FEM Composite Laminate MATLAB

FINITE ELEMENT FORMULATION FOR COMPOSITE LAMINATE PLATESSIM 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 This thesis is dedicated to my beloved parents. as well as encouragement. University Technology Malaysia. undivided support and encouragement to me during this study.iv ACKNOWLEDGEMENT First and foremost. encouragement. Apart from this. I would like to express my sincere appreciation to my supervisor of this research. My special thanks are extended to my colleagues and friends for their friendship. understanding. . I would like to express my gratitude to University Technology Malaysia for the facilities and opportunities given for me to pursue this study. I would like to thanks my family for their patience. I would like to extend my gratitude and appreciation to all the researchers of the Steel Technology Centre. They had been very helpful and patience in providing assistance throughout the work. Moreover. Faculty of Civil Engineering. continuous support. Dr Ahmad Kueh Beng Hong for his continual dedicated guidance. professional advices. support and motivation in effort to complete this research. prayers. Last but not least. 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.v ABSTRACT Composite laminate is one of the most popular materials used in various field nowadays due to its advantages and better performances. Finite element method is believed as the most powerful numerical method for analysis. . analysis and design are made achievable. However. Unfortunately. 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. This advanced technology opens a new route for the world to become less depending on the usage of conventional monolithic materials especially metal. With the introduction of finite element method in the composite field. the analysis of composite through finite element method needs a lot of large scale matrices and we cannot do it without computer. A lot of studies had been done to analyze the engineering properties of composite materials through several types of numerical method. there have not single program that specially designed for the analysis of composite laminate. Melalui pengenalan kaedah unsur terhingga dalam bidang komposit. kerja analisis tentang komposit laminasi tidak semudah bahan biasa. Pada hal yang demikian. Kajian terhadap komposit laminasi terkemas kini adalah berdasalkan Kirchhoff plate theory ataupun dikenali juga sebagai teori laminasi klasikal. Malangnya. Padahal. Banyak kajian telah dilakukan untuk menganalisis sifat-sifat kejuruteraan komposit laminasi dengan menggunakan pelbagai jenis kaedah berangka. terutamanya logam. . analisis dan rekaan dapat dijalankan. 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. Teknologi yang canggih ini berjaya membuka satu laluan baru kepada dunia supaya tidak terlalu bergantung kepada bahan monolitik yang konvensional yang semakin berkurangan. Antaranya. Hal ini disebabkan komposit laminasi menggabungkan sekurang-kurangnya sifat-sifat dua bahan asalnya dan menjadiakannya rumit sekali dikaji. komposit laminasi merupakan sejenis bahan yang paling popular dipilihgunakan dalam pelbagai bidang disebabkan kebaikan dan kelebihan atas prestasinya.vi ABSTRAK Masa terkini. kaedah unsur terhingga dipercayai merupakan kaedah berangka yang paling berkuasa dalam analisis. saban hari ini belum ada satu program yang direkakan khusus untuk menganalisis komposit laminasi. analisis terhadap komposit berdasalkan kaedah unsur terhingga memerlukan banyak matriks berskala besar dan kita tidak boleh melakukannya tanpa bantuan komputer. 1 Background of study 1 1.2 Previous research 9 2.4 Objective 6 1.1 Introduction 8 2.2 Fiber Reinforced Composite 2 1.3 Problem statement 5 1.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 1.5 Scope of study 7 LITERATURE REVIEW 2.3 Conclusion 13 . 1 Introduction 27 4.2.2.1 Conclusion 60 5.4 4.2 Laminate 29 4. w versus 47 numbers of element 4. thickness and 35 orientations of the laminae in laminate 4.3 Finite element formulation 30 4.3 Boundary conditions 44 Post-processing stage 46 4.1 Fiber and polymer matrix material 34 4.1 Lamina 28 4.3.3.3.2. K for composite 31 Analysis stage 31 4.4 Stiffness matrix.4.5 5 Second analysis 49 CONCLUSION AND RECOMMENDSTION 5.1 Introduction 15 3.3 properties 4.2 Number of layers.2 Recommendations 62 REFERENCES 63 APPENDIX A 65 APPENDIX B 68 .2.2 Related equations 18 RESULT AND ANALYSIS 4.2 Pre-processing stage 28 4.1 Normalized deflection.viii 3 4 RESEARCH METHOFOLODY 3. 1 TITLE Normalized deflection versus number of PAGE 47 element 4. 57 B11/B22.ix LIST OF TABLES TABLE NO. D11/D22 and maximum normalized deflection of the laminated plate . 4.3 The relationship between type of orientations.2 The K value for local and global coordination 56 4. K 39 4.12 Verification of K local and K global 56 4.1 Composite Laminate Plates 4 3.7 Matlab code for the element stiffness matrix.2 Finite element formulation steps 17 4.11 ABD matrix for balanced case 51 4.13 Surface plot of deflection for symmetric case 58 4.1 Pascal Triangle 30 4.x LIST OF FIGURES FIGURE NO.6 Node and element number 38 4.14 Surface plot of deflection for antisymmetric case 58 .10 ABD matrix for anti-symmetric case 51 4.9 ABD matrix for symmetric case 50 4.2 The default Matlab desktop 33 4.3 The Matlab Editor Interface 33 4.8 The normalized deflection of plate versus the numbers of element 48 4.1 Definition of the ABD matrix 16 3.5 Matlab code for the ABD matrix 37 4.4 Matlab result for ABD matrix 35 4. TITLE PAGE 1. xi 4.15 Surface plot of deflection for balanced case 59 . packing geometry. Qij - Lamina stiffness matrix . Vm - Volume fraction of fiber and matrix respectively Ef. Em - Young Modulus of fiber and matrix respectively G12f. vm - Poisson’s ratio of fiber and matrix respectively E1 - Longitudinal Young’s modulus E2 - 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.xii LIST OF SYMBOLS AND ABBREVIATIONS FRC - Fiber reinforced composite FEM - Finite element method FEA - Finite element analysis dof - Degree of freedom Vf. The value of ξ is taken as 2 for E 2 calculation while 1 for G12 calculation. Gm - Shear modulus of fiber and matrix respectively v12f. and loading conditions. w - Displacement in x. Dij - Laminate extensional stiffnesses.xiii Qij - Transformed stiffness matrix N - In plane force M - In plane moment є0 - Midplane strain κ - Curvature Aij. z direction respectively Øx. y direction respectively ζx . and laminate-bending stiffnesses respectively u. No freedom respectively [B] - Element strain matrix [K] - Element stiffness matrix F - Forces q - Transverse distributed load . y. Bij. v. Øy - Rotation about the x. laminate-coupling stiffnesses. ζy - Wrapping of the normal in x and y direction - Shape function for in-plane and out-of-plane degree of respectively Ni. usage.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. Composites are being used due to their advantages such as the costs. asphalt concrete for roadway and fiberglass. composite are materials that combine two or more conventional monolithic materials into one. They have desirable . natural or manufactured. the properties. ceramics or polymers due to its advantages and better performances. the strengths. Composites can be divided into various groups depends on their criterion such as metals and non-metals. Generally.1 CHAPTER 1 INTRODUCTION 1. and application. The most primitive composites that can be obtained easily from market are brick for building construction. Such a material contains the characteristics of both its origin materials and other new characteristics that might be useful. the durability or even the availabilities in our surroundings. (ii) matrix as the continuous phase and (iii) interphase. it is not a difficult task for us to identify all the micromechanical and macromechanical properties of the composites with reasonably simple assumptions. several numerical methods such as finite element method. Hence. the engineering constants of the materials are essentially of top priority before the usages and limitations of the materials can be determined. when we are considering other factors that might affect internally or externally the composites such as thickness. orientations. The fiber is naturally produced from the cellulosic waste streams to form high strength fiber composite materials in a polymer matrix. The . In the field of research. The method of the computation of the micromechanical properties of composites has been invented many years ago by reasonable assumptions and several equations. finite difference method. In order to solve those problems.2 properties that cannot be achieved by any other constituent materials that acting alone. distortions. mesh free method and other methods have been applied into the computation of the micromechanical and macromechanical properties of the composites. Theoretically. it is important to first obtain the micromechanical and macromechanical properties of the composites from the combination of two or more conventional monolithic materials. delaminating.2 Fiber Reinforced Composite Fiber reinforced composite is a type of composites with the combination of three components: (i) fiber as the discontinuous phase. etc. Besides. the computation could becomes complicated and tedious. However. thermal stability. number of layer. 1. the composite materials still have a great considerably potentials which are yet to be discovered. steel. thermal and electrical insulator. graphite. whereas most mammalian bone is made up of layered and oriented collagen fibrils in a proteincalcium phosphate matrix (Wainwright. Currey and Gosline. Polymer composites are very common lightweight. whereas laminate is two or more . 2007). asbestos. metal have electrical conductivity and high melting temperature that are useful in the electrical engineering composites. Polymers. Biggs. titanium and tungsten. Common fiber reinforcing agents include aluminum. tantalum. The characteristics of a composite plate are affected by the arrangement of its components in lamina or laminate state. strong and moldable products in a variety of shapes. and ceramics are the matrix materials used in composites to hold the fibers together and to protect them from damage. arranged longitudinally. glass. metals. 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. Likewise. beryllium. those materials also contain some properties such as ductility and toughness that are needed in certain fields. molybdenum. Composite manufacturing offers the benefits of producing lightweight. Among the aforementioned. polyester. glass fibers and carbon or graphite fibers are the most widely used advanced fibers. Lamina is a plane layer of unidirectional fibers in matrix. polyamide. The composite industry is maturing into an established and increasingly diversified business. Glass/epoxy and glass/polyester composites are used extensively in applications ranging from fishing rods to storage tanks and aircraft parts. while high strength carbon fibers have a tensile strength more than 6 times than that of the steel (Gibson. Besides.3 wood contains mainly of fibrous cellulose in a matrix of lignin. 1976). quartz. 4 unidirectional lamina stacked together with various orientations. flexural and impact strength properties. Figure 1. strength and other configurations as well to predict its deformation and local failure behaviors. composite can form the materials with high-stiffness. especially its strength-to-weight ratio. Micromechanics are the analysis of materials on the interactions of the microscopic structure considering the state of deformation and local failure. high-strength and low density which can have better characteristic than the monolithic materials such as metal and polymer. With different orientations and configurations. 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. However. Fiber reinforced composites such as carbon and glass reinforcement fibers have comparable densities compare to its original components. or other mixture materials like concretes and polymers. .1 Composite Laminate Plates Composite materials have many advantages over traditional metal and alloybased structures. density. Besides. composite lamina or laminate can only be analyzed based on its average characteristics such as stiffness. lower maintenance requirements and greater corrosion resistance. There are a lot of computer aided programs for design purpose which do not include the composite as their basic element. most complex engineering problem only concerns with 9 constants. Despite that. different sizes and other factors as well in a research. The values of Young’s Modulus and Poisson’s Ratio vary from the arrangement of unidirectional of fiber and its orientations. free from electrolysis and incorporate long-term benefits such as weather sustainability and ultraviolent stability as well.3 Statement of problem The research on the parameters of composite materials is not as easy as conventional monolithic materials. Besides. composites are corrosion resistant to most chemicals. orientations. Even though the results of the modeling are satisfactory. we can have over thousands of models with different combinations of materials.5 Greater strength and less weight dramatically improve its performance outcomes. the cost can be escalatingly high and . it is still a difficult task to obtain all the independent coefficients through analysis either using computer or experiment. The total number of independent elastic coefficients of anisotropic fiber reinforced composites is 81 but theoretically. Moreover. analysts have to model the composite materials independently. numbers of layer. for example each lamina in a laminate is modeled one by one and layers by layers before assembled together into one component. 1. The experiments will be done in such a way to identify all the parameters needed for further research. but it is usually time consuming. They also have to model the fiber and matrix separately. So far. If we are able to formulate the composite with the FEM. 1.4 Objective The objectives of this study are: i. It is well-known that the numerical methods can be applied to the existing researches in order to formulate the behaviors of composites and to subsequently program them into any computer software. 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. ii. To develop the Matlab program for (i) and (ii). Such a task requires detailed descriptive knowledge on both the FEM and the mechanics of the composite. . iii. it would be helpful to have a single element that represents both fibers and matrix and multilayer laminates. To analyze and compare three different orientations of fiber reinforced composite. 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 that an analysis or computation can be performed using proper assumptions and boundary conditions. Hence. we will be able to program the formulae we construct into any types of analysis software. The main theme of the current study aims to provide the link for aforementioned difficulties.6 sometimes impractical. aiming at the analysis of the composite materials just treated like any other materials. iv. there are four nodes in each element comprising 5 degree of freedoms at each node. displacement in z direction (w).5 Scope of study This study only focus on the linear elastic behaviors of fiber reinforced composites of transversely isotropic type. The lamina is unidirectional and square in term of size. The degree of freedoms are displacement in x direction (u). My study focuses on the formulation of fiber reinforced composites by using finite element method and the plate is considered thin. and rotation about the y direction (Øy). Besides. following Szilard’s theories and applications of plate analysis. All the laminae are flat plate and there are limitations in terms of the arrangement.7 1. . which is dominantly based on the classical lamination theory. displacement in y direction (v). rotation about the x direction (Øx). sport equipment. 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 . such as aircrafts and submarine structures.1 Introduction Fiber composite materials are increasingly used in variety of systems. medical prosthetic devices and electronic circuit boards. automobiles. With the increasing use of fiber reinforced composites in structures components. space structures.8 CHAPTER 2 LITERATURE REVIEW 2. The analytical study and design of composites requires knowledge of anisotropic elasticity. It is materially efficient in applications that required high strength to weight and stiffness to weight ratios. studies involving the behavior made of composites are getting considerable attention. Finite Element (FE) Method is one of the preferable methods used in the analysis of the structural and mechanical behavior of materials. structural theories and failure mode. They claimed that the first ply failure (FPF) and last ply failure (LPF) strength determined by the analysis correlate . Isoparametric finite element was introduced in the study and the eight noded serendipity and nine noded heterosis elements are used in this analysis.9 structure is not properly formulated. It is considerably powerful numerical techniques devised for solving solid. v. structural mechanics. 2. where ζx and ζy describe the wrapping of the normal in x and y direction respectively. Øx. The results were compared with classical plate theory. ζx. and ζy. They used seven degrees of freedoms per node in the finite element model for laminated composite plates. three dimensional elasticity solutions and other finite element formulations. Øy. The element was further developed by Engblom and Ochoa in 1986 with an increased interpolation function in the through thickness direction. The degrees of freedom are u. w. where Øx and Øy are rotations of the normal to the midplane about the x and y axes respectively. Tolson and Zabaras (1990) developed a two dimensional finite element failure analysis for composite plates in plate analysis with more accurate and flexible form. and even multidisciplinary problems in geometrically complicated regions. 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.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. An experimental and finite element analysis of the static deformation of natural fiber-reinforced composite beam had been done by Lim et. The Moiré fringe pattern is a sinusoidal function and is represented by intensity distribution I(x. The mechanical behavior of a new glass fiber composite post was simulated by a FE analysis on a bidimensional model. The model is quite effective in providing limits concerned and useful in the design of laminated composite plates. 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. a(x. y). y)  a( x. For the finite element . The other materials such as fiber glass composite.10 well with the actual failure strengths. al (2002). both experimentally and through FE analysis. y) is the phase at point (x. y) is background intensity variation. y)   ] where. b is the modulation strength. A ‘beating’ between two structures is observed in the form of another periodic structure. The result shows that the fiber-reinforced composite posts present quite high stresses in the cervical region due to their flexibility and also to the presence of a less stiff core material. al (2001) studied the mechanical behavior of glass fiber reinforced composite endodontic post. gold alloy and other types of metal alloy are used in the model study. They used the shadow Moirémethod for the direct measurement of whole field deformation of cantilever beam. y) cos[ ( x. There are four bidimensional model built for analysis purpose with four different types of materials. Pegoretti et. y)  b( x. The simulation results are compared with those of commercially available carbon fiber reinforced and gold alloy cast posts. and  is the amount of phase shift. The aim of the study is to analyze the mechanical behavior of a new polymeric composite post reinforced with glass fibers. The use of composite in the dentistry intervention is for treatment purpose of pulpless teeth. known as the Moiréfringe pattern. f(x. y) written in a general form as follows: I ( x. 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. Besides. the mechanical and thermal buckling loads are computed. 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 order to obtain the FEA results. the average value of the modulus of elasticity obtained in the three-point bending tests was used. the model of the cantilever was first created using SolidWorks software.11 analysis. it is also proven that finite element formulation can be applied in the analysis of fiber composites in thermal buckling. 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. This technique. al (1988) applied the finite element formulation in the study of thermal buckling of composite laminated plate. Ke is derived. The formula produced can be used in the determination of buckling behavior of the structure as well as for a parametric study. The field variables in the local coordinates are expressed as U   S q . Thermal buckling of cross-ply composite laminates is one of the important studies in the field of composite. In the end of the formulation. therefore. Mathew et. can be used as a non-contact as well as a nondestructive technique to validate the finite element model. the geometric stiffness matrix. al (1990) used one dimensional finite element analysis consisting two nodes and six degrees of freedom. Applying different boundary conditions. Kari et. 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 results obtained from the FEA are compared with the shadow Moiré’s results. and then analyzed using COSMOS Works. Even though it is not the only researches about the analysis of plate using FEM.Vz . Chakrabarti in 2001.U x . In the study. γx and γy. Øy.U z . They compare the results among two plate theories. they were considering 7 degree of freedoms such as u. Total potential energy is used in the derivation of equations for prebuckling solution. .V y . Moreover. w. they show that the different between HSDT and FSDT is small and the accuracy is increased with the increasing of the number of element. Øx. 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. Sheikh and A. cross-ply anti-symmetric laminate subjected to distributed load of sinusoidal variation and angle ply antisymmetric laminate subjected to uniformly distributed load. In the study.U yz . cross-ply rectangular laminate subjected to distributed load sinusoidal variation.U xz . 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. critical temperatures for different cases of composite laminates are obtained from the program.Vx .Vxz .H.V .W . v. In their results. the higher-order shear deformation theories (HSDT) and the first-order shear deformation theory (FSDT).V yz  T and S  is the transformations of shape functions while qis the vector of element degree of freedom. isotropic rectangular plate having different boundary conditions at four edges. The computer program COMSAP was developed and used to handle the temperature variation of both surface and thickness of fiber laminates. cross-ply square laminate subjected to uniformly distributed load.12 where U   U .U y . cross-ply skew laminate subjected to uniformly distributed load. a triangular element based on Reddy’s HSDT is developed which is able to give more precise results for the analysis using FEA. their compared both theories under different conditions such as isotropic square plate simply supported at all the edges. The accuracy and efficiency of the element against the known cases. By modeling the structures into finite element and processing the analysis with computer aided programs such as COSMOS. the results are questionable nowadays since the programs used mostly applied for isotropic materials. roughly 40. 2. Pro-Engineer. he assumed the plate are flat and quadrilateral with 3 out-of-plane degree of freedom existed at each nodes. For example. Through FEM. SolidEdge. In the beginning stage.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. he analyzed moderately thick plate elements and thick plate elements. the results can be obtained easily. However. when we are dealing with composites laminated bridge. simple plate triangular element. Nastran. Solidworks. However.3 Conclusion A lot of analysis of composite materials had been done through finite element modeling.13 Szilard had discussed about the theories and applications of plate analysis in his book published in 2004. we need to determine the engineering properties of the composite through the lamination . The setting of the program can be determined normally through the pre-determined materials properties or the existing database of the program. he analysis several types of plate element such as simple plate or thin plate rectangular element. In his book. He used finite element method as a tool of plate analysis. He also illustrated some example of calculations under his analysis in order to present his models. higher order plate elements (16 DOF) and discrete Kirchhoff triangular element. He mentioned that by mid 1990. 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 advanced stage. It shows the important of FEM in terms of numerical solution of engineering problems. we can analyze the reinforced concrete bridge with Lusas through finite element method. etc. 14 theories and apply the obtained engineering constants into the program since the material library does not have any record for a composite material. 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. it is a challenge to program a finite element model for a composite laminate into any available software. all the studies are not considering the program development after the formulation stage. Unfortunately. 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. . However. So. which is the finite element formulation of a fiber reinforced composite. 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 3.2 show the flows of the research in brief.1 Introduction This chapter explains the flow and the methods that are used in my research. the finite element method will also be applied in my study beginning with the formulation stage. Figure 3.1 and 3. Beside the general formulae used in deriving the basic composite parameters. . Qij in local coordination. Laminate State number of layers of lamina and its angles. E2f.16 Lamina Select the engineering constants E1f. Define the volume fraction of fiber and polymer. Compute the engineering constants E1. E2. Matlab Figure 3. Compute the transformed lamina stiffness matrix. G12 and v12 of fiber reinforced composite by the Rules of Mixtures and the Halphin-Tsai Equation. Gm and vm for typical polymer matrix. Select the engineering constants Em. G12f and v12f for fiber. Compute the lamina stiffness matrix.1 Definition of the ABD matrix . Compute the ABD matrix. Apply boundary condition. Post-processing Figure 3. Solve the equation. K in local coordination.17 FEM Discretization of the continuum. Assembly the K into global coordination. Matlab Develop element stiffness matrix.2 Finite element formulation steps. . Element formulation. Selection the suitable interpolation function based on degree of freedom. Rules of Mixtures E1 = E1fVf + EmVm where E1f is the Young Modulus of fiber. Vf + Vm = 1. All the engineering constants are important for calculating the stiffness matrix of lamina. It is assumed that the total of fiber volume fraction. Em is the Young Modulus of matrix.18 3. and Vm is the matrix volume fraction. E2. The engineering constants E1. First.2) . Vf is the fiber volume fraction. The equations used in this stage are: i. ii. G12 and v12 of the composite can be computed through the Rules of Mixture and the Halphin-Tsai Equation.2 Related equations The research begins from the determination of the material properties. carbon fibers and matrix. Qij in the local coordinate. we have to get the engineering constants of each component of the composite in order to compute the engineering constants of the composite lamina. Vf and matrix volume fraction. (3.1) Where Vf is the volume fraction of fiber and. (3. Vm is the volume fraction of matrix. Vm is equal to 1 since the lamina only contains two components. 19 v12 = v12fVf + vmVm (3.3) where v12f is the Poisson’s ratio of fiber.4) where. Qij Q11  Q12  E1 1  v12 v21 (3.  is the curve-fitting parameter. and  E2 f  Em (3.5) where. and Vm is the matrix volume fraction.6b) .5a) Lamina stiffness matrix.  is the curve-fitting parameter. vm is the Poisson’s ratio of matrix. iii. Halphin-Tsai Equation Em 1  V f  E2  1  V f (3.4a) E2 f  Em G12  Gm 1  V f  1  V f (3. and  iv. G12 f  Gm G12 f  Gm (3.6a) v12 E2 1  v12 v21 (3. Vf is the fiber volume fraction. Assuming the individual laminae are perfectly bonded together so as to behave as a unitary. the transformed stiffness matrix. we need to transform the lamina stiffness matrix into a global form using the transformed coefficient.7d) Q 26  Q11  Q12  2Q66  cos sin 3   Q22  Q12  2Q66  cos3  sin  (3. The equations used in this stage are as follow: Q11  Q11 cos4   Q22 sin 4   2Q12  2Q66 sin 2  cos2   Q12  Q11  Q22  4Q66  sin 2  cos2   Q12 cos4   sin 4   (3. nonhomogeneous anisotropic plate. The deformation hypothesis from classical homogeneous plate theory and the laminated forcedeformation equation can be used to define the coordinate system in developing the . lamina with different angles and thickness of each layer are computed.20 Q22  E2 1  v12 v21 (3.7a) (3. Hence.6c) Q66  G12 (3.7b) Q 22  Q11 sin 4   Q22 cos4   2Q12  2Q66 sin 2  cos2  (3.6d) After that. Qij can be calculated.7c) Q16  Q11  Q12  2Q66  cos3  sin   Q22  Q12  2Q66  cos sin 3  (3. interfacial slip is not allowed.7e)  Q 66  Q11  Q22  2Q12  2Q66 sin 2  cos2   Q66 sin 4   cos4   (3.7f) The computation of the material properties of laminate with several layers of lamina and orientations can be done using the theory of lamination plates. In global form. 21 laminated plate analysis. Hence, we can compute the force, N and moment, M 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   B12    M xy   B16 A12 A16 B11 B12 A22 A26 B12 B22 A26 B12 A66 B16 B16 D11 B26 D12 B22 B26 D12 D22 B26 B66 D16 D26 0 B16   x   0  B26   y    B66   xy0    D16   x  D26   y    D66     xy  (3.8) where N is the in-plane force, M is the in-plane moment,  0 is the midplane strain,  is the curvature, and Aij   Qij k z k  z k 1  N (3.8a) k 1 Bij  Dij  1 N Qij k z 2 k  z 2 k 1  2 k 1     1 N Qij k z 3 k  z 3 k 1  3 k 1 (3.8b) i, j = 1, 2, or 6 (3.8c) Aij, Bij and Dij are laminate extensional stiffnesses, laminate-coupling stiffnesses, and laminate-bending stiffnesses respectively. zk is the corresponding distance from middle surface to outer surface of the kth lamina, and zk-1 is the distance from the middle surface to the inner surface of the kth lamina. 22 Next, we have to do the analysis through the finite element method (FEM) from the development of a shape function which can represent all the parameter involved for a single element of the laminate. The FEM starts with the discretization of the plate into several elements. Using the interpolation function, a general shape function for an element can be formulated. By considering the engineering behavior of the laminate, we combine the element strain matrix, [B] with the ABD matrix and develop into a local stiffness matrix, K which applies for a laminate. The 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 condition and boundary condition for the laminate. It ends by solving the simultaneous equations. All the formulae related are stated below: i. In-plane shape function, u  N i1u1  N i 2 u 2  N i 3 u 3  N i 4 u 4 (3.9) v  N i1v1  N i 2 v 2  N i 3 v3  N i 4 v 4 (3.10) where u is the displacement in x direction and, v is the displacement in y direction. and,   N i1  1  x 1  y  Lx  Ly   (3.9a) Ni2  x   1 y  Lx  Ly  (3.9b) N i3  y 1  x  L y  Lx  (3.9c) N i4  xy Lx L y (3.9d) where Ni1, Ni2, Ni3 and Ni4 are the shape function for u and v, and 23 x and y are the variables in the element and, Lx and Ly are the dimension of the element in x and y direction respectively. ii. Out-of-plane shape function, w  N o1 w1  N o 21x  N o31 y  N o 4 w2  N o5 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 in z direction and, Øx is the rotation in y direction and, Øy is the rotation in x direction. and, N o1  1  3x 2  3xy Lx L y 2 Lx 2 N o 2  y  xy xy 2  3  2y Lx Ly  2y Lx L y 3 3  2x y 2 Ly 2x 2  3 Lx L y  2 xy Ly 2  3y 3 Lx 3  2 xy 2 Lx L y y 3x 2 y  3 Lx L y 3 Ly 2 2 Lx L y (3.11a) 3  xy 3 Lx L y 2 (3.11b)   2 3 2x 2 y x3 y  N o 3   x  2 x  xy  x   2 2 Lx Ly L L Lx L x L y  (3.11c) x y  N o 4  xy 3  2x y Lx L y 3 Lx L y N o 5  xy Lx  2x 3 Lx 3  2 xy  2 xy 3x 2 y  3 Lx L y 2 Lx L y 3x 2 3   xy 2 Lx L y Lx  3xy 2 Lx L y 2 (3.11d) 2 3 Lx L y 2  2  3 x2 y x3 y  N o 6    x x   2 2 Lx L L Lx L x L y  x y  (3.11e) (3.11f) 24 N o 7  xy 3  2y Ly Lx L y 3 N o8  xy  3y 3  2x y 2 Lx L y 2 Ly 3x 2 y  2 3  2 xy 3 Lx L y y 3 Ly 2  xy 2 Lx L y Lx L y 2 Lx L y 2 (3.11g) 3 3 Lx L y  3xy 2 y 2 (3.11h) Ly 2 3   N o 9   xy  2 x y  x y 2  Ly Lx L y Lx L y   N o10   xy  2 xy Lx L y 2  3x y 2 Lx L y  3xy (3.11i) 2 Lx L y 2 3  2x y 3 Lx L y N o11   xy (3.11j) 3 2 2 N o12  x y 3 Lx L y Lx L y Lx L y  xy 3 3 x y Lx L y 2 2 Lx L y (3.11k) (3.11l) where No1, No2, No3, No4, No5, No6, No7, No8, No9, No10, No11, and No12 are the shape function for w, Øx and Øy, and x and y are the variables in the element and, Lx and Ly is the dimension of the element in x and y direction respectively. iii. Element strain matrix, [B] B  N  where   is the displacement differential operator. (3.12) 14) where {F} is the nodal forces and {d} is the nodal degree of freedom In the finite element method. Stiffness matrix.15) A Matlab program is written for all stages through a proper Matlab code. a user friendly program will be available for the determination of the nodal degree of freedoms through the FEM. F    N  qA (3. Strain-displacement relationship    Bu where  is the element strain (3.14a) where q is the transverse distributed load vi.25 iv.13) where [B] is the element strain matrix and [D] is the elasticity matrix. users do not have . In this program. By the end of my study. v. force for surface. [K] K    BT DBA (3. Equilibrium of force displacement equation {F} = [K]{d} (3. in plane shear modulus. . number of element for a laminate and its corresponding dimensions. thickness of each layer. Poisson’s ratio and volume fraction of fiber and matrix respectively. They just need to include a few basic parameters such as Young Modulus. ply orientation. numbers of layer.26 to worry too much about the computation of the composite laminate properties. the analysis stage. In the analysis stage. etc. The chapter is divided into 3 parts. For the post-processing stage. and the post-processing stage and the descriptions of each stage are as follows. They are the pre-processing stage. In the pre-processing stage. the data tabulation.1 Introduction This chapter presents all the results and analysis following the original sequences stated in Chapter 3.27 CHAPTER 4 RESULT AND ANALYSIS 4. we present our results in a better way using different data processing tools such as the graph plotting. we need to determine our variables used for our analysis in order to obtain certain results. . the methods of the analysis which has been described briefly in Chapter 3 are elaborated and discussed in detail here. All the processes will be discussed in details and under preferable sequences for the convenience of the readers. E2. G12 and v12. lamina is a layer of composite which contains the fiber and the polymer in a fiber reinforced composite. we are able to obtain the engineering properties of lamina through the equations mentioned in Chapter 3 such as E1. shear modulus and Poisson’s ratio of fiber and polymer since the scope of study is constrained to the transverse isotopic type which follows the Classical Lamination Theory. E2f. In this study. it provides a clear guide for the user in understanding the stages of the analysis in sequence. Young’s modulus.1 Lamina As mentioned before. 4. Poisson’s ratio. v12f. shear strength and so on. Besides. Em. shear modulus. It is an important for us to identify our input and output of analysis before the process take place. G12f. tensile strength. compressive strength.2. From the properties of E1f.28 4.2 Pre-processing stage In the pre-processing stage. . Gm and vm. we just consider the Young’s modulus. we will describe the detail of the important data required for the analysis. either the fiber or polymer has different engineering properties such as density. The equations used in calculating the E1 and v12 are following the rules of mixture while the E2 and G12 obtained from the HalphinTsai equation have proven to be more accurate. Each of this materials. . 4.2. Unlike lamina. we can calculate the lamina stiffness matrix. in order to define the laminate stiffness matrix.29 With the pre-defined engineering properties of lamina.6 before we proceed to the analysis of laminate. nonhomogeneous anisotropic plate. For laminate. when two or more monolithic materials combined together. we are required to arrange all the laminae properly and compute it through the Theory of Laminate Plate. 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. interfacial slip is not allowed. all laminae are combined together and the product is a component called laminate. The global stiffness of lamina only represents the stiffness of each lamina in laminate form. the engineering properties of the new component will changes due to different materials combination. It exists with a set of new engineering properties. Qij for each layer of lamina in laminate through equation 3. The result will be presented in the ABD matrix. Assuming the individual laminae are perfectly bonded together so as to behave as a unitary.2 Laminate In composite materials.7 mentioned in Chapter 3. Hence. The formulation of in-plane shape function only require 1st order interpolation function since the deformation is linear. α1 α2x α4x2 α7x3 α11x4 α3 y α8x2 y α12x3y α6 y2 α5xy α9xy2 α13x2 y2 Figure 4. the laminate deforms in z direction and rotates in the x and y direction.3 Finite element formulation There are three basic fundamental considerations in using the FEM as the method of formulation. Theories and Applications of Plate Analysis). Hence. each nodal point will have three out-of-plane degree of freedoms. there are two types of deformation considered.1 Pascal Triangle α10 y3 α14xy3 α15 y4 .30 4. the plate deforms in xy plane only. the in-plane deformation and out-of-plane deformation. They are the equilibrium of forces. we have two degree of freedoms at each node. For in-plane deformation. To have an nonconforming rectangular element. For out-of-plane deformation.1 is necessary (Szilard. the compatibility of displacement and the law of material behavior. the shape function with 12 terms of polynomial derived from the Pascal’s triangle that is shown in the Figure 4. Hence.2. In order to fulfill all the fundamental requirements. 4 Stiffness matrix. In the current study. 4. . K   B T DBA (4. 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. B is the strain matrix coming from the differentiation of the shape function corresponding to its deformation whereas D is the element elasticity. As mentioned before.2.31 4. the K is integrated symbolically not by numerically method although the computation based on the latter is much more efficient and faster. we are required to define all the values of the variables in order to run the program and perform the analysis. K for composite After deriving the shape function.2) where Bi is the in-plane element strain matrix and Bo is the out-of-plane element strain matrix.1) In FE. the program used here is Matlab. The K is a 20 by 20 matrix.3 Analysis stage In this stage. we are required to develop the stiffness matrix of the laminate through equation. 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. the correlation of the ABD matrix obtained from analysis of laminate with the strain matrix is crucial. 3.2 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. Nowadays. it provides others functions such as plotting of functions and data. Matlab is an advanced computer program of Maple which is also a programming tool used for defining the mathematics related problems including the matrix. researches. It is because the Matlab has some limitations and specifications for general users. C++. and Fortran. it is believed that the development of Matlab will be improved and accepted by public. The word ‘Matlab’ stand for matrix laboratory. Moreover. Matlab contains toolbar with normal function like file. Matlab is still not a common or famous programming tool for public use. window and help. Figure 4. It is only famous in certain fields such as mathematics. edit. debug. and interfacing with programs written in other languages including C. implementation of algorithms. It is also used to show the answer or result of the work here. Even though Matlab is a program designed for the matrix operation. desktop. . However.32 Matlab is a fourth generation computer program which is used widely for a matrix operation. program development.2 and Figure 4. The interfaces of Matlab are shown in Figure 4. creation of user interfaces. etc. As usual. 33 Figure 4.2 The default Matlab desktop.3 The Matlab Editor Interface. . Figure 4. H.34 4.02mm with a lamina thickness of 0. a is set to 2mm. Chakrabakti in paper entitled ‘A new plate bending element based on higher-order shear deformation theory for the analysis of composite plate’.0067mm each. The information of fiber and polymer matrix is listed below: E1 = 174.3.3. From the equations 3.6 GPa. hence the total thickness of the composite plate is 0.1 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.4: .7 and 3. we can find the ABD matrix for this composite plate.25.2 Number of layers. G12 = 3. Sheikh and A.8 stated in Chapter 3.5 GPa and v12 = 0. A same problem case has also been studied by A. The result is shown in Figure 4. thickness and orientations of the laminae in laminate There are 3 layers of lamina with the orientation [0/90/0] in the current analysis. E2 = 7 GPa.01 where h is the total thickness of the plate and a is the width of the plate. 4. In this study. The thickness of each lamina is depending on the dimension of the plate based on the ratio h/a which is set to 0. 35 Figure 4. The result obtained from efunda is shown below: .efunda. It is done by key in the same materials properties.com.4 Matlab result for the ABD matrix The result was verified through an online computational tool provided at www. 5: . one of the objectives of this study is to develop a Matlab program by a proper code enabling user to find the engineering properties of composite laminate with some simple input command of the chosen composite materials.36 As mentioned in Chapter 1. The sample code for the ABD matrix is shown in Figure 4. Next.6 shows the discretization of the plate into 2 by 2 elements and the arrangement of every nodal point. As mentioned before. In this study.37 ABD matrix Figure 4. Figure 4. 8 by 8 elements. we are required to insert a series of information of the laminate. such as the dimensions of the laminate and the numbers of element. a square plate with a size of 2mm by 2mm is considered. The analysis of the laminate is followed by the FEA after the determination of the ABD matrix. 12 by 12 elements and 32 by 32 elements. Numbers in boxes represent the element number. the K for the composite plate is computed with Matlab by using equation 4. The plate is discretized into 2 by 2 elements. In this study. 4 by 4 elements.2. the nodal points and the elements are arranged from left to right. .5 Matlab code for the ABD matrix. 6 3 z y x Node and element number. the arrangement of the K matrix is depending on the arrangement of nodal points. As a result. Øx. Unfortunately. The result obtained from Matlab for K matrix in local is shown in the following. the size of the K matrix for each element is thus 20 by 20. v. .38 7 8 3 4 1 4 5 1 9 6 2 2 Figure 4. Øy}.7 shows the coding for the integration of the K matrix and the matrix rearrangement. By default. further coding is necessary for the rearrangement of the K matrix following the correct positions. w. since every nodal point contains 5 degree of freedoms following the sequences of {u. the K matrix we obtained first not following the correct sequences due to the different shape functions definition for inplane and out-of-plane degree of freedoms. Figure 4. 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. K with size 20 x 20 . Matrix Rearrangement Figure 4. K. Element stiffness matrix.39 Double integration of K by part.7 Matlab code for the element stiffness matrix. we are required to arrange it into a global manner for each of the elements. Besides. etc. moments. 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. . The matrix size is 45 by 45 because it contains 9 nodal points with 5 degree of freedoms at each node. The result shown below is the global stiffness matrix of the composite plate with 2 by 2 elements. strains. reactions.40 After defining the K. 41 K global with matrix size 45 x 45 . 42 . 43 . 44 4. A uniformly distributed load of 12 N/mm2 is prescribed on top of the composite plate in z direction. The results obtained from equation 3.3.14a in Chapter 3. The results for local forces and global forces are shown as follows: . K. These converted nodal loads can be computed through equation 3. we need to assemble them into a global force the way we have done for the stiffness matrix.14a are only for one element. we need to define the boundary conditions of the plate.3 Boundary conditions After defining the K global. In the FEM. the loads act only on the nodal points of the plate. Hence. fz. we have another boundary condition which relates to the type of constrain or support we applied to the composite plate. Øx and Øy at certain nodes.Øx and Øy = 0) at one of the edges of the plate and pinned (u. we can eliminate the corresponding forces.14 in Chapter 3. Mx. Besides the loading. the plate is rigid (u.45 N means local forces and moments rG means global forces and moments Notes: for local coordination. The solution obtained in the forms of the deflection. fy. the arrangement of forces and moments are in sequence of fz.w = 0) at the rest of the sides of the plate. only 26 unknown deflections and rotations have to be solved and we can analyze the plate using equation 3. Mx. the row and the column of the K matrix which are zeros. the forces and moments are arranged in sequence of fx. and My.v. In the current study. The results are shown as follows: . while for global coordination. Hence. and My.w. Finally. w and rotations. . we the results are presented in the simplest way for the clarity of the readers and the users. They are chart. Here.46 (d means deflections and rotations) Deflection at the central 4.4 Post-processing stage At this stage. There are several methods that can be applied here. the graph and the table are used as the means. graph. table and others. 6683 Sheikh & Difference(%) Chakrabakti 0. w versus numbers of element Table 4. In other words.7514 0. The relationship between numbers of element and the deflection is plotted in Figure 4.1 Normalized deflection versus number of element Number of Element 2 4 8 12 32 4 16 64 144 1024 wnd = (wh3E2/qa4)100 Normalized deflection.6708 -27.47 4. The deflections shown here are different to that shown previously because the results are normalized in terms of equation 4. .37 (4.1 shows the results obtained from the Matlab program. when the numbers of element is 64 or more.95 -0.6827 0.1 Normalized deflection.68 -0.6741 0.8. 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 results can be considered as satisfactorily good.6732 0.3) Table 4.13 0.7306 0.9588 0. wnd 0. The difference of the results when compared with the results obtained from Sheikh and Chakrabakti’s study is also decreasing with the increasing of the numbers of element.60 -5.6763 0.3. for the analysis of the laminated plate.4.6913 0. 48 Normalized deflection.8 shows the maximum deflection of the laminated plate versus the numbers of element for the plate.8 The normalized deflection of plate versus the numbers of element.9000 0.8500 0. In other words. Figure 4. the finite element method can give a result which is same or almost the same as the actual result.8000 deflection 0. finite element method is a numerical method that uses discretization in dividing the problem considered into several finite elements.0000 0. Here. the deflection is high but it decreases with the increase of the numbers of element. w versus numbers of element Normalized central deflection.w 1.6500 0. As we know.6000 0 250 500 750 1000 1250 Number of Element Figure 4.7000 0. . the difference of the results when compared with the actual results will eventually be small and approximately zero. it has been shown that the numbers of element required to have for analysis of laminated plate using FEM is 64. The graph becomes almost linear when the numbers of element is more than 100.7500 Sheikh & Chakrabakti 0. When the problem considered is discretized into more elements. At the beginning.9500 0. 4. G12f = 27 GPa.01.49 4. the finite element model will used to assess the performance of different types of laminated plate. νm = 0. hence the volume fraction of polymer matrix is 0.35. Here. ν12f = 0. Gm = 1. the selected fiber is T-300 Carbon while the polymer matrix material is Epoxy 3501-6.23. 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] . Em = 4.5 Second analysis Next.60 GPa. Three types of orientations are considered. E2f = 15 GPa. Here. the considered laminates plate has 10 layers of lamina with a thicknessdimension ratio 0.30 GPa. The volume fraction of fiber is assumed as 0.6. They are symmetric. The mechanical properties of T-300 Carbon and Epoxy 3501-6 are listed below: E1f = 230 GPa. 9 ABD matrix for symmetric case. the ABD matrix of aforementioned laminate can be determined. .50 From the Matlab program. The ABD matrices for these three different orientations are shown as follow: Figure 4. 51 Figure 4.11 ABD matrix for anti-symmetric case. ABD matrix for balanced case .10 Figure 4. In addition. the load applies to the plate is 12 N/mm2. it shown that the result by using FEM method can be considered accurate when the numbers of element is more than 64. we can see that most of the values for A matrix are remain the same. However. K for those three laminates are shown as follow: . The plate used for the analysis is square with a dimension of 2mm by 2mm. Hence. From previous analysis. D11 and D22 of symmetric and balanced cases are same while anti-symmetric and balanced share same patent of B matrix.52 From the observation base on the ABD matrices for those three cases. the laminated plate will be discritized into 12 by 12 elements. The element stiffness matrix. 53 Element stiffness matrix for symmetric case . 54 Element stiffness matrix for anti-symmetric case . we can check by showing the values of K(3.423) in the global . K(8. the K(total) is not shown here. we assemble the K of each element in a global coordination into one global K. since the size of K(total) in global coordination is too big (845 by 845).8). In order to verify the K(total) is assembled correctly or otherwise.18) of the element stiffness matrix.3). However. K and compare them with the value of K at the central or K(423.55 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.13) and K(18. K(13. 3) + K(8.13) K(423.0254e3 Anti-symmetric 3.0254e3 3.0254e3 3.12 Verification of K local and K global Similar to the previous analysis.0254e3 3. Case K(3. both of the values can be checked easily through simple coding in the Matlab as well.56 coordination since it is a contribution of adjacent 4 elements. In addition.0254e3 Balanced 3.0254e3 Figure 4.18) in local Symmetric 3. . the objective of this study is to compute the maximum deflection of the laminated plate which occurs at the center of the plate.423) in global + K(18.2 The K value for local and global coordination. The results obtained from Matlab for those three cases are listed in the table below: Table 4.8) + K(13. especially in presenting the result graphically. thickness of each layers. the laminate plate that symmetrically orientated is stiffer than the laminate plate that anti-symmetrically and balanced orientated. Table 4. The deflections are zeros at the edges and become bigger and maximum at the centre of the plate because of the boundary conditions i.9336 Anti-Symmetric -1.000 1. In order words. B11/B22. . 4. D11/D22 and maximum normalized deflection of the laminated plate. dimension of the plate.138 0. Matlab is not only used for the matrix operation.9908 Balanced Table 4.138 0. As mentioned before.14 and 4. the engineering properties of the composite materials. but it is also a powerful tool in presenting the result either in 2 dimensional or 3 dimensional plot.3 The relationship between type of orientations. forces and support applied to the plate. The results show that the normalized deflection 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.3 shows the values of maximum deflection under the same conditions i.e.3.13.0182 - 5. Figures 4.000 1.15 show the surface plot of the deflections of every nodal point for all three cases. numbers of element and the loading applied.e. numbers of layer. Note that the plate is orientated into three different cases which had mentioned before.640 5. Case B11/B22 D11/D22 Normalized deflection. w (mm) Symmetric 0.57 The results obtained from Matlab program for those three cases are listed in the Table 4. .58 Figure 4.14 Surface plot of deflection for symmetric case.13 Figure 4. Surface plot of deflection for anti-symmetric case. .15 Surface plot of deflection for balanced case.59 Figure 4. 60 CHAPTER 5 CONCLUSION AND RECOMMENDATION 5. As a result. there are few conclusions can be drawn. The general formula is shown as follows:   K   Bi  AABD Bi  Bi B ABD Bo  Bo B ABD Bi  Bo D ABD Bo A T T T T . The formulation of element stiffness matrix. the shape functions for in-plane and out-ofplane plate deformations were successfully defined. The development of the program for composite laminate was successfully done and it has been shown that this program can be applied to all type of thin composite laminate plates. K was successfully developed for the composite laminate plates. They are: 1. The in-plane deformation contains 2 degree of freedoms while the out-of-plane deformation contains 3 degree of freedoms.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. By the finite element method. most of the time. changing or sharing ideas. the results obtained can be compared and verified. Our technologies are getting advanced nowadays. . This program can be used to define the ABD matrix. we are required to design and analyze. Chakrabakti in paper entitled ‘A new plate bending element based on higher-order shear deformation theory for the analysis of composite plate’ in 2001. but it does not worth in term of time. In the fields of research. but saving time and cost as well. The whole process of analysis of composite laminate through finite element method was successfully programmed in Matlab. 3. even though we can work without computer. Sheikh and A. not only in terms of work. It shows that the current model produces a good agreement with an acceptable error when the laminated plate was discretized into 12 by 12 elements. By considering the same conditions applied to the laminated plate stated in that paper. 4. etc. Without doubt. computer does help us a lot. computer has becomes a significant tool in every field for tasks like storing data. There are one related research done by A. It shown that the laminate plate that symmetrically orientated is stiffer than the laminate plate that anti-symmetrically and balanced orientated.H. anti-symmetric and balanced FRC are analyzed and compared. Performance of symmetric. a complex problem for which this would always consume a lot of time. In other words. the element stiffness matrix and the deformation of the laminated plate in term of normalized deflection.61 2. processing large scaled computation. gaining latest information. The program is user friendly with enough information for guidance. Apart from this. It is not applicable to thick plate. . The finite element method used is quadrilateral with 4 nodes for each element. non-linear behavior is required. 2. The laminate is of transverse isotropic type and unidirectionally orientated. the program can only analyze the joint system as mentioned in Chapter 4. Hence. 7. The plate is considered thin based on the Classical Lamination Theory. 4. 12 by 12. The program should be improved for all numbers of the element. This study only considers the linear elastic behavior of the fiber reinforced composite. So far. the user needs to pre-describe it and change the code. Hence. The performing of the other elements such as triangular element can also be considered. The program can only perform the analysis with certain numbers of element such as 2 by 2. further research for higher order materials is necessary. and 32 by 32. Further coding is necessary for the computation of the stress and strain values from the analysis. 6.2 Recommendations There are several limitations in this study. 5. For other types of joints system. It is hoped that further research can be done for every type of joint system. 8 by 8. 3. some recommendations are proposed for future research: 1. For better understanding. 4 by 4.62 5. Rudolph Szilard (2004). 5. Japan: Department of Manufacturing Science. Theories and Applications of Plate Analysis. Kurashiki (2002). 2 nd Edition. Department of Mechanics – Lab of Strength Materials.Gibson (2007). Department of Mechanical Engineering. Zako. John Wiley & Sons. U. Second Edition. S. E. The National Technical University of Athens . Greece: Faculty of Applied Sciences. Composites Science and Technology 63 (2003). Mechanics of composite materials. Osaka Institute of Technology 7. Scripta Book Company.Jones (1975). Y.63 REFERENCES 1. Tolson and N. Theotokoglou and C. Department of Mechanical Engineering. M. Finite Element Analysis of Progressive Failure In Laminated Composite Plate. Robert M. Zabaras (1990).C 3. Inc 4. University of Minnesota. Finite element analysis of damaged woven fabric composite materials. J. Principle of Composite Material Mechanics.D.A.A 2.E.S. U. McGraw-Hill. Composite Strucuture 73 (2006) 370-379. Osaka University. An Introduction to Finite Element Method. Washington D. New York 6. 507-516. Ronald F. Uetsuji and T. McGraw-Hill. Vrettos (2005).S.Reddy (1993). Inc. A finite element analysis of angle-ply laminate end-notched flexure specimens.N. 32. 1117-1124. Composite Structure 88 (2009) 491-496. 5.H. India Institute of technology. M. A. Computer & Structures Vol. Canada: National Research Council of Canada (NRC). Artemev (2008). Finite element analysis of laminated composite plates using high order shear deformation theory. Institute for Aerospace Research.1 47-57. Vikram Sarabhai Space Centre. V. T. 12. K. Ramachandran (1988). Thangaratnam. G. Martinez and A. 11.64 8. K. Thermal buckling of composite laminated plates. R. Chakrabakti (2002). 10. Finite element analysis of broken fiber effects on the performance of active fiber composites. India: Department of Ocean Engineering and Naval Architecture. Tran (2007). Fibre Reinforced Plastics Research Centre. Singh and G.. Thermal buckling of cross-ply composite laminates. A new plate bending element based on higher-order shear deformation theory for the analysis of composite plates. . Mathew. 9. India Institute of Technology. Engineering College. Rao (1990). India: Department of Structural Engineering. Sheikh and A. Finite Element in Analysis and Design 39 (2003) 883-903. Structural Engineering Group. Vietnam: Thainguyen University and Hanoi University of Technology. N. No. No. Indian Institute of Technology. Ngo and I. 42. Structural Design and Analysis Division. VAST. Annamalai University. T. Department of Civil Engineering. 29. 281-287. Vietnam Journal of Mechanics. Palaninathan and J. C. Computer & Structures Vol. India: Department of Applied Mechanics. Jalpaiguri Govt. 6.1. %disp ('Please key in the thickness of lamina') t = 0.35. elseif orientation == A Angle = [0 0 45 90 90 0 0 45 90 90].layer) = t. T(1. . Gm = 1. G12f = 27.4. end % Input for thickness and layer of lamina layer = length(Angle). % Input for value of volume proportion Vm = 0.3. Vf = 1-Vm.2.65 APPENDIX A MATLAB SCRIPT FOR ANALYSIS ABD Matrix % Fiber T-300 Caron and Matrix Epoxy 3501-6 Em = 4. vm = 0. E1f = 230. elseif orientation == B Angle = [0 0 45 90 90 90 90 -45 0 0]. E2f = 15. v12f = 0. % input for value of orientation if orientation == S Angle = [0 0 45 90 90 90 90 45 0 0]. % Calculation for ABD end A = [a11 a12 a16.Q12 . Shape Function LA = 100. E2.:. a12 a22 a26. Qb(:.66 % Calculation of E1. Qb26 = (Q11 .k))*pi/180.k) = [Qb11 Qb12 Qb16.2*Q12 .16 and 32 only Lx = LA/ElementX.4.Q12 .2*Q66)*sin(ang)^2. Ni2 = (x/Lx)*(1-y/Ly).12.16 and 32 only ElementY = 12. b16 b26 b66]. %width of laminate ElementX = 12. D = (1/3)*[ d11 d12 d16.*cos(ang)^2 + Q66*(sin(ang)^4 + cos(ang)^4). % for 2.2*Q66)*sin(ang)*cos(ang)^3 + (Q12 . %N/mm2 %In-plane shape function Ni1 = (1-x/Lx)*(1-y/Ly). %Out-plane shape function No1 = 1-3*x^2/(Lx^2)-x*y/(Lx*Ly)- .Qb12 Qb22 Qb26. v21 and G12 etaE = (E2f-Em)/(E2f + 2*Em). Q22 = E2/(1-v12*v21).Q22 + 2*Q66)*sin(ang)^3. % for 2.8. Q12 = v21*Q11. v21 = E2*v12/E1 . % Calculation of Q Q11 = E1/(1-v12*v21). d16 d26 d66]. E1 = E1f*Vf + Em*Vm.4*Q66)*sin(ang)^2*cos(ang)^2 + Q12*(sin(ang)^4 + cos(ang)^4). %Length of laminate LB = 100. G12 = (Gm*(1 + etaG*Vf))/(1-etaG*Vf). Load = -12.12. Ni4 = y*(1-x/Lx)/Ly. Qb22 = Q11*sin(ang)^4 + 2*(Q12 + 2*Q66)*sin(ang)^2*cos(ang)^2 + Q22*cos(ang)^4. etaG = (G12f . Ly = LB/ElementY. b12 b22 b26.2*Q66)*sin(ang)^3.Gm)/(G12f + Gm). E2 = (Em*(1 + 2*etaE*Vf))/(1-etaE*Vf). a16 a26 a66]. B = (1/2)*[ b11 b12 b16. Ni3 = (x*y)/(Lx*Ly). v12. d12 d22 d26. Qb11 = Q11*cos(ang)^4 + 2*(Q12 + 2*Q66)*sin(ang)^2*cos(ang)^2 + Q22*sin(ang)^4. Qb16 =(Q11 . % Calculation for Qbar values for k = 1:layer ang = (Angle(1.*cos(ang).4. v12 = Vm*vm + Vf*v12f.*cos(ang) + (Q12 .Qb16 Qb26 Qb66] .8.Q22 + 2*Q66)*sin(ang)*cos(ang)^3. Q66 = G12. Qb66 = (Q11 + Q22 . Qb12 = (Q11 + Q22 . No12].No11.No8. No5 = x*y/(Lx)-2*x*y^2/(Lx*Ly)+x*y^3/(Lx*Ly^2). .No7. 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). Kd = BO*D*Bo.No9.67 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). Kb2 = BO*B*Bi. No9 = x*y/Ly-2*x^2*y/(Lx*Ly)+x^3*y/(Lx^2*Ly). Ka = BI*A*Bi.No5. No8 = x*y^2/(Lx*Ly)+y^3/(Ly^2)-x*y^3/(Lx*Ly^2)-y^2/Ly. No11 = -(x*y^2)/(Lx*Ly)+(x*y^3)/(Lx*Ly^2).No10. Kb1 = BI*B*Bo. 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). No = [No1.No2.No3. 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). No12 = -x^2*y/(Lx*Ly)+x^3*y/(Lx^2*Ly).No6. 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).No4. 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). 68 APPENDIX B RELATED DATA . 69 . Documents Similar To FEM Composite Laminate MATLABSkip carouselcarousel previouscarousel nextReport - A Digital Algorithm for Composite Laminate Analysis - Fortran (AFWAL-TR-81-4073)Analysis of Plate Using MATLABVaftsy_5_2014Composites Design Manual 3 Edanalysis of laminated composite plate using matlabFinite Element Formulation for Composite Laminate PlatesThe Finite Element Method Using MATLAB Kwon and BangDesign and Optimization of Laminated Composite MaterialsMatLab Code for Composite MaterialNumerical Analysis 9th Ed Soln Manual - Burden and FairesPrinciples of Composite Materials.(GIBSON).Mechanics of CompositesFinite Element Analysis of Composite Laminates 0792311256Mechanics of Composite Materials Solution Manual KawMechanics of Composite Materials with MatLab20150421101601643A 482383Compósitom247_ch04Bending and ShearLight Weight Hollow Sphere Composite MaterialsInTech-Mechanical Analysis of Woven Fabrics the State of the ArtComposites Properties for Analysis421239Composite Materials in Aerospace ApplicationsAIAA 96-2751Composite materials Lection_6stsm reportNano particles fibre.pdfSheet3 - Composites - Fiber ReinforcedMore From lekan4Skip carouselcarousel previouscarousel nextStrain Tensor - 2Fracture mechanics course.pdfBending and Vibration of Laminated Plates by a Layerwise Meshless FormulationAn Exact 3D Solution_elliptical PlateFundamentals of LefmOjo 2DShear Lag ModelWCE2015_pp1240-1242.pdfAnalysis of laminated structuresOjo_MOR.pdfAdomian decomposition methodBest Books About Composite MaterialWaste Materials Used in Concrete Manufacturingby Satish ChandraFinite Element Analysis and Design of Steel and Steel–Concrete Composite Bridgesby Ehab EllobodyMechanics of Composite Materials: Composite Materials, Vol. 2by Elsevier Books ReferencePolyurethane Polymers: Composites and Nanocompositesby Elsevier Books ReferenceAdvances in Concrete Slab Technology: Proceedings of the International Conference on Concrete Slabs Held at Dundee University, 3-6 April 1979by Elsevier Books ReferenceAnalysis of Composite Structuresby Christian DecolonFooter MenuBack To TopAboutAbout ScribdPressOur blogJoin our team!Contact UsJoin todayInvite FriendsGiftsLegalTermsPrivacyCopyrightSupportHelp / FAQAccessibilityPurchase helpAdChoicesPublishersSocial MediaCopyright © 2018 Scribd Inc. .Browse Books.Site Directory.Site Language: English中文EspañolالعربيةPortuguês日本語DeutschFrançaisTurkceРусский языкTiếng việtJęzyk polskiBahasa indonesiaSign up to vote on this titleUsefulNot usefulMaster your semester with Scribd & The New York TimesSpecial offer for students: Only $4.99/month.Master your semester with Scribd & The New York TimesRead Free for 30 DaysCancel anytime.Read Free for 30 DaysYou're Reading a Free PreviewDownloadClose DialogAre you sure?This action might not be possible to undo. Are you sure you want to continue?CANCELOK