Varying Thickness

7/24/2019 Varying Thickness

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FIBER REINFORCED COMPOSITE LAMINATE PLATES WITH

VARYING THICKNESSES

KHO BOON HAN

UNIVERSITI TEKNOLOGI MALAYSIA


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PSZ 19:16 (Pind. 1/07)

DECLARATION OF THESIS / UNDERGRADUATE PROJECT PAPER AND COPYRIGHT

Authors full name : ________________________________________________

Date of birth : ________________________________________________

Title : ________________________________________________________________________________________________

________________________________________________

Academic Session : ________________________________________________

I declare that this thesis is classified as :

I acknowledged that Universiti Teknologi Malaysia reserves the right as follows:

1. The thesis is the property of Universiti Teknologi Malaysia.2. The Library of Universiti Teknologi Malaysia has the right to make copies for the purpose

of research only.

3. The Library has the right to make copies of the thesis for academic exchange.

NOTES : * If the thesis is CONFIDENTAL or RESTRICTED, please attach with the letter from

the organization with period and reasons for confidentiality or restriction.

UNIVERSITI TEKNOLOGI MALAYSIA

CONFIDENTIAL (Contains confidential information under the Official SecretAct 1972)*

RESTRICTED (Contains restricted information as specified by theorganization where research was done)*

OPEN ACCESS I agree that my thesis to be published as online open access(full text)

860629-02-5521 DR AHMAD KUEH BENG HONG

KHO BOON HAN

29thJUNE 1986

FIBER REINFORCED COMPOSITE LAMINATE PLATES

WITH VARYING THICKNESSES

SEMESTER II 2009/2010

19 APRIL 2010 19 APRIL 2010

Certified by :

SIGNATURE SIGNATURE OF SUPERVISOR

(NEW IC NO. /PASSPORT NO.) NAME OF SUPERVISOR

Date : Date :


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I hereby declare that I have read through this project thesis and to my opinion this

thesis is adequate in term of scope and quality for the purpose of awarding the degree

of Bachelor of Engineering (Civil).

Signature :

Supervisor : DR. AHMAD KUEH BENG HONG

Date : 19 APRIL 2010

_________________________


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FIBER REINFORCED COMPOSITE LAMINATE PLATES WITH

VARYING THICKNESSES

KHO BOON HAN

A report submitted in partial fulfilment of the

requirements for the award of the degree of

Bachelor of Engineering (Civil)

Faculty of Civil Engineering

Universiti Teknologi Malaysia

APRIL, 2010


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I hereby declare that all materials presented in this thesis are

the results of my own research except as cited in the reference.

Signature : ______________________

Name : KHO BOON HAN

Date : 19 APRIL 2009


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For my beloved family


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ACKNOWLEDGEMENT

Completing the final year project isnt an easy task; it is the support and

encouragement of a number of people that had driven me towards accomplishing this

study.

First of all, I would like to express my immense gratitude to my supervisor,

Dr. Ahmad Kueh Beng Hong, for willingly being my supervisor and sharing his

skills, thoughts, and experiences in the topic of my study. Thank you for being very

understanding and keeping us students relaxed in times of troubles.

Next, my profound thanks to three dear friends with whom I have the great

luxury while doing the project, Seh Wai Wai, Soh Eng Pang and Sim Siang Kao. In

addition, my deepest appreciation to family and friends to whom I seek guidance, I

am ever so grateful and thank to you guys.


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ABSTRACT

This main aim of the research current study is to develop a MATLAB

program on Fiber Reinforced Composite (FRC) laminates and to investigate how

mechanical loading would affect the stress ratio and the stress & strain distribution of

the FRC. The program is verified by comparing the computed values with the

literature. The research was carried out by comparing the performance of isotropic

control material extracted from an Optical Printed Circuit Board (OPBC) and FRL

with the same varying thicknesses. FRC is then rearranged in terms of orientation

and thickness for a parametric investigation. The FRC materials are limited to T-

300/3501-6 fiber/matrix, which is used as the reference material for comparison to

the OPBC material. The study of the FRL is also restricted to only three orientations

which are 0, 90, and 45. Analysis was carried out on five thickness

arrangements: isotropic arrangement, two symmetrical arrangements and two

balanced arrangements. Results showed that the symmetrical arrangement laminate

of varying thickness produces the least maximum stress ratio, and as long as it is

arranged in symmetrical orders, they would produce the same value of stress and

strain ratios under the same mechanical loading.


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ABSTRAK

Maklamat utama kajian ini adalah untuk menghasilkan satu program

bagi kajian Fiber Reinforced Composite (FRC) laminates dengan menggunakan

MATLAB, kajian dilakukan untuk mengkaji bagaimana beban mekanikal

mempengaruhi nisbah tegasan dan tegasan dan terikan bagi FRC. Program ini telah

dibuktikan betul dengan membezakan nilai kira dengan literasi. Kajian dijalankan

dengam pembezaan antara Optical Printed Circuit Board (OPBC)denganFRCyang

mempunyai ketebalan berbeza yang sama.FRCtelah disusun mengikut orientasi dan

ketebalan bagi tujuan invastigasi parameter ini. Material FRLadalah terhad kepada

T-300/3501-6 fiber/matrix, disamping itu kajian FRC terhad kepada tiga orientasi

iaitu 0, 90, dan 45. Analisis dijalankan untuk lima kumpulan ketebalan iaitu:

susunan isotropic, dua susunan symmetrical dan dua susunan balanced. Keputusan

telah menunjukkan bahawa susunan symmetrical dengan ketebalan berbeza akan

memberikan nisbah tegasan maksimum yang paling kecil, dan sekiranya disusun

dalam symmetrical, dengan beban mekanikal yang sama, ia akan mendapat

keputusan yang sama dari segi nisbah tegasan dan terikan


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TABLE OF CONTENTS

CHAPTER TITLE PAGE

TITLE i

DECLARATIONDEDICATION

iiiii

ACKNOWLEDGEMENTS iv

ABSTRACT v

ABSTRAK vi

TABLE OF CONTENTS vii

LIST OF FIGURES x

LIST OF TABLES xii

LIST OF SYMBOLS xiii

LIST OF APPENDICES xiv

1 INTRODUCTION

1.1 Background

1.2 Problem Statement

1.3 Objectives

1.4 Scope of Study

1

5

5

6

2 LITERATURE REVIEW

2.1 Introduction

2.2 Background Study

2.3 Previous Study

2.4 Conclusion

7

7

8

9


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3 METHODOLOGY

3.1 Introduction

3.2 Determination of Engineering Constants

3.2.1 Effective axial Modulus

3.2.2 Effective axial PoissonsRatio

3.2.3 Effective Transverse Modulus

3.2.4 Effective axial Shear Modulus

3.3 Determination of Stiffness for lamina, Q

3.3.1 Transform Reduced Stiffness

3.4 Determination of Laminate Stiffness:ABD Matrix

3.5 Determination of Laminate Stresses and Strains

3.6 Determination of Stress Ratio and Strain Ratio

10

12

12

13

13

14

14

15

17

18

19

4 RESULTS AND DISCUSSIONS

4.1 Introduction

4.2 MATLAB Program

4.3 Program Procedures

4.4 Verification of Program

4.5 Isotropic Control Material

4.5.1 Optical Printed Circuit Board (OPCB)

4.5.2 Limitations

4.5.3 Problems

4.5.4 Analysis of Control Material

4.6 Fiber Reinforced Composite Material

4.6.1 Varying Thickness of Laminate Study

4.6.2 Data Analysis

4.6.2.1 Isotropic Arrangement

4.6.2.2 Symmetrical Arrangement 1


4.6.2.4 Balanced Arrangement 1


20

21

23

28

32

32

34

34

34

38

39

41

41

47

51

54

56


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5 CONCLUSION AND RECOMMENDATION

5.1 Conclusion

5.2 Recommendation

60

61

REFERENCES 62


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x

LIST OF FIGURES

FIGURE

1.1

1.2

3.1

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

4.10

4.11

4.12

4.13

4.14

4.15

TITLE

Composite Laminate Consisting of Layers With Varying

Thickness.

Varying thickness laminate from an electron micrograph.

Flow chart of research methodology

Computation of ABD Stiffness matrix

Computation of Stress ratio and Strain ratio

ABDmatrix on the command window of MATLAB

Mid Strains and Curvature (Gibson, 1994)

Global Stresses (Gibson, 1994)

Mid Strain and Curvatures using MATLAB

The Stresses as calculated using the MATLAB program.

Layers on an OPCB

Stress Distribution of the Control Material

Strain Distribution of the Control Material

Stress Ratio Curve for Isotropic Arrangement Laminate

Composite

Stress Distribution for Isotropic Arrangement Laminate

Composite

Strain Distribution for Isotropic Arrangement Laminate

Composite

Stress Ratio Curve for Symmetric Laminate Composite

Improved Stress Ratio Curve for Symmetric Laminate

Composite

PAGE

3

4

11

21

22

27

29

29

30

31

33

37

38

45

46

46

47

49


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4.16

4.17

4.18

4.19

4.20

4.21

4.22

4.23

4.24

4.25

4.26

4.27

Stress Distributions for Symmetry Laminate

Arrangement 1

Strain Distributions for Symmetry Laminate

Arrangement 1

Stress Ratio Curve for Symmetrical Laminate

Arrangement 2

Improved Stress Ratio Curve for Symmetrical Laminate

Arrangement 2

Stress Distributions for Symmetry Laminate

Arrangement2

Strain Distributions for Symmetry Laminate

Arrangement2

Stress Ratio Curve for Balanced Laminate

Arrangement 1

Stress Distributions for Balance Laminate

Arrangement 1

Strain Distribution for Balanced Laminate

Arrangement 1

Stress Ratio Curve for Balanced Laminate

Arrangement 2

StressDistributions for Balanced Laminate

Arrangement 2

Strain Distributions for Balanced Laminate

Arrangement 2

50

50

51

52

53

53

54

55

56

57

58

58


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LIST OF TABLES

TABLE TITLE PAGE

4.1

4.24.3

4.4

4.5

4.6

4.7

Properties of typical fibers materials

Properties of typical Polymer Matrix MaterialsLayer Properties

Values of stress and strain of the isotropic control material

Local Stresses of Isotropic Arrangement

Local Strains of Isotropic Arrangement

Improved Local Stresses of Symmetrical Arrangement 1

24

2533

35

42

44

48


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LIST OF SYMBOLS

E

G

v

-

-

--

-

-

-

Stress

Strain

Degree of angleShear-Strain at plane

Shear-Strain at plane

Youngs modulus

Shear modulus

Poissons ratio


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LIST OF APPENDICES

APPENDIX TITLE PAGE

A

B

Program Code

Data of Local Stresses and Local Strains

63

65


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CHAPTER 1

INTRODUCTION

1.1Background

Fiber reinforced composites are the most widely used composite materials.

Composite laminates have many applications as advanced engineering materials for

components in aircrafts, power plants, civil engineering structures, ships, cars, rail

vehicles, robots, prosthetic devices, sports equipment and others. The main reason

fiber reinforced composites have been used in the industries is due to the advantagesof composites such as follow:

i. Improved strength

ii. Improved stiffness

iii. Corrosion Resistance

iv. Light weight

v. Good thermal insulation

vi. Better wear resistance

vii. Good fatigue life

Fiber-reinforced composites have come a long way in replacing conventional

materials like metals and woods. These types of composites are derived by

combining fibrous material, which serves as the reinforcing material that primarily


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carries the load in the composite, with a matrix material, which bonds the fibers

together, supports them and is responsible for transferring the load from fiber to fiber.

The purpose of combining materials in this manner is to achieve superior

properties and performance when compared to the individual materials. As truly

engineered materials, designers of composites can select the composition to generate

particular performance specifications based on individual application needs.

Depending on the placement of fibers, individual continuous fiber lamina or

ply are arranged in different direction and stacking sequence, which can be

controlled to generate a wide range of physical and mechanical properties for the

composite laminate. Various forms of composites can be produced, they include:

i. Continuous fiber composites

ii.

Woven fiber composites

iii. Chopped fiber composites

iv. Hybrid composites

v.

Sandwich structure

Comparing the fiber reinforced composite to other material such as steel,

where the steel is homogeneous and isotropic, the fiber reinforced composites behave

differently because of its heterogeneous and anisotropic behavior. Heterogeneous

and anisotropic behavior means the properties of a composite vary from point to

point and the properties will depend on the orientation of the reinforcement within

the material. Thus, the fiber reinforced composite can be described with the stress-

strain relationship.

There are many factors that can affect the change in stress and strain, and

hence the stress and moment of the laminates. This thesis will be focusing on one of

these factors that are the laminate with varying thickness individual layers. Thestudy on the laminate with varying thickness is important because different stacking


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sequence will give a wide range of engineering properties for the material. This

means that a laminate can be designed to have higher strength with light weight

compared to a laminate with uniform thicknesses. Figure 1.1 shows the composite

laminate consisting of layers with varying thickness.

Figure 1.1 Composite Laminate Consisting of Layers With Varying Thickness.

Some industries had been using the fiber laminate of varying thickness in

their design. For instance, the aerospace industry had been using the varying

thickness fiber composite in the design of the aerofoil wing to gain benefit like

lighter weight. Meanwhile, the wind turbine design had been using the concept as

well. In the building construction, the fiber reinforced composite of varying

thickness had been used in the design of bridge. In the electronic industry, thin

electronic plates consist of laminates of varying thickness are common. Figure 1.2

shows an example of varying thickness laminate captured on an electron micrograph.


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Figure 1.2 Varying thickness laminate from an electron micrograph.

In short, as todays need for stronger, lighter and cheaper material has

increased, especially in fiber composite material, a better understanding of theperformance of the composite material is needed.


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1.2Problem Statements

Unlike the steel material which is homogeneous and isotropic, the behaviors

of the fiber reinforced composite are relatively complex. This is because the

laminated composite is composed of more than one ply where each of the plies is

fiber reinforced, and can be stacked in various orientations as well as thickness.

Because of the thickness change and the different orientation of the

composite, each ply or lamina will have different engineering properties such as

elastic modulus, thermal expansion and Poissonsratio.

As many possible arrangements of composites can be done, dealing with

multiple layers of materials glued together, can be very complex as laminate

delamination would occur under extreme heat and mechanical loads. The

delamination of composite will cause the failure of the weakest ply and hence

reduced the strength and stiffness of material substantially. Therefore, it is also

important to study the stress & strain relationship of the materials in terms of heat

and buckling.

Due to various different parameters in each ply, the computation of stress &

strain of laminate is difficult to be done manually, thus a programming approach is

required in order to ease the computation of laminate properties.

1.3Objectives

The objectives of the study are:

i. To determine the laminate stiffness: the ABD matrix of the fiber

reinforced composite laminate with a varying thickness.


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ii.

To analyze and compute the stress and strain in the fiber reinforced

composite laminate with a varying thickness of different orientation and

arrangement.

iii. To produce the MATLAB program code for objectives (i) and (ii).

1.4Scope of Study

The current study is based on the classical laminate theory. The materials areassumed to have the linear elastic behavior. In analyzing the laminate, each lamina

is assumed as a transversely isotropic thin flat plate and is consisting of multiple

layers at predetermined orientation and thickness.

In this research, number of layers is based on an isotropic control material

called Optical Printed Circuit Board (OPCB) consisting of 8 layers; meanwhile the

orientation will be restricted at 0, 45 and 90 degrees. On the other hand, the laminateis limited to one carbon fiber material and one matrix material which are T-300 and

Epoxy (3501-6). Also, the plane stress condition is assumed since the thickness of

the laminate is many times order lower than its in-plane dimensions.


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CHAPTER 2

LITERATURE REVIEW

2.1 Introduction

Fiber reinforced composites had been used in industries particularly in the

aerospace, automotive industry as well as in the infrastructure or architecture

structure mainly due to its ever expanding advantages. This is due to the highly

anisotropic properties of the unidirectional fiber reinforced composites, which have

high stiffness and strength along the fiber direction and have very low stiffness and

strength in the transverse direction. When forming laminates, the fiber reinforced

composites in the form of plate will give maximum strength depend on how it is

modelled.

2.2 Background Study

As known, composite laminated plate are modelled using variable thicknesses,

shear deformable, finite plate elements. Fiber laminate of varying thickness has been

the subject of many studies and researches, for example several researches like Joshi

&Biggers (1995)had been carried out in optimizing the composite plates in terms of

buckling load in which buckling load is an important criteria in the design of

composite plates. Optimization of composite laminated plate can be done by

specifying the material to be used, the number of plies with an orientation and a plystacking sequence, as well as the varying thickness of lamina in laminated plate.


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Meanwhile, several researches such asAndrews&Massabo (2007) had stated

that delamination of composite is associated with the function of thickness of lamina.

Since the mechanism of delaminated composites is buckling out of plane of the

group of plies above and below the delamination, which causes the reduction of the

compressive strength. Therefore, the study of fiber laminate of varying thickness is

essential.

2.3 Previous Study

Joshi & Biggers (1995)have used a feasible direction method to determine

the optimal thickness distribution over the plate, for which the thickness distributions

that maximize the buckling load are determined. Their result showed that the

buckling loads through thickness optimization decrease as transverse shear effect

increase. This showed that thickness plays an important role in the composite design.

Another study on the thickness optimization is by Khosravi & Sedaghati

(2008). In their study, optimality criteria are presented for optimum design of

composite laminates. The thickness of the layers in each element is considered as the

design variables. In their study, the varying thickness composite structures are

fabricated by ply drops and splicing. Therefore, the optimization methods are

justified only for aerospace structure where stiffness and weight is not a primary

concern.

Zineb et al. (1997)had study the influence of varying thickness glass epoxy

composite plate under the pure bending moment. The global behaviour of the

varying thickness composite plate and the local stress concentration within the

composite material were investigated. In their research, the thickness variation is

obtained by different stacking techniques, and the stress states between the stacking

sequences are study.


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Wang & Karihaloo (2006) found that the transverse tensile and shear

strengths of a fiber reinforced unidirectional lamina when situated in an angle ply

laminate, are affected by the ply angle of neighboring lamina and the thickness of

lamina. Due to the weakness of lamina in its transverse direction, transverse cracks

are likely to occur under fatigue loading which will eventually cause the

delamination of the neighboring lamina. The research showed that varying thickness

of laminate is related to the delamination of composite material.

The delamination is a prevalent form of damage that occurs in the laminated

composites and layered materials, which is induced by impact loads or the results of

manufacturing defects. Andrews & Massabo (2007) had studied the problems of

delamination in plates with varying thickness in order to investigate their influence

on fracture as well as crack. It is shown that the main effects of thickness variations

are similar to those created by the interaction of delaminations.

2.4 Conclusion

The aforementioned researches had help building a better understanding of

the effect of varying thickness laminates. It can be concluded that the behaviours of

the fiber laminate with varying thickness are related to strength and stiffness of the

composite material. While depending on the stacking sequence of varying thickness

layers, laminate may also exhibit different response in terms of stress and moment.

Therefore, this research will focus on analysing the fiber reinforced

composite laminate of varying thickness where the stress and strain of laminate will

be computed using programming approach which based on the classical laminate

theory.


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CHAPTER 3

RESEARCH METHODOLOGY

3.1 Introduction

This chapter will describe the method used to analyze the fiber reinforced

composite laminate of varying thickness. In the current study, the software

MATLAB is used. It provides an easy way to analyze lamina and laminate of fiber

reinforced composite by programming the formulae commonly used. The activities

of the research are as shown in Figure 3.1.

The study begins with insertion of the engineering constants for matrix and

fiber material,Em, Ef, Gm, Gf, m, andfas input. The engineering constants for the

composite material are computed. The equations used are the Rule of Mixture and

the Halphin-Tsai equations. The effective axial modulusE1and effective Poissons

ratio, 12 are calculated with the Rule of Mixture while the effective transverse

modulus,E2and effective axial shear modulus, G12are based on the Halphin-Tsai

equations.

The engineering constants of composite are used for the calculation of

laminas stiffness matrix Q. Next the stress-strain relations from the local 1-2-3

coordinate system have to be transformed into the global x-y-z coordinate system in

order to get the reduced stiffness () for a lamina. With a set of predeterminedorientations and ply thicknesses, the ABD matrix or the constitutive relation of a

laminate can then be computed.


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Figure 3.1 Flow chart of research methodology

Macromechanics study:Computation of composite stiffness: theABDmatrix

Computation of Stress strain, and

From theA,BandDmatrix

Plot the stress-strain variation for different

Orientation and ply of varying thickness

Input and determination of engineering constants for

E1, E2, G12, 12using the Rule of Mixture and the

Halphin-Tsai Equations

Micromechanics study:

Computation of lamina stiffness Q

Input for number of lamina,

Orientation and thickness of lamina, and T


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In the post-processing stage, by applying a fix value of forces in one direction,

the stress and strain for the composite laminates and individual lamina can be

obtained from the ABD matrix. In the current study, isotropic material is used as a

control case. The difference between those with isotropic and fiber-matrix material

are made. The main concern here is to compare the through thickness stress strain

distribution of different set of composite plates

A study on the stress and strain ratio for different orientation such as

symmetry laminates and balance laminates is performed to check which stacking

sequence would give a lowest ratio. The ratio is calculated based on the through

thickness stress and strain which is the absolute maximum over the minimum value.

The laminate with the lowest stress and strain ratio is likely of an optimized type.

The study ends with the plotting of for the varying thickness laminate at lowest stress

ratio and strain ratio.

3.2 Determination of Engineering Constants

3.2.1 Effective axial modulus

1is the Youngs modulus in the direction of the fibers. A rule of mixtureexpression is used to approximate the prediction of

1. The equation of

1is given

by:

1 = ( ) + (3.1)

From the equation, it is shown that the axial modulus and the fiber volume

fraction is a linear relationship.


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3.2.2 Effective axial Poissons Ratio

12is also called as the major Poissons Ratio. It is obtained using the same

approach as 1. The equation is given by:12 = ( ) + (3.2)

It is very similar to the rule of mixture expression for

1in that it is linear in

all of the variables.

3.2.3 Effective Transverse Modulus

2is the Youngs modulus in the transverse direction of the fibers. As thegeneral approach to predict 2using Rule of Mixture is not very accurate comparedwith experimental results. Thus, a semi empirical model is developed by Halpin andTsai to improve the original models. The Halpin-Tsai equation of 2is given by:2/ = (1 + ) / ( 1 ) (3.3)where

= 1 + (3.4) is the curve fitting parameter, which is also a measure of the degree ofreinforcement of the matrix by the fibers.


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3.2.4 Effective axial Shear Modulus

12is the effective in plane shear modulus. Similar to transverse modulus, asthe general approach to predict 2using Rule of Mixture is not very accurate whencompared with experimental results. An equation for 12can be derived using thesame approach as the transverse modulus which is the Halpin-Tsai equation. The

equation is given by:

12/

= (1 +

) / ( 1

) (3.5)

where

= 1 + (3.6)

is the curve fitting parameter, which is also a measure of the degree ofreinforcement of the matrix by the fibers.

3.3 Determination of Stiffness for lamina, Q

Considering the case of plane-stress, it is assume that in the material

coordinate system, 3=4=5=0. With {}12 = [Q] {}12, and [Q] is termed as the

reduced stiffness matrix and is given by

= 11 12 012 22 00 0

66

(3.7)


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or

= 1/ 122/ 0122/ 2/ 00 0 12 (3.8)

where= 11221

3.3.1 Transform Reduced Stiffness

By rewriting {}12 = [Q] {}12 to account for the factor of in the shear strain.

1

2

12=

11 12 0

12

22 0

0 0 266 1

2

12/2 (3.9)

Alongside with transformation matrix [T] as defined by:

[] = 2 2 22 2 2 22 (3.10)where c= cos , ands=sin .

When equating equation for both shear and strain transformation, such that {} =[]1[]{} , we get:


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=

11 12 16

12

22

26

16 26 66

(3.11)

where

= []1[]

are called the transform reduced stiffnesses or the off-axis reduced stiffness and

are defined by:

11 = 114 + 2(12 + 266)22 + 224 (3.12)12 = (11+22 466)22 + 12(4 + 4) (3.13)

16 = (1112 266)3 + (12 22 + 266)3 (3.14)22 = 114 + 2(12 + 266)22 + 224 (3.15)26 = (1112 266)3 + (12 22 + 266)3 (3.16)

66 = (

11+

22

2

12

2

66)

2

2 +

66(

4 +

4) (3.17)

where c= cos , ands=sin .


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3.4 Determination of Laminate stiffness: the ABDmatrix

The in-plane forces per unit length and the moments per unit length of a laminate are

given as follows:

= [] []0 + [][] (3.18)and

= [] []0 + [][] (3.19)0 is the reference plane strain while the is known as the laminate curvatures.The two formulas would combine to give

= 0 (3.20)or in an expanded form,

=

11

12

16

12 22 2616 26 66 11

12

16

12 22 2616 26 6611 12 1612 22 2616 26 6611 12 1612 22 2616 26 66

0

0

0 (3.21)

whereA,BandDmatrices each is respectively given by equation 3.22 to 3.24:


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= [](=1 1) (3.22)

= 12 [](2=1 12 ) (3.23) = 13 [](3=1 13 ) (3.24)

The formulas are known as theABDmatrix: Ais the extensional stiffnesses,

B is the coupling stiffnesses, and D the flexural laminate stiffnesses. The ABD

matrix defines a relationship between the stress resultants which are applied to a

laminate, and the reference surface strains and curvatures.

3.5 Determination of Laminate Stresses and Strains

Once the ABD matrix are obtained, the general force deformation equation (3.20)

can be inverted to give:

0 = 1 =1 (3.25)The stresses of a laminate in the k-th lamina are given by:

= [][]0 + [] (3.26)The mid-plane strain []0and curvature []are those given in terms of laminateforces and moments by equation (3.25)


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3.6 Determination of Stress Ratio and Strain Ratio

The stresses and strains obtained from section 3.5 are known as the global stress and

strain, in order to gain the local stress and local strain, they are transformed using

equation (3.10):

Generally, stresses in the local or 12 coordinate system can be written as:

1

212 =

(3.27)

As soon as the local stress and strain for each lamina or layer in the laminate are

obtained, the stress and strain ratio can be computed.

=

(3.28)

= (3.29)


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CHAPTER 4

RESULTS AND DISCUSSION

4.1 Introduction

This chapter presents the code used for the analysis and the results. First of

all, in order to verify and make sure that the code is correct, verification with an

example in the book will be shown. Such a comparison is made to ensure that the

written program is error-free and can be used in the analysis.

Once the code is verified, an isotropic control material with predetermined

plies and varying thickness is first selected and the stress and strain curve of the

material is calculated and investigated.

The results of the control material will hence be compared to the composite

material made of fiber and matrix. The plies and number of varying thickness follow

that of the isotropic control material. During this step, different sets of laminate with

different orientation are set up, and their stress and strain would be calculated.

Also, in order to study the behavior of the varying thickness of laminate, the

thickness arrangement of the laminate would be rearranged into random, symmetry

and balanced laminate.


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At the end of the chapter, the stress ratio and strain ratio would be calculated

from the respective stress and strain curve of these sets of laminate. Then the best

and most suitable thickness arrangement laminate can be determined.

4.2 MATLAB Program

Program for the project is done using MATLAB version 7.6.0.324 (R2008a).

The program can be divided into 2 parts. First of all, the code starts from the initialstatement of the engineering properties of the material, and then the number of plies,

following with the insertion of fiber orientation and the thickness of the laminate, the

first step ends with the computational of theABDmatrix. All equations and formulas

used are based on those as described in Chapter 3. Figure 4.1 shows the parts of the

program where theABDmatrix are computed.

Figure 4.1 Computation of ABD Stiffness matrix


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The second part of the program takes the ABDmatrix obtained from the first

part as initial input. The program continues with the definition of the mechanical

load on the laminate. With the loading applied, the program calculates the lamina

stresses and strains globally and locally of the laminate and the lamina respectively.

From the stress and strain values, the program will then calculates the stress ratio as

well as the strain ratio. Finally, a stress and a strain profile through the thickness of

the laminate are plotted. Figure 4.2 shows the code for the plotting of the stress and

strain distribution and the calculation of ratio of stress and strain.

Figure 4.2 Computation of Stress ratio and Strain ratio


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The complete coding of MATLAB, it will be shown in the appendix.

Some limitation of the program so far is, only stress and strain in local 1-

direction will be plotted. Although the calculation of the stress X and stress Y as

well as the shear in the XY-direction can be calculated by using the program. This is

not the scope of study of the thesis. So they will not be provided.

4.3 Program Procedures

In order for the users to use and run the program, it is important for them to

know the steps required. The following paragraphs provide a guideline of the

program for the users.

First of all, the users need to know the properties of fiber and matrix they are

going to use for their composite. In order to ease the users, a list of typical fibers and

matrixes had been programmed in to the MATLAB program. Table 4.1 and Table

4.2 show the material properties for fiber and matrix respectively.


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Table 4.1 Properties of typical fibers materials

Property E-Glass S-Glass AS-4

Carbon

T-300

Carbon

Longitudinal

modulus,E1f, GPa

73 86 235 230

Transverse modulus,E2f, GPa

73 86 15 15

Axial Shear

modulus, G12f, GPa

30 35 27 27

Poissons Ratio, 12f 0.23 0.23 0.2 0.2

Property IM7 Carbon Boron Kevlar 49

Aramid

Silicon

Carbide

Longitudinal

modulus,E1f, GPa

290 395 131 172

Transverse modulus,

E2f, GPa

21 395 7 172

Axial Shear

modulus, G12f, GPa

14 165 21 73

Poissons Ratio, 12f 0.2 0.13 0.33 0.2


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Table 4.2 Properties of typical Polymer Matrix Materials

Property Epoxy

(3501-6)

Epoxy

(977-3)

Epoxy

(HY6010/HT

917/DY070)

Young Modulus,Em,

GPa

4.3 3.7 3.4

Shear Modulus, Gm,

GPa

1.6 1.37 1.26

Poissons Ratio, 12f 0.35 0.35 0.36

Property Polyester VinylesterYoung Modulus,Em,

GPa

3.35 3.5

Shear Modulus, Gm,

GPa

1.35 1.3

Poissons Ratio, 12f 0.35 0.35


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If the defined fiber or matrix is not on the list, the users can insert all the

values in order to run the program. These values as have been written in the Matlab

script are:

Once the properties of both fiber and matrix are provided, the engineering

constants of the laminate composite would be calculated, the engineering properties

areE1, E2, G12, 12. The formulae of the constants are as described in Chapter 3.

Next, the users are required to provide the number of layers of the laminate,

following with the orientation in degree from top to bottom of the laminate and the

thickness in mm from top to bottom of the laminate.

E1f=input('Value of fiber longitudinal modulus in GPa? -');E2f=input('Value of fiber transverse modulus in GPa? -');G12f=input('Value of fiber axial shear modulus in GPa? -');nu12f=input('Value of fiber poisson ratio? -');

Em=input('Value of matrix young modulus in GPa? -');Gm=input('Value of matrix shear modulus in GPa? -');num=input('Value of matrix poisson ratio? -');

%The number of layerslayer=input('The number of layars in the laminate? -')

%The orientation of each laminaAng=input('The orientation(degree) of each lamina,top to bottom ie[45 -45 45 -45] -')*pi/180;

%The varying thickness of each laminaThickness=input('The thickness(mm) of each lamina,top to bottomie[0.1 0.1 0.1 0.1] -')


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Aforementioned steps are the required input for the computation of the

stiffness matrix. The result of theABDmatrix will be shown in the Matlab command

window. Figure 4.3 shows an example of ABD matrix calculated by the program.

Figure 4.3 ABDmatrix on the command window of MATLAB

In the following procedure, users will be asked to insert the mechanical loads

on the laminate;

%Unit of force in MPalmtFrc=input('The loading(Mpa) acting on laminate, ie[50 0 0 0 0 0]-')';


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The first three values the users provide represent the forces in the x, yandz-

direction, while the others three values are the moment acting on the laminate aboutx

yandzaxes respectively.

After the values of the loading are inserted the program would then calculate

the stress and strain distribution for the laminate globally and locally along with the

plotting of the distribution. Finally, the program ends with the computation of the

stress ratio and the strain ratio.

4.4 Verification of Program

In order to make sure the code of the program are reliable and correct, an

example from the literature are taken, and compared the answer with the computed

result from the program as shown in Figure 4.3. The example is based on the

problem stated in the book by Gibson (1994), page 218. The question stating theantisymmetric angle-ply laminate subjected to force of 50MPa where the resulting

stresses in each lamina are to be determined. Figure 4.4 shows the calculated mid

strain curvature on the book. Figure 4.5 shows the stresses calculated from the book.

Figure 4.6 and Figure 4.7 are the results obtained using the program.


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Figure 4.4 Mid Strains and Curvature (Gibson, 1994)

Figure 4.5 Global Stresses (Gibson, 1994)


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Figure 4.6 Mid Strain and Curvatures using MATLAB


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Figure 4.7 the Stresses as calculated using the MATLAB program.

From the figures, it can be observed that the values obtained using the

MATLAB program are the same results as shown in the example from the book.

Thus, the anent program is verified and is used for the next set of analyses.


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4.5 Isotropic Control Material

In order to study the behavior and pattern of the fiber reinforced composites,

an isotropic control material will be used for the assessment of the performance of

the fibers and matrix laminate composite.

4.5.1 Optical Printed Circuit Board (OPCB)

Printed Circuit Board is used to mechanically support and electrically

connects electronic components using conductive pathways, tracks or traces etched

from copper sheets and laminated onto a non-conductive substrate.

For the purpose of the study, a specific type of OPCB was selected. The

OPCB consists of multiple layers of materials glued together and has a dimension of

5 cm x 5 cm.

Figure 4.8 shows the layout of the distribution of the OPCB while Table 4.5

lists the properties of each layers.


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Figure 4.8 Layers on an OPCB

Table 4.3 Layer Properties

Layer Material Thickness Youngs

Modulus

Coefficient of

Thermal

Expansion

Poissons

ratio

1 SU 8 50 um 2 Gpa - /C 0.22

2 Cyclotene 50 um 2.9 Gpa - /C 0.34

3 SU 8 50 um 2 Gpa - /C 0.22

4 Solder Mask

Laminate

70 um 4.1 Gpa 30 x 10- /K 0.4

5 Copper 30 um 110 Gpa 16.5 x 10-

/K 0.34

6 FR4 (epoxy

resin), glass

transition

temp 110 to

200 C

1 mm 17 Gpa 11 x 10-

/K

lengthwise

15 x 10-6

/K

cross wise

0.136

(lengthwise)

0.116

(crosswise)

7 Copper 30 um 110 Gpa 16.5 x 10- /K 0.34

8 Solder Mask

Laminate

70 um 4.1 Gpa 30 x 10- /K 0.4


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4.5.2 Limitations

While studying the isotropic control material, there are some limitations, first

of all the material for layer 6, the thickness will be changed from 1mm to 0.05mm,

and this is to make sure that the laminate materials can be arranged in symmetrical

and balanced forms. Meanwhile, only the mechanical loading of 100MPa acting in

the X-direction would be considered.

4.5.3 Problems

The main problem with the OPCB with varying thickness plies and is made

from different materials, is when exposed to high temperature. When the OPCB was

heated to high temperature, the strength of the polymer tends to decrease. This

lowers the adhesion strength and further causes delaminate within the material. Thus

the maximum stress occur in the layer will be obtained and studied. The stress ratioand strain ratio will be obtained next. The result from the analysis of the isotropic

control material (OPCB) will then be compared to the stress and strain distribution of

that using fiber reinforced composite materials.

4.5.4 Analysis of Control Material

Analysis was carried out using the same general MATLAB program. Since it

is an isotropic material, the engineering properties of fiber and matrix are not

available as preset. Various values of materials are manually key-in into the program,

so that the analysis can be run. The program for the isotropic control material

analysis will be shown in the appendix.


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The stiffnessABDmatrix is first obtained, and it follows with the application

of the mechanical load in the X-direction with value of 100 MPa. The stress and

strain values can then be obtained. Table 4.3 shows the values obtained in the global

X-direction. The values of theABDmatrix are also shown.

Table 4.4 Values of stress and strain of the isotropic control material

m Thickness Stress_X Mpa Strain

50

0 139.2 71.1304

0.05 117.5 60.0267

50

0.05 175.8 60.0267

0.1 143.3 48.9231

50

0.1 95.7 48.9231

0.15 74 37.8194

70

0.15 161.4 37.8194

0.22 95 22.2743

30

0.22 2474.3 22.2743

0.25 1733.9 15.6121

50

0.25 258.8 15.6121

0.3 74.7 4.5084

30

0.3 500 4.5084

0.33 -240.4 -2.1538

70

0.33 -9.2 -2.1538

0.4 -75.6 -17.6989


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9.3862 3.0304 0

A = 3.0304 9.3862 0 Gpa-mm

0 0 3.1779

0.6291 0.2069 0

B = 0.2069 0.6291 0 Gpa-mm2

0 0 0.2111

0.0756 0.0248 0

D = 0.0248 0.0756 0 Gpa-mm3

0 0 0.0254

It is important to know the value of ABD matrix, as they are used in the

laminate force-deformation equation as discussed in Chapter 3.

From the data obtained from the MATLAB program, through thicknesses

stress and strain distribution are plotted. It is important to take note that all the

values are known as local stress and local strain, although there is no transformation

in isotropic material and as such the value of global stress and strain would be the

same as local stress and strain.

Figure 4.9 and Figure 4.10 show the stress distribution and the strain

distribution of the OPCB material.


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Figure 4.9 Stress Distribution of the Control Material

From the plot of stress curve, it can be observed that, there is a huge different

between the minimum and maximum values of strength. The maximum stress

happened at thickness 0.22mm with the value 2474.3MPa while the minimum stress

at 9.2MPa at 0.33mm. This means that the stress ratio is 268.9, with the not evenly

distributed stress; plies of weaker layer would likely fail first, which later causes

delamination to the material.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-500 0 500 1000 1500 2000 2500 3000

Thicknessmm

Stress MPa

Stress Distribution


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Figure 4.10 Strain Distribution of the Control Material

As for the strain curve, it can be seen that the maximum strain is 71.1303.

The curve follows a straight line showing a typical characteristic to that of isotropic

material.

4.6 Fiber Reinforced Composite Material

Based on the varying thickness plies of the isotropic material, a composite

laminate will be analyzed, the analysis would be based on how well the composite

behave as compared to the isotropic control material. The aim here is determine

whether the isotropic is better or the FRC would produce better performance under

the same loading condition.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-40 -20 0 20 40 60 80

Thicknessmm

Strain mm/mm

Strain Distribution


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Since the FRC laminate have different kind of orientations, sets of orientation

of laminates are selected. Also the varying thickness of laminates is of main concern.

Hence, these sets of orientation of laminates will be rearranged in terms of thickness

arrangement. The current study restricts the set of laminates to 5 groups. These

include the laminate in random form which following the arrangement of thickness

of the control material, two laminates are rearranged in symmetry laminate and

another two in balanced laminate.

4.6.1 Varying Thickness of Laminate Study

In studying the varying thickness of laminate composites, the specifications

and limitation will be stated first:

i. Fiber volumet fraction is 0.6

ii. Fiber material is T-300

iii.

Matrix material is epoxy (3501-6)

iv. Orientation is fixed at 0, 45 and 90 degrees only.

v. Laminate thicknesses are 30mm, 50mm and 70mm following those of the

isotropic control material.

There are a total of five set of thickness arrangements. These arrangements

comprise of:

i. Isotropic Arrangement(As similar to the control material)

ii. Symmetrical Arrangement 1(Min-Max-Max-Min)

iii.

Symmetrical Arrangement 2(Max-Min-Min-Max)

iv. Balanced Arrangement 1(Min-Max-Max-Min)

v.

Balanced Arrangement 2(Max-Min-Min-Max)


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There are only 8 layers of laminas comprising two 0.03mm laminas, four

0.05mm laminas and two 0.07mm laminas. The isotropic arrangement laminate will

be based on the exact arrangement to that of the control material, which is:

[0.05, 0.05, 0.05, 0.07, 0.03, 0.05, 0.03, 0.07]

Arrangement Min-Max-Max-Min means that the arrangement is made in such

a way that the thicknesses follow the values of the magnitude of the thickness. They

are set from minimum to maximum values and again from the maximum values back

to the minimum, top to bottom. For example such arrangement is given as:

[0.03, 0.05, 0.05, 0.07, 0.07, 0.05, 0.05, 0.03]

while for the Max-Min-Min-Max arrangement, it would become:

[0.07, 0.05, 0.05, 0.03, 0.03, 0.05, 0.05, 0.07]

Next parameter in the study of varying thickness of laminate would be the

fiber orientation. As stated in the scope of study, only 0, 45 and 90 degrees are

concerned. The fiber orientation is made such a way that it can be based on the

thickness arrangement, take case 1 as example; the 0.03mm thickness laminas are

orientated to 45 degrees, while the 0.05 and 0.07mm thickness laminas are fixed at 0

degree. Now move on to the next cases, the condition is now 0.07mm laminas

orientated at 45 degrees while the others at 0 degree. For the study, there are a total

of 18 cases for isotropic arrangement and symmetrical arrangements, while for

balanced arrangements; there are only 12 cases of orientation arrangement.


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4.6.2 Data Analysis

4.6.2.1 Isotropic Arrangement

Table 4.5 shows the sets of data computed using the MATLAB program,

there are a total of 18 cases of orientation, for example, showing on top of the case1,

0.03@45, 0.05 and 0.07@0, meaning that, the laminas with 0.03mm thickness will

be orientated at 45 degrees while the laminas of 0.05 and 0.07mm fixed at 0 degree.

The highlighted grey show the maximum values.

All the values are the local stresses correspond to loading of 100MPa acting

at X-direction. When running the cases in MATLAB, the stress ratio will also be

obtained. A plot is also given at the end of the program. The details equations used

are shown in Chapter 3.


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Table 4.5 Local Stresses of Isotropic Arrangement

Stress Ratio

1: Isotrophic arrangement

0.03@45 0.03@0 0.03@0 0.03@45 0.03@45 0.03@0 0.03@90 0.03@0 0.03@0

0.05@0 0.05@0 0.05@45 0.05@45 0.05@0 0.05@45 0.05@0 0.05@0 0.05@90

0.07@0 0.07@45 0.07@0 0.07@0 0.07@45 0.07@45 0.07@0 0.07@90 0.07@0

m Thickness case1 case2 case3 case4 case5 case6 case7 case8 case9

50 0 243.9 186.1 61.5 70.1 84.7 121.9 243.6 180.7 0.6

0.05 255.4 239.3 58.4 64.6 217.8 108.7 256.2 239.1 -2.0

50 0.05 255.4 239.3 58.4 64.6 217.8 108.7 256.2 239.1 -2.0

0.1 266.8 292.5 55.2 59.0 351.0 95.5 268.8 297.5 -4.7

50 0.1 266.8 292.5 55.2 59.0 351.0 95.5 268.8 297.5 -4.7

0.15 278.2 345.8 52.1 53.5 484.1 82.3 281.3 355.9 -7.3

70 0.15 278.2 45.8 824.3 954.6 47.8 82.3 281.3 -16.9 922.9

0.22 294.3 41.9 591.7 724.9 48.7 63.8 298.9 -12.4 649.0

30 0.22 52.0 420.3 591.7 45.7 48.7 1226.6 -23.8 437.6 649.0

0.25 46.5 452.2 492.1 42.4 49.1 1041.3 -19.3 472.7 531.6

50 0.25 301.1 452.2 45.8 42.4 750.4 55.9 306.5 472.7 -12.6

0.3 312.6 505.5 42.7 36.9 883.5 42.7 319.0 531.1 -15.2

30 0.3 37.4 505.5 326.0 36.9 49.8 732.4 -11.7 531.1 335.9

0.33 31.9 537.4 226.4 33.5 50.2 547.1 -7.2 566.1 218.5

70 0.33 319.5 35.7 226.4 363.8 50.2 34.8 326.6 -5.3 218.5

0.4 335.5 31.8 -6.2 134.1 51.1 16.3 344.2 -0.8 -55.4

Stress ratio 10.5 16.9 134.0 28.5 18.5 75.3 47.9 684.7 1467.5

0.03@90 0.03@90 0.03@0 0.03@0 0.03@0 0.03@45 0.03@45 0.03@90 0.03@90

0.05@90 0.05@0 0.05@90 0.05@45 0.05@90 0.05@0 0.05@90 0.05@0 0.05@45

0.07@0 0.07@90 0.07@90 0.07@90 0.07@45 0.07@90 0.07@0 0.07@45 0.07@0


50 0 -1.1 58.7 -0.2 129.0 -38.8 80.6 6.6 87.5 58.5

0.05 -2.3 211.8 -0.8 132.3 -43.4 216.8 -3.7 219.0 64.2

50 0.05 -2.3 211.8 -0.8 132.3 -43.4 216.8 -3.7 219.0 64.20.1 -3.4 365.0 -1.5 135.6 -48.1 352.9 -14.0 350.5 69.9

50 0.1 -3.4 365.0 -1.5 135.6 -48.1 352.9 -14.0 350.5 69.9

0.15 -4.6 518.1 -2.2 138.9 -52.7 489.1 -24.3 482.0 75.6

70 0.15 1082.6 -14.2 -2.2 -172.0 465.6 -39.5 1054.8 75.7 952.3

0.22 805.8 -9.6 -3.2 -111.4 295.7 -38.1 781.1 80.4 723.7

30 0.22 -6.1 -9.6 1514.3 1219.1 1310.1 144.6 252.5 -73.2 -116.2

0.25 -6.8 -7.6 1230.0 1041.4 1088.4 157.6 180.2 -75.7 -83.4

50 0.25 -6.8 824.4 -3.6 145.4 -61.9 761.4 -44.8 745.0 87.1

0.3 -8.0 977.5 -4.3 148.7 -66.5 897.6 -55.1 876.5 92.8

30 0.3 -8.0 -4.3 756.0 745.3 719.1 179.2 59.6 -79.7 -28.8

0.33 -8.6 -2.4 471.7 567.7 497.4 192.3 -12.8 -82.2 4.0

70 0.33 370.8 -2.4 -4.7 -16.3 28.8 -36.0 351.0 87.6 364.5

0.4 94.0 2.2 -5.7 44.3 -141.1 -34.6 77.3 92.3 135.9

Stress ratio 984.2 441.6 1892.9 74.8 33.8 25.9 285.1 12.0 240.6


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A strain distributions are calculated from the program as well. Table 4.6

shows the data of strains for the isotropic arrangement of fiber and matrix. Based on

the two tables, some observations can be made.

Firstly, the laminas where maximum stress occurs produces the maximum

strain as well, and vice-versa for the minimum values. For instance, case 1, the

maximum stress of 335.5MPa occurs at thickness 0.4mm. Similarly, the maximum

strain of 2.4mm/mm happens at the same location which is at the thickness 0.4mm.

Secondly, for the calculated stress ratio of either case, when the value is small,

the strain ratio does not followed as the stress ratio. For example, the calculated

stress ratio for case 1 is 10.5 which is the lowest in all the cases. When it comes to

the strain ratio, where computed value is 58.9, it is not the lowest in all the cases.


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Table 4.6 Local Strains of Isotropic Arrangement

Strain Ratio

1: Is otrophic arrangement

0.03@45 0.03@0 0.03@0 0.03@45 0.03@45 0.03@0 0.03@90 0.03@0 [email protected]@0 0.05@0 0.05@45 0.05@45 0.05@0 0.05@45 0.05@0 0.05@0 0.05@90

0.07@0 0.07@45 0.07@0 0.07@0 0.07@45 0.07@45 0.07@0 0.07@90 0.07@0


50 0 1.7 1.3 -2.7 -3.0 0.6 -5.6 1.7 1.3 -0.2

0.05 1.8 1.7 -2.3 -2.6 1.6 -4.9 1.8 1.7 -0.2

50 0.05 1.8 1.7 -2.3 -2.6 1.6 -4.9 1.8 1.7 -0.2

0.1 1.9 2.1 -1.9 -2.2 2.5 -4.2 1.9 2.1 -0.2

50 0.1 1.9 2.1 -1.9 -2.2 2.5 -4.2 1.9 2.1 -0.2

0.15 2.0 2.5 -1.5 -1.8 3.5 -3.4 2.0 2.5 -0.2

70 0.15 2.0 -0.2 5.9 6.8 -0.5 -3.4 2.0 -0.2 6.6

0.22 2.1 -0.4 4.2 5.2 -1.1 -2.4 2.1 -0.1 4.6

30 0.22 0.0 3.0 4.2 -1.2 -1.1 8.8 -0.2 3.1 4.6

0.25 0.0 3.2 3.5 -1.0 -1.3 7.5 -0.2 3.4 3.850 0.25 2.2 3.2 -0.6 -1.0 5.4 -2.0 2.2 3.4 -0.2

0.3 2.2 3.6 -0.2 -0.6 6.3 -1.3 2.3 3.8 -0.1

30 0.3 -0.2 3.6 2.3 -0.6 -1.6 5.2 -0.1 3.8 2.4

0.33 -0.3 3.9 1.6 -0.4 -1.9 3.9 -0.1 4.0 1.6

70 0.33 2.3 -0.8 1.6 2.6 -1.9 -0.9 2.3 -0.1 1.6

0.4 2.4 -1.1 0.0 1.0 -2.4 0.1 2.5 -0.1 -0.4

Strain ratio 58.9 21.8 151.8 18.5 11.7 67.7 27.0 47.9 44.1

0.03@90 0.03@90 0.03@0 0.03@0 0.03@0 0.03@45 0.03@45 0.03@90 0.03@90

0.05@90 0.05@0 0.05@90 0.05@45 0.05@90 0.05@0 0.05@90 0.05@0 0.05@45

0.07@0 0.07@90 0.07@90 0.07@90 0.07@45 0.07@90 0.07@0 0.07@45 0.07@0

m Thickness case10 case11 case12 case13 case14 case15 case16 case17 case1850 0 -0.2 0.4 -0.4 -6.4 -0.6 0.6 -0.2 0.6 -3.2

0.05 -0.2 1.5 -0.4 -5.4 -0.6 1.6 -0.2 1.6 -2.7

50 0.05 -0.2 1.5 -0.4 -5.4 -0.6 1.6 -0.2 1.6 -2.7

0.1 -0.2 2.6 -0.3 -4.5 -0.6 2.5 -0.3 2.5 -2.2

50 0.1 -0.2 2.6 -0.3 -4.5 -0.6 2.5 -0.3 2.5 -2.2

0.15 -0.2 3.7 -0.3 -3.5 -0.6 3.5 -0.3 3.4 -1.7

70 0.15 7.7 -0.2 -0.3 -1.4 0.1 -0.3 7.5 -0.5 6.8

0.22 5.7 -0.2 -0.2 -0.9 -0.4 -0.4 5.6 -1.1 5.2

30 0.22 -0.1 -0.2 10.8 8.7 9.3 -0.3 0.9 -0.6 -0.9

0.25 -0.1 -0.2 8.8 7.4 7.8 -0.4 0.3 -0.6 -0.7

50 0.25 -0.1 5.9 -0.2 -1.5 -0.6 5.4 -0.4 5.3 -0.8

0.3 -0.1 7.0 -0.1 -0.5 -0.6 6.4 -0.4 6.3 -0.3

30 0.3 -0.1 -0.1 5.4 5.3 5.1 -0.6 -0.7 -0.7 -0.30.33 -0.1 -0.1 3.4 4.0 3.6 -0.7 -1.3 -0.7 0.0

70 0.33 2.6 -0.1 -0.1 -0.2 -1.2 -0.4 2.5 -2.0 2.6

0.4 0.7 -0.1 0.0 0.3 -1.7 -0.4 0.6 -2.6 1.0

Strain ratio 72.5 49.9 623.9 47.1 89.9 20.8 49.7 13.8 428.4


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In order to compare the result to the isotropic control material, a plot will be

plotted so that any difference can be observed. Hence, based on the local stress

shown in Table 4.5, three values are picked where each of the value will be plotted in

lines. These values are:

For all the cases in isotropic arrangement, their stress ratios are ranging from

10.5 to 1467.5. When compared to the isotropic control material where the stress

ratio is only 268.9. This means that the fiber and matrix laminate arrangement

similar to that of the isotropic arrangement in terms of thickness is not very effective

in design. This is because the maximum stress ratio of 1467.5 is higher than the

control material. It means that difference and gap of the local stresses in the

laminated fiber matrix composite will be very high and may lead to failure of the

weaker plies. Figure 4.11 is the plotted line of stress ratio for the first study, the

isotropic arrangement. Numbers 123 in the x-axis represent the minimum, mid and

maximum stress ratio.

Figure 4.11 Stress Ratio Curve for Isotropic Arrangement Laminate Composite

Min Stress ratio = 10.5

Mid Stress ratio = 984.2

Max Stress ratio = 1467.5

ARRANGEMENT 1

10.5

984.2

1467.5

0.0

200.0

400.0

600.0

800.0

1000.0

1200.0

1400.0

1600.0

1 2 3

Stress Ratio (Arrangement 1)


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Figure 4.12 Stress Distribution for Isotropic Arrangement Laminate Composite

Figure 4.13 Strain Distribution for Isotropic Arrangement Laminate Composite

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-500.0 0.0 500.0 1000.0 1500.0 2000.0

Thicknessmm

Stress MPa

Stress Distribution 1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-2.0 0.0 2.0 4.0 6.0 8.0 10.0 12.0

Thicknessmm

Strain mm/mm

Strain Distribution 1


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Figure 4.12 and Figure 4.13 show the stress and strain distributions at which

maximum stress ratio occur. As can be seen, both plots show the same pattern of

zigzag shapes. Another thing to be noted would be the strain, although the stress

ratio is very much higher than the control material, when using fiber matrix laminate,

the strain will be lower comparing to 71.1303 of the control material.


The symmetrical arrangement 1 has the min-max-max-min arrangement. Two

tables of stress and strain data are provided as well. They are in the appendix. All

limitations and specifications are as discussed in section 4.6.2.1.

Figure 4.14 shows the stress ratio curve of symmetrical arrangement 1. The

x-axis 1, 2 and 3 represent the minimum, mid and maximum stress ratio respectively.

Figure 4.14 Stress Ratio Curve for Symmetric Laminate Composite

6.1

121.1

358.3

0.0

50.0

100.0

150.0

200.0

250.0

300.0

350.0

400.0

1 2 3



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From the curve itself, improvement in terms of design can be seen. This is

because in symmetrical arrangement, the stress ratio had been drop significantly to

maximum of 358.3.

Since the laminate composites in practice would consist of not only 0 and 90

degrees orientations, thus when excluding the cases of those comprise only of 0 and

90 degree, a new table, Table 4.7 is formed, Figure 4.15 shows the new stress ratio

curve based on the table.

Table 4.7 Improved Local Stresses of Symmetrical Arrangement 1

2: Symmetry arrangement 1

0.03@45 0.03@0 0.03@0 0.03@45 0.03@45 0.03@0 0.03@0 0.03@0 0.03@45 0.03@45 0.03@90 0.03@90

0.05@0 0.05@0 0.05@45 0.05@45 0.05@0 0.05@45 0.05@45 0.05@90 0.05@0 0.05@90 0.05@0 0.05@45

0.07@0 0.07@45 0.07@0 0.07@0 0.07@45 0.07@45 0.07@90 0.07@45 0.07@90 0.07@0 0.07@45 0.07@0

m Thickness c ase1 case2 case3 case4 case5 case6 case13 case14 case15 case16 case17 case18

30 0 46.9 361.4 450.8 38.4 34.7 1074.5 1068.6 1079.9 110.6 151.4 -61.2 -77.6

0.03 46.9 361.4 450.8 38.4 34.7 1074.5 1068.6 1079.9 110.6 151.4 -61.2 -77.6

50 0.03 286.5 361.4 34.7 38.4 450.8 56.9 117.4 -50.5 454.2 -25.6 449.1 60.9

0.08 286.5 361.4 34.7 38.4 450.8 56.9 117.4 -50.5 454.2 -25.6 449.1 60.950 0.08 286.5 361.4 34.7 38.4 450.8 56.9 117.4 -50.5 454.2 -25.6 449.1 60.9

0.13 286.5 361.4 34.7 38.4 450.8 56.9 117.4 -50.5 454.2 -25.6 449.1 60.9

70 0.13 286.5 35.4 450.8 599.8 34.7 56.9 -69.5 159.5 -28.3 609.0 60.1 596.1

0.2 286.5 35.4 450.8 599.8 34.7 56.9 -69.5 159.5 -28.3 609.0 60.1 596.1

70 0.2 286.5 35.4 450.8 599.8 34.7 56.9 -69.5 159.5 -28.3 609.0 60.1 596.1

0.27 286.5 35.4 450.8 599.8 34.7 56.9 -69.5 159.5 -28.3 609.0 60.1 596.1

50 0.27 286.5 361.4 34.7 38.4 450.8 56.9 117.4 -50.5 454.2 -25.6 449.1 60.9

0.32 286.5 361.4 34.7 38.4 450.8 56.9 117.4 -50.5 454.2 -25.6 449.1 60.9

50 0.32 286.5 361.4 34.7 38.4 450.8 56.9 117.4 -50.5 454.2 -25.6 449.1 60.9

0.37 286.5 361.4 34.7 38.4 450.8 56.9 117.4 -50.5 454.2 -25.6 449.1 60.9

30 0.37 46.9 361.4 450.8 38.4 34.7 1074.5 1068.6 1079.9 110.6 151.4 -61.2 -77.6

0.4 46.9 361.4 450.8 38.4 34.7 1074.5 1068.6 1079.9 110.6 151.4 -61.2 -77.6

Stress ratio 6.1 10.2 13.0 15.6 13.0 18.9 15.4 21.4 16.1 23.8 7.5 9.8


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Figure 4.15 Improved Stress Ratio Curve for Symmetric Laminate Composite

From the improved curve of the Symmetrical arrangement 1, the maximum

value of stress ratio drops dramatically to 23.8. It can be concluded that by using

varying thickness of fiber matrix laminate composite, one able to create and design

plates that are much effective compared to those isotropic material.

Improved Stress Ratio (Arrangement 2)


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Figure 4.16 Stress Distributions for Symmetry Laminate Arrangement 1

Figure 4.17 Strain Distributions for Symmetry Laminate Arrangement 1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-100.0 0.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0

Thicknessmm

Stress MPa


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-1.0 0.0 1.0 2.0 3.0 4.0 5.0

Thi

cknessmm

Strain mm/mm



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Showing in Figure 4.16 and Figure 4.17 are the stress and strain distributions

at which maximum stress ratio occur. Similar to the first study, both plots show the

same pattern of shape.


The symmetrical arrangement 2 has the max-min-min-max arrangement. The

stress and strain data are listed in two tables attached in the appendix. All limitations

and specifications are as discussed in section 4.6.2.1.

Similar to symmetrical arrangement 1, the stress ratio can be improved while

excluding the 0/90 degrees orientations. Showing in Figure 4.18 and Figure 4.19 are

the stress ratio curve of symmetrical arrangement 2 both original and improved.

Figure 4.18 Stress Ratio Curve for Symmetrical Laminate Arrangement 2

6.1

121.1

358.3

0.0

50.0

100.0

150.0

200.0

250.0

300.0

350.0

400.0

1 2 3



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Figure 4.19 Improved Stress Ratio Curve for Symmetrical Laminate

Arrangement 2

In the study of the varying thickness laminate with max-min-min-max

arrangement, it is found that the same values of stress ratio curve will be obtained, as

long as the laminates stay symmetrical, with the same predetermined cases of

orientation. Meanwhile, it is observed that for instance case 1 of min-max-max-min

arrangement, maximum stress happened at the plies of 0.05 and 0.07mm. For the

symmetrical arrangement 2 max-min-min-max, the same goes for the 0.05 and

0.07mm plies.

Improved Stress Ratio (Arrangement 3)


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Figure 4.20 Stress Distributions for Symmetry Laminate Arrangement 2

Figure 4.21 Strain Distributions for Symmetry Laminate Arrangement 2

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-100.0 0.0 100.0 200.0 300.0 400.0 500.0 600.0 700.0

Thicknessmm

Stress MPa


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-1.0 0.0 1.0 2.0 3.0 4.0 5.0

Thi

cknessmm

Strain mm/mm



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Showing in Figure 4.20 and Figure 4.21 are the stress and strain distributions at

which maximum stress ratio occur. Similar to the first study, both plots show the

same pattern of shape.


The balanced arrangement 1 has the min-max-max-min arrangement. The

difference of balanced to symmetrical is in terms of fibers orientation. For examplefor case 1 with 0.03@45, 0.05 and 0.07@0, the top four layers would be arrange in:

[45/0/0/0], but in the bottom four layers, the orientation would be: [90/90/90/-45].

Two tables of stress and strain data are provided as well. They are in the appendix.

All limitations and specifications are as discussed in section 4.6.2.1. Shows in Figure

4.22 will be the stress ratio curve of balanced arrangement 1.

Figure 4.22 Stress Ratio Curve for Balanced Laminate Arrangement 1

4.4

246.0

742.3

0.0

100.0

200.0

300.0

400.0

500.0

600.0

700.0

800.0

1 2 3



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From the curve, the maximum stress ratio is 742.3, which is higher than the

control material. However, the minimum value is 4.4 which is the lowest in the

study so far. It is safe to say that the balanced arrangement of varying thickness

laminates can be formed as a effective design of plates, but it only applies to certain

orientation.

Figure 4.23 Stress Distributions for Balance Laminate Arrangement 1

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-200.0 0.0 200.0 400.0 600.0 800.0 1000.0

Thicknessmm

Stress MPa



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Figure 4.24 Strain Distribution for Balanced Laminate Arrangement 1


at which maximum stress ratio occur. It showed similar pattern as other study of

thickness arrangement.


The balanced arrangement 2 will be the max-min-min-max arrangement.

Two tables of stress and strain data are provided as well. They are in the appendix.

All limitations and specifications are as discussed in section 4.6.2.1. Shows in

Figure 4.25 will be the stress ratio curve of balanced arrangement 2.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0

Thicknessmm

Strain mm/mm



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Figure 4.25 Stress Ratio Curve for Balanced Laminate Arrangement 2

Unlike symmetrical laminates, balanced laminate with varying thicknesses

does not show any identical result. It can be seen that the maximum stress ratio is

even higher at 1180.5. Thus, it can be concluded that the gap between the max and

the min stress ratio is considerably wide.

11.0

162.3

1180.5

0.0

200.0

400.0

600.0

800.0

1000.0

1200.0

1400.0

1 2 3



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Figure 4.26 Stress Distributions for Balanced Laminate Arrangement 2

Figure 4.27 Strain Distributions for Balanced Laminate Arrangement 2

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-400.0 -200.0 0.0 200.0 400.0 600.0 800.0 1000.0 1200.0 1400.0

Thicknessmm

Stress MPa


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-4.0 -2.0 0.0 2.0 4.0 6.0 8.0 10.0

Thicknessmm

Strain mm/mm



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at which maximum stress ratio occur. It can be observed that it had a similar pattern

to that of the balanced arrangement.


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CHAPTER 5

CONCLUSION AND RECOMMENDATION

5.1 Conclusion

This research has successfully accomplished its three objectives as the

MATLAB program had been developed successfully, where one is able to obtain the

stiffness matrix and the stress and strain distribution from this program. By using the

MATLAB program, analysis had been run and the conclusions that can be drawn are

as follow:

i.

A programming code for the computation of the Stiffness ABD matrix

and the stress-strain distributions have been written and proven by

validating the results from the literature. The program is able to

calculate any kind of fiber and matrix, with any layers of laminas,

orientation and the thickness of each lamina.

ii. A varying thickness isotropic material is likely to produce high stress

ratio, which mean the range of minimum and maximum stress would

be very high. This means that some layers would be able to take more

loading while others less, this might ultimately cause the delamination

of the laminate and failure of the material.

iii. Using fiber and matrix laminate of varying thickness; the stress ratio

could be reduced, whereby it is most significantly shown by the fiber

matrix laminate arranged in a symmetrical order when excluding the

cases of 0/90 degrees orientations.


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iv.

For varying thickness composite laminate, as long as it is arranged in

symmetrical orders, they would produce the same value of stress and

strain ratios under the same mechanical loading.

v. For all lamination cases the maximum stress ratio may not necessarily

gets the maximum strain ratio.

5.2 Recommendations

There are several recommendations for future research in terms of the

analysis of the varying thickness of fiber reinforced composite laminate.

i. The data input for the MATLAB program can be improved. If

possible, implementing the user interfaces where users can have more

freedom when using the program.

ii.

The development of the plotting of stress and strain curves can be

enhanced, as only stress and strains in the x-direction are focused so

far.

iii. Expand the scope of studies of the research; if possible, try increasing

the number of cases of orientation and layers and thickness so that

users would have clearer views on the behaviors of the varying

thickness composite laminate.

iv. Expand the findings of the research by exploring others parameters in

varying thickness composite laminate. As only mechanical loading are

studied, parameters such as thermal loading and moisture effect can

be implemented


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REFERENCES

R. F. Gibson,(1994).Principle of Composite Material Mechanics. U.S.A. McGraw-

Hill.

Jones, R. M., (1975).Mechanics of Composite Materials. Washington : McGraw-

Hill.

Kollar, L. P. and Springer, G. S., (2002). Mechanics of Composite Structure.

Cambridge: Cambridge University Press.

Hanselman, D. and Littlefield, B. (1997). The Student Edition of MATLAB Version

5 : User Guide. New Jersey: Prentice Hall.

Hyer, M. W., (1997). Stress Analysis of Fiber Reinforced Composite Materials.

United State of America : McGraw-Hill.

M. G. Joshi & S. B. Biggers, Jr. (1995). Composites: Part B 27B (1996) 105-

114.Thickness Optimization for Maximum buckling loads in Composite

Laminated plates.

P. Khosravi & R. Sedaghati. (2007). Struct Multidisc Optim (2008) 36:159-167.

Design of Laminated Composite Structures for Optimum Fiber Direction and

Layer Thickness, using Optimality Criteria.

T. B. Zineb et al (1998). Composites Science and Technology 58 (1998) 791-799.

Analysis of High Stress Gradients in Composite Plates with Rapidly Varying

Thickness.

J. Wang & B. L. Karihaloo (1995). Composites Structure 32 (1995) 453-466.

Fracture Mechanics and Optimization-A useful Tool for Fiber-Reinforced

Compsite Design.

M. G. Andrews & R. Massabo (2007), Composites: Part B 39 (2008) 139-150.

Delamination in Flat Sheet Geometries with Material Imperfections and

Thickness Variation.


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APPENDIX A

MATLAB Program Code

%Computation of Stress and Strain Profile of laminate%The unit in mm and Gpaclcclear

Vf =0.6;

%Engineering Properties of the fiber and carbon matrialE1f=input('Value of fiber longitudinal modulus in GPa? -');

E2f=input('Value of fiber transverse modulus in GPa? -');G12f=input('Value of fiber axial shear modulus in GPa? -');nu12f=input('Value of fiber poisson ratio? -');Em=input('Value of matrix young modulus in GPa? -');Gm=input('Value of matrix shear modulus in GPa? -');num=input('Value of matrix poisson ratio? -');

nu21=E2*nu12/E1;Q11=E1/(1-nu12*nu21);Q12=nu12*E2/(1-nu12*nu21);Q22=E2/(1-nu12*nu21);Q66=G12;Q=[Q11 Q12 0;Q12 Q22 0;0 0 Q66]

%The number of layerslayer=input('The number of layars in the laminate? -')%The orientation of each laminaAng=input('The orientation(degree) of each lamina,top to bottom ie[45 -45 45 -45] -')*pi/180;%The varying thickness of each laminaThickness=input('The thickness(mm) of each lamina,top to bottomie[0.1 0.1 0.1 0.1] -')

%Computation of the ABD matrixfork=1:layer;fori=1:3;forj=1:3;A(i,j,k)=sum(QT(i,j,k)*(Z(k+1)-Z(k))); B(i,j,k)=sum(QT(i,j,k)*((Z(k+1)^2)-(Z(k)^2)))/2; D(i,j,k)=sum(QT(i,j,k)*((Z(k+1)^3)-(Z(k)^3)))/3; end;end;end;


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%Unit of force in MPalmtFrc=input('The loading(Mpa) acting on laminate, ie[50 0 0 0 0 0]-')';

MidStrCur=invE*lmtFrc;Zmid=sum(Thickness)/2;Z(1)=-Zmid;fork=2:(layer+1);forz=1:layer;

%GloStrainGloStress

%LocalStrainLocalStress

Thick(1)=Thickness(1);fork=1:(layer-1);

fori = 1:layer;

Thickness_Z=[X1 X2 X3]'

Y1=[LocStrain(1,1,1) LocStrain(1,2,1)];Y2=[LocStress(1,1,1) LocStress(1,2,1)];

fori = 2:layer;

Strain_X=Y1'Stress_X=Y2'

%Stress and Strain Distributionsubplot(1,2,1)plot(Stress_X,Thickness_Z)xlabel('Local Stress MPa ')ylabel('Thickness mm')

subplot(1,2,2)plot(Strain_X,Thickness_Z)xlabel('Local Strain mm/mm')ylabel('Thickness mm')

%Ratio and Strain RatioStress_X= abs(Stress_X);Min_Local_Stress=min(Stress_X)Max_Local_Stress=max(Stress_X)Stress_Ratio=abs(max(Stress_X)/min(Stress_X))

Strain_X= abs(Strain_X);Min_Local_Strain=min(Strain_X)Max_Local_Strain=max(Strain_X)Strain_Ratio=abs(max(Strain_X)/min(Strain_X))


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APPENDIX B

Data of Local Stress and Local Strain

Local Stresses for Symmetry Laminate Arrangement 1


0.03@45 0.03@0 0.03@0 0.03@45 0.03@45 0.03@0 0.03@90 0.03@0 0.03@0

0.05@0 0.05@0 0.05@45 0.05@45 0.05@0 0.05@45 0.05@0 0.05@0 0.05@90

0.07@0 0.07@45 0.07@0 0.07@0 0.07@45 0.07@45 0.07@0 0.07@90 0.07@0

m Thickness c ase1 case2 case3 case4 case5 case6 case7 case8 case9

30 0 46.9 361.4 450.8 38.4 34.7 1074.5 -19.1 371.7 469.6

0.03 46.9 361.4 450.8 38.4 34.7 1074.5 -19.1 371.7 469.6

50 0.03 286.5 361.4 34.7 38.4 450.8 56.9 290.8 371.7 -7.0

0.08 286.5 361.4 34.7 38.4 450.8 56.9 290.8 371.7 -7.0

50 0.08 286.5 361.4 34.7 38.4 450.8 56.9 290.8 371.7 -7.0

0.13 286.5 361.4 34.7 38.4 450.8 56.9 290.8 371.7 -7.0

70 0.13 286.5 35.4 450.8 599.8 34.7 56.9 290.8 -9.8 469.6

0.2 286.5 35.4 450.8 599.8 34.7 56.9 290.8 -9.8 469.6

70 0.2 286.5 35.4 450.8 599.8 34.7 56.9 290.8 -9.8 469.6

0.27 286.5 35.4 450.8 599.8 34.7 56.9 290.8 -9.8 469.6

50 0.27 286.5 361.4 34.7 38.4 450.8 56.9 290.8 371.7 -7.0

0.32 286.5 361.4 34.7 38.4 450.8 56.9 290.8 371.7 -7.0

50 0.32 286.5 361.4 34.7 38.4 450.8 56.9 290.8 371.7 -7.0

0.37 286.5 361.4 34.7 38.4 450.8 56.9 290.8 371.7 -7.0

30 0.37 46.9 361.4 450.8 38.4 34.7 1074.5 -19.1 371.7 469.6

0.4 46.9 361.4 450.8 38.4 34.7 1074.5 -19.1 371.7 469.6

Stress ratio 6.1 10.2 13.0 15.6 13.0 18.9 15.2 38.0 67.1

0.03@90 0.03@90 0.03@0 0.03@0 0.03@0 0.03@45 0.03@45 0.03@90 0.03@90

0.05@90 0.05@0 0.05@90 0.05@45 0.05@90 0.05@0 0.05@90 0.05@0 0.05@45

0.07@0 0.07@90 0.07@90 0.07@90 0.07@45 0.07@90 0.07@0 0.07@45 0.07@0


30 0 -5.3 -7.0 1218.2 1068.6 1079.9 110.6 151.4 -61.2 -77.6

0.03 -5.3 -7.0 1218.2 1068.6 1079.9 110.6 151.4 -61.2 -77.6

50 0.03 -5.3 469.6 -3.4 117.4 -50.5 454.2 -25.6 449.1 60.9

0.08 -5.3 469.6 -3.4 117.4 -50.5 454.2 -25.6 449.1 60.9

50 0.08 -5.3 469.6 -3.4 117.4 -50.5 454.2 -25.6 449.1 60.9

0.13 -5.3 469.6 -3.4 117.4 -50.5 454.2 -25.6 449.1 60.9

70 0.13 637.5 -7.0 -3.4 -69.5 159.5 -28.3 609.0 60.1 596.1

0.2 637.5 -7.0 -3.4 -69.5 159.5 -28.3 609.0 60.1 596.1

70 0.2 637.5 -7.0 -3.4 -69.5 159.5 -28.3 609.0 60.1 596.1

0.27 637.5 -7.0 -3.4 -69.5 159.5 -28.3 609.0 60.1 596.1

50 0.27 -5.3 469.6 -3.4 117.4 -50.5 454.2 -25.6 449.1 60.9

0.32 -5.3 469.6 -3.4 117.4 -50.5 454.2 -25.6 449.1 60.9

50 0.32 -5.3 469.6 -3.4 117.4 -50.5 454.2 -25.6 449.1 60.9

0.37 -5.3 469.6 -3.4 117.4 -50.5 454.2 -25.6 449.1 60.9

30 0.37 -5.3 -7.0 1218.2 1068.6 1079.9 110.6 151.4 -61.2 -77.6

0.4 -5.3 -7.0 1218.2 1068.6 1079.9 110.6 151.4 -61.2 -77.6

Stress ratio 121.1 67.1 358.3 15.4 21.4 16.1 23.8 7.5 9.8


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Local Strains for Symmetry Laminate Arrangement 1


0.03@45 0.03@0 0.03@0 0.03@45 0.03@45 0.03@0 0.03@90 0.03@0 0.03@0

0.05@0 0.05@0 0.05@45 0.05@45 0.05@0 0.05@45 0.05@0 0.05@0 0.05@90

0.07@0 0.07@45 0.07@0 0.07@0 0.07@45 0.07@45 0.07@0 0.07@90 0.07@0


30 0 0.0 2.6 3.2 -1.0 -0.6 7.7 -0.2 2.7 3.3

0.03 0.0 2.6 3.2 -1.0 -0.6 7.7 -0.2 2.7 3.3

50 0.03 2.1 2.6 -0.6 -1.0 3.2 -2.1 2.1 2.7 -0.1

0.08 2.1 2.6 -0.6 -1.0 3.2 -2.1 2.1 2.7 -0.1

50 0.08 2.1 2.6 -0.6 -1.0 3.2 -2.1 2.1 2.7 -0.1

0.13 2.1 2.6 -0.6 -1.0 3.2 -2.1 2.1 2.7 -0.1

70 0.13 2.1 -0.3 3.2 4.3 -0.6 -2.1 2.1 -0.1 3.3

0.2 2.1 -0.3 3.2 4.3 -0.6 -2.1 2.1 -0.1 3.3

70 0.2 2.1 -0.3 3.2 4.3 -0.6 -2.1 2.1 -0.1 3.3

0.27 2.1 -0.3 3.2 4.3 -0.6 -2.1 2.1 -0.1 3.350 0.27 2.1 2.6 -0.6 -1.0 3.2 -2.1 2.1 2.7 -0.1

0.32 2.1 2.6 -0.6 -1.0 3.2 -2.1 2.1 2.7 -0.1

50 0.32 2.1 2.6 -0.6 -1.0 3.2 -2.1 2.1 2.7 -0.1

0.37 2.1 2.6 -0.6 -1.0 3.2 -2.1 2.1 2.7 -0.1

30 0.37 0.0 2.6 3.2 -1.0 -0.6 7.7 -0.2 2.7 3.3

0.4 0.0 2.6 3.2 -1.0 -0.6 7.7 -0.2 2.7 3.3

Strain ratio 622.7 7.6 5.4 4.4 5.4 3.7 12.1 23.1 31.4

0.03@90 0.03@90 0.03@0 0.03@0 0.03@0 0.03@45 0.03@45 0.03@90 0.03@90

0.05@90 0.05@0 0.05@90 0.05@45 0.05@90 0.05@0 0.05@90 0.05@0 0.05@45

0.07@0 0.07@90 0.07@90 0.07@90 0.07@45 0.07@90 0.07@0 0.07@45 0.07@0


30 0 -0.1 -0.1 8.7 7.6 7.7 0.0 0.0 -0.5 -0.6

0.03 -0.1 -0.1 8.7 7.6 7.7 0.0 0.0 -0.5 -0.650 0.03 -0.1 3.3 -0.2 -2.0 -0.5 3.2 -0.3 3.2 -1.1

0.08 -0.1 3.3 -0.2 -2.0 -0.5 3.2 -0.3 3.2 -1.1

50 0.08 -0.1 3.3 -0.2 -2.0 -0.5 3.2 -0.3 3.2 -1.1

0.13 -0.1 3.3 -0.2 -2.0 -0.5 3.2 -0.3 3.2 -1.1

70 0.13 4.5 -0.1 -0.2 -0.6 -1.5 -0.3 4.3 -0.6 4.3

0.2 4.5 -0.1 -0.2 -0.6 -1.5 -0.3 4.3 -0.6 4.3

70 0.2 4.5 -0.1 -0.2 -0.6 -1.5 -0.3 4.3 -0.6 4.3

0.27 4.5 -0.1 -0.2 -0.6 -1.5 -0.3 4.3 -0.6 4.3

50 0.27 -0.1 3.3 -0.2 -2.0 -0.5 3.2 -0.3 3.2 -1.1

0.32 -0.1 3.3 -0.2 -2.0 -0.5 3.2 -0.3 3.2 -1.1

50 0.32 -0.1 3.3 -0.2 -2.0 -0.5 3.2 -0.3 3.2 -1.1

0.37 -0.1 3.3 -0.2 -2.0 -0.5 3.2 -0.3 3.2 -1.1

30 0.37 -0.1 -0.1 8.7 7.6 7.7 0.0 0.0 -0.5 -0.6

0.4 -0.1 -0.1 8.7 7.6 7.7 0.0 0.0 -0.5 -0.6

Strain ratio 39.7 31.4 50.7 12.2 15.7 415.5 406.0 6.5 6.8


84/89

67

Local Stresses for Symmetry Laminate Arrangement 2


0.03@45 0.03@0 0.03@0 0.03@45 0.03@45 0.03@0 0.03@90 0.03@0 0.03@0

0.05@0 0.05@0 0.05@45 0.05@45 0.05@0 0.05@45 0.05@0 0.05@0 [email protected]@0 0.07@45 0.07@0 0.07@0 0.07@45 0.07@45 0.07@0 0.07@90 0.07@0

m Thickness c ase1 case2 case3 case4 case5 case6 case7 case8 case9

70 0 286.5 35.4 450.8 599.8 34.7 56.9 290.8 -9.8 469.6

0.07 286.5 35.4 450.8 599.8 34.7 56.9 290.8 -9.8 469.6

50 0.07 286.5 361.4 34.7 38.4 450.8 56.9 290.8 371.7 -7.0

0.12 286.5 361.4 34.7 38.4 450.8 56.9 290.8 371.7 -7.0

50 0.12 286.5 361.4 34.7 38.4 450.8 56.9 290.8 371.7 -7.0

0.17 286.5 361.4 34.7 38.4 450.8 56.9 290.8 371.7 -7.0

30 0.17 46.9 361.4

Varying Thickness

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