i
LINEAR STATIC FINITE ELEMENT ANALYSIS OF COMPOSITE
HAT-STIFFENED LAMINATED PLATES
LEE BIING CHYUAN
This Thesis is Submitted as a Partial Fulfillment of the Requirement for the
Award of the Degree of Bachelor of Mechanical Engineering (Pure)
Faculty of Mechanical Engineering
Universiti Teknologi Malaysia
MARCH, 2005
ii
“I hereby declared that this thesis entitled “Linear Static Finite
Element Analysis of Composite Hat-Stiffened Laminated Plates”
is the result of my own work excepted as cited in references.”
Signature : ____________________
Name of Author : LEE BIING CHYUAN
Date : 1 March 2005
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This book is dedicated to my beloved parents, sisters and my dearest girl friend, thanks for their
constant support and encouragement in everything.
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ACKNOWLEDGEMENT
The author would like to take this opportunity to express his gratitude to Dr.
Nazri Kamsah, his respectful supervisor who has given his guidance, patience and
invaluable advices in enabling the author to achieve the objective of this project.
The author would also like to express appreciation to En. Shukur Abu Hassan
who has unselfishly contributed his information, materials, and equipment in completing
this project. Besides, the author would also like to thank all the staffs in PUSKOM for
their kindness in helping him out by continuously contributing different ways of
improvement and sharing their experience in handling different problems.
Thank you very much.
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ABSTRACT
Laminated composite plates are extensively used in the construction of aerospace,
civil, marine, automotive and other high performance structures due to their high
specific stiffness and strength, excellent fatigue resistance, long durability and many
other superior properties compared to the conventional metallic materials. However,
high modulus and strength characteristics of composites result in structures with very
thin sections that are often prone to buckling. Stiffeners are required to increase the
bending stiffness of such thin walled members. This project is carried out to investigate
the behavior of the composite hat-stiffened laminated plates. The investigation is
restricted to linear static analysis of composite stiffened panels. Unidirectional carbon
fiber was used as reinforcement agent with epoxy resin as binder material. The plates
were arranged symmetrically in geometry about the middle surface of the structure. The
tensile and bending test had been carried out to study the stiffened panel. The
mechanical properties and behavior of the stiffened panels were recorded. The numerical
analysis has been done using finite element software and the results are compared with
the experiment values. The experiment and numerical results show that the behavior of
the composite laminated plates is depended on their fiber orientations and stiffeners give
major effects in the bending stiffness of the composite plates.
Keywords: Finite element; Stiffened plate; Hat stiffener; Composite
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ABSTRAK
Komposit laminat plat yang bersifat tinggi kekuatan, rintangan lesu yang tinggi,
panjang hayat lesu dan banyak lagi sifat-sifat yang lebih maju bahan-bahan logam telah
menjadikannya popular untuk pembinaan dalam bidang aerospace, civil, marin,
automotif, dan struktur persembahan tinggi. Walau bagaimanapun, komposit yang
bersifat modulus dan kekuatan tinggi menyebebkan struktur nipis seperti plat mudah
untuk membengkok. Struktuk penguat diperlukan untuk meningkatkan kekuatan
bengkokan struktur nipis ini. Projek ini akan menekankan analisis laminat plat komposit
yang dikuatkan dengan stiffener berbentuk topi. Analisis ini dibataskan kepada linear
statik analisis untuk plat komposit. ‘Unidirectional’ karbon fiber digunakan sebagai
bahan penguat dan epoxi sebagai pencantum. Plat adalah disusun secara simetri terhadap
permukaan tengah struktur. Plat telah dikaji dengan ujian tegangan and bengkokan.
Sifat-sifat mekanikal untuk plat itu telah direkodkan. Perisian unsur terhingga digunakan
sebagai tambahan untuk mengkaji sifat-sifat plat ini dan keputusan dibandingkan dengan
keputusan eksperimen. Keputusan eksperimen dan perisian menunjukkan bahawa sifat-
sifat ‘stiffened’ plat bergantung kepada susunan arah fiber dalam plat dan ‘stiffener’
memberi kesan utama kepada kekuatan bengkokan untuk plat komposit.
Kata-kata kunci: Kaedah unsur terhingga; ‘Stiffened’ plat; “Stiffener’ berbentuk topi;
komposit
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TABLE OF CONTENTS
CHAPTER TITLE PAGE
TITLE i
DECLARATION ii
DEDICATION iii
ACKNOWLEDGEMENT iv
ABSTRACT v
TABLE OF CONTENTS vii
LIST OF TABLE xii
LIST OF FIGURE xiii
LIST OF SYMBOLS xvi
CHAPTER I INTRODUCTION
1.1 Introduction 1
1.2 Problem Statement 2
1.3 Objective 3
1.4 Scope 3
1.5 Methodology 4
viii
CHAPTER II LITERATURE REVIEW ON
COMPOSITE MATERIAL
2.1 Introduction 7
2.1.1 Fibrous composites 11
2.1.2 Laminated Composites 13
2.1.3 Particulate Composites 14
2.2 Fiber 14
2.2.1 Glass Fiber 17
2.2.2 Carbon Fiber 18
2.2.3 Aramid Fiber (Kevlar) 19
2.2.4 Boron Fiber 20
2.3 Matrix 20
2.3.1 Polymer Matrix Composites (PMC) 21
2.3.2 Thermoplastic 22
2.3.3 Thermoset 23
2.3.3.1 Polyester 24
2.3.3.2 Epoxy 25
CHAPTER III THEORETICAL ANALYSIS OF COMPOSITE
3.1 Analysis of Lamina 27
3.1.1 Stress-strain Relations For Plane Stress 29
In Specially Orthotropic Lamina
3.1.2 Stress-strain Relations For Plane Stress 31
In Generally Orthotropic Lamina
3.2 Theory of Plate 32
3.3 Analysis of Laminate 37
3.3.1 Classical Laminated Plate Theory 38
3.3.2 Strains and Stress Variation in a 39
Laminate
ix
3.3.3 Resultant Laminate Forces and 43
Moments
3.3.4 Symmetric and Unsymmetrical 45
Laminates
3.4 Stiffened Plate 46
3.5 Bending of Simply Supported 49
Rectangular Plates
3.5.1 Governing Equations 49
3.5.2 The Navier Solution 50
CHAPTER IV FINITE ELEMENT IMPLEMENTATION
4.1 Introduction 53
4.2 Linear Static Analysis 56
4.3 Finite Element Analysis Procedures 57
4.3.1 Modeling for Unstiffened 57
Composite Laminated Plate
4.3.2 Modeling for Composite 61
Hat-Stiffened Laminated Plate
CHAPTER V EXPERIMENTAL PROCEDURES
5.1 Composite Fabrication 64
5.1.1 Hand Lay Up Method 65
5.1.2 Vacuum Bagging 68
5.2 Laminate Preparation 69
5.2.1 Hand Lay Up Procedure 70
5.3 Tensile Test Specimen Preparation 71
5.4 Bending Test Specimen Preparation 73
5.5 Tensile Test 75
x
5.5.1 Experimental Determination of 76
Strength and Stiffness
5.5.2 Testing Apparatus 80
5.5.3 Tensile Test Procedure 81
5.6 Bending Test 82
5.6.1 Testing Apparatus 83
5.6.2 Bending Test Procedure 84
CHAPTER VI RESULT AND DISCUSSION
6.1 Tensile Test Result 86
6.1.1 Discussion on Tensile Test Results 88
6.1.2 Discussion on Graph Stress versus 90
Axial Strain
6.1.3 Discussion on Graph Lateral Strain 92
versus Axial Strain
6.2 Bending Test Result 93
6.2.1 Discussion on Unstiffened Composite 96
Laminated Plate
6.2.2 Theoretical Analysis of Unstiffened 98
Composite Laminated Plate
6.2.3 Discussion on Composite Hat-Stiffened 99
Laminated Plate
6.3 FEA Simulation Result 101
6.3.1 Discussion on Unstiffened Composite 103
Laminated Plate
6.3.2 Discussion on Hat-Stiffened Composite 104
Laminated Plate
xi
CHAPTER VII CONCLUSION AND SUGGESTION
7.1 Conclusion 110
7.2 Suggestion for Future Study 113
REFERENCES 115
APPENDIX A 117
APPENDIX B 125
APPENDIX C 131
APPENDIX D 136
APPENDIX E 142
APPENDIX F 150
xii
LIST OF TABLE
Table Title Page
2.1 Fiber and wire properties 12
2.2 Properties of fiber and conventional bulk materials 16
2.3 Typical glass fiber properties 17
2.4 Properties of carbon fiber 19
2.5 Typical properties of cast resin system 26
5.1 Specimen Specification 72
5.2 Materials Specification 73
6.1 Results of tensile test 87
6.2 Summary of tensile test result 87
6.3 Results of Load and Deflection for unstiffened composite plate 93
6.4 Results of Load and Deflection for composite hat-stiffened 94
laminated plate
6.5 The analysis result of maximum deflection at 100 kg applied load 98
6.6 Comparison of experiment results and FEA value for unstiffened 102
and stiffened plate
xiii
LIST OF FIGURE
Figure Title Page
1.1 Flow Chart of Methodology 6
2.1 Comparison of specific modulus between composite and metallics 8
2.2 Comparison of stress/strain relationship between composites and 9
metallics
2.3 Classes of Composite 10
2.4 Comparison between the conventional materials and composite 11
materials
2.5 Tensile stress-strain Curve for fiber, FRP and resin 22
3.1 Two principles typical of lamina 28
3.2 Specially orthotropic lamina 29
3.3 Generally orthotropic lamina 31
3.4 Plate subjected to pure bending 33
3.5 (a) Direct stress on lamina of plate element. (b) Radii of curvature of 34
neutral surface.
3.6 Principle and structural coordinates, and lamination 38
3.7 Geometry of deformation in the xz plane 40
3.8 (a) In-plane forces on a flat laminate, (b) Moments on a flat laminate 43
3.9 Geometry of an n-layered laminate 44
3.10 Cross-sectional views of laminates 46
3.11 A hat-stiffened plate 47
3.12 Various types of stiffened panels 47
xiv
3.13 Schematic of T, J, blade, and Hat stiffener geometry 48
3.14 Plate Geometry 49
4.1 Finite element model 54
4.2 FEA model 57
5.1 Manual Lay-up process 67
5.2 Vacuum Bag mould assembly 69
5.3 A finished laminated composite plate 71
5.4 Specimen Specification 71
5.5 Tensile Test Specimen 73
5.6 Materials and tool for hand lay-up process 74
5.7 (a) Hat shaped stiffener, (b) Hat-stiffened plate 75
5.8 Tensile Specimen 76
5.9 Uniaxial loading in the 1-direction 77
5.10 Uniaxial loading in the 2-direction 78
5.11 Uniaxial loading at 45° to the 1-direction 79
5.12 Instron 4602 testing machine 80
5.13 Specimens with strain gauge 81
5.14 Location of the displacement transducers at the composite plate 82
5.15 Hydraulic Press Machine 83
5.16 Bending test rig 84
5.17 Displacement transducer (LVDT) 84
5.18 Plate specimen with strain gauge 85
6.1 Failure mode of specimen with 0 degree fiber orientation 88
6.2 Failure mode of specimen with 90 degree fiber orientation 89
6.3 Failure mode of specimen with 45 degree fiber orientation 89
6.4 Failure mode of the unstiffened composite plate 97
6.5 Failure of the Stiffened Plate 99
6.6 Location of the strain gauges at the composite hat-stiffened plate 101
6.7 Bent plate in half sinusoid wave with deformation scale of 5 103
6.8 Displacement contour for hat-stiffened plate for bottom view 105
6.9 Deformed shape of the hat-stiffened plate with deformation scale of 3 105
xv
6.10 Front view of deformed shape for the hat-stiffened plate with 106
deformation scale of 3
6.11 Side view of the critical region for composite hat-stiffened plate 107
6.12 Critical region of the hat-stiffened plate 107
6.13 Displacement contour for hat-stiffened composite plate with 109
laminate property
6.14 Side view of the deformed hat-stiffened plate with laminate property 109
xvi
LIST OF SYMBOLS
ijA - Extensional stiffness
a - Length of plate
ijB - Coupling stiffness
b - Width of plate
ijD - Bending stiffness
D - Flexural rigidity
iE , jE - Young’s modulus in i, j direction respectively
e - Tab length
12G - Shear modulus in 1-2 plane
I - Moment of inertia
k - Middle surface curvature
L - Length between the tabs
Mi - Normal moment per unit of length
Mij - Twisting moment per unit of length
Ni - Normal load per unit of length
Nij - In-plane shear load per unit of length
ijQ - Reduced stiffness
ijQ - Transformed reduced stiffness
mnQ - Load coefficient
q - Transverse load
xvii
rx , ry - Radii of curvature of the neutral surface in sections parallel to the
xz and yz planes respectively
ijv - Poisson’s ratio for transverse strain in j-direction when
subjected to a stress in the i-direction
W - Width of the tensile test specimen
w - Deflection in the z direction
u, v, w - Displacement in the x-, y-, z-direction
zk - Thickness of laminate
γ - Shear strain
τ - Shear stress
iε , jε - Strain in I, j direction respectively
θ - Angle of lamina
σ - Stress component
CHAPTER I
INTRODUCTION
1.1 Introduction
Laminated composite have found usage in aerospace, automotive, marine, civil,
and sport equipment applications. This popularity is due to excellent mechanical
properties of composites as well as their amenability to tailoring of those properties.
One of the most important structural configurations made of composite materials
is known as a plate. By definition, a plate is a planar load-carrying component spanning
two directions whose thickness is significantly less than its side lengths. Laminated
plates are one of the simplest and most widespread practical applications of composite
laminates. Laminated composite plates are extensively used in the construction of
aerospace, civil, marine, automotive and other high performance structures due to their
high specific stiffness and strength, excellent fatigue resistance, long durability and
many other superior properties compared to the conventional metallic materials.
2
Laminated composite materials provide the designer with freedom to tailor the
properties and response of the structure for given loads to obtain the maximum weight
efficiency. However, high modulus and strength characteristics of composites result in
structures with very thin sections that are often prone to buckling. Stiffeners are required
to increase the bending stiffness of such thin walled members (plates, shells). Hence, the
stiffened plates are widely used as structural components for aerospace, launch vehicles,
and other industrial applications to obtain lightweight structures with high bending
stiffness. Stiffened plates are also more tolerant to imperfections and resist catastrophic
growth of cracks. The stiffening member also provides the benefit of added load-
carrying capability with a relatively small additional weight penalty.
The present study focuses on the linear static behavior of composite hat-stiffened
laminated plate. The finite element software will be used as an aid to study the linear
static finite element analysis of laminated stiffened plates.
1.2 Problem Statement
A composite plate is extensively used in aircraft structures, ships, bridges and
other industrial applications and is loaded to varying conditions such as bending,
buckling, vibration and so on. Therefore, optimization of the plate structural is needed to
meet the working environment and gives the desired properties such high stiffness,
strength. Normally stiffeners are used to increase the stiffness of the plate especially the
bending stiffness.
3
1.3 Objective
The objective of this project is:
1) Study the effects of hat-shaped stiffeners in the deformation of the composite
laminated plates by experimentally and finite element simulation.
2) Study the different shape of stiffener in strengthening the composite plate by
finite element simulation.
1.4 Scope
The scope of this project is:
a) Literature study on composite materials, stiffener, laminate and plate structures.
b) Fabrication of the composite plate which is arranged symmetrically in both
geometry and material properties about the middle surface of the plates.
c) Determine the mechanical properties and behavior of the composite laminated
plate by carried out:
a. Tensile test
b. Bending test
d) Linear static finite element analysis of the composite laminated plate.
e) Comparison of the finite element simulation results with experiment value.
4
1.5 Methodology
The effects of stiffener will be determined through Bending Test and followed by
structural analysis. Comparison will be made in term of ultimate load at failure and
maximum deflection between the stiffened and unstiffened composite plates. The
methodologies of the project are shown as follow:
i) Identify Problem
The objective of this project is to do the analysis of the composite hat-
stiffened laminated plate by using the experimental procedures. Analysis of the
plate structure involves a lot of complex calculation and it takes plenty of time to
do it. Therefore computer software will be used as an aid to study the linear static
finite element analysis of laminated stiffened plates.
ii) Literature Study
After identify the objectives and scopes of this project, literature study
will be carried out to gather all the information needed for this project. Literature
study will focus on the topics such as follow: composite materials, plate theory,
the effects of stiffener onto composite plates and finite element analysis.
iii) Experiment Procedures
Experiment is carried out to analysis the composite laminated plate. In
this project, there are two type of experiment will be carried. The first
experiment is tensile test which is to determine the mechanical properties of the
5
composite materials. The second experiment is bending test which is to analysis
the behavior of the composite stiffened and unstiffened plate. All the specimens
are fabricated by using hand lay-up technique and the candidate materials are
unidirectional carbon fiber was used as reinforcement agent with epoxy resin as
binder material.
iv) FE Simulation
There is plenty of computer software that suit for the FE simulation such
as COSMOS/M, MSC/NASTRAN, and ABAQUS. Software that is user friendly
and provides the easy methodology in modeling and FE analysis will be chosen.
The materials properties that needed in FE simulation will be obtained from
tensile test.
v) Data Collection
The data from the test will be collected includes: ultimate applied load,
displacement and local strains.
vi) Results Comparison
After the experimental analysis and FE simulation have been done, the
comparison of the result will be done in the behavior of the composite laminated
plate and hat-stiffened plate to study the effects of the stiffener in improving the
strength of the composite plate.
6
Figure 1.1: Flow Chart of Methodology
Identify Problem Objective and Scope
Literature Study Composite, Plate Theory, Finite Element Analysis
Experimentation FE Simulation
Specimen Preparation Tensile Specimen & Plate
FEM Modeling for Laminated Plate
Finite Element Analysis (FEA)
Results Comparison
Tensile Test To get material properties, E1,
E2, v12, v21, and G12.
Bending Test for Laminated Plate
End
CHAPTER II
LITERATURE REVIEW ON COMPOSITE MATERIALS
2.1 Introduction
Historically, modern composite get its start in the aerospace community when a
need for improved material performance was voiced. A composite in its most basic
definition are those that consist two or more materials on a macroscopic scale to produce
desirable properties for a given application. It is only when the constituent phases have
significantly different physical properties and thus the composite properties are different
from the constituent properties that we have come to recognize these materials as
composite.
Composite materials can offer significant advantages over common metallics and
plastics. Figure 2.1 shows the comparison of specific modulus between composite and
conventional metallics.
8
Among the advantages of composite over metallics are:
1. High strength-to-weight ratio (a carbon lamina is 4 to 6 times greater than that of
steel or aluminum).
2. High stiffness-to weight ratio (a carbon lamina is 3 to 5 times greater than that of
steel or aluminum).
3. High fatigue endurance limit.
4. Low corrosion.
5. Excellent damping characteristics.
6. Versatile can be tailored to meet the performance needs.
The most significant advantages over a plastic are:
1. Much greater strength
2. Much greater stiffness
3. Much lighter weight
Figure 2.1: Comparison of specific modulus between composite and metallics[1]
9
One of the fundamental differences between a composite and a metal is the
stress/strain relationship, as shown Figure2.2. Composites in general show a brittle
catastrophic failure, whereas metallics generally show a yield prior to failure.
Figure 2.2: Comparison of stress/strain relationship between composites and
metallics [1]
In practice, most composites consist of a bulk material called matrix and a
reinforcement materials called fiber, added primarily to increase the strength and
stiffness of the matrix. Fibre-reinforced composite materials are the most commonly
used modern composite materials that consist of high strength and high modulus fibers
in a matrix material.
Fibers could be carbon, fiberglass, Kevlar, polyester, nylon, ceramics, and boron.
The matrix material is typically an epoxy, thermoplastic, polyester, vinyl ester, ceramics,
or even metallics. In these composite, fibers are the principal load-carrying members,
while the matrix materials keeps the fibers together, acts as a load-transfer medium
between the fibers, and protects the fibers from being exposed to the environment. The
matrix is considerably lower density, stiffness, and strength than the fibers. However,
the combination of fibers and matrix can have very high strength and stiffness yet still
have low density.
10
The fibers and matrix materials used in composites are either metallic or
nonmetallic. The common metals fibre materials in use are aluminum, copper, iron,
nickel, steel, and titanium while the organic fiber materials in use are glass, carbon,
boron, and graphite materials [2]. Composite materials are commonly formed in three
different types as shown in Figure 2.3:
1) Fibrous composites
2) Laminated composites
3) Particulate composites
Figure 2.4 shows the comparison between the conventional materials and
composite materials.
a) Fibrous b) Particulate c) Laminated Composite Composite Composite
Figure 2.3: Classes of Composite [2]
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Figure 2.4: Comparison between the conventional materials and composite materials [3]
2.1.1 Fibrous composites
A fibrous composite is the composite that consist fiber in a matrix. The stiffness
and strength of the fibrous composite comes from the fibers that are stiffer and stronger
than the same materials in bulk form. Whisker is the shorter fibers that exhibit better
strength and stiffness properties than long fibers. Long fibers are used in straight form or
woven form.
A fiber is characterized geometrically not only by its very high length-to-
diameter ratio but by its near crystal-sized diameter. Strengths and stiffness of a few
selected fibers materials are shown in Table 2.1. The strength-to density and stiffness-to-
density ratios are usually used as indicators of the effectiveness of a fiber [4]. The long
dimension reinforcement prevents the growth of the incipient cracks normal to the
reinforcement that might lead to failure. Therefore fibers are effective in improve the
fracture resistance of the matrix.
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Table 2.1: Fiber and wire properties (Source: Adapted from Dietz. By permission of the
American Society for Testing and Materials, 1965.) [4]
Fiber or wire
Density,ρ lb/in3
(kN/ m3)
Tensile strength, S 103 lb/in2 (GN/ m2)
S/ρ 105 in (km)
Tensile stiffness, E 106 lb/in2 (GN/ m2)
E/ρ 107 in (Mm)
Aluminum 0.097 (26.3) 90 (0.62) 9 (24) 10.6 (73) 11 (2.8)
Titanium 0.17 (46.1) 280 (1.9) 16 (41) 16.7 (115) 10 (2.5)
Steel 0.282 (76.6) 600 (4.1) 21 (54) 30 (207) 11 (2.7)
E-glass 0.092 (25) 500 (3.4) 54 (136) 10.5 (72) 11 (2.9)
S-glass 0.9 (24.4) 700 (4.8) 78 (197) 12.5 (86) 14 (3.5)
Carbon 0.051 (13.8) 250 (1.7) 49 (123) 27 (190) 53 (14)
Beryllium 0.067 (18.2) 250 (1.7) 37 (93) 44 (300) 66 (16)
Boron 0.093 (25.2) 500 (3.4) 54 (137) 60 (400) 65 (16)
Graphite 0.051 (13.8) 250 (1.7) 49 (123) 37 (250) 72 (18)
Whisker has essentially the same near crystal-sized diameter as fiber, but is very
short and stubby. Thus, a whisker is more perfect than a fiber and exhibits even higher
properties. Whiskers are obtained by crystallization on a very small scale resulting in a
nearly perfect alignment of crystal.
13
2.1.2 Laminated Composites
Laminated composites consist of layers of various materials. There must be at
least two different materials are bonded in laminated composites. Lamination is used to
combine the best aspects of the constituent layers in order to achieve a more useful
material. Each layer of the composite usually very thin and hence cannot be directly
used. The layers can be formed in various orientations to form a multiplayer composite
used for engineering applications [4]. The example of the laminated composites is
bimetals, clad metals, laminated glass, plastic-based laminates, and laminates fibrous
composites.
Bimetal is the laminate that combines two different metals with significantly
different coefficients of thermal expansion. Under change in temperature, bimetals warp
or deflect a predictable amount and are well suited for use in temperature measuring
devices. The cladding or sheathing of one metal with another is done to obtain the best
properties of both. This is the concept of protection of one layer of material by another.
Laminated fibrous composites are a hybrid class of composites involving both
fibrous composites and lamination techniques. The common name of this composite is
laminated fiber-reinforced composites. The layers of fiber-reinforced materials are built
up with the fiber directions of each layer typically oriented in different directions to give
different strengths and stiffness in the various directions. Therefore, the strengths and
stiffness of the laminates fiber-reinforced composites can be tailored to the specific
design requirements of the structural element being built.
14
2.1.3 Particulate Composites
Particulate Composites consist of particles of one or more materials suspended in
a matrix of another material [4]. Particle can be defined as a non-fibrous and generally
has no long dimension with the exception of platelets. The particles can be either
metallic or nonmetallic as can the matrix. Thus, there exist four possible combinations of
it as: metallic in nonmetallic, nonmetallic in metallic, nonmetallic in nonmetallic,
metallic in metallic. Metal matrix composites are an example of nonmetallic in metallic
composites. Particulate composites are differ from the fiber types composite in the
distribution of the additive constituent is usually random rather than controlled. Thus,
particulate composites are usually considered as isotropic.
The dimensions of the reinforcement determine its capability of contributing its
properties to the composites. Particles are not very effective in improving the fracture
resistance of the composite. Particles also share the load but as much smaller extent than
those fibers in fibrous composite that lies parallel to the direction of load. Particles are
effective in improving the stiffness but do not offer much strengthening to the
composites. Particles are commonly used just simply to reduce the cost of the
composites.
2.2 Fiber
Fibers are the dominant constituents of most composite system. The function of
the fibre is to produce high strength and stiffness at lowest weight in a combination with
15
matrix. One of the main objectives of any design should be able to place the fibers in
positions and orientation so that they are able to contribute efficiently to load-carrying
capability. The amount of fibre usually expressed in term of the volume fraction of fibre,
Vf or weight fraction, Wf. Properties of some common types of fibres as well as some
conventional materials is given in Table 2.2.
The functional requirements of fibers in a fiber/matrix composite are that they
should have:
1) A high modulus of elasticity to give stiffness to the composite.
2) A high ultimate strength.
3) A low variation of strength between individual fibers.
4) Stability during handling.
5) A uniform diameter.
16
Table 2.2: Properties of fiber and conventional bulk materials (*Virgin strength values.
Actual strength values prior to incorporation into composite are approximately 2.1 GPa)
Material
Tensile Modulus (E)
(GPa)
Tensile Strength (σu)
(GPa)
Density (ρ)
(g/ cm3)
Specific modulus
(E/ρ)
Specific Strength (σu /ρ)
Fibers
E-glass 72.4 3.5 2.54 28.5 1.38
S-glass 85.5 4.6 2.48 34.5 1.85
Graphite (high
modulus)
390.0 2.1 1.90 205.0 1.10
Graphite (high
tensile strength)
240.0 2.5 1.90 126.0 1.30
Boron 385.0 2.8 2.63 146.0 1.10
Silica 72.4 5.8 2.19 33.0 2.65
Tungsten 414.0 4.2 19.30 21.0 0.22
Beryllium 240.0 1.3 1.83 131.0 0.71
Kevlar 49
(aramid polymer)
130.0 2.8 1.50 87.0 1.87
Conventional
Materials
Steel 210.0 0.34 – 2.1 7.80 26.9 0.043 – 0.27
Aluminum alloys 70.0 0.14 – 0.62 2.70 25.9 0.052 – 0.23
Glass 70.0 0.7 – 2.1 2.50 28.0 0.28 – 0.84
Tungsten 350.0 1.1 – 4.1 19.30 18.1 0.057 – 0.21
Beryllium 300.0 0.7 1.83 164.0 0.38
17
2.2.1 Glass Fiber
Glass fibers account around 90% of the reinforcement used in structural
reinforced plastic application. The most common glass fibers are silica based (~50 –
60% SiO2) with addition oxides of calcium, boron, sodium, aluminium and iron. The
mechanical properties are not strongly dependent on composition, but chemical behavior,
which reflected in terms of durability and strength retention in corrosion environment, is
influenced by the details of the chemistry. Table 2.3 gives typical property values for
different glass types.
Table 2.3: Typical glass fiber properties [3]
Strength (GPa) Glass type SG Thermal espansivity
( C-1)
Tensile modulus (MPa) Undamaged Strand from
roving A-glass 2.46 7.8×10-6 72 3.5 --
E-glass 2.54 4.9×10-6 72 3.6 1.7 – 2.7
AR-glass 2.7 7.5×10-6 70 – 75 3.6 1.5 – 1.9
S/R-glass 2.5 -- 85 4.5 2.0 – 3.0
E-glass has low alkali content and is the commonest glass in the market and is
used in the construction industry. It is employed widely, especially with polyester and
epoxy resins. It has good strength, stiffness, electrical, weathering properties, and a
reasonable Modulus Young. A-glass has high alkali content and was formerly used in
the aircraft industry but is now gradually going out of production. C-glass (C for
corrosion) has a higher resistance to chemical corrosion than E-glass but is more
expensive and has lower strength properties. S-glass is produced for extra high strength
and high modulus applications in aerospace and space research. These glass strands for
18
thermosetting resins may be used in a number of different forms such as chopped strands,
chopped strand mat (CSM), continuous random mat (CRM), woven fabric with varying
architectures, and milled glass fiber powder.
2.2.2 Carbon Fiber
Carbon fibers are very thin fibers and are typified by a combination of low
density, high strength and high stiffness. They have diameters between 6 and 10 µm.
Carbon fibers consist of 99.9% of chemically pure carbon. Carbon fibers are the
predominant high strength; high modulus reinforcement used in the fabrication of high
performance resin- matrix composites. There are two general sources for the commercial
production o carbon fibers: synthetic fibers, similar to those used for making textiles,
and pitch, which is obtained by the destructive distillation of coal.
The textile fiber polyacrylonitrile (general known as PAN) is a synthetic fibre.
The high-strength bonds between carbon atoms in the layer plane results in an extremely
high modulus, while the weak van der waals-type bond between the neighboring layers
results in a lower modulus in that direction. Compare with fiberglass, advanced
composites are superior in lightweight and high stiffness but has similar strength. The
properties of the three well-known carbon fibers are given in Table 2.4.
19
Table 2.4: Properties of carbon fiber [5]
Property, units
Pitch
Rayon
PAN
Tensile strength, MPa 1550 2070 – 2760 2480 – 3100
Tensile Modulus, GPa 380 415 – 550 200 – 345
Specific gravity 2.0 1.7 1.8
Elongation, % 1 -- 0.6 – 1.2
Coefficient of thermal expansion
Axial (10-6/ C)
Transverse (10-6/ C)
-1.6 to –0.9
7.8
--
--
-0.7 to –0.5
7 – 10
Fiber diameter, µm 10 – 11 6.5 7.5
2.2.3 Aramid Fiber ( Kevlar)
The aramid fiber forming polymer, that is, the aromatic polyamides. Aramid
fibers are available in two forms: low and high modulus. The main advantage of aramid
is the very low density (lower than glass and carbon), giving high values of specific
strength and stiffness combined with excellent toughness.
20
2.2.4 Boron Fiber
Boron fibers were among the first fiber specially developed for advanced
composites. They have a density similar to glass but a tensile modulus six times greater.
Because of their large size and stiffness boron filaments cannot be woven into cloths or
handled like other fibers, so they are usually processed in parallel arrays of single
thickness sheets or tapes.
2.3 Matrix
Matrix can be taken in the form of almost any material. There are three main
materials that used as matrix in composite. That is metal matrix, polymer matrix, and
ceramic matrix. However, those that have attracted most interest are those based on
polymeric systems.
The matrix should fulfill certain function. These are:
1) To bind the fibers together and protect their surface from damage during
service life to the composite.
2) To transfer stresses to the fibers efficiently by adhesion and/ or friction.
3) To disperse the fibers and separate them.
4) To be chemically and thermally compatible with fibers.
21
2.3.1 Polymer Matrix Composites (PMC)
These are the most common and will the main area of discussion in here. Also
known as FRP - Fiber Reinforced Polymers (or Plastics), these materials use a polymer-
based resin as the matrix, and a variety of fibers such as glass, carbon and aramid as the
reinforcement. Polymers used as matrix can be divided into two main groups:
thermoplastics and thermosets. Since Polymer Matrix Composites combine a resin
system and reinforcing fibers, the properties of the resulting composite material will
combine something of the properties of the resin on its own with that of the fibers on
their own. Figure 2.5 shows the tensile stress-strain curve for fiber, FRP composite, and
resin.
Overall, the properties of the composite are determined by:
1) The properties of the fiber
2) The properties of the resin
3) The ratio of fiber to resin in the composite (Fiber Volume Fraction)
4) The geometry and orientation of the fibers in the composite
22
Figure 2.5: Tensile stress-strain Curve for fiber, FRP and resin [6]
2.3.2 Thermoplastic
Thermoplastics polymers consist of linear molecules, which are not
interconnected. This means they have no chemical linkage between the chains so they do
not undergo irreversible cross-linking reactions, but instead melt and flow on application
of heat and pressure. The chemical valency bond along the chain is extremely strong, but
the forces of attraction between the adjacent chains are weak. Because of their
unconnected chain structure, thermoplastics may be repeatedly softened and hardened by
heating and cooling respectively; with each repeated cycle, however, the materials tend
to become more brittle. Example of thermoplastics is nylon, polyehtheretherketone
(PEEK), polybutylene terephthalate, polycarbonate, polyethylene, and polysulphone.
23
Some of the advantages of the thermoplastic are:
:
1) Indefinite shelf life.
2) Good toughness.
3) High impact strength and fracture resistance.
4) Higher strains to failure.
5) Processing is concerned with physical transformations only.
However, most of the thermoplastic resins can be eliminated because of
inadequate mechanical performance at high temperatures.
2.3.3 Thermoset
Thermoset polymer is formed by a chemical reaction. In the first stage, a
substance consisting of a series of long chain polymerized molecules, similar to those
present in thermoplastics. In the second stage of the process, the chains become cross-
linked; this reaction can take place either at room temperature or under the application of
heat and pressure. The resultant materials will not flow and cannot be softened by
heating. The example of the thermosets is epoxy, melamine, phenolic, polyester,
polymide, ureas. The polymer matrix that will be used in preparation of the laminated
plate in this project comes from this group of polymer.
24
Some of the advantages of the thermoset are:
1) Low viscosity level. Hence, is efficiently fluid to allow processing without
further modification, while others need application of heat or the use of diluents
to lower the viscosity level.
2) Less creeps and stress relaxation then thermoplastics.
2.3.3.1 Polyester
Polyesters are the most commonly used of polymeric resin materials. The major
advantage of this resin is the ability for cure at room temperature. This allows large and
complex structures to be fabricated where an oven cure would not be practical. They
consist of a relatively low molecular weight unsaturated polyester dissolved in styrene.
Styrene cures the resin by polymerization and forms cross-links across unsaturated sites
in the polyester. The curing reaction is strongly exothermic. This will generate heat that
can damage the final laminate. Styrene based unsaturated polyester resins have not been
found of interest for carbon fiber laminated applications.
The popularity of the polyester cause a family of resin has been developed to
offer specific properties. A variety of products is possible with respect to the backbone
chemistry, which allows the physical, thermal and chemical properties of the cured
products to be influenced. Polyester resins are considered as a potential alternative
because it can be demonstrated that differences in the chemical nature as compared to
epoxies do not necessarily result in different composite properties.
25
2.3.3.2 Epoxy
Epoxy resins are often used for the advanced structural applications. These resins
are the primary matrix materials used in carbon fiber composites. There are generally
two parts systems consisting of an epoxy resin and a hardener, which is either an amine
or anhydride. Epoxy resins can be modified in various ways to give a broad spectrum of
properties after cure and to meet a diverse range of processing condition. The higher
performance epoxies require the application of heat during a controlled curing cycle to
achieve the best properties. Table 2.5 shows the typical properties of cast resin systems.
There are many resin curing agent combinations and the many different curing
conditions that may be employed for proper cure. Hence, this allows the modification of
the following properties:
1) Heat resistance (glass transition of the resin)
2) Moisture absorption and performance in ht wet environment
3) Fracture toughness and impact resistance
26
Table 2.5: Typical properties of cast resin system [7]
Property Polyester Epoxy
Specific gravity 1.1 – 1.5 1.2 – 1.3
Impact strength (J/m) 16 – 32 8 – 80
Density (Mgm-3) 1.2 – 1.5 1.1 – 1.4
Poisson ratio 0.37 – 0.39 0.38 – 0.4
Thermal conductivity (W/m/°C) -- 0.17 – 0.21
Tensile strength (MPa) 40 – 90 55 – 130
Compression strength (MPa) 90 – 250 100 – 200
Flexural strength (MPa) 60 – 160 125
Tensile modulus (GPa) 20 – 44 28 – 42
CHAPTER III
THEORETICAL ANALYSIS OF COMPOSITE
3.1 Analysis of Lamina
A lamina or ply is a flat (sometimes curved as in a shell) arrangement of
unidirectional or woven fibers in a matrix. It represents a fundamental building block
for composite laminates. Lamina is made of two or more constituent materials that
cannot be detected. The two typical lamina are shown in Figure 3.1 along with their
principal materials axes which are parallel and perpendicular to the fiber direction.
Unidirectional fiber-reinforced laminas exhibit the highest strength and modulus in the
direction of the fibers, but they have very low strength and modulus in the transverse
direction to the fibers. Discontinuous fiber-reinforced composite have lower strength
and modulus than continuous fiber-reinforced composite.
28
Unidirectional Fibers Woven Fibers
Figure 3.1: Two principles typical of lamina [4]
Lamina is the basic building block in a laminated fiber-reinforced composite.
Thus, the knowledge about the mechanical behavior of a lamina is essential to the
understanding of laminated fiber-reinforced structures. In formulating the constitutive
equations of a lamina we assume that [4]:
1) A lamina is a continuum, i.e., no gaps or empty spaces exist.
2) A lamina behaves as a linear elastic material.
The first assumption amounts to considering the macromechanical behavior of a
lamina. If the fiber-matrix debonding and fiber breakage, for example, are to be
included in the formulation of the constitutive equations of a lamina. The second
assumption implies that the generalized Hooke’s law is valid. It should be noted that
the two assumptions could be removed if we were to develop micromechanical
constitutive models for inelastic (e.g., plastic, viscoelastic, etc.) behavior of a lamina.
29
3.1.1 Stress-strain Relations for Plane Stress in Specially Orthotropic Lamina
If the material has a texture like wood or unidirectionally reinforced fiber
composites .The modulus E1 in the fiber direction will typically be larger than those in
the transverse directions (E2 and E3). When E1 ≠ E2 ≠ E3, the material is said to be
orthotropic. A unidirectional fiber-reinforced lamina is treated as an orthotropic
material whose material symmetry planes are parallel and transverse to the fiber
direction. The material coordinate axes x is taken to be parallel to the fiber, while the y-
axes transverse to the fiber direction in the plane of the lamina as shown in Figure 3.2.
Figure 3.2: Specially orthotropic lamina
The stress-strain relations for specially orthotropic material by taking account the
normal and shear stress and deformations are given as below:
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
12
2
1
66
2212
1211
12
2
1
0000
γεε
τσσ
QQQQQ
(3.1)
30
Where the ijQ , is the reduced stiffness, are
2112
111 1 vv
EQ
−=
2112
121
2112
21212 11 vv
Evvv
EvQ
−=
−=
(3.2)
2112
222 1 vv
EQ−
=
1266 GQ =
The 5th elastic constant ijv is a function of the others
j
ji
i
ij
Ev
Ev
= i, j = 1, 2, ….6 (3.3)
where,
iE , jE - Young’s modulus in i, j direction respectively
12G - Shear modulus in 1-2 plane
ijv - Poisson’s ratio for transverse strain in j-direction when
subjected to a stress in the i-direction
i
jijv
εε
−= (3.4)
31
3.1.2 Stress-strain Relations For Plane Stress In Generally Orthotropic Lamina
As mentioned previously, laminas are often constructed in such a manner that
the principal material directions do not coincide with the natural direction of the body.
This is not to be interpreted as that the material is itself is not longer orthotropic. We
are just looking at an orthotropic material in a coordinate system that oriented at some
finite angle to the principle material coordinate system as shown in Figure 3.3. This
lamina is called generally orthotropic lamina.
Figure 3.3: Generally orthotropic lamina
The transformation equations for expressing stress-strain relationship in an x-y
coordinate system
[ ]⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
=⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
xy
y
x
xy
y
x
Qγεε
τσσ
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
xy
y
x
xy
y
x
QQQQQQQQQ
γεε
τσσ
662616
262212
161211
(3.5)
32
In which
)cos(sincossin)22(
cossin)2(cossin)2(
cossin)2(cossin)2(
coscossin)2(2sin
)cos(sincossin)4(
sincossin)2(2cos
4466
226612221166
3662212
366121126
3662212
366121116
422
226612
41122
4412
2266221112
422
226612
41111
θθθθ
θθθθ
θθθθ
θθθθ
θθθθ
θθθθ
++−−+=
+−+−−=
+−+−−=
+++=
++−+=
+++=
QQQQQQ
QQQQQQQ
QQQQQQQ
QQQQQ
QQQQQ
QQQQQ
(3.6)
The bar over the ijQ matrix denotes that we are dealing with the transformed reduced
stiffness instead of the reduced stiffness, ijQ .
3.2 Theory of Plate
The evaluation of the fundamental equations of orthogonal-stiffened plates is
based on the following assumptions, which are accepted in the classical theory of
elasticity of thin plates [8].
a) The linear elements perpendicular to the middle plane of the plate before
bending remain straight and normal to the deflection surface of the plate after
bending.
33
b) The materials of elements follow the Hook’s law, where the materials are
elastic, continuum, homogeneous, and different elastic characteristic in both x-
and y-direction.
c) The displacements of the points of the middle plane, in normal direction to this
plane, are small in comparison to the thickness of the plate.
d) The normal stress transverse to the plane of the plate can be disregarded.
We consider a thin plate subjected to pure bending moments of intensity Mx and
My per unit length uniformly distributed along its edges. We take the xy-plane to
coincide with the middle plane of the plate before deflection and the x and y-axes along
the edges of the plate as shown in Figure 3.4. The z-axes are taken positive downward.
Figure 3.4: Plate subjected to pure bending [8]
These moments are consider positive when they produce compression in the
upper surface of the plate and tension in the lower as shown in Figure 3.5. The
thickness of plate, h is considered small in comparison with other dimension.
Let us consider an element cut out of the plate by two pairs of planes parallel to
the xz and yz planes as shown in Figure 3.5(a). Assuming that during bending, the
lateral sides of the element remain plane and rotate about the neutral axes nn to remain
34
normal to the deflected middle surface of the plate. Thus, the middle plane of the plate
does not undergo any extension during bending and is therefore a neutral plane.
(a) (b)
Figure 3.5: (a) Direct stress on lamina of plate element. (b) Radii of curvature of
neutral surface. [8]
Let rx and ry denote the radii of curvature of the neutral surface in sections
parallel to the xz and yz planes respectively as shown in Figure 3.5(b). The strain εx
and εy in the x and y direction of an element lamina abcd at a distance z from the
neutral surface are given by,
yy
xx r
zrz
== εε (3.7)
where
rx, ry - Radii of curvature of the neutral surface in sections
parallel to the xz and yz planes respectively as shown in
Figure 3.5(b)
35
z - Distance from the neutral surface.
εx, εy - Strain in the x and y direction.
The strain εx and εy in term of the normal stresses σx and σy acting on the element are
given by,
)(1
)(1
xyy
yxx
vE
vE
σσε
σσε
−=
−= (3.8)
Substituting equation (3.7) into equation (3.8), the corresponding stresses in the lamina
abcd are
⎟⎟⎠
⎞⎜⎜⎝
⎛+
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛+
−=
xyy
yxx
rv
rvEz
rv
rvEz
111
111
2
2
σ
σ
(3.9)
These stresses are proportional to the distance z of the lamina abcd from the
neutral surface and depend on the magnitude of the curvatures of the bent plate. The
normal stresses distributed over the lateral sides of the element must be equal to the
external moments Mx and My. Thus, we obtain the equations,
∫
∫
−
−
=
=
2/
2/
2/
2/
h
hyy
h
hxx
zxzxM
zyzyM
δδσδ
δδσδ
(3.10)
36
Substituting equation (3.10) for σx and σy, we obtain,
zrv
rvzEM
zrv
rvzEM
xy
h
hy
yx
h
hx
δ
δ
⎟⎟⎠
⎞⎜⎜⎝
⎛+
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛+
−=
∫
∫
−
−
11
11
2
22/
2/
2
22/
2/
(3.11)
If )1(121 2
3
2
22/
2/ vhEz
vzED
h
h −=
−= ∫
−
δ
Then,
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛+=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛+=
2
2
2
2
2
2
2
2
1
1
xwv
ywD
rv
rDM
ywv
xwD
rv
rDM
xyy
yxx
(3.12)
D is the flexural rigidity of the pate and w denotes the deflection of any point on the
plate in the z direction.
37
3.3 Analysis of Laminate
A laminate is a collection of stacked lamina with various orientations of
principle materials directions in the lamina. The major purpose of lamination is to tailor
the directional dependence of strength and stiffness to match the loading environment
of the structural element. Once lamination is complete the stack of lamina are now
referred to as a laminate as shown in Figure 3.6.
The laminate takes on a combination of properties based upon the lamina
orientation, fiber type, resin or matrix materials type, and the ratio of fiber-to-matrix
content. Depending upon the angles at which the plies are stacked, an infinite number
of physical and material properties can be produced for a given fiber and matrix
materials.
The mismatch of material properties between layers can cause shear stresses
produced between the layers, especially at the edges of a laminate. This may cause
delamination in the laminate structure. Besides, during the laminates manufacturing,
materials defects such as interlaminar voids, delamination, incorrect orientation,
damaged fibers, and variation in thickness may be introduced. Therefore, analysis and
design procedures should account for any defects.
38
Figure 3.6: Principle and structural coordinates, and lamination [1]
3.3.1 Classical Laminated Plate Theory
Classical laminate plate theory is an extension of the theory for bending of
homogeneous plates, but with an allowance for in-plane tractions in addition to bending
moments, and for the varying stiffness of each ply in the analysis. In general cases, the
determination of the tractions and moments at a given location will require a solution
of the general equations for equilibrium and displacement compatibility of plates. This
theory is treated in a number of standard texts, and will not be discussed here [4].
In the classical laminated plate theory (CLPT) it is assumed that the Kirchhoff
hypothesis holds [2]:
1) Straight lines perpendicular to the middle surface (i.e., transverse
normal) before deformation remain straight after deformation.
2) The transverse normal do not experience elongation (i.e., they are
inextensible).
39
3) The transverse normal rotate such that they remain perpendicular to the
middle surface after deformation.
The first two assumptions imply that the transverse displacement is independent
of the transverse (thickness) coordinate and the transverse normal strain εz is zero. The
third assumption implies the zero shear strains, γ xz = 0, γ yz = 0.
3.3.2 Strains and Stress Variation in a Laminate
Knowledge of the variation of stress and strain through the thickness is essential
to definite the extensional and bending stiffness of a laminate. When definition the
stiffness of the laminate, the laminate is presumed to consist of perfectly bonded
lamina. Moreover, the bonds are presumed to be infinitesimimally thin as well as non-
shear-deformable. That is, the displacements are continuous across lamina boundaries
so that no lamina can slip relative to another. Therefore, the laminate acts as a single
layer with very special properties, but nevertheless acts as a single layer of material.
The implications of the Kirchhoff or the Kirchhoff-Love hypothesis on the
laminate displacement u, v, and w in the x, y, and z- direction are derived by the used
of the laminate cross section in the xz plane as shown in Figure 3.7.
40
Figure 3.7: Geometry of deformation in the xz plane [4]
The displacement in the x-direction of point B from the undeformed to the
deformed middle surface is u0. The line ABCD remains straight under deformation of
the laminate,
βcc zuu −= 0 (3.13)
Where β is the slope of the laminate middle surface in the x-direction.
x
w∂∂
= 0β (3.14)
Then, the displacement, u, at any point z through the laminate thickness is
x
wzuu∂∂
−= 00 (3.15)
Similarly, the displacement, in the y-direction is
y
wzvv∂∂
−= 00 (3.16)
41
By virtue of the Kirchhoff-Love hypothesis where εz = γ xz = γ yz = 0, the
laminate strains have been reduced to εx, εy, and γ xy. For small strains (linear elasticity),
the remaining strains are defined in terms of displacement as
xu
x ∂∂
=ε
yv
y ∂∂
=ε (3.17)
xv
yu
xy ∂∂
+∂∂
=γ
Thus, for the derived displacement u and v in Equation (3.15) and (3.16), the strains are
20
20
xw
zx
ux ∂
∂−
∂∂
=ε
20
20
yw
zyv
y ∂∂
−∂∂
=ε (3.18)
. yx
wz
xv
yu
xy ∂∂∂
−∂∂
+∂∂
= 02
00 2γ
or
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
+⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
=⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
xy
y
x
xy
y
x
xy
y
x
kkk
z0
0
0
γεε
γεε
(3.19)
Where the middle surface strains are
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
∂∂
+∂∂
∂∂∂∂
=⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
xv
yu
yvx
u
xy
y
x
00
0
0
0
0
0
γεε
(3.20)
42
and the middle surface curvatures are
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
∂∂∂∂∂∂∂
−=⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
yxw
ywxw
kkk
xy
y
x
02
20
2
20
2
2
(3.21)
The stress-strain relations for the kth layer of a multiplayer laminate can be written as
{ } [ ] { }kkk Q εσ = (3.22)
Thus, the stresses in the kth layer can be expressed in terms of the laminate middle
surface strains and curvatures as
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
+⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
xy
y
x
xy
y
x
kkxy
y
x
kkk
zQQQQQQQQQ
0
0
0
662616
262212
161211
γεε
τσσ
(3.23)
Value ijQ is different for each layer of the laminate.
43
3.3.3 Resultant Laminate Forces and Moments
The resultant forces and moments acting on a laminate are obtained by
integration of the stresses in each lamina through the laminate thickness as given below.
∫ ∑ ∫− = ⎪
⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
=⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
=⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
−
2/
2/ 11
t
t
N
k
kxy
y
xz
zkxy
y
x
xy
y
x
dzdzNNN
k
k τσσ
τσσ
(3.24)
and
∫ ∑ ∫− = ⎪
⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
=⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
=⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
−
2/
2/ 1 1
t
t
N
k
kxy
y
xz
zkxy
y
x
xy
y
x
dzzdzzMMM
k
k τσσ
τσσ
(3.25)
Nx is a force per unit length (width) of the cross section of the laminate as shown in
Figure 3.8(a). Similarly Mx is a moment per unit length as shown in Figure 3.8(b). zk
and zk-1 are defined in Figure 3.9, noted that z0 = -t/2.
(a) (b)
Figure 3.8: (a) In-plane forces on a flat laminate, (b) Moments on a flat laminate [4]
44
Figure 3.9: Geometry of an n-layered laminate [4]
When the lamina stress-strain relations, equation (3.23), are substituted into equations
(3.24) and (3.25), we get
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
+⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
∫ ∫∑− −=
k
k
k
k
z
z
z
z
xy
y
x
xy
y
xN
K
kxy
y
x
zdzkkk
dzQQQQQQQQQ
NNN
1 10
0
0
1662616
262212
161211
γεε
(3.26)
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
+⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
=⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
∫ ∫∑− −=
k
k
k
k
z
z
z
z
xy
y
x
xy
y
xN
K
kxy
y
x
dzzkkk
zdzQQQQQQQQQ
MMM
1 1
2
0
0
0
1662616
262212
161211
γεε
(3.27)
However, 0xε , 0
yε , 0xyγ , xk , yk and xyk are not function of z but are the middle
surface values so can removed from under the summation signs. Thus equations (3.26)
and (3.27) can be rewritten as
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
xy
y
x
xy
y
x
xy
y
x
kkk
BBBBBBBBB
AAAAAAAAA
NNN
662616
262212
161211
0
0
0
662616
262212
161211
γεε
(3.28)
45
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
xy
y
x
xy
y
x
xy
y
x
kkk
DDDDDDDDD
BBBBBBBBB
MMM
662616
262212
161211
0
0
0
662616
262212
161211
γεε
(3.29)
Where
( ) ( )∑=
−−=N
kkkkijij zzQA
11
( ) ( )∑=
−−=N
kkkkijij zzQB
1
21
2
21 (3.30)
( ) ( )∑=
−−=N
kkkkijij zzQD
1
31
3
31
ijA are called extensional stiffness, ijB are called coupling stiffness, and ijD are called
bending stiffness.
3.3.4 Symmetric and Unsymmetrical Laminates
The laminate that is symmetric in geometry and materials properties about the
middle surface called symmetric laminate as shown in Figure 3.10(a). For
unsymmetrical laminate there are not symmetric about the middle surface as shown in
Figure 3.10(b). Because of the symmetric, all the coupling stiffness, that is ijB can be
shown to be zero. While for the unsymmetrical laminate, ijB is not zero.
A general laminate has layer of different orientations θ where -90° ≤ θ ≤ 90°.
The laminate [0/45/90/90/45/0] and [-45/90/90/-45] are the examples of symmetric
laminates. The laminate [0/90/0/90/0/90], [-30/60/-30/30/-30/45] are the example of the
46
unsymmetrical laminate. The numbers in the bracket denote the orientation of the
lamina from the references axes as shown in Figure 3.3.
The elimination of coupling between bending and extension has two important
practical ramifications. First, the laminates are much easier to analyze than the
laminates with coupling. Second, the laminates do not tendency to twist from the
inevitable thermally induced contractions that occur during cooling following the
curing process. Symmetric laminates are commonly used unless special cases need an
unsymmetrical laminates. Many physical applications of laminated composites require
nonsymmetrical laminates to achieve design.
(a) Symmetric (b) Unsymmetrical
Figure 3.10: Cross-sectional views of laminates [6]
3.4 Stiffened Plate
Stiffened plates have been used for many years especially in the fields of
bridges, ships, aircraft and towers. With the advancement of fiber-reinforced composite
materials, current engineering application such as high-speed aircraft designs use these
same stiffened panel concepts incorporating the newer materials. These newer
materials provide more design variables to optimize and improve the chances of
47
minimizing structural weight. By taking advantage of the beneficial tailoring capability
of the material, the panel face sheets and core sheet become orthotropic by them,
further complicating stiffness, thermal expansion, and thermal bending formulations.
Stiffeners are used when it is required to stiffen essentially flat load-bearing
panels. These stiffeners can be any geometry shape, but often “top hat” sections are
used as shown in Figure 3.11. These sections can be varied in strength and stiffness by
using different configurations. Ideally the top hats will be bonded to the load-bearing
plate rather than bolted, to enable the maximum stiffness of the overall unit to be
achieved. Figure 3.12 illustrates the typical stiffened panels.
Figure 3.11: A hat-stiffened plate [6]
Figure 3.12: Various types of stiffened panels [6]
48
Stiffeners are commonly used to increase the bending stiffness of thin-walled
members (plates and shells). The stiffeners add an extra dimension of complexity to the
model compared to unstiffened plates. They can carry more service load than
unstiffened plates for a given unit weight. Stiffened panels are quite efficient for lightly
loaded areas and applications of high temperature gradients. These qualities make them
desirable for use as hot structure on high-speed vehicles where weight reduction is a
paramount objective. Figure 3.13 represents the schematic of the typical stiffener
geometry. Stiffener can be divided into two main groups. First is the closed section
stiffener such as hat-shaped stiffener and the second is the open section such as I, T and
J-shaped stiffeners.
Figure 3.13: Schematic of T, J, blade, and Hat stiffener geometry [6]
49
3.5 Bending of Simply Supported Rectangular Plates
3.5.1 Governing Equations
Let us consider the general class of laminated rectangular plates that are simply
supported along edges x = 0, x = a, y = 0, y = b and subjected to an external transverse
load q (x,y) as shown in Figure 3.14, in the absence of thermal effects and in plane
forces.
Figure 3.14: Plate Geometry [9]
The general equation of motion governing bending deflection w for a unidirectional
laminated plate can be expressed by the following equation,
qywD
yxwDD
xwD =
∂∂
+∂∂
∂++
∂∂
4
4
2222
4
66124
4
11 )2(2 (3.31)
To obtain the solution for the deflection equation (3.31) must be solved subject to the
simply supported boundary conditions on all edges of the rectangular plate.
At x = 0 and x = a w = Mx = 0
At y = 0 and y = b w = My = 0 (3.32)
50
Where the bending moments are related to the transverse deflection by the following
equations,
yxwDM
ywD
xwDM
ywD
xwDM
xy
y
x
∂∂∂
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
−=
2
66
2
2
222
2
12
2
2
122
2
11
2
(3.33)
3.5.2 The Navier Solution
In Navier method, the displacement w is expanded in a double trigonometric
(Fourier) series in terms of unknown parameters. The choice of the trigonometric
functions in the series is restricted to those which satisfy the boundary conditions of the
problem. The load q(x, y) is also expanded in a double trigonometric series.
Substitution of the displacement and load expansions into the governing equation
should result in an invertible set of algebraic equations among the parameters of the
displacement expansions. The boundary conditions in equation (3.32) are satisfied by
the following form of the transverse deflection,
yxWyxw mnmn
βα sinsin),(11∑∑∞
=
∞
=
= (3.34)
where
b
nanda
m πβπα ==
51
Wmn are coefficients to be determined such that the governing equation (3.31) is
satisfied everywhere in the domain of the plate. The load can also be expanded in the
series form as,
yxQyxq mnmn
βα sinsin),(11∑∑∞
=
∞
=
= (3.35)
where
dxdyb
yna
xmyxqab
Qab
mn ⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛= ∫∫
ππ sinsin),(400
(3.36)
Substitute the equations (3.34), (3.35) and (3.36) into equation (3.31), yields
[ ]{ } 0sinsin)2(2 422
226612
411
11=++++−∑∑
∞
=
∞
=
yxQDDDDW mnmnmn
βαββαα
(3.37)
The equation must hold for every point (x,y) of the domain 0< x < a and 0 < y < b, the
expression inside the curl brackets should be zero for every m and n. This yields
mn
mnmn d
QW = (3.38)
where
[ ]422
2226612
44114
4
)2(2 nDsnmDDsmDb
dmn +++=π (3.39)
where s = b/a
52
Then the equation (3.34) becomes
yxdQyxw
mn
mn
mnβα sinsin),(
11∑∑∞
=
∞
=
= (3.40)
The load coefficients Qmn are different for various types of loading. In particular, for
uniformly distributed load q (x, y) = 0, a constant, on the surface of the plate, we have
mnq
Qmn 2016
π= for m, n odd. (3.41)
For a point load Q0 located at (x0, y0 ), the load coefficients are given by q (x, y) = Q0.
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛=
byn
axm
abq
Q oomn
ππsinsin
4 0 m, n = 1, 2, 3, …… (3.42)
CHAPTER IV
FINITE ELEMENT IMPLEMENTATION
4.1 Introduction
The finite element method (FEM) is a powerful computational technique for the
solution of differential and integral equations that arise in various fields of engineering
and applied science. Typical problems areas of interest in engineering and mathematical
physics that are solvable by use of the finite element method include structural analysis,
fluid flow, mass transport, and electromagnet potential [9].
The basic idea of the finite element method is to view a given domain as an
assemblage of simple geometric shapes, called finite element, for which it is possible to
systematically generate the approximation functions needed in the solution of
differential equations by any of the variation and weighted-residual methods. The ability
to represent domains with irregular geometries by a collection of finite element makes
the method a valuable practical tool for the solution of boundary, initial, and eigenvalue
problem arising in various fields of engineering.
54
The model that produced by this method represents the ideal condition of the
problem. It is because in modeling the problem, we have to consider all the factors that
will influence the analysis result. For instance, in calculating the stresses in a composite
laminated plate under bending condition, the results are affected by the properties of the
composite materials and the angle of lamination. If there is any changing of these factors,
we need to calculate the results from the basic as mentioned in chapter III.
The finite element method is applied in analysis the continuum structure. This
structure consist the individual elements that connected with nodes as shown in Figure
4.1. In this modeling, we can obtain the deflections, stresses, strain and many other
related information which depends on the analysis that we done.
Figure 4.1: Finite element model
The finite element method (FEM) can be divided into three categories depending
on the nature of the problem to be solved. The first category is equilibrium problems or
time-independent problem. The majority of applications of the FEM are included in this
category. The solution of equilibrium problems in the solid mechanics area,
displacement distribution and stress distribution can be solved by this method.
55
The eigenvalue problems of solid and fluids mechanics are fall into the second
category. These are steady-state problems whose solution often requires the
determination of natural frequencies and mode shapes of vibration of solids and fluids.
Another class of eigenvalue problems includes in the stability of structures and the
stability of laminar flows.
The third category is the multitude of time-dependent or propagation problems of
continuum mechanics. This category is composed of the problems that result when the
time dimension is added to the problems of the first and second category.
As indicated previously, the finite element method has been applied to numerous
problems, both structural and nonstructural. This method has a number of advantages
that have made it very popular. The abilities are given below [10],
1) Model irregularly shaped bodies quite easily.
2) Handle general load conditions without difficulty.
3) Model bodies composed of several different materials because the element
equations are evaluated individually.
4) Handle unlimited numbers and kinds of boundary conditions.
5) Vary the size of the elements to make it possible to use small element where
necessary.
6) Alter the finite element model relatively easily and cheaply.
7) Include dynamic effects.
8) Handle nonlinear behavior existing with large deformations and nonlinear
materials.
56
4.2 Linear Static Analysis
The linear static analysis represents most of the basic analysis. Linear means that
the computed response—displacement or stress, for example is linearly related to the
applied force. Whereas, static means that the forces do not vary with time or, that the
time variation is insignificant and can therefore be safely ignored [6].
In this part, we presume that the structures are in equilibrium. When the applied
loads are shifted, the structure will return into the undeformed shape. In some cases, the
structures undergo to deform without any additional load. In this condition, the structure
becomes instable and subjected to buckle.
The static analysis equation is:
[K]{u} = {f}
where [K] is the system stiffness matrix (based on the geometry and properties), f is the
vector of applied forces (which we specify), and u is the vector of displacements that
need to compute. Once the displacements are computed, it will be used to compute
element forces, stresses, reaction forces, and strains.
The applied forces may be used independently or combined with each other. The
loads can also be applied in multiple loading subcases, in which each subcase represents
a particular loading or boundary condition. Multiple loading subcases provide a means
of solution efficiency, whereby the solution time for subsequent subcases is a small
fraction of the solution time for the first, for a particular boundary condition.
57
4.3 Finite Element Analysis Procedures
The FEA modeling is divided into two sections, namely; Modeling for normal
composite laminated plate and modeling for composite hat-stiffened laminated plate.
4.3.1 Modeling for Unstiffened Composite Laminated Plate
In this modeling, a 250 mm square plate will be created. The plate is made of
carbon fiber with average thickness 2.14 mm and the mechanical properties of the
carbon fiber are given in Appendix A. The model is simply supported around the outer
edge and a 100g gravity load is applied normal to the plate. The plate is modeled with
flat plate elements. Nodal displacements and element stresses are computed.
Figure 4.2: FEA model
This model uses SI units: millimeters (mm) for length, Newton (N) for force, and
second (sec) for time. Below describe the procedure to create the geometry, finite
element mesh, load and constraints.
250 mm
250 mm
58
1) Modeling the geometry of the plate
Start to create the plate by following this step of common, Geometry/Curve-
Line/Rectangle. From the appearance window, enter the first corner of rectangle
and normally we enter all zero for the first corner. Then enter the diagonally
opposite corner of the rectangle. In this model, the diagonally opposite corner is
250 for X and Y while Z equal to 0. Then click OK. Resize and center the display
of the rectangle by pressing Ctrl+A.
2) Creating the applied force area
To create the applied force area on the center of the rectangle chooses
Geometry/ Curve-Circle/Center. Enter the location at the center of circle with
X and Y equal to 125 and Z equal to 0. Click OK. Then enter the radius of the
circle equal to 22.5. Click OK and Cancel.
3) Cresting the Boundary Surface
We may use the Geometry Boundary Surface command to create a boundary.
A series of lines and curves with coincident endpoints are selected. Holes can be
added by picking existing curves inside the boundary curves that form closed
holes. Boundaries are created from any number of continuous curves. These
curves must be either joined at the ends or have coincident points and be fully
enclosed. They cannot just intersect. Boundaries can contain holes, as long as the
area of the hole is completely contained within the boundary and they do not
overlap. MSC/N4W will automatically determine which curves if any represent
holes in the boundary. Because of the arbitrary geometric nature of boundaries,
many models may require you to be more careful in the mesh generation process
to obtain a good mesh.
59
4) Defining the material properties
After creating the surface, the characteristics of the materials should be defined
by using command Material under the submenu Model. MSC/N4W supports
seven types of materials - Isotropic, 2D Orthotropic, 3D Orthotropic, 2D
Anisotropic, 3D Anisotropic, Hyperelastic (Mooney-Rivlin/Polynomial form),
and Other Types. From the appearance window, click the material type button
and select 2D orthotropic. In general the 2D material types should only be used
by plane (and axisymmetric) elements and the 3D formulations should only be
used by solid elements. For some analysis programs however, the 3D
formulations are used to add transverse properties to plate elements. Then enter
the properties of the material. For bending test, the material properties that
needed are modulus Young shear modulus Young, and Poisson’s ratio.
5) Defining the Element Properties
Select the submenu property from the menu model. Click on the Elem/Property
type button and under the volume element, check the Laminate common.
Model/ Property/(Element/property type)/ Laminate. Then click OK.
Properties of this type are different than those for any other type of element. We
must specify a material ID, thickness and orientation angle for each layer or ply
in the laminate. In general, we must list all plys in the laminate. If the laminate is
symmetric, the Symmetric Layers option can be set with only enter one half of
the layers. MSC/N4W supports up to 90 plys on a property, but only 18 at a time
can be displayed in the dialog box. By pressing Next or Prev, the dialog box will
scroll to show the other plys that make up the property that we are defining.
6) Meshing the model
Mesh the model by following this step of common. Mesh/Mesh Control/Size
along curves. This command defines the number and spacing of elements along
selected curves. When setting the mesh size using this method, it overrides all
60
point and default sizes. After selecting the curves, we will see the Mesh Size
along Curve dialog box. Choose a "Number of Elements" and then every curve
that we selected will be meshed with that number of elements. After defining the
mesh size along the curves, mesh the model by using the following command
Mesh/Geometry/Surface. Select all the surfaces and then click OK to mesh the
model.
7) Defining constraint on model
After the FEA model bas been mesh, we need to put the constraint on the model
as given in his common Model/ Constraint/ Set. The constraints must be
created in sets and we can create nodal constraints, geometry based constraints
or constraint equations. In this modeling, we use geometry based constraints that
allow us to select points, curves or surfaces to constrain before or after nodes are
on them. Geometry based constraints have three options, fixed, pinned or no
rotations. The model is simply supported around the outer edge, therefore we
use pinned command around the curves at the outer edge of the model. Simply
select the curves through the standard entity selection box, and then select the
type of constraint. Nodes attached to that curve will then be constrained upon
translation or expansion.
8) Defining load on model
Similarly to the constraint, we need to set the load first. Model/ Load/ Set. We
can make a new load set or activates an existing set. Enter an ID which does not
appear in the list of available sets. Then enter a title and press OK.
Then we put the load on the model. Model/Load/On Surface. Select the surface
where the load applied. In this problem，the load will be applied on the surface
at center of the model.
61
9) Analysis the FEA model
After the previous have been done, we can now start to analyze our FEA model.
There is several type of analysis that we can do with this software. It depends on
what type of output that we need. In this problem, the linear static analysis will
be done to obtain the stress distribution and the displacement. After the model
has been analyzed, we can obtain the analysis output in the form that we need by
using this command, View/Select. We can choose the deformed style and
contour for the verities output.
4.3.2 Modeling for Composite Hat-Stiffened Laminated Plate
Similarly to the normal laminated plate, a 250 mm square plate will be created.
The plate is made of carbon fiber with average thickness 2.14 mm and the mechanical
properties of the carbon fiber are given in Appendix A. The only different is in this
modeling, the plate is stiffened by a hat-shaped stiffener.
1) Modeling of the FEA model
In order to the complexity of the structure, the stiffened plate model is not
suitable to be drawn by using the device in MSC/N4W. The more appropriate
way to prepare the FEA model is using engineering technical drawing software
such as Autocad, Solidwork, etc. In this project, the SOLIDWORK software is
used to draw the FEA model. This model uses SI units: millimeter (mm.) for
length, Newton (N) for force, and seconds (sec) for time. Note that MSC/N4W
assumes a consistent set of units, so you need to be consistent and not mix units.
The detail drawing of the FEA model is enclosed at Appendix B. After finish the
drawing, save the drawing in ACIS format (*sat).
62
2) Importing the FEA model into FEM software
Start to import the FEA model by choosing the submenu import from file.
File/Import/Geometry. A window will come out and then choose the directory
where you want to import the model. After choosing model from the directory, a
window will appear. Under the Entity Options change the geometry scale factor
to 1. Then click OK. This is important because if we not change to scale 1, the
size of the model that we import is not coinciding with our actual model size. For
instance, if the actual height of the model is 1000 mm and we use scale factor
39.37, the height of model that we imported into the FEA software will become
393700 mm which is 39.37 times bigger than our actual model size.
3) Defining the material properties
As mentioned before, the characteristics of the materials are defined by using
command Material under the submenu Model. Choose the 2D orthotropic
material and a window will appear. Key-in the modulus Young, Shear modulus,
and Poisson’s ratio into the window to define the material properties of the
model.
4) Define the element properties
Because of the improperly analysis results obtained from the laminate property,
therefore the hat-stiffened plate model will be model by using solid property.
Select the submenu property from the menu model. Click on the Elem/Property
type button and under the volume element, check the solid common. Model/
Property/(Element/property type)/ Solid. Then click OK.
5) Meshing the model
Mesh the model by following this step of common. Mesh/Geometry/ Solids. A
window will come out and then under the basic curve sizing, change the Max
63
Element size to 10. Then click OK. Another window will appear, under the
property column, select the element property title that put in at step 4. Then click
OK to mesh the model. The model is mesh by using the tetrahedral element.
The following procedures in modeling the FEA model are same as the
unstiffened composite laminated plate from step 7 to step 9. The procedures listed above
are general steps in solving FEA structural problem. The additional steps are depending
on the type of the model and analysis that need to carry out.
CHAPTER V
EXPERIMENTAL PROCEDURES
5.1 Composite Fabrication
There are various techniques for the fibre-reinforced composite fabrication and
these may be considered into two main group [3]:
a) Open mould process in which during the mould operating, the material is in
contact with the mould on one surface only.
b) Closed mould technique in which the composite is shaped between the male
and female moulds
Both the open and closed mould process can be divided into three categories:
manual, semi-manual, and automatic. The manual techniques include the hand lay-up
and pressure bag. The semi-manual techniques cover the cold press, hot press and the
resin-injection method. The automatic techniques include pultrusion, filament winding
and injection moulding.
65
The driving factors behind manufacturing considerations for composite
materials are primarily cost effectiveness, the minimization of scrap, the control of
assembly operation and the sourcing of standard parts. Furthermore, products of
nominally the same form, but manufactured different routes, could have markedly
different properties. This not only affects the stiffness and strength, but also other
attributes such as surface finish, chemical resistivity and internal damping, as well as
electrical and thermal properties. This chapter will just discus the two commonly used
method in fabricate the laminated composite, which are hand lay-up and vacuum
bagging method.
5.1.1 Hand Lay-up Method
The hand lay-up technique is one of the oldest, simplest and most commonly
used methods for manufacture of composite, or fiber-reinforced, products. This
technique is best used where production volume is low and other forms of production
would prove too expensive. In this technique only one mould is used and this may be
either male or female. This is the process wherein the application of resin and
reinforcement is done by hand onto a suitable mold surface. The resulting laminate is
allowed to cure in place without further treatment [6].
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The typical process of the Hand lay-up is listed below:
1) Mold Preparation - A mold of the part to be made is created and a release
film is applied to the molds surface.
2) Gel Coating - This consists of a specially formulated resin layer that will
become the outer surface of the laminate when it is complete. This layer is
only necessary when a good surface appearance is required.
3) Hand Lay-Up - Fiberglass is applied in the form of chopped strand mat,
cloth or woven roving. Premeasured resin and catalyst (hardener) are then
thoroughly mixed together. To ensure complete air removal and
consolidation of the excess resin, serrated rollers are used to press the
material evenly against the mold. As shown in Figure 5.1.
4) Finishing - The composite is allowed to completely harden and any
machining or assembly can be performed.
Some advantages of the Hand lay-up process are:
1) Large and complex items can be produced.
2) Relatively little equipment investment is needed.
3) The start-up lead-time and cost are minimal.
4) Tooling costs are low.
5) Semiskilled workers are easily trained.
6) Design flexibility.
7) Molded-in inserts and structural changes are possible.
8) Higher fiber contents and longer fibres than with spray lay-up.
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Some disadvantages of the Hand lay-up process are:
1) A labor-intensive process.
2) A low volume process.
3) Longer curing times, since room temperature catalysts are usually used.
4) Part quality is very dependant upon operator skill.
5) Product uniformity is difficult among parts.
6) Only one good (molded) surface is obtained.
7) Waste produced is high.
8) Health and safety considerations of resins. The lower molecular weights of hand
lay- up resins generally mean that they have the potential to be more harmful
than higher molecular weight products. The lower viscosity of the resins also
means that they have an increased tendency to penetrate clothing etc.
Figure 5.1: Manual Lay-up process [6]
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5.1.2 Vacuum Bagging
This is basically an extension of the hand lay-up process described above where
pressure is applied to the laminate once laid-up in order to improve its consolidation.
This is achieved by sealing a plastic film over the wet laid-up laminate and onto the
tool. The air under the bag is extracted by a vacuum pump and thus up to one
atmosphere of pressure can be applied to the laminate to consolidate it [6].
Some advantages of the vacuum bagging process are:
1) Higher fiber content laminates can usually be achieved than with standard wet
lay- up techniques.
2) Lower void contents are achieved than with wet lay-up.
3) Better fibre wet-out due to pressure and resin flow throughout structural fibres,
with excess into bagging materials.
4) Health and safety: The vacuum bag reduces the amount of volatiles emitted
during cure.
Some disadvantages of the vacuum bagging process are:
1) The extra process adds cost both in labor and in disposable bagging materials.
2) A higher level of skill is required by the operators.
3) Mixing and control of resin content still largely determined by operator skill.
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Figure 5.2: Vacuum Bag mould assembly [6]
5.2 Laminate Preparation
The first step in preparation of the laminate is to choose the types of fibre and
matrix that will be used to fabricate a laminate. As mentioned in the previous chapter,
there are several types of fibre and matrix in the market, so in this project the materials
that will be used are the carbon fibre and epoxy as the resin.
Second step is to design the lamina orientation in the laminate. In this project,
the lamination that used in tensile test specimens are [0/0/0/0], [90/90/90/90],
[45/45/45/45], whereas for laminate plate is [0/90/90/0]. All the laminates that will be
produced are four layers laminate and symmetric to the middle surface of the laminate.
Thus, there is no coupling between bending and extension. The fabrication method that
will be used in this project is hand lay-up method. The mechanical properties that
produced by this method are not good because the difficulty in removing the entrapped
air.
70
5.2.1 Hand Lay-up Procedure
The procedures that used in the fabrication of composite plate are as follow:
1) Water is used to clean up the surface of glass plate in order to avoid any foreign
particles or dusts remain on the surface. The cleanliness of the glass surface will
affect the quality of the finish product.
2) A plastic sheet is placed on the glass surface. Silicon sealant is used to bond the
plastic to the glass plate.
3) The surface of the plastic sheet is cleaned using tissue paper before the first
layer of carbon fiber placed on it.
4) The mixture of the resin and hardener which according to the weight ratio
mentioned before is spread and flatten on the surface of the glass table by using
a brush.
5) A layer of fibre is placed on the resin. A roller is used to wet the fibre evenly
with the resin and to remove the entrapped air.
6) Repeat the procedures (4) and (5) until the desired layer or thickness of
laminate.
7) Anther plastic sheet is placed on the surface pf the laminated composite
produced. Again silicon sealant is used to bond between the two plastic sheet to
avoid any leakage occur.
8) The laminated composite is then cured at the room temperature at least 24 hours
to ensure that it is dry enough for further processes.
9) A finished laminated composite plate is obtained and shown in figure 5.3.
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Figure 5.3: A finished laminated composite plate
5.3 Tensile Test Specimen Preparation
The specimen size is prepared according to ASTM D-3039 standard, which is a
standard test method for tensile properties of polymer matrix composite materials.
Figure 5.4: Specimen Specification
72
The details of the tensile test specimen are given in Table 5.1 and Table 5.2.
From the tables we can mention that the materials used is carbon fibre and epoxy as the
resin. The unidirectional lamina is used in producing the specimen. The weight ratio of
resin to the fibre is 50 to 50. This means that by referring to the weight of the produced
laminate, 50 % of the weight is contributed by the fibre and 50 % by the resin. There is
another weight ratio that commonly use is 60 to 40. This means 60% resin and 40 %
fibre. Besides, we can also use volume ratio to determine the ratio of matrix to
reinforcement in laminate. But weight ratio is common use because of the easy
determination of the ratio compare with volume ratio.
The aluminum tab is attached to the specimen by using epoxy adhesive. Before
attaching the tab to the specimen, we need to do sand blasting process for the tab to
roughen the smooth surface. This is because the coarse surfaces will give good holding
compare with the smooth surface. This process has been done in the casting laboratory.
Figure 5.5 shows the specimens that have been produced by using hand lay-up
method. There are three specimens for each type of lamination. We need to get the
average of the experimental data so that our results are more accurate and close to the
exact data.
Table 5.1: Specimen Specification
Specimen Orientation W (mm) e (mm) L (mm)
1 [0/0/0/0] 15 50 150
2 [90/90/90/90] 25 50 150
3 [45/45/45/45] 25 50 150
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Table 5.2: Materials Specification
Matrix Epoxy
Fibre Carbon
Number of layer 4
Type of lamina Unidirectional
Tab Aluminum
Ratio of matrix to reinforcement 50% -- Matrix
50% -- Unidirectional carbon
Figure 5.5: Tensile Test Specimen
5.4 Bending Test Specimen Preparation
There are two composite plate have been produced for the bending test, one is
for the unstiffened plate and another one is for the stiffened. The materials that used for
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the plate are also carbon fiber and epoxy. The weight ratio of resin to the fiber is 50 to
50. The orientation of the lamina for the composite is 0/90/90/0. The materials and tool
that used for the hand lay-up process are shown in figure 5.6.
Figure 5.6: Materials and tool for hand lay-up process
For the hat-stiffened plate, there is a stiffener mould has been made to produce
the composite stiffener. Then the stiffener will be attached to the composite plate by
using epoxy adhesive as shown in figure 5.7.
Carbon fiber lamina
RollerEpoxy
Hardener
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(a) (b)
Figure 5.7: (a)Hat shaped stiffener, (b) Hat-stiffened plate
5.5 Tensile Test
Tensile test is commonly performed in order to determine the in-plane tensile
properties for materials specifications, research and development, quality assurance,
structural design and analysis. In this test, we may be obtained the ultimate tensile
strength, ultimate tensile strain, tensile modulus, and Poisson’s ratio and transition
strain in the test direction.
The tensile test that will be conducted in this project is based on American
Society for Testing and Material tensile Test Method (ASTM D3039). This method is
used to determine the tensile properties for polymer matrix composite materials
reinforced by high modulus fibres. The composite forms are limited to continuous fibre
or discontinuous fibre-reinforced composites in which the lamina is balanced and
symmetric with respect to the best direction.
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A specimen having a constant rectangular cross section as shown in Figure 5.8
is mounted in the grips of a mechanical testing machine and monotonically loaded in
tension while recording load. The ultimate strength of the material can be determined
from the maximum load carried before failure. The displacement transducer is used to
monitor the strain of the specimen then the stress-strain response of the material can be
determined.
Figure 5.8: Tensile Specimen
where,
e = tab length
w = width of the specimen
t = thickness of each layer
L = length between the tab
5.5.1 Experimental Determination of Strength and Stiffness
When the tensile load is subjected to a tension load, it wills results extension in
the direction of the applied load and contraction perpendicular to the load. The basic
tenet of the experiments is that the stress-strain behavior of the materials is linear from
zero loads to the ultimate or fracture load. There are three loading condition
(longitudinal, transverse, and angle) of the tensile test will be performed to obtain the
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properties of the lamina in the principle material directions. First, consider a uniaxial
tension test in the 1-direction on a flat piece of a unidirectional reinforced lamina as
shown in Figure 5.9. The tensile properties is calculated using the following equations,
Figure 5.9: Uniaxial loading in the 1-direction
AP
X
v
E
AP
ult=
−=
=
=
1
212
1
11
1
εε
εσ
σ
(5.1)
where
ultPP, Applied load and maximum load obtained in tensile test respectively
1σ Average stress in the 1-direction
X Axial or longitudinal strength (1-direction)
21 ,εε Strain at the applied and transverse load respectively
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12v Poisson’s ratio
1E Young’s modulus in the 1-direction
A Cross-sectional area of the specimen
Second, consider a uniaxial tension test in the 2-direction on a flat piece of
unidirectional reinforced lamina as in Figure 5.10. The tensile properties is calculated
using the following equations,
Figure 5.10: Uniaxial loading in the 2-direction
AP
=2σ
2
22 ε
σ=E
2
121 ε
ε−=v (5.2)
A
PY ult=
where
2σ Average stress in the 2-direction
Y Transverse strength (2-direction)
2E Young’s modulus in the 2-direction
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At this point, the stiffness properties of the lamina should satisfy the reciprocal
relations as following,
2
21
1
12
Ev
Ev
= (5.3)
If the equation (5.3) has not been satisfied, one of these possibilities exists:
1) The data were measured incorrectly.
2) The calculations were performed incorrectly.
3) Linear elastic stress-strain relations cannot describe the material.
Third, we consider a uniaxial tension test at 45° to the 1-direction on a flat piece
of lamina as shown in Figure 5.11. The shear modulus is calculated using the following
equations,
Figure 5.11: Uniaxial loading at 45° to the 1-direction
xx
AP
Eε
= (5.4)
)2114(
1
1
12
21
12
Ev
EEE
G
x
+−−= (5.5)
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where
xE Young’s modulus in the x-direction
12G Shear modulus in the 1-2 plane
xε Strain at the applied load
5.5.2 Testing Apparatus
Instron 4602 testing machine is used to perform the tensile test as shown in
Figure 5.12. This machine consists of computer system, control panel, crosshead, load
frame panel, and load cell grip. The engineering constants that would be obtained are
Poisson’s ratio and Young’s modulus.
Figure 5.12: Instron 4602 testing machine
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5.5.3 Tensile Test Procedure
The details of the specimen materials, lamination, and specification are
mentioned earlier in section 5.4. Strain gauge is attached to the center point of the
specimen as shown in Figure 5.13 to obtain the strain reading.
Figure 5.13: Specimens with strain gauge
The crosshead displacement rate is 2mm/minute. Below is the procedure to
conduct the tensile test.
1) The specimen is fixed to the grip of the Instron testing machine and the
specimen shall be in axial alignment with the direction in pull.
2) Strain gauge is connected to the data logger and reset the initial value to
zero before load is applied.
3) Select the required test software and enter the specimen details such as
width, thickness and length at specimen menu.
4) Select the crosshead speed, maximum load, and other data.
5) Select the outputs that we want from the test.
6) Set the load balance to zero press the IEEE button to run the test.
7) All the data will be taken automatically and display on the monitor screen in
graph.
8) Finally we can printer out the data and draw the graph using plotter.
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The specimen will be loaded by pushing the specimen at a constant rate until the
specimen fail.
5.6 Bending Test
Bending test is performed in order to investigate the behavior of the simply
supported unstiffened and stiffened composite laminated plate under a distributed load
on a small area at the center of the plate. In this testing, the magnitude of the lateral
load and the deflection of the plates at various will be collected as shown in Figure 5.14.
One of the displacement transducer is located at the center of the plate and is known as
1st location. Another transducer is located at the coordinate 22 and is known as 2nd
location.
Figure 5.14: Location of the displacement transducers at the composite plate
1st location
2nd location
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5.6.1 Testing Apparatus
The hydraulic pressing machine is used to perform the bending test as shown in
Figure 5.15. This machine uses manual hydraulic system to operate and the maximum
power that can be generated is 10 ton. This machine consists two main parts which are
hydraulic arm and pressing pump.
A test rig is fabricated to test the composite plate as sown in Figure 5.16. This
test rig is made by four pieces of L-bar and bars were jointed together by using arc
welding technique. The detail drawing of the test rig is enclosed at Appendix B. This
rig is used as the base to put the composite plate on the pressing machine as shown in
Figure 5.15. Besides, two displacement transducer (LVDT) will be used to measure the
deflection of the plate as shown in Figure 5.17.
Figure 5.15: Hydraulic Press Machine
Hydraulic Pump
Bending test rig
Hydraulic Press arm
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Figure 5.16: Bending test rig
Figure 5.17: Displacement transducer (LVDT)
5.6.2 Bending Test Procedure
The details of the specimen materials, lamination, and specification are
mentioned earlier in section 5.4. Strain gauge is attached to the center point of the
specimen as shown in Figure 5.18 to obtain the strain reading.
1) Put the test rig on the pressing machine and lock the rig on the machine by
using G-clamp.
85
2) Put the plate on the center of the test rig and a LVDT is placed at the center
node in order to obtain the deflection on this node.
3) Connect the LVDT, strain gauge, and load cell to the data logger.
4) Set the initial value of the LVDT to zero.
5) Apply load on the plate by pressing the hydraulic pump.
6) Record down the deflection of the plate at every 5 kg increment of force and
stop the test once the plate failed.
Figure 5.18: Plate specimen with strain gauge
CHAPTER VI
RESULT AND DISCUSSION
6.1 Tensile Test Result
The tensile test is conducted based on the American Society for Testing and
Materials Tensile Test Method (ASTM-3039). In this project, the tensile test has been
performed for three different type of laminate orientation composite as mentioned at
chapter 5. The engineering constants have been measured in the tensile test and the full
data and the graphs of Stress versus strain for different types of specimens are shown in
APPENDIX C. In this analysis, we assumed that the composite laminate satisfy the
linear elastic stress-strain relations from zero loads to the ultimate or fracture load. Table
6.1 and 6.2 show the summary results obtained during the tensile test.
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Table 6.1: Results of tensile test
Specimen [0/0/0/0] [90/90/90/90] [45/45/45/45]
Thickness, t (mm)
Specimen 1 1.91 1.91 1.90
Specimen 2 1.91 1.90 1.90
Specimen 3 1.89 1.90 1.88
Average 1.90 1.90 1.89
Modulus Young, E (GPa)
Specimen 1 82.1526 7.1047 12.1605
Specimen 2 100.8280 7.7047 10.9659
Specimen 3 77.9243 7.0626 12.0063
Average 86.9683 7.2907 11.7109
Shear Modulus, G12(GPa)
Specimen 1 -- -- 5.4658
Specimen 2 -- -- 4.3343
Specimen 3 -- -- 5.3747
Average -- -- 5.0583
Table 6.2: Summary of tensile test result
Specimen Orientation [0/0/0/0] [90/90/90/90] [45/45/45/45]
Maximum load (KN) 27.9660 0.2882 0.5836
Ultimate Stress, σult (MPa) 981.5810 6.0882 12.2647
Modulus Young, E1 (GPa) 86.9683 -- --
Poisson Ratio,v12 0.2853 -- --
Modulus Young, E2 (GPa) -- 7.2907 --
Poisson Ratio,v21 -- 0.0096 --
Modulus Young, Ex (GPa) -- -- 11.7109
Shear Modulus, G12 (GPa) -- -- 5.0583
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6.1.1 Discussion On Tensile Test Results
By referring to Table 6.2, we can mention that the Young’s modulus for the
longitudinal direction (E1) is 86.9683 GPa, the Young’s modulus for the transverse
direction (E2) is 7.2907 GPa, the shear modulus (G12) is 5.0583 GPa, and the Poisson’s
ratio for 1-2 plane is 0.2853. In comparison, the average ultimate stress and maximum
load for the specimen with 0 degree fiber orientation is the highest among the tested
samples. The maximum recorded applied load before failure is 27.966 kN. This has been
verified that the fiber gives the highest strength at the longitudinal direction and the
failure of this specimen is shown in Figure 6.1. The figure 6.1 shows that the specimen
failed by spreading out its fiber and obviously we can see the breakage of the fibers.
Figure 6.1: Failure mode of specimen with 0 degree fiber orientation
The specimens with 90 degree fibers orientation represented the lowest strength
among the tested specimens. The maximum applied load before failure is 0.2882 kN.
This is because the fibers give the lowest strength at the transverse direction of the fibers.
For this type of orientation, only the resin is used to resist the tension caused by the
tensile load. The failure mode of this specimen is shown in Figure 6.2. From the figure,
we can mention that the specimens failed at the transverse direction of the specimen
which is also the fiber orientation.
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Figure 6.2: Failure mode of specimen with 90 degree fiber orientation
The strength for the specimen with 45 degree fibers orientation is higher than the
specimen with 90 degree fiber orientation but lower than the specimen with 0 degree
fibers orientation. The maximum applied load before failure is 0.5836 kN. The failure
mode of this type of specimen is shown in Figure 6.3. From this figure, we can mention
that the specimens failed at the fiber orientation. The modulus Young that obtained from
these specimens is used to define the shear modulus of the material and the obtained
value is 5.0583 GPa.
Figure 6.3: Failure mode of specimen with 45 degree fiber orientation
The mechanical properties listed in Table 6.1 will be used in modeling the FEA
model in numerical analysis.
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6.1.2 Discussion On Graph Stress Versus Axial Strain
Based on the data obtained during tensile test, graph stress versus axial strain has
been plotted as shown in Figure C1 to Figure C5 in Appendix C. Graphs are plotted with
Stress versus strain for three types of lamination of specimens. By referring to the Figure
C1, it represents the graph for the specimens with 0 degree fibers orientation in which
the tensile load applied at the longitudinal direction of the fibers. The graph plotted is
similarly for the three samples which show a straight line from origin with a positive
gradient. Therefore, we can conclude that the increment of the axial strain is
proportional to the applied stress. When the applied stress increases, an axial strain will
also increase accordingly. Thus, the materials have fulfilled the linear elastic stress-
strain relations. The Young’s modulus (E1) of the materials can be obtained from the
gradient of the graphs. Besides, the straight lines also represent that the fibers in general
show a brittle catastrophic failure in which it doesn’t experience plastic as metallic
materials which generally show a yield prior to failure.
The Figure C2 shows the graph for the specimen with 90 degree fibers
orientation in which the tensile load applied at the transverse direction of the fibers. The
graph plotted also similar for the three samples of specimen. From the plotted graph, it is
evident that when the applied stress increases, an axial strain will also increase
accordingly and represent a yield before the specimens show the plastic condition. When
the plastic condition occurred, the axial strain increases without any additional loads.
Therefore, the graph shows horizontal straight lines when the stress achieves until
certain level of applied loads. The maximum stress that can be afforded by the specimen
is less than 7 MPa. This value is great lesser than the specimen with 0 degree fibers
orientation which maximum stress is more than 850 MPa. The plastic condition
represented is due to the resin of the laminated composite. As mentioned before, the
strength is weak at the transverse direction of the fiber. Therefore, in this loading
condition, most of the tensile loads are carried by the resin in the samples. The resin
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used is polymer and polymer represents elastic and plastic characteristic when we apply
load on it. Hence, we can conclude that this graph is actually representing the behavior
of the resin. The Young’s modulus (E2) of the materials can be obtained from the
gradient of the graphs on linear condition.
The Figure C3 shows the graph for the specimen with 45 degree fibers
orientation in which the tensile load applied at the 45 degree direction of the fibers. The
graph plotted also similar for the three samples of specimen. The graph represents the
lines which are linear at the first and then slightly become non-linear when the applied
stress reached up to 6 MPa and it is evident that when the applied stress increases, an
axial strain will also increase accordingly and experience plastic condition before failure
occurred. As explained in the previous paragraph, the plastic condition represented is
due to the resin of the laminated composite. But for this type of orientation, the plastic
condition is not as large as the plastic condition for the specimen with 90 degree fibers
orientation. The maximum stress that can be afforded by the specimen is around 11 MPa
which is greater than the specimen with 90 degree fibers orientation. This is because the
contributions of the fibers in resisting the tensile load but is not as strong as the
specimens with 0 degree fibers orientation. The Young’s modulus (Ex) on the loading
direction can be obtained from the gradient of the graphs on linear condition then the
shear modulus (G12) in 1-2 plane is obtained by using equation (5.5).
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6.1.3 Discussion On Graph Lateral Strain Versus Axial Strain
Figure C4 shows the graph lateral strain versus axial strain for specimen with 0
degree fibers orientation. This graph shows a straight line with positive gradient crossing
the origin of the graph. This shows that it is linear relationship between the transverse
strain and the longitudinal strain. From the results obtained, the axial strain is always
positive and lateral strain is always negative. This is because when tensile load is applied
at the longitudinal direction of the specimen, it will cause elongation in this direction
and accompanied by contraction in the transverse direction. The Poisson’s ratio (v12) of
the specimen is denoted by the gradient of the graph.
Figure C5 shows the graph lateral strain versus axial strain for specimen with 90
degree fibers orientation. This graph shows a straight line with positive gradient crossing
the origin of the graph. This shows that it is linear relationship between the transverse
strain and the longitudinal strain. Similarly to the 0 degree fibers orientation specimen,
the axial strain is also always positive and lateral strain is always negative. This is
because when tensile load is applied at the longitudinal direction of the specimen, it will
cause elongation in this direction and accompanied by contraction in the transverse
direction. The result also shows that an axial strain is always higher than the lateral
strain for any load increment. The Poisson’s ratio (v21) of the specimen is denoted by the
gradient of the graph.
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6.2 Bending Test Result
In this project, the bending test has been performed for the composite hat-
stiffened laminated plate and unstiffened plate. The behavior of plates has been
measured in the bending test and the full data as well as the graphs load versus
displacement for different types of plates are shown in APPENDIX D. Table 6.3 and
Table 6.4 show the summary results obtained during the bending test. The plates are
simply supported around the outer edge and load is applied normal to the plate until the
failure occurred. Bending test is performed in order to investigate the behavior of the
simply supported unstiffened and stiffened composite laminated plate under a distributed
load on a small area at the center of the plate.
Table 6.3: Results of Load and Deflection for unstiffened composite plate
Displacement, (mm) Load
(kg) Center Coordinate 22
10 3 2 20 4 3 30 5 4 40 6 4 50 7 5 60 8 5 70 9 6 80 10 7 90 11 7
100 14 8 104 14 9
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Table 6.4: Results of Load and Deflection for composite hat-stiffened laminated plate
Displacement, (mm) Load
(kg) Center Coordinate 22
20 1 0
40 1 1
60 2 1
80 2 2
100 3 2
120 5 4
140 7 5
160 8 6
180 9 7
200 10 8
220 11 8
240 14 9
The obtained results include the magnitude of the lateral load in kg and
deflection in mm of the plates at various locations as shown in Figure 5.14. The results
of the bending test show that the composite hat-stiffened plate has the highest applied
load, which is 240 kg compare with unstiffened composite plate which is 104 kg. The
maximum deflection before failure occurred for unstiffened plate at the center and
coordinate-22 are 14 and 9 mm respectively. While the maximum deflection before
failure occurred for hat-stiffened plate at the center and coordinate-22 are 14 and 9 mm
respectively. The reason that the maximum carry load of stiffened plate is higher than
unstiffened plate is mainly because of the stiffener. Stiffener increased the stiffness and
strength of the composite by increasing the moment of inertia of the plate structure. This
can be clearly defined by using the equation of the deflection of the beam which is
simply supported as shown below;
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EI
WLw3845 3
= (6.1)
where,
w - Vertical deflection
W - Applied load
L - Distance between the supports
E - Modulus Young
I - Moment of inertia
The moment of inertia of the normal rectangular plate is defined by 12/3bdI = .
When the stiffener is attached to the plate, it will change the cross-section area of the
normal plate and then increases the moment of inertia of the plate. The more stiffeners
are added, the higher of moment inertia is increased. Therefore, it is evident that when
we fixed all the variables in equation (6.1) except I and w, the deflection will be reduced
if the value of I is increased. Hence, we can conclude that stiffened plate are quite
efficient for lightly loaded areas and also can carry more service load than unstiffened
plates for a given unit weight.
Besides that, the strength and stiffness of the composite plate is also influenced
by the orientation of fiber in the composite. In this project, the fiber orientation is same
for the stiffened and unstiffened composite plate which is (0/90)s.
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6.2.1 Discussion on Unstiffened Composite Laminated Plate
Figure D1 shows the graph applied load versus deflection for unstiffened
composite laminated plate. The deflection is recorded through data logger and the full
data of the bending test listed as in Appendix D. In this case, the deflections are
measured at every 5 kg increment of the applied load up to failure. From the results, we
can see that two lines like stairs are obtained. This is mainly because of the data logger
used cannot measure the displacements which are smaller than 1mm. By referring the
approximate line value in the graph, it shows a non-linear relationship between the
deflection and the applied load and the lines show ogif pattern. This implies that in the
early stage, less strength is required to create deflection. And slowly, more strength has
to be applied to create the deflection in the later stage.
In this testing, the plate was assumed to be deformed symmetrically to the center
of the plate. Therefore, we just examined quarter of the composite plate is convenient
and the displacement transducers are located at the position as shown in Figure 5.14.
The first location represents the center of the composite plate while the second location
is 3.5 inches away from the center which is denoted by coordinate 22. From Figure D1,
it is obviously that the deflection at the center is higher than the deflection at coordinate
22. This implies that the maximum deflection is occurred at the center of the entire plate.
The failure mode of the plate is shown in Figure 6.4. The damage of the plate is
very small compare to the entire area of the plate. As the plate is about to fail, some
cracking sound emitted from the plate. It is assumed that the noise is caused by matrix
cracking process. The matrix started to crack at the center of the plate at the lower
surface and then the crack spread from the center to all direction around the center. After
matrix cracking, delamination occurred and followed by fiber breakage.
97
Figure 6.4: Failure mode of the unstiffened composite plate
Figure D2 shows the applied load versus strain for unstiffened composite plate.
The results is obtained in 0, 90, (x- and y- direction) and 45 degree fiber direction at the
center of the plate. This graph shows a non-linear relationship between the strain and
applied load. The graph represents the lines which are linear at the first and then slightly
become non-linear when the applied stress reached up to 60 kg and it is evident that
when the applied stress increases, an axial strain will also increase accordingly and
experience plastic condition before failure occurred. From the graph, it shows that the
strain at 90 and 45 degree are always higher than the strain in 0 degree for all the applied
loads. The strain gauge in 90 degree direction gives the highest maximum strain value
and follow by the strain gauge in 45 degree; strain gauge in 0 degree direction gives the
lowest maximum strain value. This implies that the plate deform much in 90 degree than
0 degree direction.
98
6.2.2 Theoretical Analysis of Unstiffened Composite Laminated Plate
The theory lamination results are based on the macromechanical behavior of
lamina and laminate as mentioned earlier in Chapter III. The results and the steps
calculation for the stiffness matrix for the laminate [00/900/900/00] is given in Appendix
F. Besides that, the sample calculation of the maximum deflection for the unstiffened
plate is also given in Appendix F and the comparison of the maximum deflection for
experiment results and theoretical values in listed in Table F2. From this table, we can
mention that the maximum deflection at 100 kg applied load for theoretical value is
13.63 mm which approximate to the experiment results which give 14 mm. Referring to
Table 6.5, FEA analysis give the highest value of maximum deflection. The large
difference may due to the material properties in FEA analysis is not exactly same as the
specimen in experiment.
Table 6.5: The analysis result of maximum deflection at 100 kg applied load
Maximum Deflection at 100 kg, (mm)
Experiment FEA Theory
14 16.8 13.63
99
6.2.3 Discussion on Composite Hat-Stiffened Laminated Plate
Figure D3 shows the graph applied load versus deflection for hat-stiffened
composite laminated plate. The full data of the bending test is listed in Appendix D. In
this case, the deflections are measured at every 5 kg increment of the applied load up to
failure. The graph plotted is similarly to the graph in Figure D1 in which also represents
two lines like stairs. By referring to the approximate line value, the lines show that the
deflection is increased proportional to the applied load. The graph also shows that the
deflection at center is always higher than the deflection at coordinate-22.
Figure 6.5 represents the failure of the stiffened plate. From the figure, we can
mention that the plate failed at the stiffener in which cracks were produced on the
stiffener and fibers were pulled out from the matrix. This implies that the critical region
of the structure is at the stiffener at the area near the edge between stiffener and plate.
The local cracking at the failure edges shift the location of the maximum force further
into the interior of the bonded length between stiffener and plate.
Figure 6.5: Failure of the Stiffened Plate
100
Figure D4 shows the applied load versus strain for hat-stiffened composite plate.
The result is obtained in 0 and 90 (x- and y- direction) degree fiber direction at the
location as shown in Figure 6.6 and the strains are representing the strain on the surface
of the plate. This graph shows a non-linear relationship between the strain and applied
load. At low applied load, the graph is linear and then slightly become non-linear when
the applied stress reached up to 100 kg.
From the graph in Figure D4, it shows that the strain at location 2 and 4 are
represented by line that is linear from origin. This implies that the strains behave linearly
in the 0 degree fiber orientation and the strain value in this direction is greatly lower
than the strain value in 90 degree direction. The strain-2 is higher than the strain-4 and
this means that the location 2 is deformed much than location 4. The strains in 90 degree
direction are representing by the strain value at location 1, 3 and 5. Refer to the graph,
the in this direction increased very slowly at the early stage and the increment of the
strain increased when the applied stress reached up to 100 kg. This is clearly shown in
line stain-3 and strain-5. Strain-1 is lower with compare to strain-3 and strain-5. This is
because location 1 is far from the center of the plate and it is less deformation with
compare to location 3 and 5. Srain-3 is higher than strain-5 and this implies that location
3 deform much than location 5. Finally, it can be concluded that the strain in the
direction of 0 degree is lower than the strain in the direction of 90 degree.
101
Figure 6.6: Location of the strain gauges at the composite hat-stiffened plate
6.3 FEA Simulation Result
The linear static FE simulation has been carried for the unstiffened composite
laminated plate and composite hat-stiffened laminated plate. The full results of the FEA
simulation are given in Appendix E. Linear static analysis use the linear theory of
structure in which it based on the assumption to small displacements to calculate
structural deformation. Distributed loads were applied on the round surface on the plate
and it produced the bending condition with the upper layer in compression. Table 6.6
shows the comparison between the experiment results and finite element analysis for
maximum deflection at 100 kg.
From Table 6.6, it shows that the FEA value is higher than experiment value for
unstiffened plate and gives 20% of deviation. However, for hat-stiffened plate, the FEA
102
value is lower than experiment value and gives 33% of deviation which is considered
high if comparing to unstiffened plate. The difference of the deflection between the
experimental and FEA simulation is caused by the following factors:
1) The mechanical properties used in the FEA simulation are not exactly same
as the specimen in experiment.
2) The thickness of the laminates produced is not uniform.
3) There is excessive or insufficient resin between the laminas.
4) Existence of voids caused by the air bubble entrapped between the laminas.
5) The bonding between the stiffener and plate is imperfect.
Besides that, there are two more FEA analysis has been done for the square-
shaped and T-shaped of stiffened plate. From the obtained results, it shows that the
square-shaped stiffened plate give the smallest maximum deflection at 100 kg applied
load and follow by hat-stiffened plate gives the second low maximum deflection and the
T-shaped stiffened plate gives the biggest maximum deflection among the stiffened plate.
This implies that the T-shaped stiffener is less efficiency in strengthening the stiffness of
the composite plate. For the overall results, the maximum deflection of the stiffened
plate is greatly less than the unstiffened plate.
Table 6.6: Comparison of experiment results and FEA value for unstiffened and
stiffened plate
Maximum Deflection at 100 kg, (mm) Specimen
Experimental FEA
Difference,
(%) Unstiffened Plate 14 16.8 20
Hat-stiffened Plate 3 2.01 33 Square shaped stiffened
Plate -- 1.959 --
T shaped-stiffened Plate -- 3.249 --
103
6.3.1 Discussion On Unstiffened Composite Laminated Plate
FEA simulation has been done on the unstiffened composite plate and the result
is listed in Table 6.6. Figure E1 in Appendix E represents the displacement contour of
the unstiffened plate under 100 kg distributed load on the center of the plate. This figure
shows that the plate represented a symmetrical deformed shape to the center and the
regions with red color experience the highest deformations compare with the regions
with other colors. Therefore, the center of the plate is the critical part for the whole
structure and the maximum deflection occurs at the center of the plate and the
displacement reduced from the center of the plate. The maximum displacement of the
structure is 16.79988 mm at node 2548. The outer edge of the plate gives the lowest
deformation. The deformed shape of the plate is in a half sinusoid wave and is shown in
Figure 6.7.
Figure 6.7: Bent plate in half sinusoid wave with deformation scale of 5
The Von Mises stress contour for the unstiffened plate is shown in Figure E2 to
E5 in Appendix E. These figures show the Von Misses stress contour of each layer of
the laminate. The stress results show that the critical region of this structure at the center
of the plate which is also the applied area of the structure. The maximum stress is 406.4
104
MPa, 263.1 MPa, 300.8 MPa and 502 MPa for layer 1, 2, 3 and 4 respectively. From the
results, we can also know that the outer layer of the laminate is subjected to higher stress
than the inner layer where the distributed stress in layer 1 and 4 are higher than the
distributed stress in layer 2 and 3. Therefore, the critical region of the structure is on the
surface of the plate especially the area near the applied load.
6.3.2 Discussion On Hat-Stiffened Composite Laminated Plate
The FEA results for hat-stiffened composite laminated plate are shown in Figure
E6 and E7 in Appendix E. The analysis results show that the structure experiences to
bend as unstiffened plate in which the hat-stiffened plate deformed symmetrically to the
center of the plate. The center subjected to bend much than other part it is the location
where the applied load located. The maximum displacement of the plate structure is
2.009997 mm at node 5183. Figure E 6 shows the displacement contour of the hat-
stiffened plate. This figure shows that the regions with red color experience the highest
deformations compare with the regions with other colors. Figure 6.8 represents the
displacement contour of hat-stiffened plate for bottom view and it shows that the
maximum displacement is at the center of the stiffener which is same as unstiffened
plate. Besides that, it also shows that the stiffener doesn’t change much in the
deformation pattern but it helps a lot in reducing the deformation and increasing the
strength of the structure for a unit of load. Figure 6.9 and 6.10 show the deformed shape
of the structure. The FEA results for square-shaped and T-shaped stiffened plate are
shown in Figure E8 to E13 respectively in Appendix E.
105
Figure 6.8: Displacement contour for hat-stiffened plate for bottom view
Figure 6.9: Deformed shape of the hat-stiffened plate with deformation scale of 3
106
Figure 6.10: Front view of deformed shape for the hat-stiffened plate with deformation
scale of 3
Figure E7 shows the solid Von Mises stress contour of the hat-stiffened plate.
Figure 6.11 shows the front view of the structure. From this figure, we can see that the
critical region of the whole structure is located at the edge between the stiffener and the
plate. The critical regions will experience the highest stress distribution. This is because
these parts support most of the applied load on the structure. Whereas, other parts
generally experience lower stress (blue and pink color) and they are not so critical. The
maximum stress that experiences by this structure is 106.308 MPa at node 1637. This
result is same as experiment result in which the specimen is failed at the connection of
the stiffener and plate as shown in Figure 6.5. From Figure 6.12, we can see that the
maximum stress that on the upper surface of the stiffened plate is 103.8488 MPa at node
2471 which is lower than the maximum stress on the connection between the stiffener
and plate. Figure 6.11 clearly shows the stress distribution on the connection between
the stiffener and plate. Similarly to the square-shaped and T–shaped stiffened plate, the
critical region of the structure is at the connection between the stiffener and plate.
107
Figure 6.11: Side view of the critical region for composite hat-stiffened plate
Figure 6.12: Critical region of the hat-stiffened plate
108
In this analysis, the property used is solid which is different with the analysis of
unstiffened plate in which the property used is laminate. This is because it cannot be
meshed properly when using laminate property for stiffened plate. But the displacement
result of the FEA analysis for laminate property and solid property is same. The only
different for the analysis results is the distribution stress on the structure. For laminate
property, we can obtain the distribution stress for each layer of the laminate. Whereas,
for solid property; we can only get the stress distribution for the whole structure but the
critical region obtained is same as the results obtained by using laminate property.
Therefore, in order to the unavailability to mesh the stiffened plate, solid property is
used to obtain the displacement analysis and the critical region analysis as well as the
results are shown in Appendix E and has been explained in the previous paragraph.
The FEA analysis for the hat-stiffened plate by using laminate property is shown
in Figure 6.13. This figure shows the displacement contour of the stiffened plate. It is
obviously that the stiffened plate is deformed improperly as the results for solid property.
The whole structure represents pink colour which implies that the displacement is
approximate to zero. By referring to the contour bar at the side, it shows that this the
structure has a maximum displacement which is 15.47 mm but it doesn’t shown in the
figure. Besides, the maximum displacement is greatly larger than the experiment value
which gives 3 mm displacement. Hence, it can be concluded that this structure is not be
modeled properly. Figure 6.14 represents the side view of the deformed mode of the hat-
stiffened plate. From this figure, we can see that the plate is deformed through the
stiffener. This implies that there is no connection between the stiffener and plate and
stiffener doesn’t give any support to the plate in defending the applied load. This
problem is solved by defining the hat-stiffened composite plate as solid property and the
results is discussed in the previous paragraphs.
109
Figure 6.13: Displacement contour for hat-stiffened composite plate with laminate property
Figure 6.14: Side view of the deformed hat-stiffened plate with laminate property
CHAPTER VII
CONCLUSION AND SUGGESTION
7.1 Conclusion
The composite materials are increasingly important as an engineering material in
diverse applications such as aerospace, automotive, marine, civil, sport equipment
applications, and other industrial applications. A composite stiffened plate is a general
form widely used in those applications. In a unidirectional composite the longitudinal
properties are controlled by the fiber properties and give the highest strength, whereas
the transverse properties are matrix dominated and give the lowest strength. However,
high modulus and strength characteristics of composites result in structures with very
thin sections that are often prone to buckling. Stiffeners are required to increase the
bending stiffness of such thin walled members (plates, shells).
The main objectives of this project are to study the effects of hat-shaped
stiffeners in the deformation of the composite laminated plates by experimentally and
finite element simulation and study the effects of stiffener’s geometry in strengthening
the composite plate by simulation. The finite element static analysis of composite
111
stiffened plate using FEA software is presented. The deformation of the unstiffened
composite laminate pate and composite stiffened plate has been experimentally
determined and compared with the value predicted using FEA simulation. It is observed
that the deviation between the experimental and FEA values are small for unstiffened
plate but is large for composite stiffened plate and the factors influenced the
experimental value were discussed in the previous chapter.
The tensile test results show that for composite material under tensile loading,
there is a linear relationship between stress and strain as well as the transverse and
longitudinal strains before yield has started. This is due to the axial strain is directly
proportional to the applied load in the axial direction. Besides, the highest value of
ultimate stress and modulus Young can be obtained by aligning the fiber parallel to the
direction of the applied load.
The bending test results show that the maximum deflection of the stiffened plate
at the 100 kgf applied is smaller (3 mm) than the unstiffened plate (14 mm). The results
also show that the stiffened plate can carry more service load (240 kg) than unstiffened
plates (104 kg) for a given unit weight. Therefore, it can be concluded that stiffened
panels are quite efficient for lightly loaded areas. From this testing, we also found that
the failure of the unstiffened plate occurred at the area near the applied load which on
the top surface of the plate. While for the hat-stiffened plate, the failure occurred at the
area near the edge between stiffener and plate. The local cracking at the failure edges
shift the location of the maximum force further into the interior of the bonded length
between stiffener and plate.
For the unstiffened plate, the FEA simulation result shows that the outer layer of
the laminate is subjected to higher stress than the inner layer. The maximum
displacement obtained for this model is 16.8 mm which is 20 % higher than the
112
experimental value. For hat-stiffened plate, the maximum displacement obtained is 2.01
mm which is 33 % lower that the experiment value.
By comparing the maximum deflection at 100kg applied load between the
unstiffened and stiffened plate, it shows that the deflection value for unstiffened plate is
greatly higher than the stiffened plate. Hence, it can be concluded that the stiffener is
effective in increasing the strength and reducing the deformation of the composite plate.
The linear static finite element analysis also have been done for the different shape of
the stiffened plate and the comparison of results for few examples on static analysis of
laminated stiffened plate gives an overview regarding the selection of stiffener sections
in engineering designs. The FEA simulation show that different type of the stiffener will
give different effects in strengthens the stiffness of the composite plate. The results show
that the square-shaped stiffened plate is more efficient in strengthen the composite plate
in which it give the smallest maximum deflection at 100 kg applied load and follow by
hat-stiffened plate and the T-shaped stiffened plate gives the highest maximum
deflection among the stiffened plate. Besides that, through this analysis, we also know
that the critical region of the laminated stiffened plate is located at the edge between the
stiffener and plate which is same as the experiment result.
The theoretical analysis of the maximum deflection at 100 kg applied load for
unstiffened plate is 13.63 mm which is approximate to the experiment value which is 14
mm. This show that the Navier method is suitable in determining the displacement but
this method take a long estimation time.
Generally, this project has achieved it objectives based on the results from
experiments and FEA simulation.
113
7.2 Suggestion for Future Study
The objectives of this project are to investigate the deformation and stress in
composite laminated plate and to study the effects of stiffener in increasing the bending
stiffness of the composite laminated plate. Therefore, some suggestions were given in
order to improve this project in the future.
1) In this project, the composite plate that stiffened by hat shaped stiffener was
tested. This project can be extended by using the various shape of stiffener.
Different shape of the stiffener will give different effect to the bending stiffness
of the composite plate.
2) Extend the analysis to other laminated plate with different lamina orientation for
further observation of the lamina orientation in the stiffness properties.
3) In this project, the plate was tested with simply supported along all edges and
subjected to a small area distributed load at the center of the plate. Therefore, this
project can be improved further with different boundary condition and different
type of apply loads such as clamped plate and distributed load on the surface of
the plate.
4) The materials that used in this project were high strength carbon fiber and epoxy.
Analysis with other materials such as glass/vinylester and aramid/epoxy can be
used to study the effects of the material to the strength properties of the plate.
5) Various layers of lamina, width and length of the plate can be tested for the
further investigation.
114
6) Improve the hand lay up method. This is because this method is easy to cause air
bubble entrapped between the lamina and affect the accuracy of the results.
7) In this project, one sample of plate was tested for deformation and stresses
investigation. The number of the testing plate should be increased in order to
obtain the more accurate results.
115
REFERENCES
1) CompositePro for Windows TM , “Reference Guide.pdf”, Peak Composite
Innovation, United States of America, 2002.
2) Reddy, J.N. and Miravete, A., “Practical Analysis of Composite Laminates”, CRC
Press, United States of America, 1995.
3) Eckold, Geoff, “Design and Manufacture of Composite Structures”, Woodhead
Publishing Limited, England, 1994.
4) Jones, Robert M., “Mechanics of Composite Materials”, Scripta Book Company,
Washington, 1975.
5) Fitzer, Erich, “Carbon Fibers and Their Composites”, Springer-Verlag, Germany,
1985.
6) http://www.composite.about .com
7) L. Hollaway, “Polymers and Polymer Composites in Construction”, Thomas
Telford Ltd, London, 1990.
8) Timoshenko, S. and Woinowsky-Krieger, S., “Theory of Plates and Shells”,
Mcgraww-Hill International Book Company, New York, 1984.
9) Reddy, J.N., “Theory and Analysis of Composite Plates”, Universiti Putra
Malaysia, 1997.
116
10) Nazri Kamsah, “Finite Element Method”, University Technology Malaysia, 2004
11) Calcote, Lee R., “The Analysis of Laminated Composite Structures”, Van
Nostrand Reinhold Company, Canada, 1969.
12) Troitsky, M.S., “Stiffened Plates: Bending, Stability and Vibrations”, Elsevier
Scientific Publishing Company, New York, 1976.
13) Whitney, James M., “Structural Analysis of Laminated Anisotropic Plates”,
Technomic Publishing Company, United States of America, 1987.
14) Rafaat M. Hussein, “Composite Panels / Plate: Analysis and Design”, Technomic
Publishing Company, United States of America, 1986.
15) Reddy, J.N., “An Introduction to the Finite Element Method”, McGraw-Hill, Inc.,
United State of America, 1993.
16) Cook, R.D., “Concepts and Application of Finite Element Analysis”, John Wiley
& Sons, Inc, Canada, 1974.
17) John, L. Clarke, “Structural Design of Polymer Composites”, St. Edmunelsbury
Press, London, 1996.
117
APPENDIX A
AMERICAN SOCIETY FOR TESTING AND MATERIALS TEST METHOD
(ASTM-3039)
118
119
120
121
122
123
124
125
APPENDIX B
TECHNICAL DRAWING OF TEST RIG, STIFFENER, STIFFENED PLATE
126
127
128
129
130
131
APPENDIX C
RESULTS OF TENSILE TEST
132
Table C1: Specification of Specimen 1
UD0 (1) UD0 (2) UD0 (3) Number of layer 4 4 4 Lamination [0/0/0/0] [0/0/0/0] [0/0/0/0] Thickness, t (mm) 1.91 1.91 1.89 Width, W (mm) 15.00 15.01 14.88 Cross sectional area, A (mm2) 28.65 28.67 28.12
Table C2: Specification of Specimen 2
UD90 (1) UD90 (2) UD90 (3) Number of layer 4 4 4 Lamination [90/90/90/90] [90/90/90/90] [90/90/90/90] Thickness, t (mm) 1.91 1.90 1.90 Width, W (mm) 25.02 24.97 25.01 Cross sectional area, A (mm2) 47.79 47.44 47.52
Table C3: Specification of Specimen 3
UD45 (1) UD45 (2) UD45 (3) Number of layer 4 4 4 Lamination [45/45/45/45] [45/45/45/45] [45/45/45/45] Thickness, t (mm) 1.90 1.90 1.88 Width, W (mm) 25.01 25.01 25.02 Cross sectional area, A (mm2) 47.52 47.52 47.04
133
Stress VS Axial Strain
0
200
400
600
800
1000
1200
0 2000 4000 6000 8000 10000 12000 14000 16000
Axial Strain, (1E-06)
Stre
ss, (
MPa
)
Sample1 Sample2 Sample3
Figure C1: Graph stress versus axial strain for specimen with 0 degree fiber orientation
Stress VS Axial Strain
0
1
2
3
4
5
6
7
8
0 500 1000 1500 2000 2500
Axial Strain (1E-06)
Stre
ss, (
MPa
)
Sample1 Sample2 Sample3
Figure C2: Graph stress versus axial strain for specimen with 90 degree fiber orientation
134
Stress VS Axial Strain
0
2
4
6
8
10
12
14
0 500 1000 1500 2000 2500
Axial Strain, (1E-06)
Stre
ss, (
MPa
)
Sample1 Sample2 Sample3
Figure C3: Graph stress versus axial strain for specimen with 45 degree fiber orientation
Lateral Strain VS Axial Strain
0
500
1000
1500
2000
2500
3000
3500
4000
0 2000 4000 6000 8000 10000 12000 14000
Axial Strain, (u)
Lat
eral
Str
ain,
(-u)
Graph C4: Graph transverse strain versus longitudinal strain for specimen with 0 degree
fibers orientation
135
Lateral Strain VS Axial Strain
0
5
10
15
20
25
30
35
40
45
0 1000 2000 3000 4000 5000 6000
Axial Strain (u)
Lat
eral
Str
ain
(-u)
Graph C5: Graph transverse strain versus longitudinal strain for specimen with 90 degree fiber orientation
136
APPENDIX D
RESULTS OF BENDING TEST
137
Specimen : Unstiffened Composite Laminated Plate
Number of layer : 4
Lamination : 0/90/90/0
Table D1: Results of bending test for unstiffened composite laminated plate
Load(kg)
Displacement, (mm)
(center)
Displacement,(mm)
(Coordinate 22)
Strain-0 (-µ)
Strain-90 (-µ)
Strain-45 (-µ)
0 0 0 0 0 0 5 1 1 60 331 261 10 3 2 77 468 425 15 4 2 74 557 552 20 4 3 64 661 700 25 5 3 64 729 794 30 5 4 70 831 932 35 6 4 82 910 1029 40 6 4 97 977 1109 45 7 4 113 1040 1189 50 7 5 126 1104 1274 55 8 5 126 1156 1352 60 8 5 134 1201 1428 65 9 6 209 1289 1475 70 9 6 295 1405 1584 75 9 6 492 1569 1633 80 10 7 662 1733 1668 85 10 7 896 1975 1793 90 11 7 1488 2498 2124 95 12 8 2024 4596 2453
100 14 8 2666 5689 3032 104 14 9 4370 6875 5639
138
Load VS Displacement
0
20
40
60
80
100
120
0 2 4 6 8 10 12 14 16
Displacement, (mm)
Load
, (kg
)
DisplCenter Coordinatel-22 Poly. (DisplCenter) Poly. (Coordinatel-22)
Graph D1: Graph applied load versus displacement for unstiffened composite laminated plate
Applied Load VS Strain
0
20
40
60
80
100
120
0 1000 2000 3000 4000 5000 6000 7000
Strain, (-u)
App
lied
Loa
d, (k
g)
0 Degree 90 Degree 45 Degree
Graph D2: Graph strain versus applied load for unstiffened composite laminated plate
139
Specimen : Composite Hat-Stiffened Laminated Plate
Number of layer : 4
Lamination : 0/90/90/0
Table D2: Results of bending test for hat-stiffened composite laminated plate
Load(kg)
Displacement (center),
(mm)
Displacement(22), (mm)
Strain -1(-µ)
Strain -2(-µ)
Strain -3 (-µ)
Strain -4(-µ)
Strain -5(-µ)
0 0 0 0 0 0 0 0 5 0 0 5 81 40 35 22 10 0 0 11 94 58 45 40 15 0 0 26 106 84 56 61 20 1 0 40 112 104 61 79 25 1 0 65 117 134 69 107 30 1 1 82 121 155 73 127 35 1 1 100 126 179 78 148 40 1 1 115 129 199 82 164 45 1 1 118 134 224 86 184 50 2 1 131 140 250 91 206 55 2 1 146 144 274 94 226 60 2 1 162 148 297 95 245 65 2 1 181 153 325 98 267 70 2 1 198 156 349 101 287 75 2 2 219 161 381 105 312 80 2 2 238 166 406 107 330 85 3 2 251 170 430 112 347 90 3 2 272 180 463 118 371 95 3 2 297 195 491 134 389
100 3 2 337 214 530 145 423 105 4 3 361 230 559 155 447 110 4 3 393 250 578 164 474 115 5 3 448 303 642 207 516 120 5 4 455 335 697 208 592 125 6 4 474 354 761 212 730 130 6 4 543 354 834 212 831 135 6 5 585 375 892 218 902 140 7 5 614 375 1135 223 1119 145 7 5 656 392 1260 235 1218 150 7 5 684 408 1434 246 1334 155 7 6 735 423 1801 259 1566
140
Load(kg)
Displacement (center),
(mm)
Displacement(22), (mm)
Strain -1
(-µ) Strain -2
(-µ) Strain -3
(-µ) Strain -4
(-µ) Strain -5
(-µ) 160 8 6 788 436 2018 271 1691 165 8 6 843 445 2169 279 1810 170 8 6 897 464 2370 281 1918 175 9 7 959 483 2493 285 1982 180 9 7 1003 491 2596 325 2063 185 9 7 1024 510 2746 353 2106 190 9 7 1027 525 2801 376 2113 195 10 7 1068 534 2873 394 2143 200 10 8 1099 553 2945 414 2189 205 10 8 1146 562 3135 435 2197 210 10 8 1179 572 3211 453 2232 215 11 8 1191 581 3284 467 2415 220 11 8 1205 589 3300 479 2635 225 12 8 1334 596 3354 481 2801 230 12 8 1614 605 3344 487 2873 235 14 9 1911 637 3364 511 2945 240 14 9 2189 682 4326 583 3350
141
Load VS Diaplacement
0
50
100
150
200
250
300
0 2 4 6 8 10 12 14 16
Displacement, (mm)
Load
, (kg
)
Center Coordinate 22 Poly. (Center) Poly. (Coordinate 22) Graph D3: Graph applied load versus displacement for composite hat-stiffened
laminated plate
Applied Load VS Strain
0
50
100
150
200
250
0 500 1000 1500 2000 2500 3000 3500 4000 4500
Strain, (-u)
App
lied
Loa
d, (k
g)
Strain-1 Strain-2 Strain-3 Strain-4 Strain-5 Graph D4: Graph strain versus applied load for composite hat-stiffened laminated plate
142
APPENDIX E
FINITE ELEMENT METHOD ANALYSIS RESULTS
143
Figure E1: Displacement contour of unstiffened plate with simply supported around the outer edge
Figure E2: Lamina 1 Von Mises stress contour of the unstiffened plate
144
Figure E3: Lamina 2 Von Mises stress contour of the unstiffened plate
Figure E4: Lamina 3 Von Mises stress contour of the unstiffened plate
145
Figure E5: Lamina 4 Von Mises stress contour of the unstiffened plate
Figure E6: Displacement contour of hat-stiffened plate with simply supported around the
outer edge
146
Figure E7: Von Mises stress contour of the hat-stiffened plate
Figure E8: Displacement contour of square shaped-stiffened plate with simply supported
around the outer edge
147
Figure E9: Von Mises stress contour of square shaped-stiffened plate
Figure E10: Critical region of square shaped-stiffened plate
148
Figure E11: Displacement contour of T shaped-stiffened plate with simply supported
around the outer edge
Figure E12: Von Mises stress contour of T-shaped-stiffened plate
149
Figure E13: Critical region of square T-stiffened plate
150
APPENDIX F
THEORY ANALYSIS RESULTS FOR UNSTIFFENED COMPOSITE
LAMINATE PLATE
151
a) Calculation of Stiffness Matrix of Laminate [ 00 / 900 / 900 / 00 ]
1) Mechanical Properties of Carbon Fiber
E1 = 86.9683 GPa
E2 = 7.2907 GPa
v12 = 0.2853
v21 = 0.0239
G12 = 5.0583 GPa
2) Reduced Stiffness
2112
111 1 vv
EQ
−= = 87.5654
2112
121
2112
21212 11 vv
Evvv
EvQ−
=−
= = 2.0943
2112
222 1 vv
EQ
−= = 7.3408
1266 GQ = = 5.0583
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
0583.50003408.70943.200943.25654.87
Q GPa
152
3) Transformed Reduced Stiffness 3.1) 0 Degree:
[ ]
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−×
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡×
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−
−=
)0(sin)0(cos)0sin()0cos(2)0sin()0cos(2)0sin()0cos()0(cos)0(sin
)0sin()0cos()0(sin)0(cos
0583.50003408.70943.200943.25654.87
)0(sin)0(cos)0sin()0cos()0sin()0cos()0sin()0cos(2)0(cos)0(sin)0sin()0cos(2)0(sin)0(cos
22
22
22
22
22
22
0oQ
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡×
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡×
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
100010001
0583.50003408.70943.200943.25654.87
100010001
0oQ
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
0583.50003408.70943.200943.25654.87
0oQ GPa
3.2) 90 Degree:
[ ]
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−×
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡×⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−
−=
)90(sin)90(cos)90sin()90cos(2)90sin()90cos(2)90sin()90cos()90(cos)90(sin
)90sin()90cos()90(sin)90(cos
0583.50003408.70943.200943.25654.87
)90(sin)90(cos)90sin()90cos()90sin()90cos()90sin()90cos(2)90(cos)90(sin)90sin()90cos(2)90(sin)90(cos
22
22
22
22
22
22
90oQ
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−×⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡×⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−=
100001010
0583.50003408.70943.200943.25654.87
100001010
90oQ
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
0583.50005654.870943.200943.23408.7
90oQ GPa
153
4) Defining Extensional stiffness, Coupling stiffness, and Bending Stiffness
Four-ply for the plate is illustrated as below:
00
900 900 00
h0 h1 h2 h3 h4
kh -0.99 -0.495 0 0.4950 0.99 2kh 0.9801 0.2450 0 0.2450 0.9801 3kh -0.9703 -0.1213 0 0.1213 0.9703
Ply
kh - 1−kh 2
12
−− kk hh 31
3−− kk hh
1 0.4950 -0.7351 0.8490
2 0.4950 -0.2450 0.1213
3 0.4950 0.2450 0.1213
4 0.4950 0.7351 0.8490
4.1) Extensional Stiffness
( ) ( )∑=
−−=N
kkkkijij zzQA
11
[ ] [ ] [ ] [ ] [ ]( )0000 090900495.0 QQQQA +++=
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡×+
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡×=
0583.50005654.870943.200943.23408.7
20583.50003408.70943.200943.25654.87
2495.0
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
0154.100009571.931467.401467.49571.93
MN/m
Middle plane
0.495 mm 0.495 mm
0.495 mm 0.495 mm
154
4.2) Coupling Stiffness
( ) ( )∑=
−−=N
kkkkijij zzQB
1
21
2
21
( ) ( )⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+−+
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+−=
0583.50005654.870943.200943.23408.7
245.0245.00583.50003408.70943.200943.25654.87
7351.07351.021
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
000000000
kN
4.3) Bending Stiffness
( ) ( )∑=
−−=N
kkkkijij zzQD
1
31
3
31
( ) ( )⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡++
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+=
0583.50005654.870943.200943.23408.7
1213.01213.00583.50003408.70943.200943.25654.87
849.0849.031
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
2720.30002360.113547.103547.11556.50
Nm
5.0) Resultant Laminate Forces and Moments
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
0
0
0
0154.100009571.931467.401467.49571.93
xy
y
x
xy
y
x
NNN
γεε
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
xy
y
x
xy
y
x
kkk
MMM
2720.30002360.113547.103547.11556.50
155
b) Sample Calculation of Maximum Deflection for Unstiffened Composite Plate
1) Maximum deflection of the unstiffened at the 100 kg applied load
Equation of deflection,
yxdQ
yxwmn
mn
mnβα sinsin),(
11∑∑∞
=
∞
=
=
where
mnq
Qmn 2016
π= for m, n odd.
[ ]422
2226612
44114
4
)2(2 nDsnmDDsmDb
dmn +++=π , s =
b/a
b
nanda
m πβπα ==
a = 0.25 m
b = 0.25 m
Area, A = 0.25 x 0.25
= 0.0625 m2
Applied load, P = 100 kg
= 981 N
Uniform distributed load, q0 0625.0981
=
15696= N/ m2
156
Sample of calculation for the first term (m = 1, n = 1) of equation of deflection, for a = b,
s = b/a = 1,
mnq
Qmn 2016
π=
)1)(1(
15696162π×
=
40.25445=
[ ]422
2226612
44114
4
)2(2 nDsnmDDsmDb
dmn +++=π
[ ]4222444
4
)1)(236.11()1()1()1)(272.323547.1(2)1()1)(1556.50()25.0(
+×++=π
044.1924841=
α a
mπ= β
amπ
=
α 25.0)1( π
= β 25.0)1( π
=
π4= π4=
Maximum deflection of the plate,
)125.0(sin)125.0(sin)125.0,125.0(11
βαmn
mn
mn dQ
w ∑∑∞
=
∞
=
=
)125.04sin()125.04sin(044.192484140.25445
××= ππ
01322.0= m
22.13= mm
157
By retaining the first four terms (m = 1, n = 1, 3; m = 3, n = 1, 3) results in what is
essentially the exact solution of maximum deflection of the plate. The results of the first
four terms for 100 kg applied load are shown in Table F1.
Table F1: The first four terms of deflection solution at 100 kg applied load
m, n Qmn dmn α β w (mm)
1, 1 25455.40 1924841.044 4π 4π 13.22
1, 3 8481.80 27491450.13 4π 12π 0.3085
3, 1 8481.80 105133646.3 12π 4π 0.0807
3, 3 2827.27 155912124.6 12π 12π 0.0181
Maximum deflection, wmax = 13.22 + 0.3085 + 0.0807 + 0.0181
= 13.63 mm
Table F2: Comparison of the maximum deflection for experimental results and
theoretical values
Maximum Deflection, (mm) Load, (kg)
Experimental Theoretical
0 0 0 5 1 0.68
10 3 1.36 15 4 2.04 20 4 2.73 25 5 3.41 30 5 4.09 35 6 4.77 40 6 5.45
158
Maximum Deflection, (mm)
Load, (kg) Experimental Theoretical
45 7 6.13 50 7 6.81 55 8 7.49 60 8 8.18 65 9 8.86 70 9 9.54 75 9 10.22 80 10 10.90 85 10 11.58 90 11 12.26 95 12 12.95
100 14 13.63 104 14 14.17
