FEM Composite Laminate MATLAB
Short Description
Matlab implementation of finite element analysis of composite laminate...
Description
FINITE ELEMENT FORMULATION FORMULATION FOR COMPOSITE LAMINATE PLATES
SIM SIANG KAO
UNIVERSITI TEKNOLOGI MALAYSIA
FINITE ELEMENT FORMULATION FOR COMPOSITE LAMINATE PLATES
SIM SIANG KAO
A report is submitted in partial fulfillment of the requirements of the award of the degree of Bachelor of Civil Engineering
Faculty of Civil Engineering University Technology Malaysia
APRIL 2010
iii
Thi s th esi s is dedicated to my beloved parent s.
iv
ACKNOWLEDGEMENT
First and foremost, I would like to express my sincere appreciation to my supervisor of this research, Dr Ahmad Kueh Beng Hong for his continual dedicated guidance, professional advices, encouragement, support and motivation in effort to complete this research.
Apart from this, I would like to extend my gratitude and appreciation to all the researchers of the Steel Technology Centre, Faculty of Civil Engineering, University Technology Malaysia. They had been very helpful and patience in providing assistance throughout the work. My special thanks are extended to my colleagues and friends for their friendship, continuous support, understanding, as well as encouragement. Moreover, I would like to express my gratitude to University Technology Malaysia for the facilities and opportunities given for me to pursue this study. Last but not least, I would like to thanks my family for their patience, prayers, undivided support and encouragement to me during this st udy.
v
ABSTRACT
Composite laminate is one of the most popular materials used in various field nowadays due to its advantages and better performances. This advanced technology opens a new route for the world to become less depending on t he usage of conventional monolithic mater ials especially metal. The present study of composite laminate is based on Kirchhoff plate theory, or known as classical lamination theory. The analysis of composite laminate is not simple as normal materials. Composite combines behaviors of at least two constituents and they can be complex at times. A lot of studies had been done to analyze the engineering properties of co mposite materials through several types of numerical method. Finite element method is believed as the most powerful numerical method for analysis. With the introduction of finite element method in the composite field, analysis and design are made achievable. Unfortunately, the analysis of composite t hrough finite element method needs a lot of large scale matrices and we cannot do it without computer. However, there have not single program that specially designed for the analysis of co mposite laminate. This comes with the idea of the study about the formulation of composite laminate for general use in available numerical tools such as finite element method.
vi
ABSTRAK
Masa terkini, komposit laminasi merupakan sejenis bahan yang paling popular dipilihgunakan dalam pelbagai bidang disebabkan kebaikan dan kelebihan atas prestasinya. Teknologi yang canggih ini berjaya membuka satu laluan baru kepada dunia supaya tidak terlalu bergantung kepada bahan monolitik yang konvensional yang semakin berkurangan, terutamanya logam. Kajian terhadap komposit laminasi terkemas kini adalah berdasalkan Kirchhoff plate theory ataupun dikenali juga sebagai teori laminasi klasikal. Pada hal, kerja analisis tentang komposit laminasi tidak semudah bahan biasa. Hal ini disebabkan komposit laminasi menggabungkan sekurang-kurangnya sifat-sifat dua bahan asalnya dan menjadiakannya rumit sekali dikaji. Banyak kajian telah dilakukan untuk menganalisis sifat-sifat kejuruteraan komposit laminasi dengan menggunakan pelbagai jenis kaedah berangka. Antaranya, kaedah unsur terhingga dipercayai merupakan kaedah berangka yang paling berkuasa dalam analisis. Melalui pengenalan kaedah unsur terhingga dalam bidang komposit, analisis dan rekaan dapat dijalankan. Malangnya, analisis terhadap komposit berdasalkan kaedah u nsur terhingga memerlukan banyak matriks berskala besar dan kita tidak boleh melakukannya tanpa bantuan komputer. Pada hal yang demikian, saban hari ini belum ada satu program yang direkakan khusus untuk menganalisis komposit laminasi. Berdasalkan inilah kewujudan idea untuk merumuskan sifat-sifat komposit laminasi menjadi satu rumusan yang boleh digunapakai secara umum melalui kaedah yang sedia ada seperti kaedah unsur terhingga.
vii
TABLE OF CONTENTS
CHAPTER
TITLE
PAGE
DECLERATION
ii
DEDECATION
iii
ACKNOWLEDGEMENTS
iv
ABSTRACT
v
ABSTRAK
vi
TABLE OF CONTENT
vii
LIST OF TABLES
ix
LIST OF FIGURES
x
LIST OF SYMBOLS AND
xii
ABBREVIATIONS
1
2
INTRODUCTION
1.1
Background of study
1
1.2
Fiber Reinforced Composite
2
1.3
Problem statement
5
1.4
Objective
6
1.5
Scope of study
7
LITERATURE REVIEW
2.1
Introduction
8
2.2
Previous research
9
2.3
Conclusion
13
viii
3
4
RESEARCH METHOFOLODY
3.1
Introduction
15
3.2
Related equations
18
RESULT AND ANALYSIS
4.1
Introduction
27
4.2
Pre-processing stage
28
4.2.1 Lamina
28
4.2.2 Laminate
29
4.2.3 Finite element formulation
30
4.2.4 Stiffness matrix, K for composite
31
Analysis stage
31
4.3.1 Fiber and polymer matrix material
34
4.3
properties 4.3.2 Number of layers, thickness and orientations
of
the
laminae
35
in
laminate
4.4
4.3.3 Boundary conditions
44
Post-processing stage
46
4.4.1 Normalized deflection,
w
versus
47
numbers of element 4.5 5
Second analysis
49
CONCLUSION AND RECOMMENDSTION
5.1
Conclusion
60
5.2
Recommendations
62
REFERENCES
63
APPENDIX A
65
APPENDIX B
68
ix
LIST OF TABLES
TABLE
4.1
NO.
TITLE
Normalized deflection versus number of
PAGE
47
element 4.2
The K value for local and global coordination
56
4.3
The relationship between type of orientations,
57
B11 /B22, D11 /D22 and
maximum normalized
deflection of the laminated plate
x
LIST OF FIGURES
FIGURE NO.
TITLE
PAGE
1.1
Composite Laminate Plates
4
3.1
Definition of the ABD matrix
16
3.2
Finite element formulation steps
17
4.1
Pascal Triangle
30
4.2
The default Matlab desktop
33
4.3
The Matlab Editor Interface
33
4.4
Matlab result for ABD matrix
35
4.5
Matlab code for the ABD matrix
37
4.6
Node and element number
38
4.7
Matlab code for the element stiffness matrix, K
39
4.8
The normalized deflection of plate versus the numbers of element
48
4.9
ABD
matrix for symmetric case
50
4.10
ABD
matrix for anti-symmetric case
51
4.11
ABD
matrix for balanced case
51
4.12
Verification of K local and K global
56
4.13
Surface plot of deflection for symmetric case
58
4.14
Surface plot of deflection for antisymmetric case
58
xi
4.15
Surface plot of deflection for balanced case
59
xii
LIST OF SYMBOLS AND ABBREVIATIONS
FRC
-
Fiber reinforced composite
FEM
-
Finite element method
FEA
-
Finite element analysis
dof
-
Degree of freedom
V f , V m
-
Volume fraction of fiber and matrix respectively
E f , E m
-
Young Modulus of fiber and matrix respectively
G12f , Gm
-
Shear modulus of fiber and matrix respectively
v12f , vm
-
Poisson’s ratio of fiber and matrix respectively
E 1
-
Longitudinal Young’s modulus
E 2
-
Transverse Young’s modulus
G12
-
In-plane shear modulus
υ12
-
Poisson’s ratio
ξ
-
Measure of fiber reinforcement of the composite that depends on the fiber geometry, packing geometry, and loading conditions. The value of ξ is taken as 2 for E 2 calculation while 1 for G12 calculation.
Qij
-
Lamina stiffness matrix
xiii
Qij
-
Transformed stiffness matrix
N
-
In plane force
M
-
In plane moment
є
0
-
Midplane strain
κ
-
Curvature
Aij, Bij, Dij
-
Laminate extensional stiffnesses, laminate-coupling stiffnesses, and laminate-bending stiffnesses respectively
u, v, w
-
Displacement in x, y, z direction respectively
Ø x, Ø y
-
Rotation about the x, y direction respectively
ζ x , ζ y
-
Wrapping of the normal in x and y direction
-
Shape function for in-plane and out-of-plane degree of
respectively N ,i N o
freedom respectively [ B]
-
Element strain matrix
[ K ]
-
Element stiffness matrix
F
-
Forces
q
-
Transverse distributed load
1
CHAPTER 1
INTRODUCTION
1.1
Background of study
Composite materials have been widely applied to various fields of work as the replacement of traditional monolithic materials such as metals, ceramics or polymers due to its advantages and better performances. Generally, composite are materials that combine two or more conventional monolithic materials into one. Such a material contains the charact eristics of both its origin materials and other new characteristics that might be useful.
Composites can be divided into various groups depends on their criterion such as metals and non-metals, natural or manufactured, usage, and application. The most primitive composites that can be obtained easily from market are brick for building construction, asphalt concrete for roadway and fiberglass. Composites are being used due to their advantages such as the costs, the strengths, the properties, the durability or even the availabilities in our surroundings. They have desirable
2
properties that cannot be achieved by any other constituent materials that acting alone. Besides, the composite materials still have a great considerably potentials which are yet to be discovered.
In the field of research, the engineering constants of the mat erials are essentially of top priority before t he usages and limitations of the materials can be determined. Hence, it is important to first obtain the micromechanical and macromechanical properties of the composites from the combination of two or more conventional monolithic materials. The method of the computation of the micromechanical properties of composites has been invented many years ago by reasonable assumptions and several equations. Theoretically, it is not a difficult task for us to identify all the micromechanical and macromechanical properties of the composites with reasonably simple assumptions. However, when we are considering other factors that might affect internally o r externally the composites such as thickness, number of layer, orientations, thermal stability, distortions, delaminating, etc, the computation could becomes complicated and tedious. In order to solve those problems, several numerical methods such as finite e lement method, finite difference method, mesh free method and other methods have been applied into the computation of the micromechanical and macromechanical properties of the composites.
1.2
Fiber Reinforced Composite
Fiber reinforced composite is a type of composites with the combination of three components: (i) fiber as the discontinuous phase, (ii) matrix as the continuous phase and (iii) interphase. The fiber is naturally produced from the cellulosic waste streams to form high strength fiber composite materials in a polymer matrix. The
3
wood contains mainly of fibrous cellulose in a matr ix of lignin, whereas most mammalian bone is made up of layered and oriented collagen fibrils in a prot eincalcium phosphate matrix (Wainwright, Biggs, Currey and Gosline, 1976).
The primary function of the fibers is to carry the loads along their longitudinal directions and to obtain maximum tensile strength and stiffness of a material when held together in a structural unit with binder or matrix material. Common fiber reinforcing agents include aluminum, asbestos, beryllium, graphite, glass, molybdenum, polyamide, polyester, quartz, steel, tantalum, titanium and tungsten. Among the aforementioned, glass fibers and carbon or graphite fibers are the most widely used advanced fibers. Glass/epoxy and glass/polyester composites are used extensively in applications ranging from fishing rods to storage t anks and aircraft parts, while high strength carbon fibers have a t ensile strength more than 6 times than that of the steel (Gibson, 2007).
Polymers, metals, and ceramics are t he matrix materials used in composites to hold the fibers together and to protect them from damage. Besides, those materials also contain some properties such as ductility and toughness that are needed in certain fields. Likewise, metal have electrical conductivity and high melting temperature that are useful in the electrical engineering composites. Polymer composites are very common lightweight, thermal and electrical insulator. The composite industry is maturing into an established and increasingly diversified business. Composite manufacturing offers the benefits of producing lightweight, strong and moldable products in a variety of shapes.
The characteristics of a composite plate are affected by the arrangement of its components in lamina or laminate state. Lamina is a plane layer of unidirectional fibers in matrix, arranged longitudinally; whereas la minate is two or more
4
unidirectional lamina stacked together with various orientations. With different orientations and configurations, composite can form the mater ials with high-stiffness; high-strength and low density which can have better charact eristic than the monolithic materials such as metal and polymer, or other mixture materials like concretes and polymers. Micromechanics are the analysis of materials on the interactions of the microscopic structure considering the state of deformation and local failure. However, composite lamina or laminate can only be analyzed based on its average characteristics such as stiffness, density, strength and other configurations as well to predict its deformation and local failure behaviors.
Figure 1.1
Composite Laminate Plates
Composite materials have many advantages over traditional metal and alloy based structures, especially its strength-to-weight ratio, lower maintenance requirements and greater corrosion resistance. Besides, composites exhibit a higher strength to weight ratio than steel or aluminum and can be modified based on its configuration and different components to provide a wide range of tensile, flexural and impact strength properties. Fiber reinforced composites such as carbon and glass reinforcement fibers have comparable densities compare to its original components.
5
Greater strength and less weight dramatically improve its performance outcomes. Moreover, composites are corrosion resistant to most chemicals, free from electrolysis and incorporate long-term benefits such as weat her sustainability and ultraviolent stability as well.
1.3
Statement of problem
The research on the parameters of composite materials is not as easy as conventional monolithic materials. The values of Young’s Modulus and Poisson’s Ratio vary from the arrangement of unidirectional of fiber and its orientations. The total number of independent elastic coefficients of anisotropic fiber reinforced composites is 81 but theoretically, most complex engineering problem only concerns with 9 constants. Despite that, it is still a difficult task to obtain all the independent coefficients through analysis either using computer or experiment.
There are a lot of computer aided programs for design purpose which do not include the composite as their basic element. So far, analysts have to model the composite materials independently, for example each lamina in a laminate is modeled one by one and layers by layers before assembled to gether into one component. They also have to model the fiber and matrix separately. The experiments will be done in such a way to identify all the para meters needed for further research. Even though the results of the modeling are satisfactory, but it is usually time consuming. Besides, we can have over thousands of models with different combinations of materials, orientations, numbers of layer, different s izes and other factors as well in a research, the cost can be escalatingly high and
6
sometimes impractical. Hence, it would be helpful to have a single element that represents both fibers and matrix and multilayer laminates.
It is well-known that the numerical methods can be app lied to the existing researches in order to formulate the behaviors of composites and to subsequently program them into any computer software, aiming at the analysis of the composite materials just treated like any other materials. One of the methods of formulation that are widely used in analysis is the finite element method (FEM) which discretizes a model into several elements so t hat an analysis or computation can be performed using proper assumptions and boundary conditions. If we are able to formulate the composite with the FEM, we will be able to program the formulae we construct into any types of analysis software. Such a task requires detailed descriptive knowledge on both the FEM and the mechanics of the composite. The main theme of the current study aims to provide the link for aforementioned difficulties.
1.4
Objective
The objectives of this study are: i. ii.
To formulate an element stiffness matrix for a composite laminate. To make a comparison between such an element with the existing ones in terms of displacement.
iii.
To develop the Matlab program for (i) and (ii).
iv.
To analyze and compare three different orientations of fiber reinforced composite.
7
1.5
Scope of study
This study only focus on the linear elastic behaviors o f fiber reinforced composites of transversely isotropic type. The lamina is unidirectional and square in term of size. All the laminae are flat plate and there are limitations in terms of the arrangement. My study focuses on the formulation of fiber reinforced composites by using finite element method and the plate is considered thin, which is dominantly based on the classical lamination theory. Besides, there are four nodes in each element comprising 5 degree of freedoms at each node. The degree of freedoms are displacement in x direction (u), displacement in y direction (v), displacement in z direction (w), rotation about the x direction (Ø x), and rotation about the y direction (Ø y ), following Szilard’s theories and applications of plate analysis.
8
CHAPTER 2
LITERATURE REVIEW
2.1
Introduction
Fiber composite materials are increasingly used in variety of systems, such as aircrafts and submarine structures, space structures, automobiles, sport equipment, medical prosthetic devices and electronic circuit boards. It is materially efficient in applications that required high strength to weight and stiffness to weight ratios. With the increasing use of fiber reinforced composites in structures components, studies involving the behavior made of composites are getting considerable attention. The analytical study and design of co mposites requires knowledge of anisotropic elasticity, structural theories and failure mode.
Finite Element (FE) Method is one of the preferable methods used in the analysis of the structural and mechanical behavior of materials, especially when the researchers are dealing with the non-linear structure or a complex structure which cannot be analyzed with merely hand calculation or even a computer program if the
9
structure is not properly formulated. It is considerably powerful numerical techniques devised for solving solid, structural mechanics, and e ven multidisciplinary problems in geometrically complicated regions.
2.2
Previous Research
The first finite element based failure analysis of composites was studied by Lee (1982) incorporating a direct mode in determining the failure criterion and the standard laminate strength of plates with circular holes. Reddy and Pandey (1987) formulated a two dimensional plate element with a first ply failure analysis of composite laminates based on the first order shear deformation plate theory. The element was further developed by Engblom and Ochoa in 1986 with an increased interpolation function in the t hrough thickness direction.
Tolson and Zabaras (1990) developed a two dimensional finite element failure analysis for composite plates in plate analysis with more accurat e and flexible form. They used seven degrees of freedoms per node in the finite element model for laminated composite plates. The degrees of freedom are u, v, w, Ø x, Ø y , ζ x , and ζ y, where Ø x and Ø y are rotations of the normal to the midplane about the x and y axes respectively, where ζ x and ζ y describe the wrapping of the normal in x and y direction respectively. Isoparametric finite element was introduced in the study and the eight noded serendipity and nine noded heterosis elements are used in this analysis. The results were compared with classical plate theory, three dimensional elasticity solutions and other finite element formulations. They claimed that the first ply failure (FPF) and last ply failure (LPF) strength determined by the analysis correlate
10
well with the actual failure strengths. The model is quite effective in providing limits concerned and useful in the design of laminated composite plates.
Pegoretti et. al (2001) studied the mechanical behavior of glass fiber reinforced composite endodontic post. The use of composite in the dentistry intervention is for treatment purpose of pulpless teeth. The aim of the study is to analyze the mechanical behavior of a new polymeric composite post reinforced with glass fibers, both experimentally and through FE analysis. There are four bidimensional model built for analysis purpose with four different types of materials. One of the materials is the natural tooth treated as the reference model where the stiffness of the model is equal to enamel and dentine. The other materials such as fiber glass composite, gold alloy and other types of metal alloy are used in the model study. The simulation results are compared with those of commercially available carbon fiber reinforced and gold alloy cast posts. The mechanical behavior of a new glass fiber composite post was simulated by a FE analysis on a bidimensional model. The result shows that the fiber-reinforced composite posts present quite high stresses in the cervical region due to their flexibility and also to t he presence of a less stiff core material.
An experimental and finite element analysis of the st atic deformation of natural fiber-reinforced composite beam had been done by Lim et. al (2002). They used the shadow Moiré method for the direct measurement of whole field deformation of cantilever beam. A ‘beating’ between two structures is observed in the form of another periodic structure, known as the Moiré fringe pattern. The Moiré fringe pattern is a sinusoidal function and is r epresented by intensity distribution I ( x, y) written in a general form as follows:
I ( x, y) a( x, y) b( x, y) cos[ ( x, y) ] where, a( x, y) is background intensity variation, b is the modulation strength, f( x, y) is the phase at point ( x, y), and
is the amount of phase shift. For the finite element
11
analysis, the average value of the modulus of elasticit y obtained in the three-point bending tests was used. In order to obtain the FEA results, the model of the cantilever was first created using SolidWorks software, and then analyzed using COSMOS Works. The results obtained from the FEA are compared with the shadow Moiré’s results. The comparison of the predictions from the FEA and the optical measurement shows a maximum difference of 10% at the free end of the cantilever. This technique, therefore, can be used as a non-contact as well as a nondestructive technique to validate the finite element model.
Thermal buckling of cross-ply composite laminates is one of the important studies in the field of composite. Mathew et. al (1990) used one dimensional finite element analysis consisting two nodes and six degrees of freedom. A cross-ply laminate having many orthotropic layers with different thickness was subjected to constant temperature where the formulation of the structure is done using finite element method. In the end of the formulation, the geometric stiffness matrix, K e is derived. Applying different boundary conditions, the mechanical and thermal buckling loads are computed. The formula produced can be used in the determination of buckling behavior of the structure as well a s for a parametric study. The results proved that the laminates with immovable end conditions have higher buckling resistance and buckling parameter reduces with a decrease in the slenderness ratio. Besides, it is also proven that finite element formulation can be applied in the analysis of fiber composites in thermal buckling.
Kari et. al (1988) applied the finite element formulation in the study of thermal buckling of composite laminated plate. They used semiloof shell element formulation with 43 degrees of freedom in the analysis but 11 degrees of freedom were eliminated based on the Kirchhoff shear constrains. The field variables in the local coordinates are expressed as
U S q
12
where
U T U ,V ,W ,U x ,U y ,V x ,V y ,U z ,V z ,U xz ,V xz ,U yz ,V yz
and S is the transformations of shape functions while qis the vector of element degree of freedom. Total potential energy is used in the derivation of equations for prebuckling solution. The computer program COMSAP was developed and used to handle the temperature variation of both surface and thickness of fiber laminates. The accuracy and efficiency of the element aga inst the known cases; critical temperatures for different cases of composite laminates are obtained from the program.
Another study on the composite plate bending element based on a higher order shear deformation theory through finite element analysis had been done by A.H. Sheikh and A. Chakrabarti in 2001. In the study, they were considering 7 degree of freedoms such as u, v, w, Ø x, Ø y, γ x and γ y. They compare the results among two plate theories, the higher-order shear deformation theories (HSDT) and the first-order shear deformation theory (FSDT). In their results, their compared both theories under different conditions such as isotropic square plate simply supported at all the edges, isotropic rectangular plate having different boundary conditions at four edges, cross-ply square laminate subjected to uniformly distributed load, cross-ply rectangular laminate subjected to distributed load sinusoidal variation, cross -ply skew laminate subjected to uniformly distributed load, cross-ply ant i-symmetric laminate subjected to distributed load of sinusoidal variation and angle ply antisymmetric laminate subjected to uniformly distributed load. In the study, they show that the different between HSDT and FSDT is small and the accuracy is increased with the increasing of the number of element. Moreover, a triangular element based on Reddy’s HSDT is developed which is able to give more precise results for the analysis using FEA. Even though it is not the only researches about the analysis of plate using FEM, it is a very good study for the verification of the currently formulated model through the comparison of the results since one of the case studies is set as the same as the current investigation.
13
Szilard had discussed about the theories and applications of plate analysis in his book published in 2004. He used finite element method as a tool of plate analysis. He mentioned that by mid 1990, roughly 40,000 papers and more than one hundred books had been publish about the FEM but only 87 are listed in his book which is related with his plate analysis. It shows the important of FEM in terms of numerical solution of engineering problems. In his book, he assumed the plate are flat and quadrilateral with 3 out-of-plane degree of freedom existed at each nodes. Through FEM, he formulates a quadrilateral and a triangular element for the plate for a comparison with the results obtained with a prescribed polynomial order. In the beginning stage, he analysis several types of plate element such as simple plate or thin plate rectangular element, simple plate triangular element, higher order plate elements (16 DOF) and discrete Kirchhoff triangular element. In the advanced stage, he analyzed moderately thick plate elements and thick plate elements. He also illustrated some example of calculations under his analysis in order to present his models.
2.3
Conclusion
A lot of analysis of composite materials had been done t hrough finite element modeling. By modeling the structures into finite element and processing the analysis with computer aided programs such as COSMOS, Solidworks, Pro-Engineer, SolidEdge, Nastran, etc, the results can be obtained easily. However, the results are questionable nowadays since the programs used mostly applied for isotropic materials. For example, we can analyze the reinforced concrete bridge with Lusas through finite element method. The setting of the program can be determined normally through the pre-determined materials properties or the existing database of the program. However, when we are dealing with composites laminated bridge; we need to determine the engineering properties of the composite through the lamination
14
theories and apply the obtained engineering constants into the program since the material library does not have any record for a composite material. Unfortunately, the structure is still considered as isotropic when the materials are prescribed and the behaviors of laminates such as delamination of composite are always being neglected.
There are a lot of researches on the fiber reinforced composite related to the finite element method besides the studies that have mentioned above. All the papers and reference books mentioned are interrelated to my research, which is the finite element formulation of a fiber reinforced composite. However, all the studies are not considering the program development after the formulation stage. So, it is a challenge to program a finite element model for a composite laminate into any available software. By referring to previous finite element formulations it is hoped that a finite element for a composite laminate can be produced for an extensive use in any commercial FE software.
15
CHAPTER 3
RESEARCH METHODOLOGY
3.1
Introduction
This chapter explains the flow and the methods that are used in my research. Beside the general formulae used in deriving the basic composite parameters, the finite element method will also be applied in my study beginning with the formulation stage. Figure 3.1 and 3.2 show the flows of the research in brief.
16
Lamina
Select the engineering constants E 1f, E 2f, G12f and v12f for fiber.
Select the engineering constants E m, Gm and vm for typical polymer matrix.
Define the volume fraction of fiber and polymer.
Compute the engineering constants E 1 , E 2 , G12 and v12 of fiber reinforced composite by the Rules of Mixtures and the Halphin-Tsai Equation.
Compute the lamina stiffness matrix, Qij in local coordination.
Laminate
State number of layers of lamina and its angles.
Compute the transformed lamina stiffness matrix.
Compute the ABD matrix.
Matlab
Figure 3.1
Definition of the ABD matrix
17
FEM
Discretization of the continuum.
Selection the suitable interpolation function based on degree of freedom.
Element formulation.
Matlab
Develop element stiffness matrix, K in local coordination.
Assembly the K into global coordination.
Apply boundary condition.
Solve the equation.
Post-processing
Figure 3.2
Finite element formulation steps.
18
3.2
Related equations
The research begins from the determination of the material properties. The engineering constants E 1, E 2, G12 and v12 of the composite can be computed through the Rules of Mixture and the Halphin-Tsai Equation. First, we have to get the engineering constants of each component of the composite in order to compute the engineering constants of the composite lamina. It is assumed that the total of fiber volume fraction, V f and matrix volume fraction, V m is equal to 1 since the lamina only contains two components, carbon fibers and matrix. All the engineering constants are important for calculating the stiffness matrix of lamina, Qij in the local coordinate. The equations used in this stage are:
i.
V f + V m = 1,
(3.1)
Where V f is the volume fraction of fiber and, V m is the volume fraction of matrix.
ii.
Rules of Mixtures E 1 = E 1f V f + E mV m where E 1f is the Young Modulus of fiber, E m is the Young Modulus of matrix, V f is the fiber volume fraction, and V m is the matrix volume fraction.
(3.2)
19
v12 = v12f V f + vmV m
(3.3)
where v12f is the Poisson’s ratio of fiber, vm is the Poisson’s ratio of matrix, V f is the fiber volume fraction, and V m is the matrix volume fraction.
iii.
Halphin-Tsai Equation
E 2
E m 1 V f 1 V f
(3.4)
where, is the curve-fitting parameter, and
E 2 f E m
(3.4a)
E 2 f E m
G12
Gm 1 V f 1 V f
(3.5)
where, is the curve-fitting parameter, and
iv.
G12 f Gm G12 f Gm
(3.5a)
Lamina stiffness matrix, Qij
Q11
Q12
E 1 1 v12 v 21
(3.6a)
v12 E 2 1 v12 v 21
(3.6b)
20
Q22
E 2 1 v12 v 21
(3.6c)
Q66 G12
(3.6d)
After that, we need to transform the lamina stiffness matrix into a global form using the transformed coefficient. In global form, lamina with different angles and thickness of each layer are computed. Hence, the transformed stiffness matrix, Qij can be calculated. The equations used in this stage are as follow:
Q11 Q11 co s4 Q22 sin 4 2Q12 2Q66 sin 2 cos2 Q12 Q11 Q22 4Q66 sin cos Q12 2
2
(3.7a)
cos sin 4
4
(3.7b)
Q 22 Q11 sin 4 Q22 cos 4 2Q12 2Q66 sin 2 cos2
Q16 Q11 Q12 2Q66 cos sin Q22 Q12 2Q66 cos sin 3
(3.7c)
3
(3.7d)
Q 26 Q11 Q12 2Q66 cos sin Q22 Q12 2Q66 cos sin 3
3
Q 66 Q11 Q22 2Q12 2Q66 sin cos Q66 2
2
sin
4
(3.7e)
(3.7f)
cos4
The computation of the material properties of laminate with several layers of lamina and orientations can be done using the theory of lamination plates. Assuming the individual laminae are perfectly bonded together so as to behave as a unitary, nonhomogeneous anisotropic plate, interfacial slip is not allowed. The deformation hypothesis from classical homogeneous plate theory and the laminated forcedeformation equation can be used to define the coordinate system in developing the
21
laminated plate analysis. analysis. Hence, we can compute the the force, N force, N and and moment, M moment, M per per unit length for the laminate, which can be compute as follows:
N x A11 N y A12 N xy A 16 M x B11 M y B 12 M xy B16
A12
A16
B11
B12
A22
A26
B12
B22
A26
A66
B16
B26
B12
B16
D11 D12
B22 B26 D12 D22 B26 B66 D16 D26
0 B16 x 0 B26 y
D16 x D26 y D66 xy 0 B66 xy
(3.8)
where N is is the in-plane force, M is is the in-plane moment,
0 is the midplane strain, is the curvature, and
Aij
N
Q z k 1
Bij
1
k
z k
Q z 2 1
2
ij k
Q z 3
1
(3.8b)
3
k 1
z k 2
k
N
ij k
(3.8a)
1
N
k 1
Dij
ij k
z k 3
k
1
i, j = 1, 2, or 6
(3.8c)
Aij, Bij and D and Dij are laminate extensional stiffnesses, laminate-coupling stiffnesses, and laminate-bending stiffnesses respectively. z k is the corresponding cor responding k is distance from middle surface to outer surface of the k th th lamina, and z and z k-1 k-1 is the distance from the middle surface to the t he inner surface of the k th th lamina.
22
Next, we have to do the analysis analysis through the finite element method (FEM) from the development of a shape function which can represent all the para meter involved for a single single element of the laminate. laminate. The FEM starts with the discretization of the plate into into several elements. Using the interpolation interpolation function, a general general shape function for an element element can be formulated. By considering considering the engineering behavior behavior of the laminate, we combine the element strain matrix, [ B] B] with the ABD the ABD matrix matrix and develop into a local stiffness matrix, K matrix, K which which applies for a laminate. laminate. The following following steps are similar to the conventional FEM, for example the assembly of the element stiffness matrix into global stiffness matrix, before an application of the initial initial condition and boundary condition condition for the laminate. laminate. It ends by by solving the simultaneous equations. equations. All the the formulae related related are stated below: below:
i.
In-pl In-plane ane shap shapee fun functi ction on,,
u N i1u1 N i 2 u 2 N i 3u3 N i 4 u 4 v N i1v1 N i 2 v 2 N i 3 v3 N i 4 v 4
(3.9) (3.10)
where u is the displacement in x in x direction direction and, v is the displacement in y in y direction. direction. and,
y N i1 1 x L 1 L x y
(3.9a)
N i 2 x 1 y L x L y
(3.9b)
1 x L y L x
(3.9c)
y N i 3 N i 4
xy L x L y
(3.9d)
where N i1 and N i4 i1 , N i2 i2 , N i3 i3 and N i4 are the shape function for u and v, and
23
x and x and y y are the variables in the element and, L x and L and L y are the dimension of the element in x in x and and y y direction direction respectively.
ii.
Out-of-plane shape function,
w N o1w1 N o 2 1 x N o 3 1 y N o 4 w2 N o 5 2 x N o 6 2 y
N o 7 w3 N o8 3 x N o9 3 y N o10 w4 N o11 4 x N o12 4 y
(3.11)
where w is the displacement d isplacement in z in z direction direction and, Ø x is the rotation in y in y direction direction and, Ø y is the rotation in x in x direction. direction. and,
N o1
1 3 x
3 xy
2
L x
2
L x L y
N o 2 y
2
xy
2
xy
2 y
L x L y
3
L y
3
2 x
3 y
2
L y
2 x
2
3
y 3
L x L y
3
2 xy
L x
3
3 x
2
y
2
L x L y
3
L x L y
(3.11a)
3
2 2 2 y 2 xy y 3 xy 3 2 2 L x L y L x L y L y L x L y
(3.11b)
2 2 x 2 3 2 x y xy x 3 y x L N o 3 x 2 2 L x L x L y L L L y (3.11c) y x x
N o 4
2 x
xy L x L y
3
y
N o 5
3
L x L y xy L x
2
3
2 xy
3
N o 6 x
2 x
2 xy
L x
3
L x L y 2
L x L y
3 x
3
2
y 2
L x L y
3 x
3 xy
2
L x L y
2
2
L x
2
xy 3 L x L y
(3.11d)
2
x 2 y x 3 y x 3 2 2 L x L x L y L x L x L y
(3.11e)
(3.11f)
24
N o 7
2 y
xy
2
2
2
3 3
L y
N o8
2
3 y 3 x y 3 xy L x L y L y L x L y L x L y
2 x
3
y
3
L x L y
xy 2 L x L y
y 3 L y
2
2
2 xy
2
(3.11g)
3
L x L y
3
xy 3 L x L y
2
y 2
(3.11h)
L y
2 xy 2 x y x 3 y N o 9 2 L y L x L y L L x y
xy
N o10
2 xy
L x L y
3 x
2
y 2
L x L y
3 xy
(3.11i)
2
L x L y
2
2 x
3
y
3
L x L y
x 2 y
(3.11j)
3
xy 2 N o11 N o12
3
L x L y
3
xy 2 L x L y L x L y
(3.11k)
x 3 y 2 L x L y L x L y
(3.11l)
where N o1 and N o12 o1, N o2 o2, N o3 o3, N o4 o4, N o5 o5, N o6 o6 , N o7 o7 , N o8 o8, N o9 o9, N o10 o10, N o11 o11, and N o12 are the shape function for w, Ø x and Ø y, and x and x and y y are the variables in the element and, L x and L and L y is the dimension of the element in x in x and and y y direction direction respectively.
iii.
Element strain matrix, [ B] B]
B N where is the displacement differential operator.
(3.12)
25
iv.
Stiffness matrix, [ K ]
K BT D BA
(3.13)
where [ B] is the element strain matrix and [ D] is the elasticity matrix.
v.
Equilibrium of force displacement equation { F } = [ K ]{d }
(3.14)
where { F } is the nodal forces and {d } is the nodal degree of freedom
In the finite element method, force for surface,
F N qA
(3.14a)
where q is the transverse distributed load
vi.
Strain-displacement relationship
Bu
(3.15)
where is the element strain
A Matlab program is written for all stages through a proper Matlab code. By the end of my study, a user friendly program will be available for the determination of the nodal degree of freedoms through the FEM. In this program, users do not have
26
to worry too much about the computation of the composite laminate properties. They just need to include a few basic parameters such as Young Modulus, in plane shear modulus, Poisson’s ratio and volume fraction of fiber and matrix respectively, ply orientation, numbers of layer, thickness of each layer, number of element for a laminate and its corresponding dimensions.
27
CHAPTER 4
RESULT AND ANALYSIS
4.1
Introduction
This chapter presents all the results and analysis following the original sequences stated in Chapter 3.
The chapter is divided into 3 parts. They are the pre-processing stage, the analysis stage, and the post-processing stage and the descriptions of each stage are as follows. In the pre-processing stage, the methods of the analysis which has been described briefly in Chapter 3 are elaborated and discussed in detail here. In the analysis stage, we need to determine our variables used for our analysis in order to obtain certain results. For the post-processing stage, we present our results in a better way using different data processing tools such as the grap h plotting, the data tabulation, etc. All the processes will be discussed in details and under preferable sequences for the convenience of the readers.
28
4.2
Pre-processing stage
In the pre-processing stage, we will describe the detail of the important data required for the analysis. It is an important for us to identify our input and output of analysis before the process take place. Besides, it provides a clear guide for the user in understanding the stages of the analysis in sequence.
4.2.1
Lamina
As mentioned before, lamina is a layer of composite which contains the fiber and the polymer in a fiber reinforced composite. Each of this materials, either the fiber or polymer has different engineering properties such as density, Young’s modulus, shear modulus, Poisson’s ratio, tensile str ength, compressive strength, shear strength and so on. In this study, we just consider the Young’s modulus, shear modulus and Poisson’s ratio of fiber a nd polymer since the scope of study is constrained to the transverse isotopic t ype which follows the Classical Lamination Theory. From the properties of E 1 ,f E 2 ,f G12f , v12f, E m , Gm and v m , we are able to obtain the engineering properties of lamina through the equations mentioned in Chapter 3 such as E 1 , E 2 , G12 and v12. The equations used in calculating the E 1 and v12 are following the rules of mixture while the E 2 and G12 obtained from the HalphinTsai equation have proven to be more accurate.
29
With the pre-defined engineering properties of lamina, we can calculat e the lamina stiffness matrix, Qij for each layer of lamina in laminate through equation 3.6 before we proceed to the analysis of laminate.
4.2.2
Laminate
In composite materials, all laminae are combined together and the product is a component called laminate. Unlike lamina, when two or more monolithic materials combined together, the engineering properties of the new component will changes due to different materials combination. It exists with a set of new engineering properties. For laminate, we consider only the global stiffness of the lamina which differs due to the orientation of each lamina and its thickness. The computation of the global stiffness of the laminate is performed using the equation 3.7 mentioned in Chapter 3.
The global stiffness of lamina only represents the stiffness of each lamina in laminate form. Hence, in order to define the laminate stiffness matrix, we are required to arrange all the laminae properly and compute it through the Theory of Laminate Plate. Assuming the individual laminae are perfectly bonded together so as to behave as a unitary, nonhomogeneous anisotropic plate, interfacial slip is not allowed. The result will be presented in the ABD matrix.
30
4.2.3
Finite element formulation
There are three basic fundamental considerations in using the FEM as t he method of formulation. They are the equilibrium of forces, the compatibility of displacement and the law of material behavior. In order to fulfill all the fundamental requirements, there are two types of deformation considered, the in-plane deformation and out-of-plane deformation. For in-plane deformation, the plate deforms in xy plane only. Hence, we have two degree of freedoms at each node. The formulation of in-plane shape function only require 1 st order interpolation function since the deformation is linear. For out-of-plane deformation, the laminate deforms in z direction and rotates in the x and y direction. Hence, each nodal point will have three out-of-plane degree of freedoms. To have an nonconforming rectangular element, the shape function with 12 terms of polynomial der ived from the Pascal’s triangle that is shown in the Figure 4.1 is necessar y (Szilard, Theories and Applications of Plate Analysis).
α1 α2x α4x α7x α11x
α3 y α5xy
α8x y α12x y
α6 y α9xy
α13x y
Figure 4.1 Pascal Triangle
α10 y α14xy
α15 y
31
4.2.4
Stiffness matrix,
K for
composite
After deriving the shape function, we are required to develop the stiffness matrix of the laminate through equation, K
B DBA T
(4.1)
In FE, B is the strain matrix coming from the differentiation of the shape function corresponding to its deformation whereas D is the element elasticity. In the current study, the correlation of the ABD matrix obtained from analysis of laminate with the strain matrix is crucial. The result of the element stiffness matrix is computed as the equation below: K
Bi T A ABD Bi Bi T B ABD Bo Bo T B ABD Bi Bo T D ABD Bo A
(4.2)
where Bi is the in-plane element strain matrix and Bo is the out-of-plane element strain matrix. The K is a 20 by 20 matrix. It is in a local coordination which needs to be rearranged and assembled into the global stiffness matrix later depending on the numbers of element. For the current study, the K is integrated symbolically not by numerically method although the computation based on the latter is much more efficient and faster.
4.3
Analysis stage
In this stage, we are required to define all the values of t he variables in order to run the program and perform the analysis. As mentioned before, the program used here is Matlab.
32
Matlab is a fourth generation computer program which is used widely for a matrix operation. The word ‘Matlab’ stand for matrix laboratory. The interfaces of Matlab are shown in Figure 4.2 and Figure 4.3. Figure 4.2 shows the Matlab ’s command window normally used for presenting the work with the pre-described coding in the workplace that is shown in Figure 4.3. It is also used to show the answer or result of the work here. As usual, Matlab contains toolbar with normal function like file, edit, debug, desktop, window and help.
Even though Matlab is a program designed for the matr ix operation, it provides others functions such as plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages including C, C++, and Fortran. Moreover, Matlab is an advanced computer program of Maple which is also a programming tool used for defining the mathematics related problems including the matrix.
Nowadays, Matlab is still not a common or famous programming tool for public use. It is only famous in certain fields such as mathematics, researches, program development, etc. It is because the Matlab has some limitations and specifications for general users. However, it is believed that the development of Matlab will be improved and accepted by public.
33
Figure 4.2
The default Matlab desktop.
Figure 4.3
The Matlab Editor Interface.
34
4.3.1
Fiber and polymer matrix material properties
The first part of the analysis is focuses on a cross-ply square laminate that is subjected to a uniformly distributed load. A same problem case has also been studied by A.H. Sheikh and A. Chakrabakti in paper entitled ‘A new plate bending element based on higher-order shear deformation theory for the analysis of composite plate’. The information of fiber and polymer matrix is listed below: E 1 = 174.6 GPa, E 2 = 7 GPa, G12 = 3.5 GPa and v12 = 0.25.
4.3.2
Number of layers, thickness and orientations of the laminae in laminate
There are 3 layers of lamina with the orientation [0/90/0] in the current analysis. The thickness of each lamina is depending on the dimension of the plate based on the ratio h/a which is set to 0.01 where h is the total thickness of the plate and a is the width of the plate. In this study, a is set to 2mm, hence the total thickness of the composite plate is 0.02mm with a lamina thickness of 0.0067mm each. From the equations 3.7 and 3.8 stated in Chapter 3, we can find the ABD matrix for this composite plate. The result is shown in Figure 4.4:
35
Figure 4.4
Matlab result for the ABD matrix
The result was verified through an online computational tool provided at www.efunda.com. It is done by key in the same materials properties. The result obtained from efunda is shown below:
36
As mentioned in Chapter 1, one of the objectives of this study is to develop a Matlab program by a proper code enabling user to find t he engineering properties of composite laminate with some simple input command of the chosen composite materials. The sample code for the ABD matrix is shown in Figure 4.5:
37
ABD matrix
Figure 4.5
Matlab code for the ABD matrix.
The analysis of the laminate is followed by the FEA after the determination of the ABD matrix. Next, we are required to insert a series of information of the laminate, such as the dimensions of the laminate and the numbers of element. In this study, a square plate with a size of 2mm by 2mm is considered. The plate is discretized into 2 by 2 elements, 4 by 4 elements, 8 by 8 elements, 12 by 12 elements and 32 by 32 elements. As mentioned before, the K for the composite plate is computed with Matlab by using equation 4.2. Figure 4.6 shows the discretization of the plate into 2 by 2 elements and the arrangement of every nodal point. In this study, the nodal points and the elements are arranged from left to right. Numbers in boxes represent the element number.
38
7
8 3
4
1
4 5
1
9
6
x
2 2
Figure 4.6
z
3
y
Node and element number.
The arrangement of the K matrix plays a crucial role in the finite element method as the definition and the computation of the correct results in the correct position is important. Normally, the arrangement of the K matrix is depending on the arrangement of nodal points. By default, since every nodal point contains 5 degree of freedoms following the sequences of {u, v, w, Ø x , Ø y}, the size of the K matrix for each element is thus 20 by 20. Unfortunately, the K matrix we obtained first not following the correct sequences due to the different shape functions definition for in plane and out-of-plane degree of freedoms. As a result, further coding is necessary for the rearrangement of the K matrix following the correct positions. Figure 4.7 shows the coding for the integration of the K matrix and the matrix rearrangement. The result obtained from Matlab for K matrix in local is shown in the following.
39
Double integration of K by part. Matrix Rearrangement
Figure 4.7
Matlab code for the element stiffness matrix, K .
Element stiffness matrix, K with size 20 x 20
40
After defining the K , we are required to arrange it into a global manner for each of the elements. Besides, we also need to assemble the entire element stiffness matrices into one which representing the whole composite plate before we can solve the degree of freedoms of the composite plate at each nodes and eventually other engineering behaviors such as stress, strains, reactions, moments, etc. The result shown below is the global stiffness matrix of the composite plate with 2 by 2 elements. The matrix size is 45 by 45 because it contains 9 nodal points with 5 degree of freedoms at each node.
41
K global with matrix size 45 x 45
42
43
44
4.3.3
Boundary conditions
After defining the K global, we need to define the boundary conditions of the plate. A uniformly distributed load of 12 N/mm2 is prescribed on top of the composite plate in z direction. In the FEM, the loads act only on the nodal points of the plate. These converted nodal loads can be computed through equation 3.14a in Chapter 3. The results obtained from equation 3.14a are only for one element. Hence, we need to assemble them into a global force the way we have done for the stiffness matrix, K . The results for local forces and global forces are shown as follows:
45
N means local forces and moments
rG means global forces and moments
Notes: for local coordination, the arrangement of forces and moments are in sequence of f z , M x, and M y, while for global coordination, the forces and moments are arranged in sequence of f x, f y, f z , M x, and M y.
Besides the loading, we have another boundary condition which relates to the type of constrain or support we applied to the composite plate. In the current study, the plate is rigid (u,v,w,Ø x and Ø y = 0) at one of the edges of the plate and pinned (u,w = 0) at the rest of the sides of the plate. Hence, we can eliminate the corresponding forces, the row and the column of the K matrix which are zeros. Finally, only 26 unknown deflections and rotations have to be solved and we can analyze the plate using equation 3.14 in Chapter 3. The solution obtained in the forms of the deflection, w and rotations, Ø x and Ø y at certain nodes. The results are shown as follows:
46
(d means deflections and rotations)
Deflection at the central
4.4
Post-processing stage
At this stage, we the results are presented in the simplest way for the clarity of the readers and the users. There are several methods that can be applied here. They are chart, graph, table and others. Here, the graph and the table are used as the means.
47
4.4.1
Normalized deflection, w versus numbers of element
Table 4.1
Normalized deflection versus number of element Number of Element
2 4 8 12 32
4 16 64 144 1024
wnd = (wh3 E 2 /qa4)100
Normalized deflection, wnd 0.9588 0.7306 0.6827 0.6741 0.6683
Sheikh & Difference(%) Chakrabakti 0.7514 0.6913 0.6763 0.6732 0.6708
-27.60 -5.68 -0.95 -0.13 0.37 (4.3)
Table 4.1 shows the results obtained from the Matlab program. The deflections shown here are different to that shown previously because the results are normalized in terms of equation 4.3. The results show that the deflection at the center of the plate is converging to the published results when the numbers of element is increased. The difference of the results when compared with the results obtained from Sheikh and Chakr abakti’s study is also decreasing with the increasing of the numbers of element. In other words, for the analysis of the laminated plate, when the numbers of element is 64 or more, the results ca n be considered as satisfactorily good. The relationship between numbers of element and the deflection is plotted in Figure 4.8.
48
Normalized deflection, w versus numbers of element 1.0000 w , 0.9500 n o i t 0.9000 c e l f e d 0.8500 l a r t 0.8000 n e c d 0.7500 e z i l a 0.7000 m r o 0.6500 N
deflection Sheikh & Chakrabakti
0.6000 0
250
500
750
1000
1250
Number of Element
Figure 4.8
The normalized deflection of plate versus the numbers of element.
Figure 4.8 shows the maximum deflection of the laminated plate versus t he numbers of element for the plate. At the beginning, the deflection is high but it decreases with the increase of the numbers of element. The graph becomes almost linear when the numbers of element is more than 100. As we know, finite element method is a numerical method that uses discretization in dividing the problem considered into several finite elements. When the problem considered is discretized into more elements, the difference of the results when compared with the actual results will eventually be small and approximately zero. In other words, the finite element method can give a result which is same or almost the same as the actual result. Here, it has been shown that the numbers of element required to have for analysis of laminated plate using FEM is 64.
49
4.5
Second analysis
Next, the finite element model will used to assess the performance of different types of laminated plate. Here, the selected fiber is T-300 Carbon while the polymer matrix material is Epoxy 3501-6. The mechanical properties of T-300 Carbon and Epoxy 3501-6 are listed below: E 1f
= 230 GPa,
E 2f
= 15 GPa,
G12f
= 27 GPa,
ν12f
= 0.23.
E m
= 4.30 GPa,
Gm
= 1.60 GPa,
νm
= 0.35.
The volume fraction of fiber is assumed as 0.6; hence the volume fraction of polymer matrix is 0.4.
Here, the considered laminates plate has 10 layers of lamina with a thicknessdimension ratio 0.01. Three types of orientations are considered. They are symmetric, anti-symmetric and balanced. The arrangements of the orientations in each layer are listed below: Symmetric:
[0/0/45/90/90/90/90/45/0/0]
Anti-symmetric:
[0/0/45/90/90/0/0/45/90/90]
Balanced:
[0/0/45/90/90/90/90/-45/0/0]
50
From the Matlab program, the ABD matrix of aforementioned laminate can be determined. The ABD matrices for these three different orientations are shown as follow:
Figure 4.9
ABD matrix for symmetric case.
51
Figure 4.10
ABD matrix for anti-symmetric case.
Figure 4.11
ABD matrix for balanced case
52
From the observation base on the ABD matrices for those three cases, we can see that most of the values for A matrix are remain the same. However, D11 and D22 of symmetric and balanced cases are same while anti-symmetric and balanced share same patent of B matrix.
The plate used for the analysis is square with a dimension of 2mm by 2mm. From previous analysis, it shown that the result by using FEM method can be considered accurate when the numbers of element is more than 64. Hence, the laminated plate will be discritized into 12 by 12 elements. In addition, the load 2
applies to the plate is 12 N/mm . The element stiffness matrix, K for those three laminates are shown as follow:
53
Element stiffness matrix for symmetric case
54
Element stiffness matrix for anti-symmetric case
55
Element stiffness matrix for balanced case
Assembly is the step where we change the local coordination of an element into global coordination and after that; we assemble t he K of each element in a global coordination into one global K . However, since the size of K (total) in global coordination is too big (845 by 845), the K (total) is not shown here. In order to verify the K (total) is assembled correctly or otherwise, we can check by showing the values of K (3,3), K (8,8), K (13,13) and K (18,18) of the element stiffness matrix, K and compare them with the value of K at the central or K (423,423) in the global
56
coordination since it is a contribution of adjacent 4 elements. In addition, both of the values can be checked easily through simple coding in the Matlab as well. The results obtained from Matlab for those three cases are listed in the table below:
Table 4.2
The K value for local and global coordination. Case
K (3,3) + K (8,8) + K (13,13)
K (423,423) in global
+ K (18,18) in local Symmetric
3.0254e3
3.0254e3
Anti-symmetric
3.0254e3
3.0254e3
Balanced
3.0254e3
3.0254e3
Figure 4.12
Verification of K local and K global
Similar to the previous analysis, the objective of this stud y is to compute the maximum deflection of the laminated plate which occurs at the center of the plate.
57
The results obtained from Matlab program for those three cases are listed in the Table 4.3.
Table 4.3
The relationship between type of orientations, B11 /B22, D11 /D22 and
maximum normalized deflection of the laminated plate. Case
B11 /B22
D11 /D22
Normalized deflection, w (mm)
Symmetric
0.640
5.138
0.9336
Anti-Symmetric
-1.000
1.000
1.0182
-
5.138
0.9908
Balanced
Table 4.3 shows the values of maximum deflection under the same conditions i.e. the engineering properties of the composite materials, t hickness of each layers, numbers of layer, dimension of the plate, numbers of element and the loading applied. Note that the plate is orientated into three different cases which had mentioned before. The results show that the normalized deflect ion of symmetric laminate plate with the highest values of B11 /B22 and D11 /D22 obtains the lowest w while the anti-symmetric laminate plate has lowest values of B11 /B22 and D11 /D22 obtains the highest w. In order words, the laminate plate that symmetrically orientated is stiffer than the laminate plate that anti-symmetrically and balanced orientated.
As mentioned before, Matlab is not only used for the matrix operation, but it is also a powerful tool in presenting the result e ither in 2 dimensional or 3 dimensional plot, especially in presenting the result graphically. Figures 4.13, 4.14 and 4.15 show the surface plot of the deflections of every nodal point for all three cases. The deflections are zeros at the edges and become bigger and maximum at the centre of the plate because of the boundary conditions i.e. forces and support applied to the plate.
58
Figure 4.13
Figure 4.14
Surface plot of deflection for symmetric case.
Surface plot of deflection for anti-symmetric case.
59
Figure 4.15
Surface plot of deflection for balanced case.
60
CHAPTER 5
CONCLUSION AND RECOMMENDATION
5.1
Conclusion
This study focuses on the formulation of composite laminate by the finite element method and the programming of the equations needed for the analysis of composite laminate. The development of the program for composite laminate was successfully done and it has been shown that t his program can be applied to all type of thin composite laminate plates. As a result, there are few conclusions can be drawn. They are:
1. By the finite element method, the shape functions for in-plane a nd out-of plane plate deformations were successfully defined. The in-plane deformation contains 2 degree of freedoms while the out-of-plane deformation contains 3 degree of freedoms. The formulation of element stiffness matrix, K was successfully developed for the composite laminate plates. The general formula is shown as follows: K
B
i
T
A
Bi
ABD
B B T
i
Bo
ABD
B B T
o
Bi
ABD
B D T
o
Bo
ABD
A
61
2. The whole process of analysis of composite laminate t hrough finite element method was successfully programmed in Matlab. The program is user friendly with enough information for guidance. This program can be used to define the ABD matrix, the element stiffness matrix and the deformation of the laminated plate in term of normalized deflection. 3. There are one related research done by A.H. Sheikh and A. Chakrabakti in paper entitled ‘A new plate bending element based on higher -order shear deformation theory for the analysis of composite plate’ in 2001. By considering the same conditions applied to the laminated plate st ated in that paper, the results obtained can be compared and verified. It shows that the current model produces a good agreement with an acceptable erro r when the laminated plate was discretized into 12 by 12 elements.
4. Performance of symmetric, anti-symmetric and balanced FRC are analyzed and compared. It shown that the laminate plate that symmetrically orientated is stiffer than the laminate plate that anti-symmetrically and balanced orientated.
Our technologies are getting advanced nowadays. Without doubt, computer has becomes a significant tool in every field for tasks like storing data, processing large scaled computation, gaining latest information, changing or sharing ideas, etc. Apart from this, computer does help us a lot, not only in terms of work, but saving time and cost as well. In the fields of research, we are required to design and analyze, most of the time, a complex problem for which this would always consume a lot of time. In other words, even though we can work without computer, but it does not worth in term of time.
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5.2
Recommendations
There are several limitations in this study. Hence, some recommendations are proposed for future research:
1. This study only considers the linear elastic behavior of t he fiber reinforced composite. For better understanding, non-linear behavior is required. 2. The laminate is of transverse isotropic type and unidirectionally orientated. Hence, further research for higher order materials is necessary. 3. The plate is considered thin based on the Classical Lamination Theory. It is not applicable to thick plate. 4. The finite element method used is quadrilateral with 4 nodes for each element. The performing of the other elements such as triangular element can a lso be considered. 5. The program can only perform the analysis with certain numbers of element such as 2 by 2, 4 by 4, 8 by 8, 12 by 12, and 32 by 32. The program should be improved for all numbers of the element.
6. So far, the program can only analyze the joint system as mentioned in Chapter 4. For other types of joints system, the user needs to pre-describe it and change the code. It is hoped that further research can be done for every type of joint system.
7. Further coding is necessary for the computation of the stress and strain values from the analysis.
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REFERENCES
1. Ronald F.Gibson (2007), Principle of Composite Material Mechanics, 2 nd Edition, McGraw-Hill, Inc, U.S.A 2. Robert M.Jones (1975), Mechanics of composite materials, Scripta Book Company, Washington D.C 3. Rudolph Szilard (2004), Theories and Applications of Plate Analysis, John Wile y & Sons, Inc 4. S. Tolson and N. Zabaras (1990), Finite Element Analysis of Progressive Failure In Laminated Composite Plate, Department of Mechanical Engineering, University of Minnesota, U.S.A. 5. J.N.Reddy (1993), An Introduction to Finite Element Method, Second Edition, McGraw-Hill, New York 6. M. Zako, Y. Uetsuji and T. Kurashiki (2002), Composites Science and Technology 63 (2003), 507-516, Finite element analysis of damaged woven fabric composite materials, Japan: Department of Manufacturing Science, Osaka University; Department of Mechanical Engineering, Osaka I nstitute of Technology 7. E.E. Theotokoglou and C.D. Vrettos (2005), Composite Strucuture 73 (2006) 370-379, A finite element analysis of angle-ply laminate end-notched flexure specimens, Greece: Faculty of Applied Sciences, Department of Mechanics – Lab of Strength Materials, The National Technical University of Athens
64 8. A.H. Sheikh and A. Chakrabakti (2002), Finite Element in Analysis and Design 39 (2003) 883-903, A new plate bending element based on higher-order shear deformation theory for the analysis of composite plates, India: Department of Ocean Engineering and Naval Architecture, India Institute of technology; Department of Civil Engineering, Jalpaiguri Govt., E ngineering College. 9. M. Martinez and A. Artemev (2008), Composite Structure 88 (2009) 491-496, Finite element analysis of broken fiber effects on the performance of active fiber composites, Canada: National Research Council of Canada (NRC), Institute for Aerospace Research. 10. N. K. Ngo and I. T. Tran (2007), Vietnam Journal of Mechanics, VAST, 29, No.1 47-57, Finite element analysis of laminated composite plates using high order shear deformation theory, Vietnam: Thainguyen University and Hanoi University of Technology. 11. T. C. Mathew, G. Singh and G. V. Rao (1990), Computer & Structures Vol. 42, 281-287, Thermal buckling of cross-ply composite laminates, India: Department of Structural Engineering, Annamalai University; Structural Design a nd Analysis Division, Structural Engineering Group. Vikram Sarabhai Space Centre. 12. K. R. Thangaratnam, Palaninathan and J. Ramachandran (1988), Computer &
Structures Vol. 32, No. 5, 1117-1124, Thermal buckling of composite laminated plates, India: Department of Applied Mechanics, Indian Institute of Technology; Fibre Reinforced Plastics Research Centre, India Institute of Technology.
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APPENDIX A
MATLAB SCRIPT FOR ANALYSIS
AB D Matrix
% Fiber T-300 Caron and Matrix Epoxy 3501-6 Em = 4.3; Gm = 1.6; vm = 0.35; E1f = 230; E2f = 15; G12f = 27; v12f = 0.2; % Input for value of volume proportion Vm = 0.4; Vf = 1-Vm; % input for value of orientation if orientation == S Angle = [0 0 45 90 90 90 90 45 0 0]; elseif orientation == A Angle = [0 0 45 90 90 0 0 45 90 90]; elseif orientation == B Angle = [0 0 45 90 90 90 90 -45 0 0]; end % Input for thickness and layer of lamina layer = length(Angle); %disp ('Please key in the thickness of lamina') t = 0.1; T(1,layer) = t;
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% Calculation of E1, E2, v12, v21 and G12 etaE = (E2f-Em)/(E2f + 2*Em); etaG = (G12f - Gm)/(G12f + Gm); E1 = E1f*Vf + Em*Vm; E2 = (Em*(1 + 2*etaE*Vf))/(1-etaE*Vf); v12 = Vm*vm + Vf*v12f; G12 = (Gm*(1 + etaG*Vf))/(1-etaG*Vf); v21 = E2*v12/E1 ; % Calculation of Q Q11 = E1/(1-v12*v21); Q12 = v21*Q11; Q22 = E2/(1-v12*v21); Q66 = G12; % Calculation for Qbar values for k = 1:layer ang = (Angle(1,k))*pi/180; Qb11 = Q11*cos(ang)^4 + 2*(Q12 + 2*Q66)*sin(ang)^2*cos(ang)^2 + Q22*sin(ang)^4; Qb12 = (Q11 + Q22 - 4*Q66)*sin(ang)^2*cos(ang)^2 + Q12*(sin(ang)^4 + cos(ang)^4); Qb22 = Q11*sin(ang)^4 + 2*(Q12 + 2*Q66)*sin(ang)^2*cos(ang)^2 + Q22*cos(ang)^4; Qb16 =(Q11 - Q12 - 2*Q66)*sin(ang)*cos(ang)^3 + (Q12 - Q22 + 2*Q66)*sin(ang)^3.*cos(ang); Qb26 = (Q11 - Q12 - 2*Q66)*sin(ang)^3.*cos(ang) + (Q12 - Q22 + 2*Q66)*sin(ang)*cos(ang)^3; Qb66 = (Q11 + Q22 - 2*Q12 - 2*Q66)*sin(ang)^2.*cos(ang)^2 + Q66*(sin(ang)^4 + cos(ang)^4); Qb(:,:,k) = [Qb11 Qb12 Qb16;Qb12 Qb22 Qb26;Qb16 Qb26 Qb66] ; % Calculation for ABD end A = [a11 a12 a16; a12 a22 a26; a16 a26 a66]; B = (1/2)*[ b11 b12 b16; b12 b22 b26; b16 b26 b66]; D = (1/3)*[ d11 d12 d16; d12 d22 d26; d16 d26 d66];
Shape Function LA = 100; %Length of laminate LB = 100; %width of laminate ElementX = 12; % for 2,4,8,12,16 and 32 only ElementY = 12; % for 2,4,8,12,16 and 32 only Lx = LA/ElementX; Ly = LB/ElementY; Load = -12; %N/mm2 %In-plane shape function Ni1 = (1-x/Lx)*(1-y/Ly); Ni2 = (x/Lx)*(1-y/Ly); Ni3 = (x*y)/(Lx*Ly); Ni4 = y*(1-x/Lx)/Ly; %Out-plane shape function No1 = 1-3*x^2/(Lx^2)-x*y/(Lx*Ly)-
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3*y^2/(Ly^2)+2*x^3/(Lx^3)+3*x^2*y/(Lx^2*Ly)+3*x*y^2/(Lx*Ly^2)+2*y^3/ (Ly^3)-2*x^3*y/(Lx^3*Ly)-2*x*y^3/(Lx*Ly^3); No2 = y-x*y/Lx-2*y^2/Ly+2*x*y^2/(Lx*Ly)+y^3/(Ly^2)-x*y^3/(Lx*Ly^2); No3 = x-2*x^2/Lx-x*y/Ly+x^3/(Lx^2)+2*x^2*y/(Lx*Ly)-x^3*y/(Lx^2*Ly); No4 = x*y/(Lx*Ly)-2*x^3/(Lx^3)-3*x^2*y/(Lx^2*Ly)3*x*y^2/(Lx*Ly^2)+2*x^3*y/(Lx^3*Ly)+2*x*y^3/(Lx*Ly^3)+3*x^2/(Lx^2); No5 = x*y/(Lx)-2*x*y^2/(Lx*Ly)+x*y^3/(Lx*Ly^2); No6 = -x^2/Lx+x^3/(Lx^2)+x^2*y/(Lx*Ly)-x^3*y/(Lx^2*Ly); No7 = x*y/(Lx*Ly)+3*y^2/(Ly^2)-3*x^2*y/(Lx^2*Ly)-3*x*y^2/(Lx*Ly^2)2*y^3/(Ly^3)+2*x^3*y/(Lx^3*Ly)+2*x*y^3/(Lx*Ly^3); No8 = x*y^2/(Lx*Ly)+y^3/(Ly^2)-x*y^3/(Lx*Ly^2)-y^2/Ly; No9 = x*y/Ly-2*x^2*y/(Lx*Ly)+x^3*y/(Lx^2*Ly); No10 = -x*y/(Lx*Ly)+3*x^2*y/(Lx^2*Ly)+3*x*y^2/(Lx*Ly^2)2*x^3*y/(Lx^3*Ly)-2*x*y^3/(Lx*Ly^3); No11 = -(x*y^2)/(Lx*Ly)+(x*y^3)/(Lx*Ly^2); No12 = -x^2*y/(Lx*Ly)+x^3*y/(Lx^2*Ly); No = [No1;No2;No3;No4;No5;No6;No7;No8;No9;No10;No11;No12]; Ka = BI*A*Bi; Kb1 = BI*B*Bo; Kb2 = BO*B*Bi; Kd = BO*D*Bo;
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APPENDIX B
RELATED DATA
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