functional specification is the Quality Functional Deployment (QFD) method. Step 2 – Materials and Manufacturing Selection is like opening Pandora’s box. There are many fibre and resin materials to select from, as well as fibre/resin system combinations. Most material vendors have a preferred list of the fibre/resin system combinations to guide this selection, but the first requirement is the fibre type, fibre form and resin type for the structure. Typically the fibre selection is either a glass or carbon type depending on the application and of course cost. Other fibres should be investigated where

Design Specification

Materials & Manufacturing Selection

Initial Sizing (Design Charts)

Detail Sizing

Design Review

1. External Loads 2. Geometric Constraints 3. Environment 4. Stability/Stiffness 5. Etc.

1. Manufacturing Process 2. Material Type 3. Material Form 4. Etc.

Design Detail

Design Review

Optimization

1. Joints 2. Sandwich Structure 3. Cuts-outs 4. Damage Analysis 5. Repair Design 6. Interlaminar Stresses

Detailed Design

Initial Costing

necessary. The selection of either unidirectional (tape, rovings, etc) or woven cloth is next. The selection of the fibre form is generally based on either fabrication convenience or shape complexity. Noted that there is a major impact on the strength-to-weight ratio and stiffness-to-weight ratio when selecting cloth over tape forms. Resin selection is typically an operational environment issue. The operational environment, including vapour/fluid absorption characteristics, temperature range (hot and cold) and the corrosive condition, dominate the resin selection. Also give heed to the fibre/resin interfacial issues with respect to the environment. The other part of the step is manufacturing choices. This part of the selection process comes down to what you already have available usually, and time/cost of manufacturing the part. Also impacting the manufacturing process selection is quality control of the product performance requirements and the number of parts to be produced (refer to Step 1). Step 3 – Initial Sizing of the composite structural is used to determine the number of plies required in the various orientations. This step can be done using simple sizing charts as shown in Figure A-2 from the work of John Hart-Smith (Boeing). Here for example when the laminate axial stiffness or strength requirement is known as a ratio of the single ply equivalent property, then the percentage of plies in the 0, ±45, and 90 degree directions can be determined. WARNING – do not rely on these results to finalize a design as other factors need to be considered when the final configuration is set. Note that this is just one of several types of design charts to be used to initially size the composite structure. Step 4 – Detailed Sizing of the composite structure requires a computer program to develop the engineering properties of the composite laminate. When determining the in-plane properties a relatively simple program can be used, including ones that run as a spreadsheet. But this step must be done to evaluate the complete in-plane stiffness properties and estimate the strength of the laminate in the principle directions. These simple programs also will estimate many of the other physical and mechanical properties, such as Poisson’s ratio and the coefficient of thermal expansion. The analysis should also give an estimation of the initial failure condition (referred to as First Ply Failure) and ultimate strength of the structure. In detailed sizing the stacking sequence of the plies becomes important particularly with estimation of the flexural rigidity (bending) resistance and strength characteristics of the laminate. Step 5 – First Design Review is used to check on the development process of the laminate design. Here the outcomes of the design, the mechanical and physical properties, are compared against the design functional specification. Here the rationale of meeting, exceeding or falling short of the requirements can guide the designer in changing either one or all of the materials used, manufacturing process selected or the actual ply orientation and stacking sequence chosen. Step 6 – Initial Costing of the product can now give a good idea if the project is financial worthwhile. The previous steps defined the volume of the structure, the materials to be used and the manufacturing method employed. With this information, knowledge of the consumables to be used, and the production quantity the cost per part can be estimated. Compare this with the other competing products and the Functional

Figure A-2: Laminate Simple Sizing Chart (courtesy Hart-Smith, 1989)

Specification to determine project continuation. Step 7 – Design Detail covers a multitude of activities such as joint design, inclusion of holes and cut-out, flexural stiffening with cores, investigation of potential interlaminar stress areas and the effect of damage and repairability of the design. Such activities will lead to local ply build-up or thinning where required. Good stress analysis techniques, applicable to composite materials, must be used to complete this phase of the project design work. Step 8 – Second Design Review considers the detailed design work just completed. This review further assists in considering improvements to the design of the composite structure in meeting the design goals with a cost reduction. Typical cost reduction is achieved through reduced materials usage, learner manufacturing steps and faster part fabrication. Step 9 – Optimization of the structure follows the second design review. Here small, incremental changes can be made to individual ply materials, material form, ply orientation and stacking sequences, etc. Note that it is strongly recommended that you only change one aspect and re-run the analysis. Making too many changes at once will not indicate which parameter change was contributing to the overall improvement or degradation of structural performance. Step 10 – Detailed Design typical means some form of complex stress analysis such as finite element analysis. This detailed design analysis will look for further structural improvements without cost gains. Note that detailed stress analysis of composite structures requires significant processing time. This is especially so if the analysis is to be done thoroughly.

B. Basic Stress Types The basic stress types to be considered in composite structure design are:

1. Axial and Transverse In-plane Tensile Stresses (1-2 plane) 2. Axial and Transverse In-plane Compressive Stresses (1-2 plane)

3. In-plane Shear Stresses (1-2 plane)

4. Interlaminar or Through-the-Thickness Stresses (1-3 and 2-3 planes)

Figure B-1 defines these directions.

Figure B-1: Stress Designation

Axial and Transverse In-plane Tensile Stresses. The in-plane tensile stresses in a composite structure act in the principal directions of the laminate. These stresses are predominately reacted by the fibres in the 0 degree and the 90 degree laminate directions. The axial stresses are primarily supported by the 0 degree plies, or the warp fibres in a woven cloth ply. The 90 degree plies, or the fill direction for woven cloth, support the transverse stresses. These stresses are designated V1 and V2. Axial and Transverse In-plane Compressive Stresses. The in-plane compressive stresses in a composite are essentially the same as the in-plane tensile stresses. However, whilst the fibres take the compressive stresses, the resin plays a much more important role in compression. The fibres have a tendency to

3

2 1

V1 V2

V3

W12

W13 W23

microbuckle and it is the role of the resin (matrix) to support the fibres and resist the fibre microbuckling failure mode. The compressive strength properties are therefore controlled by the resin. When the resin is operated at a high temperature or has a degree of absorbed moisture the resin becomes more rubbery and the maximum compressive strength will reduce. The compressive stresses are designated -V1 and -V2. In-plane Shear Stresses. The in-plane shear stresses are designated W12. Based on the well known Mohr’s circle of stress transformation a pure shear stress field can be transformed into the biaxial principal tensile and compressive stresses along the ±45 degree lines in the 1-2 plane. Hence, we can see that the ±45 plies play a crucial role in supporting the shear stresses in a composite structure. Interlaminar Stresses. The interlaminar stresses are also known as the out-of-plane or through-the-thickness stresses. There are three interlaminar stress components; the interlaminar normal stress V3, and the two interlaminar shear stresses W13 and W23. For conventional two-dimensional composite structures with the fibres in the 1-2 plane the interlaminar stresses are typically taken by the resin. Since the resin is a relatively weak material in the composite structure there is much effort in designing the laminate to reduce the interlaminar stresses as much as possible. Interlaminar stresses are typically concentrated at free edges and where there are ply variations as shown in Figure B-2.

Free Edge

Notch (Hole)

Ply Drop-Off

Bonded Joint

Delamination Buckle

V

W W

3

13 23

Figure B-2: Out-of-Plane Stresses Induced in Composites Structures

C. Laminate Theory The basic theory of laminate analysis is referred to as Classical Laminate Plate Theory. The theory is essentially based on the development of the engineering and physical properties of a ply based on the fibre angle to the principal loading direction. Then simply summing the properties in a particular direction with respect to the ply thickness and dividing by the total thickness. If you assume that for a high performance carbon fibre unidirectional composite then the ply transverse and shear properties are approximately 10% of the axial ply properties, Figure C-1. Hence, the more 0 degree plies the greater is the contribution to strength and stiffness in the axial direction.

Figure C-1: Ply Property Changes with Ply Orientation (Courtesy Hart-Smith, 1989)

Generally speaking a unidirectional ply or woven cloth with the principal loading direction aligned to the ply fibre direction behaves in an orthogonal manner. That is to say the material behaviour has mutually perpendicular degrees of symmetry. When the ply axis of symmetry is not aligned to the loading axis the ply behaves anisotropically. This means that the deformation behaviour of the ply is influenced by both in-plane extensorial and flexural forces. When anisotropically behaving plies is subject to in-plane loads it will twist and bend, as well as have in-plane deformations. By laminating with specific ply orientations we can produce a laminate where the in-plane properties are uniform in all directions. Uniform properties in a material are termed as having isotropic behaviour. When the in-plane properties are uniform the laminate is referred to a quasi-isotropic laminate. These special quasi-isotropic laminate are when there are equal numbers of plies in specific directions, i.e. S/3 laminate [0/+60/-60], S/4 laminate [0/+45/90/-45], S/6 laminate [0/+30/-30/+60/-60/90], etc. The order is not important for the in-plane properties but as long as there is an equal number of plies in each direction for these types of laminates then the laminate performance will be quasi-isotropic. Pure isotropic behaviour (include the through-the-thickness properties) is achieved with fibres running in the 3-direction. This typically occurs with random chopped fibre composites. The overall in-plane behaviour of a structural laminate is a function of the in-plane stiffness and the flexural stiffness properties. The in-plane stiffness is referred to as the A-matrix or [A]. Where the flexural stiffness of a laminate is referred to as the D-matrix or [D]. In some laminate configuration there is coupling between in-plane and flexural behaviour and this results with non-zero terms in the B-matrix [B].

The in-plane or extensorial stiffness matrix relates the in-plane strains (axial, transverse and shear) of the laminate to the load per unit width:

1 11 12 16 1

2 21 22 26 2

6 61 62 66 6

N A A AN A A AN A A A

HHH

­ ½ ª º ­ ½° ° ° °« » ® ¾ ® ¾« »° ° ° °« »¯ ¿ ¬ ¼ ¯ ¿

The flexural behaviour relates the radii of curvatures to the D-matrix to obtain the unit moments as:

1 11 12 16 1

2 21 22 26 2

6 61 62 66 6

M D D D kM D D D kM D D D k

­ ½ ª º ­ ½° ° ° °« » ® ¾ ® ¾« »° ° ° °« »¯ ¿ ¬ ¼ ¯ ¿

The extensorial/flexure couple stiffness relates the radii of curvatures with unit loads or the in-plane strains with unit moments by:

1 11 12 16 1

2 21 22 26 2

6 61 62 66 6

N B B B kN B B B kN B B B k

­ ½ ª º ­ ½° ° ° °« » ® ¾ ® ¾« »° ° ° °« »¯ ¿ ¬ ¼ ¯ ¿

and 1 11 12 16 1

2 21 22 26 2

6 61 62 66 6

M B B BM B B BM B B B

HHH

­ ½ ª º ­ ½° ° ° °« » ® ¾ ® ¾« »° ° ° °« »¯ ¿ ¬ ¼ ¯ ¿

As a total representation of the in-plane/flexural behaviour for given deformations we have (note that diagonal symmetry is assumed):

1 11 12 16 11 12 16 1

2 22 26 21 22 26 2

6 66 61 62 66 6

1 11 12 16 1

2 22 26 2

6 66 6

...

... ...

... ... ...

... ... ... ...

... ... ... ... ...

N A A A B B BN A A B B BN A B B BM D D D kM D D kM D k

HHH

­ ½ ª º ­ ½° ° « » ° °° ° « » ° °° ° « » ° °° ° ° ° « »® ¾ ® ¾

« »° ° ° °« »° ° ° °« »° ° ° °« »° ° ° °¯ ¿ ¬ ¼ ¯ ¿

For greater detail in the application of the classical laminate plate theory refer to Tsai and Hahn, 1980, Tsai, 1988, and Jones, 1975.

D. Rule of Mixtures The rule of mixtures approach is a useful method in determining engineering and physical properties of plies in a laminate. The rule of mixtures approach is based on knowledge of the fibre and resin properties and the volume of fibres and resin that make up a ply. The principle term used in the rule of mixtures approach is the Fibre Volume Ratio (Vf). That is the physical volume of fibres to the total ply volume. The total expression is: Vf + Vresin + Vvoid = 1 The following provides some of the engineering and physical properties that can be determined using the rule of mixtures approach. A more extensive listing can be found in ASM Handbook - Composites, 2001.

Density U = Vf Uf + Vm Um Longitudinal Stiffness (Modulus) E = Vf Ef + Vm Em Poisson's Ratio

� X�= Vf Xf + Vm Xm � Xf is difficult to measure

Shear Modulus

G =

»»¼

º

««¬

ª�

m

m

f

f

GV

G

V1

Tensile Strength (Longitudinal) V = Vf Vf + Vm Vm Transverse Modulus

1 1y

my z

mf

f

EE EEVE

ª º§ ·« »¨ ¸� �

¨ ¸« »© ¹¬ ¼

Interlaminar Shear

Gyz=

»»

¼

º

««

¬

ª

¸¸

¹

·

¨¨

©

§��

yzf

mf

m

GG

V

G

11

Tensile Strength (Transverse)

VyT= � �

»»

¼

º

««

¬

ª

¸¸

¹

·

¨¨

©

§���

yf

mff E

EVV 11 VmT

where: f - fibre m – matrix (resin) Notes: 1. The void content is assumed to be zero. 2. Voids reduce the fibre volume ratio 3. Voids are a result of the manufacturing method and the cure process used.

In some cases the fibre or resin property can not be easily determined. However, by measuring the ply property and determining the fibre and resin volume ratios, the individual fibre or resin properties can be estimated. A complete listing of the various testing methods for composites materials can be found in ASTM Standards and Literature References for Composite Materials, 1990.

Generally there is a difficulty in determining the measured value of the fibre volume ratio in a fabrication process. Here the weight ratios of the fibre and resin are used, but weight ratios can not be used directly to determine the engineering properties. To determine the fibre volume ratio from weight ratios of the fibre

and resin you need knowledge of the fibre and resin densities of a ply. The densities can be found in the MSDS or MDS documentation. If the individual fibre and resin densities are not known, but the ply density and fibre volume ratio is provided, then use: U = Vf Uf + Vm Um. The expression that determines the fibre volume ratio from the fibre and resin weight ratios is (Heslehurst, July 2006):

1

resin

resin

11ffibre

fibre

VWW

UU

�ª º« »« » �« »§ ·§ ·« »¨ ¸¨ ¸¨ ¸« »© ¹© ¹¬ ¼

where W is the weight of the fibre and resin systems

An example of the properties of some common composite systems is shown in Table D-1.

Table D-1: Common Composite System Fibre/Resin Properties

Type CFRP BFRP CFRP GFRP KFRP BFRA

Fibre T300 Boron(4) AS E-Glass Kevlar 49 Boron

Matrix N5208 5505 3501 Epoxy Epoxy Al

Fibre Relative Density 1.750 2.600 1.750 2.600 1.440 2.600

Matrix Relative Density 1.200 1.200 1.200 1.200 1.200 3.500

Uf/Um 1.458 2.167 1.458 2.167 1.200 0.743

Void Ratio, Vv 0.005 0.005 0.005 0.005 0.005 0.000

Fibre Volume Ratio 0.700 0.500 0.666 0.450 0.700 0.450

Matrix Volume Ratio 0.295 0.495 0.329 0.545 0.295 0.550

Composite Rel Density 1.579 1.894 1.560 1.824 1.362 3.095

E. Modeling and Finite Element Analysis The modelling of composite laminated structures varies from the relatively simple to very complex. All modelling of composite structure requires computer support due to the detail in the lamination of the individual plies. Each ply has its own individual mechanical and physical properties in 3-dimensions. However, some simple assumptions allow for less complex analysis to be performed and allows for initial sizing of composite structures to be performed. The assumptions that can be made for ease of analysis is that the structure is orthotropic and homogenous initially and that through-the-thickness effects are ignored. A slightly more detailed approach is to analysis the individual in-plane state of the laminate and uses these results to interpret the total structure performance. An expansion of this approach is to assume that the through-the-thickness performance is purely based on the resin properties. But for a complete analysis of the structure the 3-dimensional stress analysis of the plies must considered anisotropic behaviour of the laminate. A review of each approach follows. The in-plane orthotropic, homogenous laminate approach is the same analysis methods used with traditional isotropic materials. The basic assumption is that the entire laminate is homogenous thus the overall properties of the structure are smeared throughout the laminate. The individual ply stress state is ignored. However, the laminate does possess orthotropic properties in that the laminate properties vary along the principal axes. Note that the material in-plane axes of symmetry are aligned with the principal loading axes. Beware that the off-axis properties are not modelled well as they are typically assumed to vary linearly between the axes of symmetry, and not like that shown in Figure C-1. The next level of complexity in modelling composite structures is to look at the in-plane orthotropic behaviour on a ply-by-ply basis. One important aspect of this analysis is that the laminate as a whole is also orthotropic such that all angled plies come as a matched pair of + and – plies. This results in a balanced laminate that eliminates the in-plane anisotropic effects. Note that the out-of-plane deformations due to bending are not truly balanced. This is explained later in this chapter. The analysis interrogates the stress state of the individual ply and the combines all of the plies to provide the total laminate performance. This approach required detailed knowledge of the ply stacking sequence. The method also allows for the use of different fibre/resin layers or hybrid composite structures. One word of caution is that the laminate properties are typical a weighted average over the thickness and this can lead to erroneous laminate results when say using sandwich structures. The most complex approach is to use 3-dimensional anisotropic ply-by-ply analysis. The difficulty in using this approach is more related to what material properties are known. For example if the behaviour of a laminate in 3-dimensions was analysed for a given applied strain the following expression would be used. The stiffness matrix [Cij], where i = j = 1 … 6, needs to be evaluated. Assuming diagonal symmetry, i.e. C12 = C21, etc, this required 21 tests to be performed to provide these constant of proportionality.

1 11 12 13 14 15 16 1

2 21 22 23 24 25 26 2

3 31 32 33 34 35 36 3

4 41 42 43 44 45 46 4

5 51 52 53 54 55 56 5

6 61 62 63 64 65 66 6

C C C C C CC C C C C CC C C C C CC C C C C CC C C C C CC C C C C C

V HV HV HV HV HV H

­ ½ ª º ­ ½° ° « » ° °° ° « » ° °° ° « » ° °° ° ° ° « »® ¾ ® ¾

« »° ° ° °« »° ° ° °« »° ° ° °« »° ° ° °¯ ¿ ¬ ¼ ¯ ¿

However, when the plane of symmetry is assumed through the thickness (mid-plane symmetry or the ply is symmetrical through the thickness) i.e. parallel to the 1-2 plane, then the out-of-plane shear behaviour is decoupled from the in-plane behaviour and only 14 constants of proportionality need to be evaluated. 1 11 12 13 16 1

2 21 22 23 26 2

3 31 32 33 36 3

4 44 45 4

5 54 55 5

6 61 62 63 66 6

0 00 00 0

0 0 0 00 0 0 0

0 0

C C C CC C C CC C C C

C CC C

C C C C

V HV HV HV HV HV H

­ ½ ª º ­ ½° ° « » ° °° ° « » ° °° ° « » ° °° ° ° ° « »® ¾ ® ¾

« »° ° ° °« »° ° ° °« »° ° ° °« »° ° ° °¯ ¿ ¬ ¼ ¯ ¿

When there is three mutually perpendicular planes of symmetry then the shearing behaviour is decoupled from the normal stresses and the shear stresses in the three planes are independent then only 9 constants of proportionality need to be evaluated. 1 11 12 13 1

2 21 22 23 2

3 31 32 33 3

4 44 4

5 55 5

6 66 6

0 0 00 0 00 0 0

0 0 0 0 00 0 0 0 00 0 0 0 0

C C CC C CC C C

CC

C

V HV HV HV HV HV H

­ ½ ª º ­ ½° ° « » ° °° ° « » ° °° ° « » ° °° ° ° ° « »® ¾ ® ¾

« »° ° ° °« »° ° ° °« »° ° ° °« »° ° ° °¯ ¿ ¬ ¼ ¯ ¿

With in-plane (1-2 plane) behaviour the primary concern then the stress-strain behaviour is further simplified to:

1 11 12 1

2 21 22 2

6 66 6

00

0 0

Q QQ Q

Q

V HV HV H

­ ½ ª º ­ ½° ° ° °« » ® ¾ ® ¾« »° ° ° °« »¯ ¿ ¬ ¼ ¯ ¿

, note that V6 = W12 and H6 = J12

where: Q11 represents the ply axial modulus (equivalent to the axial Young’s modulus) Q22 represents the ply transverse modulus (equivalent to the transverse Young’s modulus) Q66 represents the ply shear modulus Q12 = Q21 represents the Poisson’s ratio effect of transverse strain due to axial stress

3

1

2

3

1

2

3

1

2

Thus for the in-plane behaviour only 4 constants are required to be evaluated in tests, the axial and transverse Young’s moduli, the shear modulus and the Poisson’s ratio. Finite Element Analysis (FEA) allows very detailed stress and deformation solutions to complex shapes. The analysis is based on dividing the structure into a large number of small elements and assigning each element the appropriate material properties, then solving a very large number of simultaneous equations in the form of the load versus deflection relationship in terms of the known stiffness of the element. The element stiffness is based on the Young’s modulus of the material and the element cross-sectional area and length: P = kG where: P = applied or internal load G = deformation k = element stiffness = EA/L E = Young’s modulus A = cross sectional area L = element length The solution can be visualized as a plot of the stresses in a colour contour map on the structure and visualized deformation, as seen in Figure E-1.

a. Maximum Positive Loading b. Maximum Negative Loading

Figure E-1: FEA of WWII Fighter Wing Fabricated with Composite Materials In composite materials the main issue in the application of FEA is what material properties to use. The level of complexity is the ultimate choice. At the lowest level the structural element has global orthotropic properties. In other words the structural components use the laminate properties that have mutually perpendicular property variation. The through-the-thickness properties are still an issue here. The next level allows orthotropic properties of the individual plies with orthotropic properties and again the issue of through-the-thickness properties needs to be addressed. The sub-division of the plies requires a significant increase in the number of elements and processing time increases accordingly (more simultaneous equations to solve). The most complex approach, where

more time is required in undertaking the analysis and model building, is to include the full 3D properties of the individual layers and allow for anisotropic behaviour in the analysis. The closed form solution approach in the stress analysis of composite structures allows quicker results than with FEA. The closed form solution approach is not a accurate as FEA, but in many cases is within 5% of the more accurate solution. The closed form solution approach is excellent for undertaking the initial sizing stress analysis and then provides a good initial basis for further detailed stress analysis such as FEA. The only issue with the closed form stress analysis approach is that you have to have a very good understanding of the behaviour and anisotropic relationships in composite materials. The case study at the end of this chapter provides an example of a closed form solution approach. Further reading of the complex closed form solutions and FEA approach can be found in Whitney, 1987, Reddy J.N. & Miravete A., 1995, and Gibson, 1994.

F. Lay-up Notation The basic principal of lay-up notation is to ensure that the lay-up pattern achieved in fabrication of the composite laminate is the same as the engineering requirements. The engineering lay-up requirements are set from the engineering stress and stiffness calculations. The basic patterns of ply orientation can be illustrated with a simple diagram showing the ply principal directions in a laminate. The ply lay-up pattern will also aid in distinguishing between unidirectional plies and woven cloth. The basic reference for ply lay-up notation is taken from the Boeing Drafting Standard BDS-1330. However, there are a number of ways of specifying the ply lay-up patterns, just ensure that a key to the notation is provided and that there is logic and consistency with the notation used. To assist in determine the difference between plies laid up at angles to the principal direction we use what is termed the warp clock (with permission from Abaris Training, www.abaris.com). The recognized industry standard for use of this symbol states that the counter-clockwise (CCW) symbol will be used on the main views in the manufacturing drawing to show the relative fiber or warp yarn direction, normally with the 0q axis parallel to the primary load direction of the part-laminate or assembled structure.

The counter clockwise (CCW) warp clock is drawn from the manufacturing point of view, whereas the fiber angles are viewed from the inside of the panel looking toward the tool surface. This would be the case during lay-up. Notice that the 0º axis is parallel to the long axis of the rectangular shape and the +45º is located by moving in a counterclockwise direction from 0º to 90º.

The clockwise (CW) warp clock is drawn from the engineering-design standpoint, whereas the fiber angles are viewed from the outside of the structure, or from the tool surface, looking in. Notice that the 0º axis is parallel to the long axis of the rectangular shape and the +45º is located by moving in a clockwise direction from 0º to 90º. This symbol is a mirror image of the CCW warp clock.

0q

+45q

-45q

90q

0q

+45q

-45q

90q

Tool surface

Some basic rules for ply lay-up notation are:

x Plies of plus/minus orientation can be shown as:

o ±45 → [+45/-45] stack, or

o B 45 → [-45/+45]

x Multiple plies can be shown with a subscript: o [0/0/0/90/90] → [03/902]

x A lay-up with mid-plane symmetry uses a subscript outside the brackets:

o [0/0/90/90/0/0] → [02/90]s

x Symmetry with odd number of plies:

o [0/0/90/0/0] → [02/ 90 ]s

x Combined tape and fabric:

o [0/±45fabric/0] → [0/+45F/0] ,

o Noting that the warp direction for the 45o fabric is to be orientated +45o to the laminate’s 0o axis.

x When the orientations of the warp directions of fabrics to the laminate’s 0o axis are optional:

o [0/±45F/90/(0 or 90)F]s

x When the warp direction of the angled fabric ply is essential:

o [0/+45F/-45F/0]s

o with warp parallel to +45° → +45F and

o warp parallel to −45° → −45F.

x For hybrid laminate lay-ups the individual plies are coded with the material of that ply:

o [(0 or 90)FFG/(+45/0/–45/90)C/(0 or 90)FFG],

o where FG – fibreglass and C – carbon.

For example is the following laminate lay-up sequence and notation:

[±45F FG/((0 or 90)F/02/90/±45/+45F)C]s

45 fabric (Fibreglass)

0 or 90 fabric

0 tape (Carbon)

0 tape (Carbon)

90 tape (Carbon)

+45 tape (Carbon)

-45 tape (Carbon)

+45 fabric (Carbon)

+45 fabric (Carbon)

+45 tape (Carbon)

90 tape (Carbon)

0 tape (Carbon)

0 tape (Carbon)

0 or 90 fabric

45 fabric (Fibreglass)

-45 tape (Carbon)

G. Symmetry and Balanced Laminates The basic lay-up properties of a composite laminate are defined by the symmetry about the mid-plane and the balance of any angled plies in the laminate. These properties of the lay-up configuration have major impact on the loading and deformation behaviour of the laminate. First the definitions:

x A symmetric laminate has the ply orientations mirror imaged about the structural mid-plane of the laminate as show below with unidirectional tape:

x The in-plane deformation balanced laminate is defined that for every +angled ply there

is a –angled ply. This definition does NOT include 0 and 90 degree orientations. By this definition a 45 degree orientated plain weave cloth is effectively balanced (assuming the same number of filaments in the warp and fill directions). Note that other weave patterns can have slight variations in the warp and fill directions so they are not really balanced by themselves.

x Out-of-plane behaviour is a function of ply position away from the mid-plane. To

achieve a flexural behaviour balanced laminate the angled plies must be co-mingled to have equal distances from the mid-plane. Generally speaking only the plain weave cloth 45 degree orientation plies achieve this, but not in a purest sense as warp/fill direction ply count can be different.

In terms of structure behaviour the following deformation phenomenon will occur:

x A symmetric laminate does not possess coupling between in-plane (extensorial) and flexural behaviour. Thus symmetric laminate show the extensorial/flexural coupling matrix as [B] = 0. Asymmetric cross-plies (0 and 90 degrees) result in B11 ≠ 0, B12 = B21 ≠ 0 and B22 ≠ 0. With the angled plies in an asymmetric configuration B16 ≠ 0, B26 ≠ 0 and B66 ≠ 0.

x In-plane balanced laminate show no coupling between in-plane axial deformation and

in-plane shear deformation. In other words, A16 = A26 = 0. However, this does not constitute flexural balance as D16 = D26 ≠ 0.

+45 -45

0 90 90

0 -45 +45

Structural Mid-plane

x What is termed a specially orthotropic laminate has flexural behaviour balance, so D16 = D26 = 0

Figure F-1 summarizes the behaviour of the laminates based on mid-plane symmetry and angled ply balance.

Figure F-1: Stiffness Coupling Phenomenon (Davis, 1987)

Tsai, 1988, covers many of these issues in detail.

H. Cracking in Composites There are three major categories of cracking in composite structures and each is illustrated in Figure H-1:

a. fibre fracture, b. fibre/resin interfacial fracture, and

c. resin or matrix cracking

a. Fibre Fracture b. Interfacial Fracture c. Matrix Cracking

Figure H-1: Categories of Cracks in Composite Structures Fibre fracture will most likely occur as final fracture in well designed composite structures. If fibre fracture occurs early in the composite fracture process then it is likely that the matrix is too compliant for the fibre system. For example if carbon fibres where incorporated in a sealant material and loaded to failure then the fibre would more than likely fail first. Interfacial fracture is a poor material selection and fabrication process problem. Sizing the fibre for the resin system inhibits the fracture between fibre and resin. Typically interfacial failures are an indication of poor surface preparation of the fibres. Often the interfacial failure mode can be associated with fibre/resin interfacial degradation due to the absorption of some fluid, i.e. moisture or hydraulic oils. The most common low load cracking mechanism in composite structures is matrix cracking. This is often what is termed ‘First Ply Failure’, where the matrix strength is exceeded. Matrix cracking is not typically an ultimate failure mode but the early stages of a fracture process. There are basically two types of matrix cracking; intralaminar and interlaminar or delaminations. Figure H-2 delineates

the difference between intralaminar cracking and interlaminar cracking of the matrix. By definition intralaminar cracks are matrix cracking within a ply and interlamination cracks and matrix cracking between adjacent plies.

a. Intralaminar Cracking b. Interlaminar (Delaminations) Cracking

Figure H-2: Types of Matrix Cracks in Composite Structures (from Agarwal and Broutman, 1990)

The Fracture Process. The fracture and cracking of well designed composite structures is first by intralaminar matrix cracks. The intralaminar matrix cracks will typically appear in a high concentration but often localised, see Figure H-3. These intralaminar matrix cracks will be seen as cracking parallel to the fibres in unidirectional composite plies. Intralaminar cracks generally initiate delaminations. The delaminations are the matrix cracks between the adjacent plies, Figure H-3. Finally fibres will fracture as the ultimate failure mode, Figure H-3. Note that in Figure H-3 that the various ply orientations define individual cracking mechanisms. The 90 degree plies will fail early by pure intralaminar cracks. The angled plies (in this case 45 degree plies) show all three failure modes of cracking in the sequences just described. The 0 degree plies will have intralaminar cracking and likely fibre fracture with little or no delamination occurring. Fabric composite structure will have limited widespread intralaminar cracking as the transverse fibre tend to minimise these matric cracks, but can have extensive delaminations and then finally fibre fracture. Figure H-4 illustrates the fracture of a composite structure made with 8-harness satin glass cloth and epoxy resin.

Intralaminar Cracks Intralaminar Crack

Interlaminar Crack

Figure H-3: Composite Structure Fracture Process

Figure H-4: Fabric Composite Structure Fracture

Intralaminar Cracks

Interlaminar Cracking

Fibre Fracture

Delaminations

Fibre Fracture

I. Vibration and Damping The vibration and damping of composite structures is another beneficial property over existing engineering materials. The vibration performance is enhanced by the stiffness properties (a fibre and ply orientation property) and the good damping characteristics are attributed to the polymeric matrix. This section comments on both the vibration characteristics (natural frequencies and modes of vibration), and the natural damping behaviour, particularly at high frequencies, of composite structures. As a simple example if we examine a flat rectangular plate of dimensions length a and width b. The plate thickness t is substantially smaller than the plate in-plane dimensions. The frequency response of the plate is an out-of-plane behaviour and thus is influence by three factors:

1. The plate aspect ratio R = a/b. 2. The plate density (U) and is thus influenced by the fibre and resin densities and the fibre

volume ratio (Vf).

3. The flexural stiffness or D-matrix [Dij] of the plate and thus depends on ply orientation and ply through-the-thickness position in the plate.

There are several modes of vibration in a structure and the mode shape is defined by integer values in the longitudinal direction (m) and the transverse direction (n) in the plane of the plate. The natural frequency over a range of vibration modes for a specially orthotropic laminated plate (i.e. the laminate is symmetric and specially balanced such that A16 = A26 = D16 = D26 = 0) is given by:

� �� �2

4 2 2 2 4 411 12 66 22

1 2 2mn D m D D m n R D n RRbSZ

U§ · � � �¨ ¸© ¹

from Whitney, 1987

Note for first mode of vibration m = n = 1. Hence, there are a lot of potential variables in the development of the plate natural frequency. The panel aspect ratio has a major role to play in determining the natural frequency of the structure, as does the panel density (controlled by fibre/resin type and the fibre volume ratio to some extent). However the key factor is the flexural rigidity of the panel and thus the position of the contributing plies. Damping of composite structures is more complex in analysis, but is fundamentally a function of matrix (resin), fibre volume ratio and ply orientation. The natural damping characteristics of composite structures are of great benefit in design. The damping properties can be tailored, as are most composite properties, though variations in the fibre and resin selection, fibre volume ratio and ply orientation. The most significant feature of the composite structure that contributes to the

damping properties is the viscoelastic behaviour of the resin. Figure I-1 illustrates the typical characteristic behaviour of viscoelastic materials. Here the increasing load will store strain energy (area under the curve). As the load is removed the deformation path is not similar to the load application path, hence the strain energy released is the area under the lower curve. The difference in energy stored and release is absorbed energy or energy loss. This concept allows some degree of analysis to be undertaken that will determine the damping characteristics of composite materials.

Figure I-1: Viscoelastic Behaviour of Polymer Materials The damping characteristic of materials is typically given as the loss factor (K), which is a function of the applied frequency. The loss factor is a ratio of the loss modulus (E”) and the storage modulus (E’), also a function of frequency. Both the storage and loss modulii are derived by relationship between the individual fibre and resin modulii and the fibre volume ratio. For example the longitudinal loss factor of a unidirectional ply can be estimated by:

� �mmff

mmff

VEVEVEVE

EE

)(')(')(")("

)(')("

1

1

1

11 ZZ

ZZZZ

ZK�

� from Gibson, 1994

Thus, using similar micromechanics expressions for the storage (stiffness) and loss modulii in the shear and transverse directions the loss factor can be calculated for a variety of laminate configurations. Note, the loss factor of a unidirectional ply is typically dominated by the stiff fibre, but for matrix dominated directions (45 to 90 degrees) the loss factor increases significantly and is dominated by the viscoelastic matrix material. Therefore, when attempting to design a highly damped structure a good proportion of ±45 degree plies is important.

J. Smart Structures Due to the fabrication process composite materials can easily allow the imbedding of other materials that make the structure inform or react in certain situations. With the imbedded devices the structure becomes ‘smart’. Essential the smart structure allows the structure to sense threats, heal itself or adapt in form and/or function. The application of smart structures falls into the following three categories: 1. Health monitoring:

a. Imbedded sensor can identify local damage; whether it is overload, extreme temperature or removed material.

b. Environmental degradation changes to mechanical properties can also be identifies in the smart structure.

c. Likewise changes to the physical properties through the absorption of fluids and gases can be identified by imbedded sensors.

2. Adaptive structures:

a. With the use of imbedded actuators the structural shape can be changed either by external signals or reactive feedback.

b. The various operation signatures (radar, infrared, visual, etc), can be modified by the imbedded sensor to change or reduce the signature of the structure.

c. The inclusion of electronics in the structural skin of a system will improve operational performance such as conformal antennas.

3. Life cycle improvement:

a. Active imbedded sensors can eliminate vibration through damping and thus extend the fatigue life of a structure.

b. The smart repair patch will indicate if the patch adhesion has degraded thus allowing the repair to be changed.

c. Self healing of matrix cracks where imbedded capsules break and fill the cracks in the resin material or rebond the fibre and resin.

A detailed discussion of smart structures can be found in Baker, Dunn and Kelly, 2004.

K. Fatigue The fatigue damage associated with composite materials is either a cracking of the fibres, cracking at the fibre/resin interface (hereafter called interfacial cracking) seen as fibre pull-out, or cracking of the resin system (referred hereafter as matrix cracking). Each of these failure conditions is typically associated with the number of fatigue cycles. Low cycle fatigue (high stress levels) typically results in fibre fracture and interfacial cracking and high cycle fatigue (low stress levels) will more commonly result in matrix cracking. Figure K-1 illustrates this issue of low and high cycle fatigue characteristics of composite materials.

Figure K-1: Low and High Cycle Fatigue Characteristics of Composite Materials

Low cycle fatigue is dominated by the matrix stress levels in composite materials and thus fatigue crack inspection is associated with looking for the small micro-cracking in the resin. The behaviour of composite fatigue cracking is very different to that in metals where small cracks can be critical in size. Well designed composite laminate can tolerate large matrix cracks and thus provide a higher level of damage tolerance that do metals. Figure K-2 highlights the difference between damage size and inspection requirements between metals and composites. Here we see the large damage size in a composite material before it becomes critical. A further comparison of the fatigue behaviour of composite to metals is shown in the typical S-N curves based on Specific Alternating Stresses (alternating stress divided by material density). This comparative view is shown in Figure K-3. From Figure K-3 we see the typically lower strength loss

of the composites. Of particular interest is that Boron fibre composites perform better than glass fibre composites. This can be explained in Figure K-4 where the global stresses are proportioned to the fibres and the resin (matrix) based on the ratio of the fibre and resin individual stiffness.

Figure K-2: Comparison of Fatigue Crack Behaviour

(redrawn from Jones, 1975)

FATIGUE CYCLES OR TIME

DA

MA

GE

SIZE

Initial Imperfections

Inspection Threshold

Fracture

InitiationPropagation

Critical Damage Size

Fracture

COMPOSITES

METALS

Figure K-3: Comparative Typical Specific S-N Curves

(redrawn from Jones, 1975)

K-4: Stress Distribution Effects Based on Fibre Stiffness vs. Matrix Stiffness

10 10 10 104 5 6 7

CYCLES TO FAILURE

100

200

300

400

500

600

700

ALT

ERN

ATI

NG

ST

RES

S/D

ENSI

TY

(ksi

/lb/c

u.in

)

S-Glass/Epoxy

Boron/Epoxy

Titanium 8-1-1

Alloy Steel 4130Aluminium 2024

The form of the fibre system also impacts the fatigue cracking of the matrix. Figure F-5 illustrates this effect. Cracking in the matrix increases when the fibres are not aligned with the loading direction. Thus unidirectional fibres perform better that woven fabrics and random fibre forms tend to have a great fatigue cracking problem. Included in this effect is the reduced fibre volume ratio with fibre form.

K-5: Stress Distribution Effects Based on Fibre Stiffness vs. Matrix Stiffness (redrawn from Agarwal and Broutman, 1990)

10 10 10 104 5 6 7

CYCLES TO FAILURE

10

20

30

40

50

60

70

ALT

ERN

ATI

NG

STR

ESS

AM

PLIT

UD

E (k

si)

Random

Non-woven 85% Unidirectional

Non-woven UnidirectionalNon-woven Cross-Ply

Woven Fabric

103 108

Non-woven bias ±5Þ

L. Comparison to Metals A direct comparison of composite properties to those of metals does not normally show the complete story. A better method of comparing the properties is to take advantage of the material densities. This shows the distinct advantage of composite materials over metals due to the lower densities of composite fibre/resin systems. Another issue to be addressed in the comparison of composites to metals is that there is a wide range of composite material properties with various fibre orientations. The comparative charts that follow show the composite system with all plies orientated in the 0 degree direction and the quasi-isotropic laminate configuration. The unidirectional composite represents the upper limit of the composite properties and the quasi-isotropic laminate represents the other practical lower limit. Figures L-1 to L-6 illustrates a range of properties that compare four typical composite systems with three common metals. The composite systems represented in the comparison are a glass/epoxy, kevlar/epoxy, graphite/epoxy and boron/epoxy fibre/resin systems. Aluminium alloy 7075-T6, titanium alloy 6Al 4V and stainless steel PH 17-4 are the common aerospace quality metal alloys to compare the composite systems with.

Figure L-1: Specific Tensile Modulus

There is a clear stiffness-to-weight advantage for the graphite and boron fibre/resin composite systems over metals for both the upper and lower limits of the material modulus property. Glass and Kevlar are on a nearly equal footing with metals with respect to specific modulus.

Figure L-2: Specific Shear Modulus The specific shear modulus does not do as well in the comparison for composite systems with metals. The shear properties are the lower limit with unidirectional fibre orientations and perform poorly against the metals. The quasi-isotropic laminates of graphite and boron are slightly better than metal in the shear modulus-to-weight ratio.

Figure L-3: Specific Tensile Strength

The unidirectional specific tensile strength performance of the composite fibre/resins clearly is superior than the metals and in particular the kevlar and graphite fibre composites systems. In the quasi-isotropic laminate configuration only glass, graphite and boron preform better than the metals for the tensile strength-to-weight ratios with graphite fibre systems being the best of the composites.

Figure L-4: Specific Compression Strength

We can note the reduce performance of glass and kevlar fibre systems under the specific compression strength behavior against metals. However, boron fibre composites have a clear compression property advantage over the entire range of practical compression laminate configuration properties.

Figure L-5: Specific Shear Strength

The shear strength performance property of most composite laminates is poor between the unidirectional and quasi-isotropic configurations. Shear strength of a composite system is only improved with more ±45 degree plies with the pure ±45 angled ply configuration giving the best shear strength properties. Only graphite and boron quasi-isotropic configurations show an on-par performance with metals.

Figure L-6: CTE Comparison Whilst the metals typical have a higher CTE value than most of the composite systems the extreme difference between the aluminium alloy and all the composite systems should be noted. The major issue that arise with the differences between the CTE’s of metals and composites is the residual stresses that arise in joint regions, particularly heat cured adhesive joints.

M. Residual Stresses Residual stresses in composite structure are observed at two levels. The individual ply can have residual stresses between the fibres and the resin (matrix), and there can be residual stresses between adjacent plies in the laminate. In both cases the residual stresses typically results in some form of warping of the composite structure. The residual stresses originate from thermal variation between the cure and operating temperatures of the laminate, and the difference between the coefficient of thermal expansion (CTE) of fibres, resin and each ply orientation. Moisture absorption can also lead to expansion variations in the resin and thus residual stresses in the laminate. On some occasions the fibres are pre-stressed prior to infiltration and curing of the resin, and this will induce residual stresses between the resin and the fibre at the interface. Residual Thermal Stresses The residual thermal stresses arise because of the variation between the CTE of the fibres and the resin, and the difference between the cure temperature and the operating temperature (referred to as the delta T or 'T). Initially the selection of the fibre/resin combination and the curing temperature needs to be considered so that thermal residual stresses do not cause fibre/resin interfacial shear failures. In unidirectional composite plies this fibre/resin CTE difference will produce in-plane direction variations of the ply CTE. For example, looking at four typical unidirectional fibre/resin systems the longitudinal (fibre direction) CTE is often much smaller than the transverse direction, see Table M-1:

Table M-1: Common Composite System Fibre/Resin CTE Properties (adapted from Tsai, 1988)

Type BFRP CFRP GFRP KFRP

Fibre Boron (4) Graphite E-Glass Kevlar 49

Matrix Epoxy Epoxy Epoxy Epoxy

Longitudinal CTE per deg F x 10-6 3.38 -0.17 4.78 -2.22

Transverse CTE per deg F x 10-6 16.83 15.61 12.28 43.89

Resin/Fibre CTE % Difference 80% 101% 61% 105% With such the large variations in ply orthogonal CTE values there can be a significant effect on laminate post-cure residual deformations. When a laminate does not have mid-plane symmetry the room temperature deformation of the laminate can be extreme, especially for thin laminates (less than 12 plies). Take for example a series of four ply laminates in carbon/epoxy unidirectional prepreg tape cured at 350 deg F. Consider the following lay-up configurations, symmetry and post-cured room temperature thermal stress relief shape, Table M-2.

Table M-2: Common Composite System Fibre/Resin CTE Properties

Configuration [0/0/0/0] [0/90/90/0] [0/90/0/90] [0/0/90/90] [0/0/90/0] [0/0/0/90]

Symmetric Yes Yes No No No No

Post-Cured Shape Flat Flat Slightly warped

Warped

(oil canning)

Significantly warped

Grossly warped

Whilst the symmetric panels are flat they still have residual stresses within them. However, the residual stresses are balanced in the symmetric lay-up. If the laminate [0/90/90/0] is split along the mid-plane the two panels are no longer symmetric and will be warped. Likewise if the laminate [0/0/90/90] is split along the mid-plane the two panels are individually symmetric and thus are flat. Review Figure M-1 to understand his behaviour. This behaviour has a very important impact on repair using pre-cured or co-bonded repair patches.

Figure M-1: Effect of Thermal Stresses in Symmetric and Asymmetric Laminates

0

0

9090

0

0

9090

0

0

9090

00

9090

00

9090

00

9090

STRESS FREE STATE (Cure Temperature)

CONSTRAINED STATE (Room Temperature)

UNCONSTRAINED STATE (Room Temperature)

H

Hx

y

V

Thermal Stress Couple Balanced

Warping Results to Relieve Thermal

Stresses

N - Case Study – Torque Tube Problem Description

A torque tube is to be designed from graphite/epoxy composite materials at minimal weight. The torque tube has a maximum diameter of 3” and is 4ft in total length between the end constraints. The tube centrally loaded at 1,350 in.lb ultimate design torque, see Figure N-1. The torsional deflection of the tube at design load is to be no greater than 1 degree.

Figure N-1: Torque Tube Geometric Description

Determine the following:

a. Tube wall thickness. b. Ply configuration.

As this design project is a prototype one of the simplest and cost effective manufacturing process available to you is by hand lay-up on a deflatable tubular mold. As a general rule the allowable design shear strain is restricted to 2,000 Pstrain, which reduces the chance of first ply failure condition in the tube as intralaminar matrix cracking. Now the basic design equations are:

Maximum Shear Stress PHW 000,2GJ

TRallowable d

where: T = 1,350 in.lb R = 1.5 in

Polar Moment of Area � �> @44

2tRRJ ��

S

T = 1,350 in.lb

I = 3”l = 4 ft

T = 1,350 in.lb

I = 3”l = 4 ft

Angle of Twist D1d GJTLI

where: L = 4 ft = 48 in

Therefore: 26 lb.in 10713.3)180/(1

481350 xx

xTLGJ d SI

A major step in the design of the tube is to estimate the shear stiffness (G). This can be done by first looking at the range of shear stiffness values for a given composite material. For a typical graphite/epoxy unidirectional prepreg composite material: G0 = 1.22 msi This is the lowest value of shear stiffness for a 0 degree ply G±45 = 7.55 msi This is the highest value of shear stiffness for a ±45 degree ply GQI = 4.39 msi This is the quasi-isotropic value of shear stiffness A relatively large value of shear stiffness is sort for this design, but we do not want to have all 45 degree plies in the configuration as there are often axial loads that will need 0 degree plies. Assume the value of the shear stiffness as a starting point: G = 6.0 msi The unidirectional axial modulus is E0 = 29.46 msi, which gives the ratio with design shear stiffness:

204.00

EG

Using Figure N-2, taken from Hart-Smith, 1989, requires the percentage of ±45 degree plies as 76%. From the angle of twist expression the effective tube thickness is determined:

� �> @ 46

44 619.010713.32

inG

xtRRJ �� S

Therefore with R = 1.5 in, then t = 0.03 in Assuming that a plies thickness is 0.005 in, then only 6 plies are required to achieve the design stiffness goal. We require 76% of the laminate to be ±45 degree plies, which means that 5 of the 6 plies is required. As the angled plies need to be balanced 6 plies are required. As a general design

rule there should also be at least one plies in every direction of the [0/90/±45] combinations the final laminate lay-up could be: [±45/0/±45/90/±45]T

Figure N-2: Angle Ply Percentage Chart This ply configuration is based on the following reasoning:

x The ±45 degree plies on the outer surface will maximize the shear stiffness. x The interspersion of the plies will minimize interlaminar stresses and reduce the risk of

delaminations. The detailed laminate analysis of the configuration thus produces the following laminate properties: In-plane Properties E1 = 8.33 msi G12 = 5.97 msi Q21 = 0.4848 D1 = 0.8866 x 10-6 /degF Flexural/Torsion Properties

0 20 40 60 80 100

Percentage ±45° Plies

0

0.1

0.2

0.3

0.4

0.5

Non

-dim

ensi

onal

ised

In-P

lane

She

ar S

tiffn

ess (

G/E

0 )

0.028

0.264

Eo = Longitudinal Young's Modulus for Unidirectional Tape Modulus

0 20 40 60 80 100

Percentage ±45° Plies

0

0.1

0.2

0.3

0.4

0.5

Non

-dim

ensi

onal

ised

In-P

lane

She

ar S

tiffn

ess (

G/E

0 )

0.028

0.264

Eo = Longitudinal Young's Modulus for Unidirectional Tape Modulus

E1 = 9.13 msi G12 = 6.07 msi Torsional Strength @2,000 Pstrain W12 = 11.94 ksi Wult = 26.15 ksi Since the design problem is a torsional issue then the torsional shear modulus of 6.07 msi meets the design goals. Now check the results against the design equations: Geometry: R = 1.5 in t = 0.005 x 8 plies = 0.04 in A = 0.372 sqin J = 0.8149 in4

Shear Stresses: ksixJ

TRapplied 48.2

8149.05.11350 W

%380%148.294.11

¸¹·

¨©§ � MofS

Twist: DD 175.08149.01007.6

4813506 d xxx

GJTLI

Hence, for strength and performance the design is successful. Other issues to consider further are:

x End fixtures and appropriate joint analysis x Load application fitting and appropriate joint analysis

x Areas of local reinforcement

x Damage tolerance

x Long term manufacturing approach

Chapter Summary The design of composite structures requires a very good understanding of the constituent materials (fibre and resin), their interaction (fibre volume ratio and interfacial strength), the effects of ply anisotropy and the effects of curing conditions. This is because the resulting design mechanical and physical properties are ultimately related to the fabrication process. When the composite structure is in its final fabricated form you obtain the mechanical and physical properties at the that time. This understanding, and the resulting structural behaviour, of composite structures allows them to be designed for their unique characteristics that can be used to overcome the limitations of traditional materials. An excellent example of this is the Grumman X-29 Forward Swept Wing demonstrator aircraft. A forward swept wing has several aerodynamics and performance advantages over aft swept wings. However, there are significant structural limitations due to an aeroelastic phenomenon call torsional divergence. Too overcome torsional divergence a forward swept wing configuration has to increase the structural torsional rigidity. This adds weight and not desirable for an aircraft. However, with the application of the unique deformation behaviour of off-axis composite laminates the adverse torsional deformation is overcome by opposing structural deformation.

To utilize the special feature of composite materials and fabricated structures … know your materials and processes.

Questions 1. State the basic stress types and indicate which stress types are the domain of the fibres in 2-

dimension composite structures? 2. Why is the elimination of interlaminar stress important to the design of composite structures? 3. Define symmetric and balanced laminate configurations. 4. Referring to the [A] and [B] matrices, if a laminate possesses mid-plane symmetry, but the

angle plies are not balanced what are the zero component values in the both matrices? 5. What is the main driving parameter in the determination of ply properties using the rule of

mixtures approach?

6. What are the benefits of analyzing composite structures in the 1-2 plane only? 7. Where is the most likely site of cracking in well designed composite laminates? If cracking is

an interfacial type what does this tell you about the fibre/resin selection? 8. What properties of a composite laminate provide the natural frequency and damping

characteristics? 9. How does fatigue loading affect the performance of a structure graphite/epoxy composite? 10. How are thermal residual stresses developed in a composite laminate? How can the thermal

residual stresses be eliminated?

References Agarwal, B.D. and L.J. Broutman, 1990, Analysis and Performance of Fibre Composites. 2nd edition ed. NY: John Wiley & Sons

ASM Handbook - Composites, 2001, ed. D. Donaldson. Vol. 21, Materials Park, OH: ASM International.

ASTM Standards and Literature References for Composite Materials, 1990, Philadelphia, Pa.: ASTM.

Baker, Dunn and Kelly, 2004, Composite Materials for Aircraft Structures, 2nd Edition, AIAA, Reston VA.

Davis M.J., April 1987, Mechanics of Composite Materials, International Conference and Workshop on Composites in Manufacturing, Melbourne.

Gibson R.F., 1994, Principles of Composite Material Mechanics, McGraw-Hill, NY.

Hart-Smith L.J. , 1989, A New Approach to Fibrous Composite Laminate Strength Prediction, Douglas Paper 8366, presented to MIL-HDBK-17 Committee Meeting, Singer Island, Palm Beach, Florida.

Heslehurst R.B., July 2006, Composite Structures Engineering Design vs. Fabrication Requirements, Invited Paper, ACUN-5: International Composites Conference, Developments in Composites: Advanced, Infrastructural, Natural, and Nano-composites, UNSW Sydney, 11-14 July 2006.

Jones R.M., 1975, Mechanics of Composite Materials, Scripta Book Company, Washington, DC.

Reddy J.N. & Miravete A., 1995, Practical Analysis of Composite Laminates, CRC Press, NY.

Tsai S.W. & Hahn H.T. , 1980, Introduction to Composite Materials, Technomic, Lancaster PA.

Tsai S.W., 1988, Composite Design, 4th Ed., Think Composites, Dayton OH.

Whitney J.M., 1987, Structural Analysis of Laminated Anisotropic Plates, Technomic Publishing Co., Lancaster PA.